Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies using Ultrasound, Volume 24, Second Edition (Studies in Interface Science)

Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies Using Ultrasound

Studies in Interface Science Vol. 1 Dynamics of Adsorption at Liquid Interfaces. Theory, Experiment, Application. By S.S. Dukhin, G. Kretzschmar and R. Miller Vol. 2 An Introduction to Dynamics of Colloids. By J.K.G. Dhont Vol. 3 Interfacial Tensiometry. By A.I. Rusanov and V.A. Prokhorov Vol. 4 New Developments in Construction and Functions of Organic Thin Films. Edited by T. Kajiyama and M. Aizawa Vol. 5 Foam and Foam Films. By D. Exerowa and P.M. Kruglyakov Vol. 6 Drops and Bubbles in Interfacial Research. Edited by D. Mo¨bius and R. Miller Vol. 7 Proteins at Liquid Interfaces. Edited by D. Mo¨bius and R. Miller Vol. 8 Dynamic Surface Tensiometry in Medicine. By V.M. Kazakov, O.V. Sinyachenko, V.B. Fainerman, U. Pison and R. Miller Vol. 9 Hydrophile-Lipophile Balance of Surfactants and Solid Particles. Physicochemical Aspects and Applications. By P.M. Kruglyakov Vol. 10 Particles at Fluid Interfaces and Membranes. Attachment of Colloid Particles and Proteins to Interfaces and Formation of Two-Dimensional Arrays. By P.A. Kralchevsky and K. Nagayama Vol. 11 Novel Methods to Study Interfacial Layers. By D. Mo¨bius and R. Miller Vol. 12 Colloid and Surface Chemistry. By E.D. Shchukin, A.V. Pertsov, E.A. Amelina and A.S. Zelenev Vol. 13 Surfactants: Chemistry, Interfacial Properties, Applications. Edited by V.B. Fainerman, D. Mo¨bius and R. Miller Vol. 14 Complex Wave Dynamics on Thin Films. By H.-C. Chang and E.A. Demekhin Vol. 15 Ultrasound for Characterizing Colloids. Particle Sizing, Zeta Potential, Rheology. By A.S. Dukhin and P.J. Goetz Vol. 16 Organized Monolayers and Assemblies: Structure, Processes and Function. Edited by D. Mo¨bius and R. Miller Vol. 17 Introduction to Molecular-Microsimulation of Colloidal Dispersions. By A. Satoh Vol. 18 Transport Mediated by Electrified Interfaces: Studies in the linear, non-linear and far from equilibrium regimes. By R.C. Srivastava and R.P. Rastogi Vol. 19 Stable Gas-in-Liquid Emulsions: Production in Natural Waters and Artificial Media Second Edition By J.S. D’Arrigo Vol. 20 Interfacial Separation of Particles. By S. Lu, R.J. Pugh and E. Forssberg Vol. 21 Surface Activity in Drug Action. By R.C. Srivastava and A.N. Nagappa Vol. 22 Electrorheological Fluids: The Non-aqueous Suspensions. T. Hao Vol. 23 Nanocomposite Structures and Dispersions - Science and Nanotechnology - Fundamental Principles and Colloidal Particles. Edited by: I. Capek

Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies Using Ultrasound 2nd edition

Andrei S. Dukhin and Philip J. Goetz

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2002 Copyright # 2010, Elsevier B.V. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333, E-mail: permissions@elsevier. com. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53621-1 ISSN: 1383-7303 For information on all Elsevier publications visit our Web site at www.elsevierdirect.com Print and bound in Great Britain 10 11 9

8 7 6 5

4 3 2 1

Contents

Preface to the Second Edition Preface to the First Edition List of Symbols

vii ix xi

Chapter 1. Introduction

1

Chapter 2. Fundamentals of Interface and Colloid Science 21 Chapter 3. Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

91

Chapter 4. Acoustic Theory for Particulates

127

Chapter 5. Electroacoustic Theory

187

Chapter 6. Experimental Verification of the Acoustic and Electroacoustic Theories

239

Chapter 7. Acoustic and Electroacoustic Measurement Techniques

261

Chapter 8. Applications for Dispersions

303

Chapter 9. Applications for Nanodispersions

343

Chapter 10. Applications for Emulsions and Other Soft Particles

369

Chapter 11. Titrations

401

Chapter 12. Applications for Ions and Molecules

425

Chapter 13. Applications for Porous Bodies

441

Bibliography

467

Subject Index

497 v

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Preface to the Second Edition

Ultrasound for Characterizing Colloids was first published in 2002. We decided to work on the second edition after two print runs were sold out and the book became unavailable. By the beginning of 2008, we gathered substantial justifications to prepare a new edition of the book instead of running an extra print of the first edition. First of all, about 40 papers had been published in various scientific journals by users of our instruments. There were, in addition, about a dozen of our own papers and several papers published by other scientists in the field. These papers demonstrated the wide spectrum of applications for characterization methods by ultrasound. Second, we discovered promising ways of using ultrasound that were known in specific scientific areas but not to us. These had not been mentioned in the first edition and we wanted to rectify the situation. One of the most striking examples was “seismoelectric current.” This electroacoustic and electrokinetic phenomenon had been known in geology for 70 years. However, there is practically no information about it in the major books on colloid and interface science despite the fact that it can be defined as “streaming current that is nonisochoric at ultrasound frequencies.” Frenkel’s theory of this effect is the first electroacoustic theory (1944) known to us, whereas Ivanov’s experiment is dated about the same time (1940) along with the first electroacoustic experiments by Hermans (1938). We present detailed descriptions, both theoretical and experimental, of this little known electrokinetic effect in Chapters 5 and 13. Seismoelectric current may become important in characterizing porous bodies. It clearly offers a simple way to measure z-potential in pores. In addition, it has the potential to characterize porosity and pore sizes. Another important example of discoveries in old works is “longitudinal rheology” and a host of related applications. Interpretation of acoustic data in rheological terms opens many new ways for applying ultrasound for characterizing complex systems. Mason predicted this 50 years ago for polymer solutions. There were other enthusiasts, but the lack of instruments and systematic reviews brought this field to a complete stop. We present here a review of the many works in “longitudinal viscosity” and provide some applications of our acoustic spectrometer as a “longitudinal rheometer.” One

vii

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Preface to the Second Edition

of the most important applications is the measurement of the “bulk viscosity,” an obscure second-viscosity coefficient of regular Newtonian liquids. Another example is Passynski’s theory (1938) that relates the speed of sound with the size and solvation number of ions. This theory was used by Bockris and others 40 years ago but had since been neglected. There is much discussion about measuring nanoparticles with sizes down to 1 nm using dynamic light scattering. This was done many decades ago using sound speed measurements with appropriate theoretical treatments. We discuss this in Chapter 12. There are several other new sections in the second edition. Most of them are on applications described in Chapters 8–13 but there are some new theoretical developments discussed in Chapters 3–5 as well. This field has become so wide-ranging in methods and applications that we have started talking more about the versatility of ultrasound for characterization purposes. This is explored in Chapter 1. This second edition is almost double the size of the first and has 50% more references. In preparing the first edition, we made the statement that it “marked the end of ultrasound’s ‘childhood’ in the field of colloids”. This second edition marks the beginning of adulthood. The number of groups working with our instruments has increased from 100 to 350. Geography has become much more diverse. In addition to groups in the United States, Germany, Japan, Taiwan, China, United Kingdom, Belgium, Finland, Singapore, South Korea, Mexico, and Canada, we have now users of our instruments in the Netherlands, Brazil, Thailand, Malaysia, India, Russia, Lithuania, Austria, Switzerland, Italy, Spain, France, Czech Republic, Columbia, Kuwait, Australia, and South Africa. We would also like to express our gratitude to our international agents: Quantachrome GmbH (Europe), Nihon Rufuto (Japan), Horiba USA, Acil Weber (Brazil), Advantage Scientific (China), and LMS (SE Asia). Without their dedication, we would not have been able to achieve so much in such a short time. They learned these new methods and now actively promote and teach them to others. We would like also to mention the contributions from our colleagues at Dispersion Technology: Betty Rausa, Lazlo Kovacs, Ross Parrish, Kenneth Schwartz, and Arthur Sigel. By taking proper care of everyday problems and the company’s needs, they gave us time to write this second edition. Andrei Dukhin President, Dispersion Technology, Inc.

Preface to the First Edition

The roots of this book go back 20 years. In the early 1980s Philip Goetz, then President of Pen Kem, Inc., was looking for new ways to characterize z-potential, especially in concentrated suspensions. His scientific consultants, Bruce Marlow, Hemant Pendse, and David Fairhurst, pointed towards utilizing electroacoustic phenomena. Several years of effort resulted in the first commercial electroacoustic instrument for colloids, the PenKem-7000. In the course of their work, they learned much about the potential value of ultrasound and collected a large number of papers published over the last two centuries on the use of ultrasound as a technique to characterize a diverse variety of colloids. From this it became clear that electroacoustics was only a small part of an enormous new field. Unfortunately, by that time Pen Kem’s scientific group broke apart: Bruce Marlow died, David Fairhurst turned back to design his own electrophoretic instruments, and Hemant Pendse concentrated his efforts on the industrial online application of ultrasound. In order to fill the vacuum of scientific support, Philip Goetz invited me to the United States, motivated by my experience and links to the well-established group of my father, Professor Stanislav Dukhin, then Head of the Theoretical Department in the Institute of Colloid Science of the Ukrainian Academy of Sciences. Whatever the reasons, I was fortunate to become involved in this exciting project of developing new techniques based on ultrasound. I was even luckier to inherit the vast experience and scientific base created by my predecessors at Pen Kem. Their hidden contribution to this book is very substantial, and I want to express here my gratitude to them for their pioneering efforts. It took 10 years for Phil and me to reach the point when, overwhelmed with the enormous volume of collected results, we concluded that the best way to summarize them for potential users was in the form of a book. There were many people who helped us during these years of development and later in writing this book. I would like to mention here several of them personally. First of all, Vladimir Shilov helped us tremendously with the electroacoustic theory. I believe that he is the strongest theoretician alive in the field of electrokinetics and related colloidal phenomena. Although he is very well known in Europe, he is less recognized in the United States. More recently, David Fairhurst joined us again. He brought his extensive expertise in the field of particulates, various colloids-related applications, and techniques; his comments were very valuable.

ix

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Preface to the First Edition

In our company, Dispersion Technology, Inc., Ross Parrish, Manufacturing and Service Manager, and Betty Rausa, Office Manager, extended their responsibilities last year, thereby allowing Phil and me the time to write this book. We are very grateful for their patience, reliability, and understanding. We trust that publishing this book will truly mark the end of ultrasound’s “childhood” in the field of colloids; it is no longer a “new” technique for colloids. Currently, there are more than 100 groups in the United States, Germany, Japan, Taiwan, China, United Kingdom, Belgium, Finland, Singapore, South Korea, Mexico, and Canada using Dispersion Technology’s ultrasound-based instruments. We do not know the total number of competing instruments from Malvern, Matec, Colloidal Dynamics, and Sympatec which are also being used in the field, but we estimate that there are over 200 groups working with them worldwide. We strongly believe that this is only the beginning. Ultrasonics has “come of age” and we hope that you will also share this view after reading this book and learning of the many advantages ultrasound can bring to the characterization of colloid systems. Andrei Dukhin President, Dispersion Technology, Inc.

List of Symbols

a b bsh C ci Cp Cdl Cs Cext Cabs Csca Di Du d e E hEi f F Fh Fhook k

particle radius cell radius radius of the shell in the core-shell model sound speed concentration of the ith ion species heat capacity at constant pressure DL capacitance electrolyte concentration extinction cross-section absorption cross-section scattering cross-section diffusion coefficient of the ith ion species Dukhin number particle diameter elementary electric charge electric field strength macroscopic electric field strength frequency of ultrasound Faraday constant hydrodynamic friction force Hookean force Bolzmann constant

kT ks K0 K Ir hIi I It Ii Is

thermal wavenumber shear wavenumber limiting conductances of cations and anions conductivity attributed with index local current in the cell macroscopic current intensity of the sound or light intensity of the transmitted sound or light intensity of the incident sound or light intensity of the scattered wave

xi

xii

List of Symbols

j jm (ka)

complex unit Bessel function

h H li L Lw mi M* NA Np N nm (ka) P pind

special function (see Special Functions) special function (see Special Functions) cell layer thickness gap in the electroacoustic chamber mass load mass of the solvated ions stress modulus Avogadro number number of particles number of the volume fractions Neumann function pressure induced dipole moment

R

gas constant

Robs R0 R1 r t t T Sexp v r Vdd Vhr Vcr Wext Wabs Wsca w u Um Udd

observation distance Rayleigh distance radius of the first Fresnel zone spherical radial coordinate transport numbers of cations and anions time absolute temperature measured electroacoustic signal volume of the solvated ions radial velocity corresponding to the dipole-dipole interaction radial velocity corresponding to the electro-hydrodynamic interaction radial velocity corresponding to the concentration interaction extinction energy absorption energy scattered energy width of the sound pulse speed of the motion attributed according to the index amplitude of the oscillation velocity energy of the dipole-dipole interaction

X

particle size

x

distance

List of Symbols

zi

valency of the ith ion species

z Z yi a b bp

valencies of the cations and anions acoustic impedance activity coefficient of the ith ion species attenuation specified with index thermal expansion attributed with index compressibility

d
dT dm e e0 f fc fT f
FIVI

viscous depth thermal depth scattering phase angles dielectric permittivity dielectric permittivity of the vacuum electric potential potential of the compression wave potential of the thermal wave potential of the shear wave phase of the Ion Vibration Current

Fe g gh gind
’ Fw k ks l m md mi m0i n n y r sd s sl t o ¼ 2pf

flow resistance hydrodynamic friction coefficient specific heat ratio Particle electric polarizibility dynamic viscosity volume fraction weight fraction reciprocal Debye length surface conductivity wave length electrophoretic mobility dynamic electrophoretic mobility electrochemical potential standard chemical potential kinematic viscosity dissociation numbers for cations and anions spherical angular coordinate density attributed according to the index electric charge of the diffuse layer electric surface charge standard deviation of the log-normal PSD heat conductance attributed according to the index frequency

xiii

xiv

oMW z c cdl (x) cd Ps Oe O S

List of Symbols

Maxwell-Wagner relaxation frequency electrokinetic potential angle in the polar coordinates Electric potential in the double layer Stern potential total scattered power by rigid sphere porosity drag coefficient entropy production

Indexes i p m s r y vis th sc int ef in out rod sur

index of the particle fraction or ion species particles medium dispersion radial component tangential component viscous thermal scattering intrinsic effective medium acoustic input acoustic output delay rod properties surface layer

Chapter 1

Introduction

1.1. Historical Overview 4 1.2. Versatility of UltrasoundBased Characterization Techniques 9 1.3. Comparison of UltrasoundBased Methods with Traditional Techniques 11 1.3.1. General Features 11

1.3.2. Particle Sizing 1.3.3. Measurements of z-Potential 1.3.4. Longitudinal and Shear Rheology 1.3.5. Characterization of Porous Bodies References

12 14 14 15 15

Several key words define the scope of this book’s second edition. All the words are mentioned in the title: ultrasound, liquids, nano- and microparticulates, and porous bodies. The key word, ultrasound, refers to characterization techniques described in this book, while all the others indicate the types of objects (such as dispersions and emulsions) that are studied by methods based on ultrasound. Each word is a key to a major scientific discipline. Ultrasound establishes acoustics as the main scientific basis for the measuring techniques presented here. The other key words define hydrodynamics, rheology, porosimetry, and colloidal and interfacial science as disciplines that deal with particulates and porous bodies. Historically, there has been curiously little real communication between acoustics and the scientific disciplines mentioned above. There is a large body of literature devoted to ultrasound phenomena in liquids, particulates, and porous bodies, but it has been mostly written from the perspective of scientists in the field of acoustics. There is limited recognition of ultrasound phenomena as of real importance for learning the properties of liquids, particulates, and porous bodies, and, in turn, developing applications. Scientists in these fields have not embraced acoustics as an important tool for characterizing their objects of interest. The lack of serious dialogue between these scientific fields is perhaps best demonstrated by the fact that there are no references to

Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23001-6 Copyright # 2010, Elsevier B.V. All rights reserved.

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CHAPTER

1

ultrasound or acoustics in the major handbooks on colloid and interface science [1, 2], rheology [3, 4], hydrodynamics [5–8], and porosimetry [9]. One may ask, “Perhaps this link does not exist because it is not important?” To answer this question, let us consider the potential place of ultrasound-related effects within an overall framework of nonequilibrium phenomena. It will be helpful to first classify nonequilibrium phenomena in two dimensions, as outlined in Table 1.1: the first is determined by whether the relevant disturbances are electrical, mechanical, or electromechanical in nature; the second is based on whether the time domain of that disturbance can be described as stationary, low frequency, or high frequency. The lowand high-frequency ranges are separated on the basis of the relationship between either the electrical or mechanical wavelength, l, and some system dimension, L. Light scattering clearly represents electrical phenomena in colloids at high frequency (the wavelength of light is certainly smaller than the system’s dimensions). However, until very recently, there was no mention in textbooks of mechanical or electromechanical phenomena in the region where l is shorter than the system’s dimensions. This would appear to leave two empty spaces in Table 1.1. Such mechanical wavelengths are produced by sound or, when the frequency exceeds our hearing limit of 20 kHz, ultrasound. Ultrasound wavelengths lie in the range from 10 µm to 1 mm, whereas the system’s dimensions are usually in the range of centimeters. For this reason, we consider ultrasound-related effects to lie within the high-frequency range. One of the empty spaces in Table 1.1 can be filled by acoustic measurements at ultrasound frequencies that characterize nonequilibrium phenomena of a

TABLE 1.1 Colloidal Phenomena Electrical nature

Electromechanical

Mechanical nature

Stationary

Conductivity, surface conductivity

Electrophoresis, electro-osmosis, sedimentation potential, streaming current/ potential, electro-viscosity

Viscosity, stationary colloidal hydrodynamics, osmosis capillary flow

Low frequency (l > L)

Dielectric spectroscopy

Electro-rotation, dielectrophoresis

Shear rheology

High frequency (l < L)

Optical scattering, X-ray spectroscopy

Empty? Electroacoustics!

Empty? Acoustics!

Introduction

3

mechanical nature at high frequency. The second empty space can be filled by electroacoustic measurements that permit characterization of electromechanical phenomena at high frequency. This book helps fill these gaps and demonstrates that acoustics (and electroacoustics) can provide useful knowledge to various scientific disciplines. As an aside, we do not consider the use of high-power ultrasound for modifying systems here. We only undertake the use of low-power sound as a noninvasive investigation tool that has unique capabilities. Several questions may come up when starting to read this book. We think it is important to deal with these questions right away, with some preliminary answers, which will be later clarified and expanded upon in the main text. Here are these questions and the short answers. Why should one care about acoustics if generations of scientists have successfully worked on a particular field without it? While it may be true that the usefulness of acoustics is currently not widely understood, it seems that earlier generations of scientists had a somewhat better appreciation. Many well-known scientists applied acoustics for characterization purposes, as will be described in a detailed historical overview in the next section. Briefly, we mention the names of Stokes, Rayleigh, Maxwell, Henry, Tyndall, Reynolds, and Debye. An interesting but not well-known fact is that Lord Rayleigh, the first author of a scattering theory, titled his major book as Theory of Sound. He developed the mathematics of scattering theory for both sound and light, but apparently considered sound to be more deserving of a book title than light. If acoustics is so important, why has it remained almost unknown as a characterization tool for such a long time? We think that the failure to exploit acoustic methods might be explained by a combination of factors: the advent of the laser as a convenient source of monochromatic light; technical problems with generating monochromatic sound beams over a wide frequency range; the mathematical complexity of the theory; and complex statistical analysis of the raw data. In addition, acoustics is more dependent on mathematical calculations than other traditional instrumental techniques. Many of these problems have now been solved mostly by the advent of fast computers and the development of new theoretical approaches. As a result, there are a number of commercially available instruments—developed by Matec, Malvern, Sympatec, Colloidal Dynamics, and Dispersion Technology—that utilize ultrasound for characterization of colloids. What information do ultrasound-based instruments yield? The versatility of ultrasound-based characterization methods deserves a special section in this chapter, as seen below. Over the last decade, we have personally thought on several occasions that we have exhausted new ideas and have finished developing a family of ultrasound-based characterization instruments. Reality has proved us wrong every

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1

time. New possibilities suddenly appear that we had not thought about earlier. One of the most astounding examples is the application of electroacoustics in studying porous bodies. We recently discovered a paper published 60 years ago that presents a theory of the electroacoustic effect in a porous body. This was the first electroacoustic theory that was published by Frenkel in 1944 [10]. What was even more surprising is that this old theory was initiated by an even older experiment of seismoelectric current, observed by Ivanov in 1940 [11]. These works create the basis for applying electroacoustics for studying properties of porous bodies. Where can one apply ultrasound? It can be applied for any liquid-state system that is Newtonian or nonNewtonian and as diverse as pure water to pastes. The following list gives some idea of the existing applications for which the ultrasound-based characterization technique is appropriate: aggregative stability, cement slurries, ceramics, chemical-mechanical polishing, coal slurries, coatings, cosmetic emulsions, environmental protection, flotation, ore enrichment, food products, latex, emulsions and micro emulsions, mixed dispersions, nanosized dispersions, nonaqueous dispersions, paints, photo materials.

This list is not complete. A table in Chapter 8 summarizes all experimental studies currently known to us. What are the advantages of ultrasound over traditional characterization techniques? There are so many advantages of ultrasound that we will discuss for a particular application in following sections of this chapter. Here is a striking example: ultrasound eliminates the need for sample dilution and special preparation for particle-sizing and z-potential measurements. Finally, we would like to stress that this book primarily targets scientists who consider ultrasound for characterization purposes. We will emphasize those aspects of acoustics that are important for these goals and neglect many others.

1.1. HISTORICAL OVERVIEW When preparing the first edition of this book almost 10 years ago, we did an extensive search of the relevant literature. Since that time, we found only a single old article of great importance that was missing from our original review. It turns out that the first electroacoustic theory was developed 7 years earlier than thought by a completely different person in a completely different place. There are also recent developments that deserve to be mentioned. These changes are incorporated in the text below. The roots of our current understanding of sound go back to more than 300 years when Newton suggested the first theory for calculating the speed of sound [12]. Newton’s work is still interesting today because it demonstrates the importance of thermodynamic considerations when trying to adequately

5

Introduction

describe ultrasound phenomena. Newton assumed that sound propagated while maintaining a constant temperature, that is, an isothermal case. Laplace later corrected this misunderstanding by showing that it was actually adiabatic in nature [12]. This thermodynamic aspect of sound is a good example of the importance of keeping a historical perspective. In at least two incidences in the past 200 years, the thermodynamic contribution to various sound-related phenomena was initially neglected and only later found to be important. The first neglect of thermodynamics happened in the nineteenth century when Stokes’s purely hydrodynamic theory for sound attenuation [13, 14] was later corrected by Kirchhoff [15, 16]. The second neglect occurred in the twentieth century when Sewell’s hydrodynamic theory for sound absorption in heterogeneous media [17] was later extended by Isakovich [18] with the introduction of a mechanism for thermal losses. Table 1.2 lists the important steps in the development of our understanding of sound. From the beginning, sound was considered to be a rather simple

TABLE 1.2 Milestones in Understanding Sound in Relation to Colloids Year

Author

Topic

1687

Newton [12]

Sound speed in fluid, theory, erroneous isothermal assumption

Early 1800s

Laplace [12]

Sound speed in fluid, theory, adiabatic assumption

1820

Poisson [19, 20]

Scattering by atmosphere, arbitrary disturbance, first successful theory

1808

Poisson [19, 20]

Reflection from rigid plane, general problem

1845–1851

Stokes [13, 14]

Sound attenuation in fluid, theory, viscous losses

1842

Doppler [21]

Alternation of pitch by relative motion

1868

Kirchhoff [15, 16]

Sound attenuation in fluid, theory, thermal losses

1866

Maxwell [22]

Kinetic theory of viscosity

1870–1880

Henry, Tyndall, Reynolds [23–26]

First application to colloids—sound propagation in fog

1871

Rayleigh [27, 28]

Light scattering theory

1875–1880

Rayleigh [12, 29, 30]

Diffraction and scattering of sound, Fresnel zones in Acoustics Continued

6

CHAPTER

TABLE 1.2 Milestones in Understanding Sound in Relation to Colloids— Cont’d Year

Author

Topic

1878

Rayleigh [12]

Theory of Sound, vol. II

1910

Sewell [17]

Viscous attenuation in colloids, theory

1933

Debye [31]

Electroacoustic effect, introduction for ions

1936

Morse [32]

Scattering theory for arbitrary wavelengthsize ratio

1938

Hermans [33]

Electroacoustic effect, introduced for colloids

1938

Passynski [34]

Theory relating compressibility of liquid with solvating numbers of ions

1940

Ivanov [35]

Discovery of seismoelectric phenomena

1944

Frenkel [9]

First electroacoustic theory. Application— seismoelectric phenomena

1944

Foldy [36, 37]

Acoustic theory for bubbles

1948

Isakovich [18]

Thermal attenuation in colloids, theory

1946

Pellam, Galt [38]

Pulse technique

1947

Bugosh, Yaeger [39]

Electroacoustic theory for electrolytes

1951–1953

Yeager, Hovorka, Derouet, Denizot [40–43]

First electroacoustic measurements

1951–1952

Enderby, Booth [44, 45]

Electroacoustic theory for particulates

1953

Epstein and Carhart [46]

General theory of sound attenuation in dilute colloids

1958–1959

Happel, Kuwabara [47–49]

Hydrodynamic cell models

1962

Andreae et al. [50, 51]

Multiple frequencies attenuation measurement

1967

Eigen et al. [52, 53]

Nobel Prize, acoustics for chemical reactions in liquids

1971

Bockris, Saluja [54]

Measurement solvating numbers of ions with ultrasound

1972

Allegra, Hawley [55]

ECAH theory for dilute colloids Continued

1

7

Introduction

TABLE 1.2 Milestones in Understanding Sound in Relation to Colloids— Cont’d Year

Author

Topic

1973

Cushman [56]

First patent for acoustic particle sizing

1974

Levine, Neale [57]

Electrokinetic cell model

1978

Beck [58]

Measurement of z-potential by ultrasonic waves

1981

Shilov, Zharkikh [59]

Corrected electrokinetic cell model

1983

Marlow, Fairhurst, Pendse [58]

First electroacoustic theory for concentrates

1983

Uusitalo [60]

Mean particle size from acoustics, patent

1983

Oja, Peterson, Cannon [61]

ESA electroacoustic effect

1988

Harker, Temple [62]

Coupled phase model for acoustics of concentrates

1987

Riebel [63]

Particle size distribution, patent for the large particles

1988–1989

O’Brien [64, 65]

Electroacoustic theory, particle size, and z-potential from electroacoustics

1990

Anson, Chivers [66]

Materials database

1999

Shilov and others [67, 68]

Electroacoustic theory for CVI in concentrates

1990 to present

McClements, Povey [69–71]

Acoustics for emulsions

1996 to present

Dukhin, Goetz [72–74]

Combining together acoustics and electroacoustics for particle sizing, rheology, and electrokinetics

2002

Dukhin, Goetz [75]

First Edition of this book

2004

Shilov, Borkovskaya, Dukhin [76]

Electroacoustic theory for overlapped DLs, nonaqueous systems, nanoparticulates

2005

ISO committee on particle characterization

ISO standard 20998/1 for acoustic particle sizing

2009

Dukhin, Goetz, and Thommes [77]

Electroacoustics for characterizing porous bodies

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1

example of a wave on which a general theory of wave phenomena was developed. Later, the new body of understanding for sound was extended to other wave phenomena, such as light. Tyndall, for example, used references to sound to explain the wave nature of the light [23, 24]. Newton’s corpuscular theory of light was first opposed both by the celebrated astronomer, Huygens, and the equally celebrated mathematician, Euler. They both believed that light, like sound, was a product of wave motion. In the case of sound, the velocity depended upon the relation between elasticity and density in the body that transmitted the sound. The greater the elasticity of the body, the greater is the velocity; the lower the density of the body, the greater is the velocity of the sound. To account for the enormous velocity of propagation in the case of light, the substance that transmitted it was assumed to have both extreme elasticity and extreme density. The dominance of sound over light as examples of the wave phenomena continued even with Lord Rayleigh. He developed his theory of scattering mostly for sound and paid much less attention to light [12, 27–30]. At the end of the nineteenth century, sound and light parted ways because further investigation was directed more on the physical roots of each phenomenon instead of on their common wave nature. The history of light and sound in colloid science is very different. Light has been an important tool since the first microscopic observations of Brownian motion and the first electrophoretic measurements. It became more important in middle of the twentieth century because of the use of light scattering to determine particle size. In contrast, sound remained unknown in colloid science, despite a considerable amount of work in the field of acoustics where fluids that were essentially colloidal in nature were used. The goal of these studies was to learn more about acoustics than about colloids. This was the spirit in which the ECAH theory (Epstein-Carhart-Allegra-Hawley [46, 55]) for ultrasound propagation through dilute colloids was developed. Although acoustics was not specifically used for colloids, it was a powerful tool for other purposes [89, 90]. For instance, it was used to learn more about the structure of pure liquids and the nature of chemical reactions in liquids. These studies, associated with the name of Prof. Eigen who received a Nobel Prize in Chemistry in 1967 [52, 53, 78], are described in more detail in the chapter, “Fundamentals of Acoustics.” Curiously, the penetration of ultrasound into colloid science began with electroacoustics, which is more complex than traditional acoustics. An electroacoustic effect was predicted for ions by Debye in 1933 [31], and later extended to colloids by Hermans and, independently, Rutgers in 1938 [33]. Recently, we have learned that at about the same time, electroacoustic phenomena were observed in geology by Ivanov in 1940 [35]. Theory of this effect, called seismoelectric current, became the first ever electroacoustic theory by Frenkel in 1944 [9]. Early experimental electroacoustic work

Introduction

9

is associated with Yeager and Zana who conducted many experiments in the 1950s and 1960s with various coauthors [39–42]. This work was later continued by Marlow, O’Brien, Ohshima, Shilov, and the authors of this book [58, 64, 65, 67, 68, 79, 80, 91]. As already mentioned, there are now several commercially available electroacoustic instruments for characterizing the z-potential. Acoustics has only very recently attained some recognition in the field of colloid science. It was first suggested as a particle-sizing tool by Cushman and others in 1973 [56], and later refined by Uusitalo and others [50]. Acoustics for sizing was suggested for large particles by Riebel [63]. Development of a commercial instrument that could measure a wide range of particle sizes was begun by Goetz, Dukhin, and Pendse in the 1990s [72, 81–83]. At the same time, a group of British scientists including McClements, Povey, and others [66, 69–71, 84, 85] actively promoted acoustics, especially for the study of emulsions. As mentioned earlier, there are now several commercially available acoustic spectrometers, manufactured by Sympatec, Matec, and Dispersion Technology. In conclusion of this short historical review, we would like to mention a development that we consider of great importance for the future—the combination of both acoustic and electroacoustic spectroscopies. The synergism of this combination is described in the papers and patents by Dukhin and Goetz [68, 73, 74, 80–83, 86, 87].

1.2. VERSATILITY OF ULTRASOUND-BASED CHARACTERIZATION TECHNIQUES There are two distinctively different ultrasound-based measuring methods. One is called acoustics and the other one is known as electroacoustics. They measure completely different parameters. Acoustics is simpler because a single field is involved—field of the mechanical stress. The measured parameters are usually sound speed and attenuation coefficient. The electroacoustic method is more complicated because it is based on the coupling of two fields—electrical and mechanical. The measured parameters are the magnitude and phase of the electroacoustic signal. Four raw parameters can be used as a fingerprint of a particular liquid system. l l l l

Sound speed Attenuation coefficient (usually at multiple frequencies) Magnitude of the electroacoustic signal Phase of the electroacoustic signal

Alternatively, the raw data could be theoretically treated so that a multitude of other properties can be characterized. The theoretical treatment is the main reason for ultrasound’s versatility for characterization purposes. It turns out

10

CHAPTER

1

that there are two very different sets of calculated parameters that can be extracted from the raw data, depending on degree of our a priori knowledge about the system. If our prior knowledge is limited and we are forced to model a system as a homogeneous medium with unknown viscoelastic properties, then the ultrasound-based method allows us to calculate following parameters: l l l l l l l l l l l l

Viscosity longitudinal within the 1–100 MHz frequency range Viscous longitudinal modulus G00 Bulk viscosity for Newtonian liquids Elastic longitudinal modulus G0 Compressibility Newtonian liquid test in the megahertz range Isoelectric point—range of aggregative instability Optimum dose of surfactant Volume fraction of the dispersed phase from sound speed Kinetics of dissolution, crystallization Kinetics of sedimentation Verification of large particle presence in opaque systems

We would like to stress here that the system might be very complex and intuitively heterogeneous, but it should not prevent the application of the homogeneous mode to its description. For instance, milk can be treated as a homogeneous liquid when we ignore the fact that it is actually a collection of fat droplets, proteins, and sugars in water. Basically, any liquid system can be modeled as homogeneous or heterogeneous. These models are creations in our minds for adequate characterization of various physical and chemical properties. In the case of ultrasound-based techniques, we can first model a system as homogeneous and calculate the set of parameters presented above. Then, as the next step, we can apply the heterogeneous model and theoretically treat the same set of experimental raw data for extracting another set of parameters, given below: l l

l l l l l l l l

Particle size distribution of solid particles with known density Particle size distribution of soft particles (droplets) when thermal expansion coefficient is known Volume fraction of solids at the submicrometer range Hook’s parameter for particle bonds in structured systems Microviscosity x-Potential in particulates Surface conductivity Debye length when conductivity is known Ion size from compressibility in aqueous solution Ion size from electroacoustics in aqueous and nonaqueous solutions

Introduction

l l l

11

Electric charge of macromolecules Porosity, pore size, and x-potential in porous bodies Properties of deposits and sediments

Application of the heterogeneous model usually requires a priori information about the volume fraction of the dispersed phase. This parameter may be extracted from the raw data in some special cases, for instance when particles are rigid and their sizes range from 0.1 to 1 µm. Sometimes, it can be calculated from the sound’s speed as well. However, for the most reliable results, volume (weight) fraction of the dispersed phase must be treated as an independently known input parameter. In terms of traditional measuring techniques, ultrasound can combine the descriptions of rheology, particle size distribution, and z-potential into one unit instead of three independent instruments. This has substantial advantages over traditional methods. We discuss some of them below.

1.3. COMPARISON OF ULTRASOUND-BASED METHODS WITH TRADITIONAL TECHNIQUES The versatility of ultrasound-based methods makes them comparable with several traditional techniques, such as particle sizing by light scattering, electrophoretic light scattering, microelectrophoresis, and shear rheology.

1.3.1. General Features A major distinction of ultrasound-based techniques, in contrast to traditional light-based characterization methods, is that sound can propagate through concentrated, opaque liquid systems and even porous bodies. This peculiar feature opens up the opportunity for tremendous simplification in sample preparation. In particular, ultrasound-based methods applied to concentrated dispersions and emulsions eliminate the need for dilution. Systems can be characterized as they are. Dilution required by traditional techniques can destroy aggregates or flocs. The corresponding measured particle size distribution for the diluted system would not be correct for the original concentrated sample. Eliminating dilution is especially critical for z-potential characterization because the parameter is a property of both the particle and the surrounding liquid; dilution changes the suspension medium and the z-potential. This feature of eliminating dilution is applicable to both the measurements of particle size and z-potential. Sample handling for characterizing these parameters becomes as simple as in traditional rheological methods. Acoustic methods are very robust and precise [73, 74]. They are much less sensitive to contamination in comparison to traditional light-based techniques because the high concentration of particles in a fresh sample dominates any small residue left from the previous sample.

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Ultrasound-based methods are relatively fast. A single measurement can normally be completed within a few minutes. This feature, together with the ability to measure flowing systems, makes acoustic attenuation very attractive for on-line characterization.

1.3.2. Particle Sizing The numerous advantages of using ultrasound for characterizing particle size are summarized in Table 1.3. Detailed analysis of ultrasound-based techniques is given later in this book. The following is a short summary. There are several advantages of ultrasound-based over light-based instruments because of the longer wavelength used. The wavelength of ultrasound in water, at the highest frequency typically used (100 MHz), is about 15 µm. It increases even further to 1.5 mm at the lowest frequency (1 MHz). In contrast, light-based instruments typically use wavelengths on the order of 0.5 µm. If the particles are small compared to the wavelength, the Rayleigh long-wavelength requirement is said to be satisfied. Particle

TABLE 1.3 Features and Benefits of Acoustics over Traditional Particle-Sizing Techniques Feature

Benefit

No requirement of dilution

Less sensitive to contamination

No calibration with the known particle size

More accurate

Particle size range from 5 nm to 1000 µm with the same sensor

Simpler hardware, more cost effective

Simple decoupling of sound adsorption and sound scattering

Simplifies theory

Possible to eliminate multiple scattering even at high volume fractions up to 50% vol

Simplifies theory for large particle size

Existing theory for ultrasound absorption in concentrates with particles interaction

Possible to treat small particles in concentrates

Data availability over wide range of wavelengths

Allows use of simplified theory and reduces particle shape effects

Innate weight basis, lower power of the particle size dependence

Better for broad polydisperse systems, highly sensitive to nanoparticles

Particle sizing in dispersions with several dispersed phases (mixed dispersions)

Real-world, practical systems

Particle sizing in structured dispersions

13

Introduction

sizing in this long-wavelength range is more desirable than in the intermediateor short-wavelength ranges because of the lower sensitivity to shape factors and a simpler theoretical interpretation. So, applying the longer wavelengths available by acoustics allows us to characterize a greater range of particle sizes while still meeting the long-wavelength requirement. Nature has provided one more significant advantage of ultrasound over light which relates to the wavelength dependence. As the wave travels through the colloid, the combined effects of both scattering and absorption cause the extinction of both ultrasound and light [32, 88]. As most light scattering experiments are performed at a single wavelength, it is not possible to experimentally separate the two contributions of scattering and absorption to the total extinction. More often than not, the absorption of light is simply neglected in most light scattering experiments, which can lead to errors. In the case of ultrasound, the absorption and scattering are distinctively separated on the wavelength scale. Figure 1.1 illustrates the dependence of ultrasound attenuation as a function of relative wavelength, ka, as defined by: 2pa ð1:1Þ l where a is the particle radius and l is the wavelength of ultrasound. The attenuation curve has two prominent ranges. The lower frequency region corresponds to absorption; the higher frequency region corresponds to scattering. It is obvious from Figure 1.1 that both contributions can be easily separated because there is very little—indeed almost negligible—overlap. ka ¼

0.3 µm 0.5 µm 1 µm 5 µm 10 µm

9

attenuation [dB/cm/MHz]

8 7 6 5 4 3 2 1 0 10−4

10−3

10−2

10−1

ka

100

101

FIGURE 1.1 Scattering attenuation and viscous absorption of ultrasound.

102

103

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This peculiar aspect of ultrasound frequency dependence tremendously simplifies the theory. In the majority of cases, absorption and scattering can be considered separately. Except for the cases with very high volume fractions and some special nonaqueous systems with soft particles [49], this simplification is valid.

1.3.3. Measurements of z-Potential In contrast to acoustics, electroacoustics is a relatively new technique. In principle, it provides information for both particle-sizing and z-potential characterization. However, we believe that acoustics is better suited to particle sizing than electroacoustics. For this reason, as justified later in Chapters 4, 5, and 7, we consider electroacoustics to be primarily a technique for characterizing only the electric surface properties, such as the z-potential. In this sense, electroacoustics competes with microelectrophoresis and other traditional electrokinetic methods. However, electroacoustics has many advantages over traditional electrokinetic methods, which can be summarized as follows: l l l l l l l l l l

No dilution required, volume fraction up to 50% vol Less sensitive to contamination, easier to clean Higher precision ( 0.1 mV) Low surface charges (down to 0.1 mV) Electrosmotic flow not important Convection not important Measurement at high ionic strength exceeding 1 M level Measurement at very low ionic strength in nonpolar liquids Small sample volume, as little as 1 ml Faster

In addition, as explained later in Chapter 8, electroacoustic probes can be used for various titration experiments.

1.3.4. Longitudinal and Shear Rheology Ultrasound-based methods are similar to rheology in that they rely on applying stress to the system to learn about some of its properties. The difference is the type of stress applied. Traditional rheology uses shear stress, whereas ultrasound is a wave of longitudinal stress. Thus, there are two different rheological methods—shear rheology and longitudinal rheology. Chapter 3 presents detailed definitions and comparison of these two measurement techniques. Here we will only mention the main differences. First of all, the two different rheological methods work at very different frequency ranges. Shear rheology is applicable at a low frequency range up to roughly 100 kHz. Longitudinal rheology functions only at the megahertz frequency scale. This makes both methods complimentary, not competitive.

Introduction

15

Second, longitudinal rheology is nondestructive. Longitudinal stress does not break bonds between particles. It only causes the bonds to expand and collapse. Third, longitudinal rheology provides information about the bulk viscosity of Newtonian liquids. Bulk viscosity is a more sensitive probe of any structural feature in a Newtonian liquid but it is impossible to measure using shear-based techniques. Finally, measurement of the ultrasound attenuation at multiple frequencies helps in assessing whether a particular liquid can be described as Newtonian at the megahertz frequency range.

1.3.5. Characterization of Porous Bodies This is a relatively new field for ultrasound-based methods and still is under development. However, there is sufficient amount of data to be optimistic about the method’s future. Characterization of porous bodies usually includes measurements of the porosity, pore size, and the electric charging of the walls of the pores. The measurement of porosity and pore size can be achieved using gas adsorption and mercury porosimetry. The characterization of electric charging usually relies on streaming current/potential measurement. The propagation of ultrasound through a porous body generates a host of different effects that can be used for characterization purposes. Detailed analysis of the seismoelectric current may be the most promising: it simply is a streaming current at high ultrasound frequency in the nonisochoric mode. If this development can be confirmed, this method could compete with mercury porosimetry and minimize usage of this environmentally dangerous material. The seismoelectric current method would allow characterization of the electric surface properties of materials with very low hydrodynamic permeability because of small pore size. Many materials of this kind are impossible to be measured with classical electrokinetic methods.

REFERENCES [1] J. Lyklema, Fundamentals of Interface and Colloid Science, vol. 1, Academic Press, New York, 1993. [2] R.J. Hunter, Foundations of Colloid Science, Oxford University Press, Oxford, 1989. [3] H.A. Barnes, A Handbook of Elementary Rheology, University of Wales, Institute of NonNewtonian Fluid Mechanics, Aberystwyth, 2000. [4] C.L. Yaws, Handbook of Viscosity, vol. 4, Inorganic Compounds and Elements, Gulf Publishing Company, Houston, TX, 1997. [5] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965. [6] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959. [7] H. Lamb, Hydrodynamics, sixth ed., Dover Publications, New York, NY, 1932.

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[8] J.O. Hirschfelder, C.F. Curtis, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, NY, 1964. [9] S. Lowell, J.E. Shields, M.A. Thomas, M. Thommes, Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density, Kluwer, The Netherlands, 2004. [10] A.G. Ivanov, Bull. Aca. Sci. USSR, Ser. Geogr. Geophys. 5 (1940) 699. [11] M.C. Potter, D.C. Wiggert, Mechanics of Fluids, Prentice Hall, Englewood Cliffs, NJ, 1997. [12] L. Rayleigh, The Theory of Sound, vol. 2, second ed., Macmillan, New York, NY, 1896; first ed., 1878. [13] Stokes, On a difficulty in the Theory of Sound, Philos. Mag. Nov. (1848). [14] Stokes, Dynamic theory of diffraction, Camb. Philos. Trans. IX (1849). [15] Kirchhoff, Pogg. Ann. CXXXIV (1868) 177. [16] Kirchhoff, Vorlesungen uber Mathematische Physik, (1876). [17] C.T.J. Sewell, The extinction of sound in a viscous atmosphere by small obstacles of cylindrical and spherical form, Philos. Trans. R. Soc. Lond. 210 (1910) 239–270. [18] M.A. Isakovich, Zh. Exp. Theor. Phys. 18 (1948) 907. [19] Poisson, Sur l’integration de quelques equations lineaires aux differnces prtielles, et particulierement de l’equation generalie du mouvement des fluides elastiques, Mem., de l’Institut t.III (1820) 121. [20] Poisson, J. de l’ecole Polytech. t.VII (1808). [21] Doppler, Theorie des farbigen Lichtes der Doppelsterne, Prag (1842). [22] Maxwell, On the viscosity or internal friction of air and other gases, Philos. Trans. 156 (1866) 249. [23] J. Tyndall, Light and Electricity, D. Appleton & Comp., New York, NY, 1873. [24] J. Tyndall, Sound, Philos. Trans. third ed. (1874). [25] Henry, Report of the Lighthouse Board of the United States for the year 1874. [26] O. Reynolds, Proc. R. Soc. XXII (1874) 531. [27] J.W. Rayleigh, On the light from the sky, Philos. Mag. (1871). [28] J.W. Rayleigh, On the scattering of light by small particles, Philos. Mag. (1871). [29] J.W. Rayleigh, Acoustical observations, Philos. Mag. IX (1880) 281. [30] J.W. Rayleigh, On the application of the principle of reciprocity to acoustics, R. Soc. Proc. XXV (1876) 118. [31] P. Debye, J. Chem. Phys. 1 (1933) 13. [32] P. Morse, U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, NY, 1968. [33] J. Hermans, Philos. Mag. 25 (1938) 426. [34] A. Passynski, Acta Physicochim. 8 (1938) 385. [35] J. Frenkel, On the theory of seismic and seismoelectric phenomena in a moist soil, J. Phys., USSR 3 (5) (1994) 230–241. Re-published in J. Eng. Mech. (2005). [36] L.L. Foldy, Propagation of sound through a liquid containing bubbles, OSRD Report No.6.1sr1130-1378, 1944. [37] E.L. Carnstein, L.L. Foldy, Propagation of sound through a liquid containing bubbles, J. Acoust. Soc. Am. 19 (3) (1947) 481–499. [38] J.R. Pellam, J.K. Galt, Ultrasonic propagation in liquids: application of pulse technique to velocity and absorption measurement at 15 Megacycles, J. Chem. Phys. 14 (10) (1946) 608–613. [39] J. Bugosh, E. Yeager, F. Hovorka, J. Chem. Phys. 15 (1947) 592. [40] E. Yeager, F.J. Hovorka, Acoust. Soc. Am. 25 (1953) 443. [41] E. Yeager, H. Dietrick, F. Hovorka, J. Acoust. Soc. Am. 25 (1953) 456. [42] R. Zana, E.J. Yeager, Phys. Chem. 71 (1967) 4241.

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[43] B. Derouet, F.C.R. Denizot, Acad. Sci. Paris 233 (1951) 368. [44] F. Booth, J. Enderby, On electrical effects due to sound waves in colloidal suspensions, Proc. Am. Phys. Soc. 208A (1952) 32. [45] J.A. Enderby, On electrical effects due to sound waves in colloidal suspensions, Proc. R. Soc. Lond. A207 (1951) 329–342. [46] P.S. Epstein, R.R. Carhart, The absorption of sound in suspensions and emulsions, J. Acoust. Soc. Am. 25 (1953) 553–565. [47] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, Dordrecht, The Netherlands, 1973. [48] J. Happel, Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles, AICHE J. 4 (1958) 197–201. [49] S. Kuwabara, The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers, J. Phys. Soc. Jpn. 14 (1959) 527–532. [50] J. Andreae, P. Joyce, 30 to 230 megacycle pulse technique for ultrasonic absorption measurements in liquids, Br. J. Appl. Phys. 13 (1962) 462–467. [51] J. Andreae, R. Bass, E. Heasell, J. Lamb, Pulse technique for measuring ultrasonic absorption in liquids, Acustica 8 (1958) 131–142. [52] M. Eigen, Determination of general and specific ionic interactions in solution, Faraday Soc. Discuss. 24 (1957) 25. [53] M. Eigen, L. deMaeyer, in: Weissberger (Ed.), Techniques of Organic Chemistry, vol. VIII, Part 2, Wiley, New York, NY, 1963. [54] J.O’M. Bockris, P.P.S. Saluja, Ionic salvation numbers from compressibilities and ionic vibration potential measurements, J. Phys. Chem. 76 (1972) 2140–2151. [55] J.R. Allegra, S.A. Hawley, Attenuation of sound in suspensions and emulsions: theory and experiments, J. Acoust. Soc. Am. 51 (1972) 1545–1564. [56] Cushman, et al., US Patent 3,779,070, 1973. [57] S. Levine, G.H. Neale, The prediction of electrokinetic phenomena within multiparticle systems. 1. Electrophoresis and electroosmosis, J. Colloid Interface Sci. 47 (1974) 520–532. [58] Beck, et al., Measuring zeta potential by ultrasonic waves, Tappi 61 (1978) 63–65. [59] V.N. Shilov, N.I. Zharkih, Yu.B. Borkovskaya, Theory of nonequilibrium electrosurface phenomena in concentrated disperse system. 1. Application of nonequilibrium thermodynamics to cell model, Colloid J. 43 (1981) 434–438. [60] S.J. Uusitalo, G.C. von Alfthan, T.S. Andersson, V.A. Paukku, L.S. Kahara, E.S. Kiuru, Method and apparatus for determination of the average particle size in slurry, US Patent 4,412, 451, 1983. [61] T. Oja, G. Petersen, D. Cannon, Measurement of electric-kinetic properties of a solution, US Patent 4,497,208, 1985. [62] A.H. Harker, J.A.G. Temple, Velocity and attenuation of ultrasound in suspensions of particles in fluids, J. Phys. D. Appl. Phys. 21 (1988) 1576–1588. [63] U. Riebel, et al., The fundamentals of particle size analysis by means of ultrasonic spectrometry, Part. Part. Syst. Charact. 6 (1989) 135–143. [64] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190 (1988) 71–86. [65] R.W. O’Brien, Determination of particle size and electric charge, US Patent 5,059,909, Oct. 22, 1991. [66] L.W. Anson, R.C. Chivers, Thermal effects in the attenuation of ultrasound in dilute suspensions for low values of acoustic radius, Ultrasonic 28 (1990) 16–25.

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[67] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. New theory and CVI experiment, Langmuir 15 (20) (1999) 6692–6706. [68] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. Effect of the surface conductivity, Langmuir 16 (2000) 2615–2620. [69] M. Povey, The application of acoustics to the characterization of particulate suspensions, in: V. Hackley, J. Texter (Eds.), Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, Am. Ceramic Soc., Ohio, 1998. [70] J.D. McClements, Ultrasonic determination of depletion flocculation in oil-in-water emulsions containing a non-ionic surfactant, Colloids Surf. 90 (1994) 25–35. [71] D.J. McClements, Comparison of multiple scattering theories with experimental measurements in emulsions, J. Acoust. Soc. Am. 91 (2) (1992) 849–854. [72] A.S. Dukhin, P.J. Goetz, Acoustic and electroacoustic spectroscopy, Langmuir 12 (19) (1996) 4336–4344. [73] A.S. Dukhin, P.J. Goetz, Acoustic and electroacoustic spectroscopy for characterizing concentrated dispersions and emulsions, Adv. Colloid Interface Sci. 92 (2001) 73–132. [74] A.S. Dukhin, P.J. Goetz, New developments in acoustic and electroacoustic spectroscopy for characterizing concentrated dispersions, Colloids Surf. 192 (2001) 267–306. [75] A.S. Dukhin, P.J. Goetz, Ultrasound for Characterizing Colloids, Elsevier, The Netherlands, 2002. [76] V.N. Shilov, Y.B. Borkovskaya, A.S. Dukhin, Electroacoustic theory for concentrated colloids with overlapped dls at arbitrary ka. Application to nanocolloids and nonaqueous colloids, J. Colloid Interface Sci. 277 (2004) 347–358. [77] A.S. Dukhin, P.J. Goetz, M. Thommes, Method for determining porosity, pore size and zeta potential of porous bodies, US Patent, pending, 2009. [78] L. DeMaeyer, M. Eigen, J. Suarez, Dielectric dispersion and chemical relaxation, J. Am. Chem. Soc. 90 (1968) 3157–3161. [79] V.N. Shilov, A.S. Dukhin, Sound-induced thermophoresis and thermodiffusion in electric double layer of disperse particles and electroacoustics of concentrated colloids, Langmuir (submitted for publication). [80] A.S. Dukhin, V.N. Shilov, Yu. Borkovskaya, Dynamic electrophoretic mobility in concentrated dispersed systems. Cell model, Langmuir 15 (10) (1999) 3452–3457. [81] A.S. Dukhin, P.J. Goetz, Characterization of aggregation phenomena by means of acoustic and electroacoustic spectroscopy, Colloids Surf. 144 (1998) 49–58. [82] A.S. Dukhin, P.J. Goetz, Method and device for characterizing particle size distribution and zeta potential in concentrated system by means of acoustic and electroacoustic spectroscopy, US Patent 09/108,072, 2000. [83] A.S. Dukhin, P.J. Goetz, Method and device for determining particle size distribution and zeta potential in concentrated dispersions, US Patent, pending. [84] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth study of ultrasound velocity and attenuation in suspensions: comparison of theory with experimental measurements, J. Colloid Interface Sci. 156 (1993) 261–269. [85] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth ultrasonic study of suspensions: the variation of velocity and attenuation with particle size, J. Colloid Interface Sci. 168 (1994) 339–348. [86] A.S. Dukhin, P.J. Goetz, Acoustic spectroscopy for concentrated polydisperse colloids with high density contrast, Langmuir 12 (1996) 4987–4997. [87] T.H. Wines, A.S. Dukhin, P. Somasundaran, Acoustic spectroscopy for characterizing heptane/water/AOT reverse microemulsion, JCIS 216 (1999) 303–308.

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[88] C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, NY, 1983. [89] L. Kinsler, A. Frey, A. Coppens, J. Sanders, Fundamentals of Acoustics, Wiley, New York, NY, 2000. [90] D. Blackstock, Fundamentals of Physical Acoustics, Wiley, New York, NY, 2000. [91] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65.

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Chapter 2

Fundamentals of Interface and Colloid Science

2.1. Real and Model Systems 2.2. Particulates and Porous Systems 2.3. Parameters of the Model Dispersion Medium 2.3.1. Gravimetric Parameters 2.3.2. Rheological Parameters 2.3.3. Acoustic Parameters 2.3.4. Thermodynamic Parameters 2.3.5. Electrodynamic Parameters 2.3.6. Electroacoustic Parameters 2.3.7. Chemical Composition 2.3.8. Electrochemical Composition of Aqueous and Nonaqueous Solutions 2.4. Parameters of the Model Dispersed Phase 2.4.1. Rigid Versus Soft Particles 2.4.2. Solid Versus Fractal and Porous Particles

23 25 26 26 26 27 28 28 30 30

31 34 36

2.4.3. Porous Body 39 2.4.4. Particle Size Distribution 40 2.5. Parameters of the Model Interfacial Layer 44 2.5.1. Flat Surfaces 46 2.5.2. Spherical DL, Isolated and Overlapped 47 2.5.3. Electric Double Layer at High Ionic Strength 49 2.5.4. Polarized State of the Electric Double Layer 51 2.6. Interactions in Colloid Interface Science 54 2.6.1. Interactions of Colloid Particles in Equilibrium: Colloid Stability 54 2.6.2. Biospecific Interactions 58 2.6.3. Interaction in a Hydrodynamic Field: Cell and CoreShell Models— Rheology 60

37

Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23002-8 Copyright # 2010, Elsevier B.V. All rights reserved.

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CHAPTER

2.6.4. Linear Interaction in an Electric Field: Electrokinetics and Dielectric Spectroscopy 66 2.6.5. Nonlinear Interaction in the Electric Field: Nonlinear

Electrophoresis, Electrocoagulation, and Electrorheology 2.7. Traditional Particle Sizing 2.7.1. Light Scattering: Extinction = Scattering + Absorption References

2

72 77

78 82

The goal of this chapter is to provide a general description of the objects that are referred to in the title of this book, and related phenomena. We introduce here some terms that will be used to characterize these objects, together with a brief overview of corresponding theories and traditional measuring techniques. There are several changes compared to the first edition. We added “porous” and “fractal” particles to the list of objects that can be studied with ultrasound. There is also a section on electrochemistry and double-layer structure in nonpolar liquids. We also included a section on “biospecific interactions.” It has gained major importance lately because of nanotechnology as well as ecological and human health concerns. There are also some minor modifications made to several other sections, such as electrophoresis of particulates with overlapped double layers, gravity as a factor of aggregative stability, etc. In writing this section, we follow Lyklema’s definition of a colloid as “an entity, having at least in one direction a dimension between 1 nm and 1 mm, that is, between 10–9 and 10–6 m. The entities may be solid, liquid, or, in some cases, even gaseous. They are dispersed in the medium” [1]. This medium may be also solid, liquid, or gaseous. We have found it more useful in this book to refer to these as “entities” instead as the “dispersed phase,” and the “medium” as the “dispersion medium.” There is an enormous variety of systems satisfying this wide definition of a colloid. Given three states for the dispersed phase and three states for the dispersion medium allows, altogether, nine possible varieties of colloids. For the purpose of this book, we restrict ourselves to just three of these possibilities, namely a collection of solid, liquid, or gas particles dispersed in a single liquid medium. All our colloids can, therefore, be defined as “a collection of particles immersed in a liquid”; the particles can be solid (a suspension or dispersion), liquid (an emulsion), or gas (a foam). These three types of dispersed systems play an important role in all kinds of applications: paints, latices, food products, cements, minerals, ceramics, blood, and many others. All these disperse systems have one common feature; because of their small size, they all have a high surface area relative to their volume. It is surface-related phenomena, therefore, that primarily determine their behavior in various processes and justify consideration of colloids as effectively a different state of matter.

Fundamentals of Interface and Colloid Science

23

What follows are the combined terms and methods that have been created over the course of two centuries to characterize these very special systems. In addition to Lyklema’s book, we also incorporate terminology from Nomenclature in Dispersion Science and Technology published by the US National Institute of Standards and Technology [2].

2.1. REAL AND MODEL SYSTEMS Real heterogeneous systems, by their very nature, are quite complex. Two of these complexities are the variation of the particle size and shape within a particular dispersion. Furthermore, the particles might not even be separate, but linked together to form some network, or the liquid might contain complex additives. This inherent complexity makes it difficult, perhaps impossible, to fully characterize a real heterogeneous system except in the simplest cases. However, we do not always need a complete, detailed characterization; a somewhat simplified picture can often be sufficient for a given particular purpose. There are two approaches to obtaining a simplified picture of any real system under investigation. The first approach is “phenomenological” [1], in that it relates only experimentally measured variables, and usually is based on simple thermodynamic principles. Usually, we model real system as “homogeneous” using this approach. For instance, we can characterize milk or blood as imaginary homogeneous liquids if we are looking for, let us say, viscosity of these liquids. This can be a very powerful approach, but, typically, it does not yield very much insight into any underlying structure within the system. The second approach is “modeling,” in which we replace the practical system with some approximation based on an imaginary “model system.” This model of the system is an attempt to describe a real liquid or colloid in terms of a set of simplified model parameters including, of course, the characteristics that we hope to determine. The model, in effect, makes a set of assumptions about the real world in order to reduce the complexity of the colloid, and thereby also simplifies the task of developing a suitable prediction theory. It is important to recognize that in substituting a model system for the real system, we introduce a certain “modeling error” into the characterization process. Sometimes, this error may be so large as to make this substitution inadequate or, perhaps, even worthless. For example, most particle-size measuring instruments substitute a model that assumes that the particles are spherical. In this case, a complete geometrical description of a single particle is specified using just one parameter, its diameter d. Such a model, obviously, would not adequately describe a dispersion of long, thin carpet fibers that have a high aspect ratio, and any theory based on this oversimplified model might well give very misleading results. This is an example of an extreme case in which the modeling error makes the model system completely inappropriate. It is important to acknowledge that some “modeling error” almost always exists. We can reduce it by adopting a more complicated model in response to the complexities of the real-world colloid, but we can seldom make it vanish.

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In turn, the selected model then limits, or increases, the complexity of the theory that must be brought into play to adequately predict any acoustic properties based on the chosen model. Having chosen a model for our colloid, one can then select and develop a more appropriate theory. In the course of such a development, certain theoretical assumptions, or approximations, which introduce additional “theoretical errors,” may be made. These theoretical errors reflect our inability to completely predict even the behavior of our already simplified model system. However, it would be pointless to use some overly elaborate and complex theory in an effort to reduce these theoretical errors to values much less than the already unavoidable modeling errors. In performing experimental tests to confirm various theories, it is of course necessary to minimize the difference between the real system used for the tests and the model system on which the proposed theory is based. For example, the model colloid mentioned by Lyklema [1] (a collection of spherical particles with a certain size in the liquid) describes quite precisely a real colloid consisting of a suspension of monodisperse latex particles. The design of any model liquid or colloid is also strongly related to the notion of a “dimensional hierarchy,” which we introduce here to generalize the rather widely accepted idea of macroscopic and microscopic levels of characterization. For instance, Lyklema described macroscopic laws as dealing on a large scale with “gigantic numbers of molecules.” In contrast, the microscopic level seeks an ab initio interpretation based on a very small scale or a molecular picture. In dealing with colloids, we further need to expand this simple picture of a merely two-dimensional hierarchy. Certainly, we can speak of a molecular, microscopic level, and the colloid as a whole can be viewed on a macroscopic level. But there are intermediate levels of importance as well. For instance, the size of a typical colloid particle, which lies in between the molecular and macroscopic dimension ranges, is also an important dimension in its own right, and we refer to this as the “single-particle level.” For many colloids, other key dimensions might also be important. For example, in a polymer system, we may wish to speak of a dimension level related to the length of the polymer chains. In a flocculated system of particles, a dimension related to the floc size may be useful. For a given colloid, we refer to the complete set of all important dimensions as its “dimensional hierarchy.” The design of a suitable “model colloid,” and the related theory, must somehow reflect the “dimensional hierarchy” of the real system. The model might place more emphasis on one dimension than on another; it depends on the final goal of the characterization. As an example, consider a dispersion of biological cells in saline. We might create an elaborate model at the single-particle level by modeling the cells as multilayer spheres with built-in ionic pumps, etc., while very simply at the “microscopic level” by modeling the saline dispersion medium as a continuum with a certain viscosity and density.

Fundamentals of Interface and Colloid Science

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However, there is one important dimension common to all colloids, namely the thickness of the interfacial layer. The behavior of this boundary layer plays a key role in many colloid-related phenomena and must also be adequately reflected in the design of the “model colloid.” The complexity of real colloids, as well as the corresponding diversity in the dimensional hierarchy necessary to depict these colloids, makes it practically impossible to provide a description of all the colloid models that have been proposed over the years. Historically, the more common models include the microscopic level to describe the dispersion medium, the single-particle level to characterize the disperse phase, and the interfacial level to describe the interface boundary layers. We present below some general models that have been developed during the last two centuries for each of these three levels. It is important to stress that there are two distinctively different models of liquids and colloids: “homogeneous” and “heterogeneous.” The differences between homogeneous and heterogeneous models are results of our perception and goals of the characterization procedure. Practically, any liquids and colloids can be homogeneous from some aspects and heterogeneous from the others. For instance, water becomes heterogeneous if we are interested in the interaction of water molecules between themselves. We will show that acoustics yields the possibility of describing complex systems using both homogeneous and heterogeneous models.

2.2. PARTICULATES AND POROUS SYSTEMS Complex heterogeneous systems represent the main class of objects that are being considered in this book. These systems have distinctively different phases—dispersion medium and dispersed phase. There is possibility of splitting this group of objects into two groups depending on the state of the continuous phase. If continuous phase is liquid, then corresponding systems could be referred to with a common name of “particulates.” Dispersed phase in this case is finitely divided on small pieces—particles, which float in the liquid dispersion medium. An alternative situation exists in porous bodies with a continuous solid matrix, which represents essentially a solid dispersion medium. This matrix could have voids with dimensions in the above-specified range. These voids (pores) can be filled with a gas or liquid. There are also systems that bear features of both particulates and porous bodies. Dispersion of porous chromatographic silica particles is an example of particulate dispersion with porous particles. There is also the possibility of smooth transition from a particulate system to a porous system. For instance, this occurs when sedimenting particles form a deposit. A deposit is essentially a porous body. Sedimentation transforms a particulate system into a porous one. Ultrasound offers the possibility of studying both of them simultaneously and learning important information from such a transition.

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2.3. PARAMETERS OF THE MODEL DISPERSION MEDIUM Although we have restricted our attention to colloids having only liquid media, we need not limit ourselves to pure liquids. The dispersion medium might be a mixture of different liquids, a liquid with many dissolved ion species, or some dissolved polymers. We can even include in our medium definition the possibility that the medium itself might be a colloid. For example, consider the case of kaolin clay dispersed in water and stabilized by the addition of latex particles. In reality, this suspension contains two types of dispersed particles: kaolin and latex. However, an alternative view would be to consider the kaolin particles dispersed in a new latex/water “effective dispersion medium.” This reduces a complex, three-phase colloid to a much simpler two-phase colloid, but the price that we must pay is a more complex model for this “effective dispersion medium.” Basically, we are playing here with two concepts: a homogeneous medium and a heterogeneous medium. Depending on the dimensional scale of the effect of interest, we can consider everything with lesser dimensions as the homogeneous medium. At the same time, the colloid system is described as a heterogeneous medium. Let us assume first that the dispersion medium is a simple, pure liquid. For clarity, we will organize the parameters that characterize this pure liquid into six classes: gravimetric, rheological, acoustic, thermodynamic, electrodynamic, and electroacoustic. To some extent, all of these parameters are temperature dependent; this dependence is strong for some parameters and negligible for others. Some of these parameters also depend on the frequency of the ultrasound, which we will discuss in the following text. The effect of the chemical composition of the liquid on these properties is considered in the last section of this chapter.

2.3.1. Gravimetric Parameters The density of the dispersing medium, rm, is an intensive (i.e., independent on the sample volume) parameter that characterizes the liquid mass per unit volume and is expressed in units of kg/m3. Since the volume of the medium changes with temperature, the density is also temperature dependent. There are some empirical expressions for this temperature dependency [3]. The density of the medium is assumed to be independent of the ultrasound frequency.

2.3.2. Rheological Parameters Viscosity and elasticity are rheological parameters of general interest for any material. Most pure liquids are primarily viscous, having little elasticity. Many traditional rheological instruments cannot measure the elastic

Fundamentals of Interface and Colloid Science

27

component, or, if possible, can measure it only at low frequencies. Elasticity yields information about structure. Small elasticity implies a weak structure. Acoustics provides a means to measure the elastic properties of liquids and to characterize their structure at much higher frequencies than those available with more traditional methods [4–6]. The elastic properties of the liquid are defined by a bulk storage modulus, Mm, which is the reciprocal of the liquid compressibility, bpm . This compressibility is the change in the liquid volume due to a unit variation of pressure. It is usually assumed that the liquid exhibits no yield stress. The SI unit of the bulk modulus is Pascal (Pa), which is equal to 0.1 dyne cm2 in CGS units. The viscous properties of the liquid are characterized by two dynamic viscosity coefficients: a shear dynamic viscosity,
m, and a volume dynamic viscosity, hvm [7]. The volume viscosity appears only in phenomena related to the compressibility of the liquid. All effects that can be described assuming an incompressible liquid are independent of the volume viscosity. This is why the volume viscosity is hard to measure and, therefore, is usually assumed to be zero. Ultrasonic attenuation is the only known way to measure volume viscosity [6]. It turns out that all liquids exhibit a volume viscosity. For example, the volume viscosity of water is almost three times larger than the shear viscosity. The volume viscosity depends on the structure of the liquid. Hence acoustic attenuation measurements potentially provide important information about the liquid structure. The SI unit for these dynamic viscosities is [Pa s], which is equal to 1000 centipoise (cP) in CGS units. Rheological parameters usually depend on temperature; some empirical expressions for this dependency can be found in the Handbook of Chemistry and Physics [3]. These parameters are independent of the frequency of the ultrasound for simple Newtonian liquids.

2.3.3. Acoustic Parameters Rheological and acoustic properties are closely related; each provides a different perspective of the viscoelelastic nature of the fluid. There are two key acoustic parameters: the attenuation coefficient, am, and the sound speed, cm. We have already described two key rheological parameters: the bulk storage modulus, Mm, and the dynamic viscosity,
m. The bulk modulus and sound speed both characterize the elastic nature of the liquid; the dynamic viscosity and attenuation both characterize its viscous, or dissipative, nature. Elastic and dissipative characteristics can be combined into one complex parameter. In acoustics, this complex parameter is referred to as a “compression complex wave number,” kc,m. The attenuation and sound speed are

28

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related to the imaginary and real parts of this complex wave number by the expressions: am ¼  Imkc,m cm ¼

o Rekc,m

ð2:1Þ ð2:2Þ

The unit of sound speed is [m/s]. The attenuation can be expressed in several different ways as discussed in Chapter 3. In all of the following text, we define attenuation in units of [dB/cm MHz]. Another complex parameter, also widely used in acoustics, is the acoustic impedance, Zm. It is defined as  
cm am  am l ð2:3Þ ¼ cm rm 1  j Zm ¼ cm rm 1  j o 2p where l is a wavelength of the ultrasound in the liquid. The attenuation is almost independent of variations in temperature, whereas the sound speed is very sensitive to temperature. For example, the sound speed of water changes 2.4 m/s per degree Celsius at room temperature. This is an important distinction between these two parameters. The frequency dependence of these two parameters is also quite different. For simple Newtonian liquids, the sound speed is practically independent of frequency, whereas the attenuation is roughly proportional to frequency, at least over the frequency range of 1–100 MHz (see Chapter 3 for details).

2.3.4. Thermodynamic Parameters There are three thermodynamic parameters that are important in characterizing the thermal effects in a colloid: thermal conductivity, tm [mW/cm  C], heat –4   capacity Cm p [J/g C], and thermal expansion bm [10 1/ C]. Fortunately, it turns out that for almost all liquids, except water, tm is virtually the same, and Cm p can be approximated by a simple function of the material density. Figures 2.1 and 2.2 illustrate the variation of both properties for more than 100 liquids; the data is taken from the paper of Anson and Chivers [8] as well as our own database. This reduces the number of required thermodynamic properties to just one, namely the thermal expansion coefficient. In principle, all thermodynamic parameters are temperature dependent, but independent of the ultrasound frequency.

2.3.5. Electrodynamic Parameters The two essential electrodynamic parameters that characterize the response of the liquid to an applied electric field are conductivity, Km, and dielectric permittivity, e0em.

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Fundamentals of Interface and Colloid Science

5 water

Cp [J/g k]

4 3 2 1 0 0.5

1.0

1.5

2.0 density [g/cu.cm]

2.5

3.0

3.5

FIGURE 2.1 Heat capacity at constant pressure for various liquids, Anson and Chivers [8].

thermal conductivity [mW/cm K]

7 water

6 5 4 3 2 1 0

0.5

1.0

1.5

2.0 density [g/cu.cm]

2.5

3.0

3.5

FIGURE 2.2 Thermal conductance for various liquids, Anson and Chivers [8].

The SI unit of conductivity is S/m [C/(s V m)] and of dielectric permittivity [C2/(Nm2)]. The relative dielectric permittivity, em, (dielectric constant) of the liquid is the dielectric permittivity of the medium divided by that of vacuum, e0, and is dimensionless. The dielectric permittivity of a vacuum, e0, is 8.8542  1012 C2/N m2. In principle, both the conductivity and dielectric permittivity are frequency dependent, even for a pure liquid. However, for the typical frequency range of acoustic measurements (1–100 MHz), both parameters are usually assumed to be independent of frequency.

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The temperature dependence of the dielectric permittivity for some liquids is given in The Handbook of Chemistry and Physics [3]. These two parameters can be combined into one complex number (either a  complex dielectric permittivity, em , or a complex conductivity, Km ). This simplifies the calculations with respect to an applied alternating electric field:  ¼ Km  joe0 em Km

  Km em ¼ em 1 þ j oem e0

ð2:4Þ ð2:5Þ

2.3.6. Electroacoustic Parameters There are two reciprocal electroacoustic effects that can be differentiated depending on the nature of the driving force. For ultrasound as the driving force, we speak of the ion vibration current as representative of the electroacoustic effect. The ion vibration current is an alternating current generated by the motion of ions in the sound field and is characterized by a magnitude, IVI, and phase, Fivi. The IVI and phase, when normalized by the magnitude and phase of the acoustic pressure, can be considered to be the electroacoustic properties of the dispersion medium. In the reverse case, when an electric field is the driving force, the resultant ultrasound signal, called electrosonic amplitude [9, 10], represents the electroacoustic effect. It is also characterized by a magnitude, ESA, and phase, Fesa. These also can be used as electroacoustic properties of the dispersion medium. The magnitude of both electroacoustic effects is much less sensitive to temperature than to phase. Both IVI and ESA can be considered to be frequency independent within the megahertz frequency range. Both electroacoustic effects are strongly dependent on the chemical composition of the liquid.

2.3.7. Chemical Composition The dispersion medium may contain many different chemical species. The collection of specified amounts of these species is called its chemical composition. Each species is characterized with an electrochemical potential, mi, given by: mi ¼ m0i þ 2RT lnyi ci þ zi Ff

ð2:6Þ

where m0i is a standard chemical potential that describes the interaction between species i and the liquid, ci is the concentration of that species, yi is an activity coefficient, which depends on the interaction between species, R is the Faraday constant, T is the absolute temperature, zi is the valence of that species if electrically charged, and f is an electric potential. Finally, there are three additional parameters that characterize the dynamic response of the solution. The best known is the diffusion coefficient, Di.

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In addition, we will need the mass, mi, and volume, Vi, of any solvated particles, especially for electrically charged species. These last two parameters play an important role in the electroacoustics of solvated ions. Although all properties of the dispersion medium depend on this chemical composition, the degree of this dependence is not the same for each property. In many cases, we can neglect the chemical composition. For instance, the viscosity of a liquid depends on the concentration of ions. The continuous adjustment of the hydration layer upon ion displacement leads to friction and hence to energy dissipation. This process is known as dielectric drag. For very small ions, this friction may well be stronger than for larger ions. Consequently, the viscosity,
s, of an aqueous electrolyte solution increases with ion concentration, c, and generally follows the Jones-Dole law:
s ¼ 1 þ Ac0:5 þ Bc þ   
w where
w is the viscosity of water. However, the parameters A and B are rather small and we can usually neglect this effect when the concentration of ions is low. For example, the viscosity of 1 M KCl (aq.) decreases only by 1% compared to that of pure water [3]. This does not mean that we can always neglect the influence of the chemical composition on the viscosity of the medium. Many chemicals strongly affect the viscosity of the medium, and the dimension of these viscositymodifying species is very important. In calculating particle size from acoustic data, the relevant viscosity to be used is the one experienced by the particle as it moves in response to the sound wave. In the case of gels, or other structured systems, this “microviscosity” may be significantly less than the “macroviscosity” measured with conventional rheometers. This example of viscosity thus highlights the complexity regarding the influence of chemical composition on a particular property of the dispersion medium. In general, there is no theoretically justified answer that will be valid in all cases. However, in many situations empirical experience does provide a guide. From our experience, the chemical composition is always important when considering sound speed, conductivity, and the electroacoustic signal. As a result, these are good candidates for studying the properties of ions, molecules, and other simple species in liquids. Conversely, attenuation is the least sensitive to chemical composition. This makes the attenuation measurement a very good candidate for a robust particle sizing technique.

2.3.8. Electrochemical Composition of Aqueous and Nonaqueous Solutions On reviewing the existing literature, we have discovered that despite giving a quite detailed model of aqueous ionic solutions, modern electrochemistry still faces problem with low- and nonpolar liquids. For instance, the main reviews

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on this subject [11–18] do not provide a simple instruction on how to control ionic composition of the nonpolar liquids, how to measure ions concentration, and how to characterize ionic properties that would be needed for describing the double-layer structure. Complexity of these questions has been recognized for a long time. For instance, Lyklema wrote 40 years ago that “. . .electrolyte concentrations in apolar solvents are low and ill-defined. . ..” [17]. This statement remains valid to date to a certain degree. Uncertainty in the electrolyte concentration creates a profound problem for electrokinetcs in nonpolar liquids. For instance, it creates problem in calculating the Debye length, 1/k. Major differences in electrochemistry of aqueous and nonaqueous solutions are related to dielectric permittivity. It is known that water is an abnormal liquid in many aspects. It has very high dielectric permittivity, around 80. Conductivity of water-based solutions is also high. It varies from roughly 10–3 to 10 S/m depending on the concentration of the electrolyte. These peculiarities of water are related to the high polarity of its molecules. There are many other liquids with much lower dielectric permittivity, around 2, and much smaller conductivity, on the scale of 10–10 S/m. Here are some examples: diesel, kerosene, choroform, cyclohexane, decane, hexane, heptane, hexadecane, toluene, and all varieties of oils. These hydrocarbon liquids are practically nonpolar. Lyklema and van der Hoeven [12] suggested four classes of liquids based on the value of dielectric permittivity: l l l l

nonpolar if e < 5 low-polar 5 < e < 12 semipolar e > 12 polar e ffi 80

Dielectric permittivity affects strongly the ionization of various chemical species in liquid. Existence of ions in a liquid is the result of the balance between electrostatic attractive energy and thermal motion, as stated by the Bjerrum theory [14, 16]. This theory simply compares the attractive energy of the cation-anion interaction, E, with the energy of thermal motion, kT, where k is the Boltzman constant and T is the absolute temperature. This theory predicts that ions might exist separately only if their size aion exceeds a certain value, the so-called Bjerrum radius aB. Otherwise, they would build cationanion pairs because the thermal motion would not be able to prevent their aggregation at small distances. This condition can be formulated as the following nonequality: aion > aB ¼

e2 8pee0 kT

ð2:7Þ

Fundamentals of Interface and Colloid Science

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Statistically, some ions would exist even if their sizes are below the Bjerrum radius because there is some equilibrium distribution between free ions and pairs. There is a theory describing this distribution by Fuoss [15]. It yields the following equation for the dissociation constant aion:  2 2  4pd3 ze 2 n0 f exp ð2:8Þ aion ¼ 1  3 dee0 kT where d is the center-to-center distance in the ion pair, n0 is the number of ions per unit volume, z is valency, and f is the activity coefficient which is close to 1 in dilute systems. Simple calculations led Morrison [13] to the conclusion that the critical ion diameter is 0.7 nm in water and 28 nm in low-conducting media with a low dielectric permittivity of around 2. In both cases, the critical Bjerrum radius exceeds significantly the size of the low molecular weight atoms. In aqueous systems, the size of the ions builds because of the adsorption of water molecules. They are polar with substantial dipole moment, which can build a structure around the ion under the influence of its electric field. This hydration shell around the ions prevents them in coming close enough for building an ionic pair. They are sterically stabilized, as mentioned by Morrison [13]. A low-conducting nonpolar medium does not allow such steric stabilization because molecules of the liquid do not have dipole moment and are incapable of building protective shells around the ions. There are two opportunities for the ions to exist in these conditions. According to Kitahara [11], ions can be present in the nonpolar media if l l

soluble molecules dissociate into large ions; dissociation occurs with further solubilization of the simple, small ions in larger micelles.

Both approaches are used for controlling conductivity of nonaqueous solutions. The first one is usually associated with ionic surfactants. The second approach opens an interesting opportunity of using nonionic surfactants for building a protective solvating shell around ions. Figure 2.3 illustrates this idea. The surfactant’s polar parts would be attracted by the ions. This attraction builds up a surfactant shell around the ions. Ions become encapsulated into the reverse micelle. It is important to mention here that this micelle is quite different from a regular reverse micelle built by the attraction of the polar groups. It has an ion at the center. This could make this type of micelles quite different. It is possible also that there would be difference between micelles with cations and anions. They require opposite orientation of the polar group. It might happen that the surfactant structure would not allow either cations or anions reaching the required pole of the surfactant dipole moment. In this

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Cation micelle in oil

Ion in water

−

−

2

− + +

−

+ +

+

−

−

+

−

+

+

− −

+

Typical micelle in oil Anion micelle in oil

− −

+ + + +

−

−

− −

+ +

−

+ +

−

−

FIGURE 2.3 Various solvation structures in water and oil.

case, the surfactant would stabilize only one type of ions. The other type of ions remains bare. It would be possible for it to exist in small size because of the lack of accessible ions with the opposite charge. They are screened by a large shell of the surfactant. This hypothesis has found some experimental confirmation [19]. This field is far from complete in our understanding. It is clear that properties of ions in nonpolar liquids are rather difficult to get. There is practically no data on their diffusion coefficients. Very little is known about the means of controlling their concentration. It turns out that electroacoustics can be useful for shedding some light on this problem, as will be discussed in chapters dedicated to its applications.

2.4. PARAMETERS OF THE MODEL DISPERSED PHASE First, any model for particles in the dispersed phase must describe the properties of a single particle: its shape, physical and chemical properties, etc. The model must also reflect any possible polydispersity, that is, the variation in properties from one particle to another. This is a rather difficult task. Fortunately, at least with particle shape, there is one factor that helps— Brownian motion. Colloidal particles are generally irregular in shape, as illustrated in Figure 2.4. There are of course a small number of exceptions such as

35

Fundamentals of Interface and Colloid Science

+

Real System

- roughness - heterogeneous (mosaic) surface - hairness

-

-

-

+

-

-

- nonspherical

+

-

-

+

+

system modeling error

-

-

Model System

-

- spherical - smooth - homogeneous surface

-

-

-

FIGURE 2.4 Real and model spherical particle.

emulsions or latices. All colloidal particles experience Brownian motion which constantly changes their position. A natural averaging occurs over time; the resulting time-averaged particle looks like a sphere with a certain “equivalent diameter,” d. It is for this reason that a sphere is the most widely and successfully used model for particle shape. Obviously, a spherical model does not work well for very asymmetric particles. Here, an ellipsoidal model can yield more useful results [20–22]. This adds another geometrical property: an aspect ratio, the ratio of the longest dimension to the shortest dimension. For such asymmetric particles, the orientation also becomes important, bringing further complexity to the problem. Here we will simply assume that the particles can be adequately represented by spheres. This assumption affects ultrasound absorption much more difficult than ultrasound scattering, as is the case with light scattering. According to both our experience and preliminary calculations for ultrasound absorption [22], this assumption of a sphere is valid for aspect ratios less than 5:1. The material of the dispersed phase particles is characterized with the same set of gravimetric, rheological, thermodynamic, electrodynamic, acoustic, and electroacoustic parameters as the material of the dispersion medium described in the previous section. We will use the same symbols for these parameters, changing only the subscript to p. The fact that we define so many properties for both the medium and the particles does not mean that in order to derive useful information from the

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measured acoustic or electroacoustic measurement, we need to know this complete set for a particular dispersion. Later, in Chapter 4, we will outline the set of input properties that are necessary to constructively use ultrasound measuring techniques. The exact composition of this required set depends on the nature of the particles, particularly on their rheology. In this regard, it turns out that it is very useful to divide all particles into just two classes, namely rigid and soft.

2.4.1. Rigid Versus Soft Particles Historically, Morse and Ingard [5] introduced these two classes of rigid and nonrigid (soft) particles to the acoustic theory for describing ultrasound scattering. In the case of scattering, the notion of a rigid particle serves as an extreme but not real case. It turns out that this classification is even more important in the case of ultrasound absorption. In the case of ultrasound absorption, the notion of rigid particles corresponds to real colloids, such as all kinds of minerals, oxides, as well as inorganic and organic pigments: indeed, virtually everything except emulsions, latices and biological material. Rigid particles dissipate (absorb) ultrasound without changing their shape under either an applied sound field or an electric field. For interpreting ultrasound absorption caused by rigid particles, we need to know, primarily, the density of the material. There is no need to know any other gravimetric or acoustic properties or, in fact, any rheological, thermodynamic, electrodynamic, or electroacoustic properties at all. Furthermore, the sound speed in these rigid particles is, usually, close to 6000 m/s. Thus, in most cases, we can simply assume that this same value is appropriate. The sound attenuation inside the body of these particles is usually negligible. We do not need to know the electrodynamic properties, except in rare cases of conducting and semiconducting particles. It is important to mention here that submicrometer metal particles are not truly conducting. There is not much difference in electric potential between the particle poles to conduct an electrochemical reaction across the small distance between these poles. Such submicrometer particles represent a special case of the so-called “ideally polarized particles” [23]. These empirical observations reduce the number of parameters required to characterize rigid particles to just one, namely the density. It is also important to mention that most often the density of these particles significantly exceeds that of the liquid medium. The situation with soft particles is very different. First, the density of these softer particles quite often closely matches that of the dispersion medium. As a result, all effects that depend on the density contrast are minimized. Instead, the thermodynamic effects are much more important, and the thermal expansion coefficient of the particle and suspending medium replace the density contrast as a critical parameter for ultrasound absorption.

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Fundamentals of Interface and Colloid Science

So far, we have talked about particles that are internally homogeneous, constructed from the same material. However, there are some particles that are heterogeneous inside, consisting of different materials at different points within their bodies. Usually a “shell model” is used to describe the properties of these complex particles [24, 25]. Occasionally, a simple averaging of properties over the particle volume suffices, as for instance, using an average density.

2.4.2. Solid Versus Fractal and Porous Particles There are three different models that can be used for describing dispersed particles depending on their morphology. The most obvious and widely used is simply a model of “separate homogeneous solid particles.” This is the usual model for characterizing dispersions and emulsions. We simply assume that particles consist only of the dispersed phase material. This assumption yields a simple relationship between densities and weight fractions: rp ¼ rs wp ¼ ws where the index p corresponds to the particles and index s to the solid material. An alternative model of porous particles takes into account the possibility that part of the liquid would be trapped inside the particles with complex shape. This might occur either for naturally porous materials or for aggregate particles that are built up from some primary solid particles. This model introduces an additional parameter for characterizing particles—porosity, w. It is the volume fraction of the dispersion medium trapped inside of the particle. This model yields the following equations for the relationship between densities and weight fractions: rp ¼ ð1  Oe Þðrs  rm Þ þ rm fp ¼

fs w s rm ¼ 1  w ð1  wÞ½ws rm þ ð1  ws Þrs  

wrm wp ¼ w s 1 þ ð1  wÞrs

ð2:9Þ ð2:10Þ

 ð2:11Þ

The nature of this model eliminates possibility of advection—liquid motion through the particle, because we consider liquid inside of the pores trapped and moving with the particle. There is another model for porous particles that assumes certain symmetry in their structure. It is the “fractal model.”

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Fractals are aggregates of primary dispersed particles with a certain structural symmetry that reproduces itself from one spatial level to another. The notion of fractals was introduced almost 30 years ago by Mandelbrot [26–28] and is a widely accepted model for describing coagulation phenomena. There is a simple relationship between the radius of an aggregate that contains i particles (ai) and radius of the primary particles a1 given by ai ¼ kf i1=df a1

ð2:12Þ

The two parameters in this equation describe various types of aggregate structures. The parameter kf is less important, as its value is typically very close to 1, and we therefore omit considering it in our further analysis. The parameter df is more important and usually goes by the name of “fractal dimension.” For typical dispersions, this number varies from roughly 1 to 3. It equals 3 for coalescing emulsion droplets. The range of 2.1–2.2 corresponds to reaction-limited coagulation, which results in rather compact aggregates. Smaller values of this fractal dimension, less than 1.8, are typical of diffusion-limited coagulation with loose flocs. Finally, a fractal dimension equal to 1 represents linear chain aggregates [29–32]. Fractal aggregates contain a certain amount of trapped liquid inside. The amount of such trapped liquid can be quantified by one of three different parameters. The first way to describe this trapped liquid is by the porosity, w, which represents the volume ratio of the liquid and solid phases inside the aggregate:  3þdf ai w ¼ 1  fin ¼ 1  ð2:13Þ a1 where ’in is volume fraction of the solid phase inside the aggregate. An alternative parameter that can be used to describe the trapped liquid is the density of the aggregate, ragg, which is different from the density of the primary particles, rp, or the density of the liquid medium, rm, and is given by  3þdf ai ragg ¼ ðrp  rm Þ þ rm ð2:14Þ a1 A larger volume of aggregates leads to a larger volume fraction of the dispersed phase, ’agg, in the dispersion compared to the volume fraction of solids ’solids:  3df fagg ai ¼ ð2:15Þ fsolids a1 The weight fraction is given by the following equation: wp ¼

fp rp rm þ fp ðrp  rm Þ

ð2:16Þ

Fundamentals of Interface and Colloid Science

39

Of the three defined volume fractions, only ’solids is easily measurable with a pycnometer. Neither the aggregate volume fraction ’agg nor the volume fraction of solids inside the aggregate, ’in, is measurable with a pycnometer. The aggregates density, ragg, is also not easily measurable. However, there are known means for measuring the aggregates size. Light scattering offers one opportunity, as described in the literature [10, 20]. Acoustic attenuation spectroscopy is also suitable for this purpose as will be discussed in detail in this chapter. We will show that acoustics yields information on the aggregate size practically independently on the fractal number. There are many theoretical and experimental works dedicated to the hydrodynamic and mechanical properties of fractals, but very little is known about their electrical and, especially, their electrokinetic properties. This model ignores advection as well. This model requires also information on the primary size. It is possible to use this parameter as adjustable, instead of a standard deviation. This means that for the purpose of the fitting attenuation spectra we consider a monodisperse collection of aggregates with the same size and the same primary size. These two numbers are adjustable parameters instead of the median size and standard deviation of the lognormal particle size distribution (PSD).

2.4.3. Porous Body A porous body differs from a porous particle because of its large external dimension. It is assumed that it contains voids—pores with much smaller dimensions on micro and nanoscales. According to Lowell and others [33], pores can be classified as “micropores,” “mesopores,” and “macropores.” The internal width of micropores is less than 2 nm, mesopores between 2 and 50 nm, and macropores above 50 nm. This most known theory of sound propagation through a wet, porous body was developed by Biot in 1956 [34, 35], who mentioned the earlier work 10 years previously of Frenkel [36] as forming the essential scientific basis. The Biot theory is quite general and uses a number of input parameters. The total set consists of following physical properties of the solid matrix and liquid: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Density of sediment grains Bulk modulus of the grains Density of the pore fluid Bulk modulus of the pore fluid Viscosity of the pore fluid Porosity Pore size parameter Dynamic permeability Structure factor Complex shear modulus of the frame Complex bulk modulus of the frame

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The last four properties present the most problem in the application of the Biot theory. This was discussed in detail by Ogushwitz [37], who proposed several empirical and semiempirical methods for calculating these parameters. None of these methods is general. In some cases, it involves simply the substitution of one parameter with another unknown constant. This problem was also discussed by Barret-Guitepe et al. [38] in their study of compressibility of colloids. They talk about the importance of the “skeleton effect” and difficulty of measuring the required input parameters independently.

2.4.4. Particle Size Distribution Colloidal systems are generally of a polydisperse in nature, that is, the particles in a particular sample vary in size. Aggregation, for instance, usually leads to the formation of polydisperse sols, mainly because the formation of new nuclei and the growth of established nuclei occur simultaneously, and so the particles finally formed are grown from nuclei formed at different times. In order to characterize this polydispersity, we use the well-known concept of PSD. We define this here following Leschonski [39] and Irani [40]. There are multiple ways to represent the PSD depending on the physical principles and properties used to determine the particle size. A convenient general scheme proposed by Leschonski is reproduced in Table 2.1. The independent variable (abscissa) describes the physical property chosen to characterize the size, whereas the dependent variable (ordinate) characterizes the type and measure of the quantity. The different relative amounts of particles measured in certain size intervals form a so-called density distribution, qr(X), which represents the first derivative of the cumulative distribution, Qr (X). The subscript, r, indicates the type of the quantity chosen. Figures 2.5 and 2.6 illustrate the density and cumulative PSDs. The parameter, X, characterizes a physical property uniquely related to the particle size.

TABLE 2.1 Representation of PSD Data Type of quantity

Measure of quantity Cumulative Qr

r ðXÞ Density qr ðXÞ ¼ dQdX

r ¼ 0: number

Cumulative PSD on number basis

Density PSD on number basis

r ¼ 1: length

Figure 2.5

Figure 2.4

r ¼ 2: area

Cumulative PSD on area basis

Density PSD on area basis

r ¼ 3: volume, weight

Cumulative PSD on weight basis

Density PSD on weight basis

41

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Fundamentals of Interface and Colloid Science

qr (X)

DQr = qr (X) DX

DX

Xmin

Xmax

size [length]

FIGURE 2.5 Density particle size distribution.

1.0

Qr

Qr(X)

0.5

DQr(X1,X2) Qr(X2) Qr(X1)

0.0

DX Xmin

X1 X2 X50 size

Xmax

FIGURE 2.6 Cumulative particle size distribution.

It is assumed that a unique relationship exists between the physical property and a one-dimensional property unequivocally defining “size.” This can be achieved only approximately. For irregular particles, the concept of an “equivalent diameter” allows one to characterize irregular particles. This is the diameter of a sphere that yields the same value of any given

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physical property when analyzed under the same conditions as the irregularly shaped particle. The cumulative distribution Qr (X) in Figure 2.6 is normalized. This distribution determines the particles (in number, length, area, weight, or volume depending on r) that are smaller than the equivalent diameter X. Figure 2.7 shows the normalized discrete density distribution or histogram qr(Xi, Xiþ1). This distribution specifies the amount of particles having diameters larger than Xi and smaller than Xiþ1 and is given by: qr ðXi , Xiþ1 Þ ¼

Qr ðXiþ1 Þ  Qr ðXi Þ Xiþ1  Xi

ð2:17Þ

The shaded area represents the relative amount of the particles. The histogram transforms to a continuous density distribution when the thickness of the histogram column tends to zero. It is presented in Figure 2.7 and can be expressed as the first derivative from the cumulative distribution: dQr ðXÞ ð2:18Þ dX A histogram is suitable for presentation of the PSD when the value of each particle size fraction is known. It is the so-called full PSD. It can be measured using either counting or fractionation techniques. However, in many cases, full information about the PSD is either not available or not even required. For instance, Figure 2.8 represents the histograms with a nonmonotonic and nonsmooth variation of the column size. This might be related to the fluctuations and have nothing to do with the statistically representative PSD

qr

qr ðXÞ ¼

qr(Xi,Xi+1)

Xmin

Xi Xi+1

size

FIGURE 2.7 Discrete density particle size distribution.

Xmax

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Fundamentals of Interface and Colloid Science

qr (X)

modified log-normal distribution

histogram

Xmin

Xmax

size

FIGURE 2.8 Transition from the discrete to the continuous distribution.

for this sample. That is why in many cases histograms are replaced with various analytical PSDs. There are several analytical PSDs that approximately describe empirically the determined PSDs. One of the most useful, and widely used, distributions is the lognormal distribution. In general [40], it can be assumed that the size, X, of a particle grows, or diminishes, according to the following relationship: dX ðX  Xmin ÞðXmax  XÞ ¼ const  dt ðXmax  Xmin Þ

ð2:19Þ

This equation reflects the obvious fact that, when d approaches either Xmax, or Xmin, it becomes independent of time. Using the assumptions of the normal distribution of growth and destruction times, and combining with Equation (2.19), leads to: 0 2 1 ðX  Xmin ÞðXmax  Xmin Þ32 B 6 ln 1 ðXmax  XÞX 7 C pffiffiffi qr ðXÞ ¼ pffiffiffiffiffiffi ð2:20Þ expB 4 5 C @ A 2p lns 2 lns l

l

! where X and sl are two unknown constants that can have certain physical meaning, especially for the standard lognormal PSD (Equation 2.21). The distribution given by Equation (2.20) is called the modified lognormal distribution. It is not symmetrical on a logarithmic scale of particle sizes. One example of a modified lognormal PSD is shown in Figure 2.8. For the case when Xmin ¼ 0 and Xmax ¼ 1, the modified lognormal PSD reduces to the standard lognormal PSD: 8 2 39 X 2> > > ln = < 6 !7> 1 X 7 p ffiffi ffi ð2:21Þ exp 6 qr ðXÞ ¼ pffiffiffiffiffiffi 4 5 > 2 lnsl > 2p lnsl > > ; :

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! The parameters X and sl are then the geometrical median size and geometrical standard deviation. The median size corresponds to the 50% point on the cumulative curve. The standard deviation characterizes the width of the distribution. It is smallest (sl ¼ 1) for a monodisperse PSD. It can be defined as the ratio of the size at 15.87% cumulative probability to that at 50%, or the ratio at 50% probability to that at 84.13%. These points, at 16% and 84% roughly, are usually reported together with the median size at 50%. Lognormal and modified lognormal distributions require that we extract only a few parameters (two and four, respectively) from the experimental data. This is a big advantage in many cases when the amount of experimental data is restricted. These two distributions plus a bimodal PSD as the combination of two lognormals are usually sufficient for characterizing the essential features of the vast majority of practical dispersions.

2.5. PARAMETERS OF THE MODEL INTERFACIAL LAYER In the previous two sections we have introduced a set of properties to characterize both the dispersion medium and the dispersed particles. It is clear that the same properties might have quite different values in the particle as compared to the medium. For example, in an oil-in-water emulsion, the viscosity and density inside the oil droplet would be quite different from their values in the bulk of the water medium. However, this does not mean that these parameters change stepwise at the water-oil interface. There is a certain transition (or interfacial) layer, where the properties vary smoothly from one phase to the other. Unfortunately, there is no thermodynamic means to decide where one phase or the other begins. A convention suggested by Gibbs resolves this issue [1], but the details of this rather complicated thermodynamic problem are beyond the scope of this book. We wish to discuss here only the electrochemical aspects of this interfacial layer and these are generally combined as the concept of the “electric double layer” (DL). Lyklema [1] made a most comprehensive review of this concept, which plays a very important role in practically all aspects of colloid science. We present here a short overview of the most essential features of the DL, with particular emphasis on those that can be characterized using ultrasound-based technology. We will consider the DL in two states: equilibrium and polarized. We use term “equilibrium” as a substitute for the term “relaxed” used by Lyklema [1]. (The term relaxed is somewhat more general, and might include situations when the DL is created with nonequilibrium factors. For instance, the DL of living biological cells might have a component that is related to the nonequilibrium transmembrane potential [25, 41].) For the first equilibrium state, the DL exists in an undisturbed dispersion characterized by a minimum value of the free energy. The second “polarized” state reflects a deviation from this equilibrium state due to some external disturbance such as an electric field.

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Fundamentals of Interface and Colloid Science

According to Lyklema: “. . .the reason for the formation of a ‘relaxed’ (‘equilibrium’) double layer is the nonelectric affinity of charge-determining ions for a surface. . .” This process leads to the buildup of an “electric surface charge,” s, expressed usually in mC/cm2. This surface charge creates an electrostatic field which then affects the ions in the bulk of the liquid (Figure 2.9). This electrostatic field, in combination with the thermal motion of the ions, creates a countercharge and thus screens the electric surface charge. The net electric charge in this screening, diffuse layer is equal in magnitude to the net surface charge but has the opposite polarity. As a result, the complete structure is electrically neutral. Some of the counterions might specifically adsorb near the surface and build an inner sublayer, the so-called Stern layer. The outer part of the screening layer is usually called the “diffuse layer.” What about the difference between the surface charge ions and the ions adsorbed in the Stern layer? Why should we distinguish them? There is a thermodynamic justification [1], but we think a more comprehensive reason is kinetic (ability to move). The surface charge ions are assumed to be fixed to the surface (immobile); they cannot move in response to any external disturbance. In contrast, the Stern ions, in principle, retain some degree of freedom, almost as high as ions of the diffuse layer [42–46]. We give below some useful relationships of the DL theory from Lyklema’s review [1].

Stern layer Ψs

negatively charged diffuse layer Ψd

bulk of the liquid

ζ

positively charged particle

electric potential κ-1 Debye length

slipping plane Stern plane

FIGURE 2.9 Illustration of the structure of the electric double layer.

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2.5.1. Flat Surfaces The DL thickness is characterized by the so-called Debye length 1/k defined by: X ci zi k2 ¼ F2 ð2:22Þ e0 em RT i where the valences have the sign included; for a symmetrical electrolyte, zþ ¼ –z– ¼ z. Equation (2.22) allows the calculation of the Debye length only when the concentrations of the ion species are known. This information usually is not available. This justifies the introduction of approximate methods in evaluating this parameter. One suggests using the easily measurable conductivity of the liquid, Km, and then the following equation: Km ð2:23Þ e0 em Deff Conductivity of the dispersion can be used as an approximate value for the conductivity of the liquid. The main complicating error in this equation comes from the unknown effective diffusion coefficient Deff. However, this parameter varies over a limited range. For instance, the diffusion coefficients of most ions in aqueous solutions are similar and have values at room temperature in the range of 0.6  10–9 to 2  10–9 m2/s. The square root of this variation would lead to uncertainty in the scale of only a few tens of percent. The same equation can be used for estimating the Debye length in nonaqueous sultions as well. The diffusion coefficient of ions in nonaqeous systems usually is much smaller than in water as discussed in Section 2.3.8 due to the much larger ion size. It is possible to account for this variation using Equation (2.7) for Bjerrum size as an estimate of the ion size. This would imply that reverse ionic micelles form the diffuse layer in the particular nonaqueous solution. This might be not true if the hypothesis of the nonequal ion sizes in nonpolar liquids discussed in Section 2.3.8 is valid. For a flat surface and a symmetrical electrolyte of concentration, Cs, there is a straightforward relationship between the electric charge in the diffuse layer, sd, and the Stern potential, cd, namely: k2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fcd 8e0 em Cs RT sinh ð2:24Þ 2RT If the diffuse layer extends right to the surface, Equation (2.24) can then be used to relate the surface charge to the surface potential. Sometimes it is helpful to use the concept of a differential DL capacitance. For a flat surface and a symmetrical electrolyte, this capacitance is given by: sd ¼ 

Cdl ¼

ds Fcd ¼ e0 em k cosh dcc¼cs 2RT

ð2:25Þ

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Fundamentals of Interface and Colloid Science

For a symmetrical electrolyte, the electric potential, c, at the distance, x, from the flat surface to the DL is given by: zFcðxÞ 4RT expðkxÞ ¼ ð2:26Þ zFcd tanh 4RT The relationship between the electric charge and the potential over the diffuse layer is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   sd ¼ ðsigncd Þ 2e0 em Cs RT ðnþ ℓzþ cd þ n ℓz cd  nþ  n Þ1=2 ð2:27Þ tanh

where n is the number of cations and anions produced by dissociation of a  d is a dimensionless potential given by: single electrolyte molecule, and c d

 d ¼ Fc c ð2:28Þ RT In the general case of an electrolyte mixture, there is no analytical solution. However, some convenient approximations have been suggested [1, 42, 47, 48].

2.5.2. Spherical DL, Isolated and Overlapped There is only one geometric parameter in the case of a flat DL, namely the Debye length 1/k. In the case of a spherical DL, there is an additional geometric parameter, namely the radius of the particle, a. The ratio of these two parameters (ka) is a dimensionless parameter which plays an important role in colloid science. Depending on the value of ka, two asymptotic models of the DL exist. A “thin DL” model corresponds to colloids in which the DL is much thinner than the particle radius, or simply: ka  1

ð2:29Þ

The vast majority of aqueous dispersions satisfy this condition, except for very small particles in low ionic strength media. If we assume an ionic strength greater than 10–3 mol/l, corresponding to the majority of most natural aqueous systems, the condition ka  1 is satisfied for virtually all particles having a size above 100 nm. The opposite case of a “thick DL” corresponds to systems where the DL is much larger than the particle radius, or simply: ka 1

ð2:30Þ

The vast majority of dispersions in hydrocarbon media, having inherently very low ionic strength, satisfy this condition.

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These two asymptotic cases allow one to picture, at least approximately, the DL structure around spherical particles. A general analytical solution exists only for low potential: RT 25:8 mV ð2:31Þ F This so-called Debye-Hu¨ckel approximation yields the following expression for the electric potential in the spherical DL, c(r), at a distance, r, from the particle center: cd

a ð2:32Þ cðrÞ ¼ cd expðkðr  aÞÞ r The relationship between the diffuse charge and the Stern potential is then:   1 ð2:33Þ sd ¼ e0 em kcd 1 þ ka This Debye-Hu¨ckel approximation is valid for any value of ka, but this is somewhat misleading since it covers only isolated double layers. The approximation does not take into account the obvious probability of an overlap of double layers as in a concentrated suspension, that is, high volume fraction. A simple estimate of this critical volume fraction, ’over, is the volume fraction for which the Debye length is equal to the shortest distance between the particles. Thus: ’over

0:52

ð2:34Þ

1 3 1 þ ka

This dependence is illustrated in Figure 2.10. 0.5

vfr of DL overlap

0.4 0.3 0.2 0.1 0.0 0.01

0.10

1.00 ka

10.00

100.00

FIGURE 2.10 Estimation of the volume fraction of the overlap of the electric double layer.

Fundamentals of Interface and Colloid Science

49

It is clear that for ka  10 (thin DL) we can consider the DLs as isolated entities even up to volume fractions of 0.4. However, the model for an isolated DL becomes somewhat meaningless for small ka (thick DL) because DL overlap then occurs even in very dilute suspensions. The DL overlap becomes very important for electrophoresis of nanoparticulates and in nonaqueous systems. The electrophoretic theory which takes into account the overlap of DLs was formulated by Overbeek and coworkers [49, 50] and is discussed briefly in Section 2.5.3. Theoretical treatments have been suggested for these two extreme cases of DL thickness. Briefly they are: Thin DL (ka > 10): Loeb, Dukhin, and Overbeek [51–53] proposed a theory to describe the DL structure for a symmetrical electrolyte by applying a series expansion in terms of powers of 1/(ka). The final result is: d!  d 4 tanh zc 2FC z z c s 4 sd ¼  þ 2 sinh ð2:35Þ 2 ka k Thick DL (ka < 1): No theory for an isolated DL is necessary since DL overlap must be considered even in the dilute case. A theory that does include DL overlap has been proposed only very recently [54]: ! !! d  d 4 tanh zc d d 2FCs z zc 3B zc zc d 4 þ þ 2 sinh exp  sinh s ¼ 2 ka 2 2 k ka ð2:36Þ where A expðkðb  aÞÞ ðkb  1Þ 2ka The constant, A, is obtained by matching the asymptotic expansions of the long-distance potential distribution with the short-distance distribution and b is the radius of the cell. This new theory suggests, at last, a way to develop a general approach to the various colloidal-chemical effects in nonaqueous dispersions. B¼

2.5.3. Electric Double Layer at High Ionic Strength According to classical theory, the DL should essentially collapse and cease to exist when the ionic strength approaches 1 M. At this high electrolyte concentration, the Debye length becomes comparable with the size of the ions, implying that the counterions should collapse onto the particle surface. However, there are indications [55–62] that, at least in the case of hydrophilic surfaces, the DL still exists even at ionic strengths exceeding 1 M.

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The DL at high ionic strength is strongly controlled by the hydrophilic properties of the solid surfaces. The affinity of the particle surface for water creates a structured surface water layer, and this structure changes the properties of the water considerably from that in the bulk. Figure 2.11 illustrates the DL structure near a hydrophilic surface. Note the difference between the much lower dielectric permittivity within the structured surface layer, esur, compared to that of the bulk liquid, em. There are two effects caused by this variation of the dielectric permittivity [60]: l l

The structured surface layer becomes insolvent, at least partially, to ions; The structured surface layer may gain electric charge because of the difference in the standard chemical potentials of the ions.

Both effects influence the electrokinetic behavior of colloids at high ionic strength. The first effect is the more important one because it leads to a separation of electric charges in the vicinity of the particle. The insolvent structured layer repels the screening electric charge from the surface to the bulk and makes electrokinetics possible at high ionic strength if the structured layer retains a certain hydrodynamic mobility in the lateral direction. A theory of electro-osmosis for the insolvent and hydrodynamically mobile DL at high ionic strength has been proposed by Dukhin and Shilov in a paper which is not available in English. There is a description and

structured water layer

-

bulk

+

0

surface charge

particle

electroosmotic flow

+ ψ(X)

- +

ψsur

+ + +

-

εm

structured layer electric charge

εsur(x)

-

+ screening charge

-

+ insolvent hydrodynamically mobile

FIGURE 2.11 Electric double layer near hydrophilic surface with structured water layer.

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Fundamentals of Interface and Colloid Science

reference to this theory in an experimental paper that is available in English [61]. We discuss these effects in more detail in Chapter 11 with regard to the experimental study published in the article [63].

2.5.4. Polarized State of the Electric Double Layer External fields may affect the DL structure. These fields might be of various origins: electric, hydrodynamic, gravity, concentration gradient, acoustic, etc. Any of these fields might disturb the DL from its equilibrium state to some “polarized” condition [1, 42]. Such external fields affect the DL by moving the excess ions inside it. This ionic motion can be considered as an additional electric surface current, Is (Figure 2.12). This current is proportional to the surface area of the particle and to a parameter called the surface conductivity Ks [1]. This parameter reflects the excess conductivity of the DL due to the excess ions attracted there by the surface charge [143]. The surface current adds to the total current. But there is another component that depends on the nature of the particle, which has the opposite effect. For example, a nonconducting particle reduces the total current going through the conducting medium, simply because the current cannot pass through the nonconducting volume of the particle. In the important case of charged,

Electric field

Polarization electric field DL layer with higher conductivity

-

negatively charged particle

+

Bulk electric current

Surface current Polarization dipole charges

FIGURE 2.12 Mechanism of polarization of the electric double layer.

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nonconducting particles, these two opposing effects influence the total current value. The surface current increases the total current, whereas the nonconducting character reduces it. The amplitude of the surface current is proportional to the surface conductivity Ks. The amplitude current arising from the nonconductive character depends on the conductivity of the liquid, Km; the higher the value, the more the current lost, because the conducting liquid is replaced with nonconducting particles. The balance of these two effects depends on the dimensionless number, Du, given by: Du ¼

Ks aKm

ð2:37Þ

The particle size, a, is in the denominator of Equation (2.37) because the surface current is proportional to the surface area of the particle (a2), whereas the reduction of the total current due to the nonconducting particles is proportional to the volume of the particles (a3). The abbreviation, Du, for this dimensionless parameter was introduced by Lyklema. He called it the “Dukhin number” [1] after S. S. Dukhin, who explicitly used this number in his analysis of electrokinetic phenomena [23, 42]. The surface electric current redistributes the electric charges in the DL. One example is shown in Figure 2.12. It is seen that this current produces an excess of positive charge at the right pole of the particle and a deficit at the left pole. Altogether, this means that the particle gains an “induced dipole moment” (IDM) [23, 42] due to this polarization in the external field. A calculation of the IDM value for the general case is a rather complex problem. It can be substantially simplified in the case of the thin DL (ka > 10). There is a possibility to split the total electric field around the particle into two components: a “near-field” component and a “far-field” component. The near field is located inside the DL. It maintains each normal section of the DL in a local equilibrium. This field is shown as a small arrow inside the DL in Figure 2.12. The far field is located outside the DL, in the electro-neutral area. It is simply the field generated by the IDM. Let us assume now that the driving external field is an electric field. The polarization of the DL affects all the other fields around the particles: hydrodynamic, electrolyte concentration, etc. However, all other fields are secondary in relationship to the driving electric field applied to the dispersion. The IDM value, pidm, is proportional to the value of the driving force, which is the electric field strength, E, in this case. In a static electric field, these two parameters are related by the following equation: pind ¼ gind E ¼ 2pe0 em a3 where gind is the polarizability of the particle.

1  Du E 1 þ 2Du

ð2:38Þ

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Fundamentals of Interface and Colloid Science

The IDM value depends not only on the electric field distribution but also on the distribution of the other fields generated near the particle. For example, the concentration of electrolyte shifts from its equilibrium value in the bulk of the solution, Cs. This effect, called concentration polarization, complicates the theory of electrophoresis and is the main reason for the gigantic values found for dielectric dispersion in a low-frequency electric field. Fortunately, this concentration polarization effect is not important for ultrasound, since it appears only in the low-frequency range where the frequency, o, is smaller than the characteristic concentration polarization frequency, ocp: o ocp ¼

Deff d2

ð2:39Þ

where Deff is the effective diffusion coefficient of the electrolyte. Typically, this concentration polarization frequency is below 1 MHz, even for particles as small as 10 nm. The IDM value becomes a complex number in an alternating electric field, as well as polarizability. According to the Maxwell-Wagner-O’Konski theory [64–66], confirmed later by DeLacey and White numerical calculations [144], the IDM value for a spherical, nonconducting particle is: pidm ¼ gidm E ¼

ep  em E 2em þ ep

ð2:40Þ

where the complex permittivity and conductivity of the medium are defined by Equations (2.4) and (2.5). The complex permittivity of the charged nonconducting particle is given by: ep ¼ ep þ j

2Km Du oe0

ð2:41Þ

The influence of the DL polarization on the IDM decreases as a function of the frequency (following from Equation 2.40). There is a certain frequency above which the DL influence becomes negligible because the field changes faster than the DL can react. This is the so-called Maxwell-Wagner frequency, oMW, which is defined as: oMW ¼

Km e0 em

ð2:42Þ

So far, we have described the polarization of the DL mostly as it relates to an external electric field. Indeed, most theoretical efforts over last 150 years have been expended in this area. However, we have already mentioned that other fields can also polarize the DL. The major interest of this book is to describe in some detail the polarization caused not by this electric field but a hydrodynamic field, since this gives rise to the electroacoustic phenomena in which we are interested.

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2.6. INTERACTIONS IN COLLOID INTERFACE SCIENCE The characterization of interactions is strongly connected to the idea of a “dimension hierarchy.” We can speak of several specific dimensional levels in the colloid and each level has its own specific interaction features. Interactions that originate at the microscopic level, where we deal with molecules and ions, are the so-called chemical interactions. They are usually related to the exchange of electron density between various microscopic bodies. There are many indications that ultrasound techniques are well suited for characterizing such chemical interactions. Although not the main subject of this book, this aspect will be covered briefly in Chapter 3. We are most interested in effects that occur at the single-particle level or, in other words effects with a characteristic spatial parameter on the order of the size of a typical particle. This means that many thousands of molecules and ions are involved in this interaction. We observe the result of some averaging of individual chemical interactions. This averaging masks, and practically eliminates, any recognizable features of the original chemical interactions and microscopic objects. It justifies the introduction of a completely new set of concepts to describe effects at this higher structural level. We call all interactions that happen on this level “colloidal-chemical interactions,” and further subdivide these reactions into two categories: equilibrium and nonequilibrium reactions. Equilibrium interactions, as the name suggests, corresponds to interactions where the particles are in equilibrium with the dispersion medium. These interactions, subject to the classical DLVO theory, include: 1. Dispersion Van der Waals forces 2. Electrostatic forces of the overlapped double layers 3. Various steric interactions. Nonequilibrium interactions correspond to cases where the equilibrium between the particle and the dispersed phase is disturbed. For example, any external field, whether it is electric, hydrodynamic, gravitational or magnetic, might create such a disturbance. Living biological cells present a special case in which the equilibrium is disturbed by an electrochemical exchange between the cell and the dispersion medium. We briefly consider here both equilibrium and nonequilibrium colloidalchemical interactions. The purpose of this overview is merely to determine what features of these colloidal-chemical interactions may affect ultrasound in the dispersions and, consequently, to learn which of these features might possibly be characterized with ultrasound-based techniques.

2.6.1. Interactions of Colloid Particles in Equilibrium: Colloid Stability The affinity between the particles and the dispersion medium allows us to separate colloids into two broad classes. “Lyophobic” colloids represent systems with very low affinity between particle and the liquid, whereas for “lyophilic”

Fundamentals of Interface and Colloid Science

55

colloids this affinity is high. From this definition, it follows that particles of a lyophobic colloid tend to aggregate, in an effort to reduce the contact surface area between the particle and the medium. Such colloids are, usually, thermodynamically unstable in the strict sense, but might actually be kinetically stable for a long period. In contrast, lyophilic colloids are, often, thermodynamically quite stable, and consequently their kinetics are not as important. It is possible to convert one colloid class to the other by applying appropriate surface modifications. We can coagulate a lyophilic dispersion even though it was initially thermodynamically stable, or we can stabilize a lyophobic dispersion even though initially unstable. It is simply a question of our overall objective. We will show later that ultrasound-based techniques might be very helpful for optimizing these procedures. In order to explain the possibilities open to ultrasound characterization, we first need to give a short description of the particles’ interaction and coagulation. There are several mechanisms of particle-particle interaction which are described in detail in the DLVO theory [67–70], and these can be found in most handbooks on colloid science [1, 47, 71]. The presentation of the DLVO theory usually starts with the simplest case, in which the potential energy, Upp, on bringing two surfaces close together can be expressed as the sum of the Van der Waals attraction and DL repulsion. These two basic mechanisms exhibit quite different dependences with respect to the separation distance, xpp, between the surfaces. The Van der Waals attraction decays quite rapidly as the inverse power of the separation distance. The DL repulsion, on the other hand, typically decreases more slowly, more or less exponentially with this distance, and, typically, has a range on the order of the Debye length 1/k. As a result, the Van der Waals attraction is dominant for small separations, whereas the DL repulsion may be dominant at larger distances. A general potential energy plot illustrating qualitatively the main features of this interaction is given in Figure 2.13. The key feature of this potential energy plot is that there is often a welldefined potential energy barrier at a distance comparable to the Debye length. A particle that approaches this barrier, perhaps due to its random Brownian motion, must have sufficient energy to overcome this barrier if it is to fall into the primary minimum and result in an aggregated particle. This barrier is therefore one important key to achieving stability in a colloid system. This potential barrier exists when the DL repulsion exceeds the Van der Waals attraction. We cannot do much to change the attractive forces, since these are fixed by the Hamaker constants of the material. We can only manipulate the repulsive forces by changing either the potential at the Stern plane, or by changing the rate at which this potential decreases with distance (as determined by the Debye length). The Stern potential can be affected strongly by the addition of potential-determining ions to the medium. The Debye length can be decreased by increasing the ionic strength. Consequently, the height of this potential barrier depends on the Stern potential of the particles

56

CHAPTER

Upp

2

potential barrier

Xpp secondary minimum

primary minimum

FIGURE 2.13 Typical DLVO interaction energy.

and electrolyte concentration or ionic strength. An increase in the ionic strength reduces this barrier height by shifting it closer to the surface where the Van der Waals attraction is stronger. There is a critical concentration of electrolyte, Cccc, at which this barrier disappears altogether, and the particles might therefore coagulate. !4 d zc 1 tanh 4 3 5 5 2 9:85 10 e R T m Cccc ½moll1  ¼ ð2:43Þ !4 F6 A2 z6 d zc tanh þ1 2 A more detailed and accurate calculation of this critical coagulation concentration, as well as other stability criteria, can be found in the literature [1, 70]. In addition to ionic strength, aggregation stability is also sensitive to the concentration of DL potential-determining ions that control the value of the Stern potential. In principle, these two parameters (overall ionic strength and concentration of potential-determining ions) might be considered as independent parameters, which bring us to the conclusion that both aggregation stability and DL potentials should be considered in some multidimensional space. Contours of constant B-potential would then appear as a “fingerprint” of the particular colloid. Indeed, this idea of fingerprinting is the logical

57

6

10 12 14 16 18 20 22 24 26 28 30 32 34 0.0 0.5 % o, 1.0 rati olin -ka

7 8 pH

FIGURE 2.14 solution.

9 10

11 2.0

zeta [mV]

Fundamentals of Interface and Colloid Science

1.5 H SHP

Electrokinetic three-dimensional plot for kaolin slurry in sodium hexametaphosphate

way to characterize a colloidal system, and Marlow and Rowell have successfully applied this principle [72, 73] for various applications. We show, as one example, the three-dimensional plot of 40 wt% kaolin slurry that was titrated with sodium hexametaphosphate (Figure 2.14). From an examination of this, it is clear that the highest surface charge occurs at a pH value of 9.5 and at a hexametaphosphate to kaolin ratio of 0.6 wt%. This, then, is the optimum condition for gaining aggregative stability. Hexametaphosphate affects the stability of the kaolin particles through electrostatic interactions. There are different types of stabilizing agents that might enhance stability via other mechanisms, such as steric interactions. For instance, various polymers adsorb on the particle surface and change its affinity to the solvent from lyophobic to lyophilic. However, there is a danger, if the stabilizing agent is overdosed, of flocculating the colloid through the bridging effect. This is the reason why surface characterization is important. It allows one to determine the optimum dose of the stabilizing agent and thus eliminate the dangers of flocculation. There is one more factor that can affect strongly the stability of the heterogeneous system—gravitation. It is nonequilibrium and its principle will be mentioned in another section. However, it always exists and competes with other equilibrium factors mentioned here. Gravity affects both pair particle interaction and kinetics of the PSD evolution. There is a recent published review [74] that describes in detail several works dedicated to this subject. One of the main conclusions is that gravity becomes as important as the stability factor for particles with sizes roughly of 1 mm. It is a very

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approximate number that must be used to alert that gravity must be considered if the size exceeds this critical value.

2.6.2. Biospecific Interactions Intensive development of nanotechnology has generated concerns regarding nanoparticles’ impact on human health. The level of interest is so high that Nature published recently a review article on this subject [75]. Unfortunately, authors of this review missed one major peculiar feature of the living biological cells. These objects are not in equilibrium with the surrounding media. This prevents us from transposing directly knowledge collected in colloid science for describing these object and, in particular, for understanding their interaction with nanoparticles. As a result, the review [75] is applicable only for “dead” biological surfaces. Some very important aspects associated with the living biological cells’ functioning remained outside the scope of the review [75]. The authors of the review are not familiar with a large body of work that was carried out on cell-particle interactions during the last few decades of the twentieth century. The authors referred only two papers from among 96 citations that came from that time period. The earliest is dated 1995. This would not have been a problem had the more recent works properly reflected the observations and conclusions of the old studies. Unfortunately, it is not the case. The review overlooks some very important facts and mechanisms that may dominate cell-particle interactions. We hope to bring to your attention those forgotten papers. The earliest relevant work in this field, in my mind, was done by H.A. Pohl, the famous inventor of dielectrophoresis. He monitored the motion of small particles in the vicinity of biological cells for verifying the hypothesis that biocells generate an AC electric field. The earliest publication is dated 1980 [76]. His short essay published in 1987 [77] cited several more publications in major journals, and Ref. [78] is one of them. He observed indeed unusually long-range interactions between biological cells and dielectric particles, in scale of micrometers. In the 1990s, experiments on the same subject were performed by several groups in (the then) U.S.S.R. and some of them were published in international journals [79–84]. The Soviet investigators’ group led by Z. Ulberg and V.Karamushka had discovered some peculiar features of bacterial interactions with gold nanoparticles and dramatically distinguished them from typical interactions between nano- and micro-sized equilibrium objects. Some bacteria collected large numbers of gold nanoparticles on their surfaces (Figure 2.15). The interaction occurred in two stages: the nanoparticles were initially bound to the cell very weakly and reversibly, but the reversible aggregation gradually became irreversible over 1 h.

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Fundamentals of Interface and Colloid Science

FIGURE 2.15 Bacillus subtilis cells in the presence of gold particles. Scale bar 1.8 mm. (Adapted with permission from Ref. [79]).

The first stage of reversible aggregation was the most interesting and peculiar. It was controlled by the cell’s metabolism and, in particular, by the activity of the ion pumps and the transmembrane potential. Earlier authors proved this mechanism by using special chemicals that switched on and off specific ion pumps. Turning the ion pumps on and off created cycles of aggregation and release of the nanoparticles. Results of one of these experiments are shown in Figure 2.16. Pentachlorophenol, an inhibitor of metabolism, was injected at different times to release 1

+ +

Ct/Co

0.8

+

Cit

*

*

*

+

*

3 +

2

0.6 *

0.4

4

*

0.2 A 0

0

1

Ct 20

40

t, min

60

80

5

100

FIGURE 2.16 Kinetic curves of the gold particles adsorption by bacteria, where Ct is the relative concentration of gold particles in the solution. Curve 1 is uninterrupted adsorption. Curve 2 is adsorption when pentachlorophenol is added initially to the solution. Curves 3, 4, and 5 represent kinetics after injection of pentachlorophenol at different time moments. (Adapted with permission from Ref. [79]).

60

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gold particles back into the solution. The amount of released particles decayed with time, suggesting that the aggregation went from reversible to irreversible. This and other similar experiments indicated the existence of interaction mechanisms that solely existed for living cells as a result of their metabolism and ion exchange. We are not aware of the complete physicochemical theory for this interaction. Biospecific mechanism of a DL formation [83–85] was a first step in explaining these experiments as well as the observed metabolismdependent components of the cell electrophoresis [86–88]. The biospecific mechanism predicts a relationship between the living cell’s z-potential and its transmembrane potential DU. It postulates that transmebrane potential becomes redistributed between capacity of the membrane Cm and capacity of the cell DL, Cdl. This model yields the following expression for the fraction of the cell z-potential (Dz) that is associated with the transmembrane potential and cell metabolism: Dz ¼

Dse DUCm þ Cdl þ Cm Cdl þ Cm

ð2:44Þ

The first term in this equation takes into account possible change in the surface charge under the influence of the intramembrane field that can affect the conformation of intramembrane macromolecules. This additional component of the z-potential can be eliminated by reducing transmembrane potential. This correlation was observed experimentally in electrophoretic experiments [86–88]. The additional component of the z-potential would affect electrostatic component of the cell-nanoparticle interaction energy. This particular mechanism of cell-particle interactions is intriguing and opens new possibilities of controlling biological cell interaction with nanoparticles.

2.6.3. Interaction in a Hydrodynamic Field: Cell and Core-Shell Models—Rheology A hydrodynamic field exists when a particle moves relative to the liquid medium. There are a number of effects caused by this motion (see Ref. [89]). When a particle moves relative to a viscous liquid, it creates a sliding motion of the liquid in its vicinity. The volume over which this disturbance exists is usually referred to as the hydrodynamic boundary layer. As the volume fraction of the sample is increased, the average distance between particles becomes smaller. When this distance becomes comparable to the thickness of the hydrodynamic layer, these layers overlap and, as a result, one moving particle affects its immediate neighbors. The particles interact through their respective hydrodynamic fields, and such interactions can be defined mathematically. The motion of the liquid in the hydrodynamic layer surrounding a particle is described in the terms of a local liquid velocity, u, and a hydrodynamic

Fundamentals of Interface and Colloid Science

61

pressure, P. Liquids are usually assumed to be incompressible, described mathematically as: Divu ¼ 0

ð2:45Þ

We will see later that this assumption can be used even for ultrasound-related phenomena in the long-wavelength range. The assumption of fluid incompressibility then allows one to use a simplified version of the NavierStokes’ equation [89], namely: du ¼
rot rot u þ grad P ð2:46Þ dt Equation (2.46) neglects terms related to the volume viscosity, hvm , of the liquid. These last two equations require a set of boundary conditions in order to calculate the liquid velocity and hydrodynamic pressure. The boundary conditions at the surface of the particle are rather obvious: rm

ur ðr ¼ aÞ ¼ up  um

ð2:47Þ

uy ðr ¼ aÞ ¼ ðup  um Þ

ð2:48Þ

where r and y are spherical coordinates associated with the particle. The other set of boundary conditions must specify the velocity of the liquid somewhere in the bulk. For a dilute system this presents no problem. Particles do not interact and this point can be selected infinitely far from the particles. This yields, for instance, the well-known Stokes’ law [90] for the frictional force, Fh, exerted on particles moving relative to the liquid: Fh ¼ 6p
aOðup  um Þ

ð2:49Þ

where O is the drag coefficient. In the case of a concentrated colloid, this approach is not viable because the particles interact and the hydrodynamic field of one particle affects that of its neighbors. The concept of a “cell model” [91, 92] allows one to take into account this interaction and resolve the hydrodynamic problem for a collection of interacting particles. The basic feature of a cell model is that each particle in the concentrated system is considered separately inside the spherical cell of liquid that is associated only with the given individual particle (Figure 2.17). In the past, the cell model has been applied only to monodisperse systems. This restriction allows one to define the radius of the “cell.” Equating the solid volume fraction of each cell to the volume fraction, ’, of the entire system yields the following expression for the cell radius, b: a b¼p ffiffiffiffi 3 ’

ð2:50Þ

62

CHAPTER

2

cell surface

particle

b, cell radius fraction of the liquid associated with the particle

FIGURE 2.17 Illustration of the cell model.

The cell boundary conditions formulated on the outer boundary, at r ¼ b, of the cell reflect the particle-particle interaction. The two most widely used versions of these boundary conditions are named after their authors: the Happel cell model [91] and the Kuwabara cell model [92]. Both of them are formulated for an incompressible liquid. For the Kuwabara cell model, the cell boundary conditions are given by the following equations: rot ur¼b ¼ 0

ð2:51Þ

ur ðr ¼ bÞ ¼ 0

ð2:52Þ

In the case of the Happel cell model they are: uy @ 1 @ur ðr ¼ bÞ ¼ þr r ¼0 ry @r r @y

Y

ur ðr ¼ bÞ ¼ 0

ð2:53Þ ð2:54Þ

The general solution for the velocity field contains three unknown constants C, C1, and C2:    Z b b3 x3 ð2:55Þ ur ðrÞ ¼ C 1  3 þ 1:5 1  3 hðxÞdx r r r

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Fundamentals of Interface and Colloid Science

   Z b b3 x3 uy ðrÞ ¼ C 1 þ 3  1:5 1 þ 3 hðxÞdx 2r 2r r

ð2:56Þ

b~ ~  C2 h2 ðbÞÞ ~ C ¼  ðC1 h1 ðbÞ 3

ð2:57Þ

where hðxÞ ¼ C1 h1 ðxÞ þ C2 h2 ðxÞ 2 0 13 expðxÞ 4x þ 1 x þ 1 h1 ðxÞ ¼ sinx  cosx þ i@ cosx þ sinxA5 x x x 2 0 13 expðxÞ 4x þ 1 1  x sinx  cosx þ i@ cosx þ sinxA5 h2 ðxÞ ¼ x x x

ð2:58Þ

The drag coefficient can be expressed, in a general form, for both Kuwabara and Happel cell models: O¼

  a~2 dðC1 h1 þ C2 h2 Þ C1 h1 þ C2 h2 4j~ a2  þ a~ 3 9 dx x¼~ a

ð2:59Þ

a=a; and x is normalized in the same manner where a~2 ¼ a2 wrm =2hm ; b~ ¼ b~ as a˜. Coefficients C1 and C2 are different in the two cell models (see Table 2.2). There are several special functions used in this theory. They are defined as follows: ~  Ið~ I ¼ IðbÞ aÞ I1, 2 ¼ j

ℓ

xð1þjÞ

x

TABLE 2.2 Parameters of the Cell Model Solutions Kuvabara

Happel

C1

~ h2 ðbÞ I

~  2I23 ~ 2 ðbÞ bh ~ bI þ 2ðI2 I13  I1 I23 Þ

C2



~ h1 ðbÞ I



~  2I13 ~ 1 ðbÞ bh ~ bI þ 2ðI2 I13  I1 I23 Þ

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2

" !# x2 3x 1 xð1þjÞ 3ð1  xÞ ~ IðxÞ ¼ h1 ðbÞℓ þ j 3  ... 3  3 x 2b~ b~ 2b " !# 2 3ð1 þ xÞ x 3x 1 xð1þjÞ ~ℓ h2 ðbÞ þj 3þ 3 3 x b~ 2b~ 2b~

I13 ¼ 

I13 ¼ 

ℓ

xð1þjÞ



3 b~ ℓ

xð1þjÞ



  3 3 2 ð1 þ xÞ þ j x þ x 2 2

  3 3 ð1 þ xÞ þ j x2 þ x 2 2

3 b~ The Happel cell model is better suited to acoustics because it describes energy dissipation more adequately, whereas the Kuwabara cell model is better suited to electroacoustics because it automatically yields the Onsager relationship [42, 47]. In the case of a polydisperse system, the introduction of the cell is more complicated, because the total liquid can be distributed between size fractions in an infinite number of ways. However, the condition of mass conservation is still necessary. Each fraction can be characterized by particles having radii ai, cell radii bi, thickness of the liquid shell in the spherical cell li ¼ biai, and volume fraction ’i. The mass conservation law ties these parameters together as:  N  X li 3 1þ ’i ¼ 1 ð2:60Þ ai i¼1

This expression might be considered as an equation with N unknown parameters, li. An additional assumption is still necessary to determine the cell properties for the polydisperse system. This additional assumption should define for each fraction the relationship between the particle radii and shell thickness. We have suggested the following simple relationship [47]: li ¼ lani

ð2:61Þ

This assumption reduces the number of unknown parameters to only two, related by: N X

ð1 þ lan1 Þ3 ’i ¼ 1 i

ð2:62Þ

i¼1

The parameter n is referred to as a “shell factor.” Two specific values of the shell factor correspond to easily understood cases. A shell factor of 0 depicts the case in which the thickness of the liquid layer is independent of the

65

Fundamentals of Interface and Colloid Science

particle size. A shell factor of 1 corresponds to the normal “superposition assumption,” giving the same relationship between particles and cell radii as in the monodisperse case, that is, each particle is surrounded by a liquid shell which provides each particle with the same volume concentration as the volume concentration of the overall system. In general, the “shell factor” might be considered to be an adjustable parameter because it adjusts the dissipation of energy within the cells. However, our experience using this cell model with acoustics for particle sizing [93, 94] indicates that a shell factor of unity is almost always more suitable. There is another approach to describe the hydrodynamic particle interaction; it is the so-called effective media approach. It was used originally by Brinkman [95] for hydrodynamics, and later by Bruggeman for dielectric spectroscopy [23]. This approach and its relationship to the cell model are fully described in a recent review [96]. The effective media approach is very useful for interpreting acoustic spectroscopy results, and will be discussed later in Chapter 8. Recently, a new “core-shell model” has become quite popular [96]. It attempts to combine the cell model and “effective media” approaches as illustrated in Figure 2.18. Here, each particle is surrounded by a spherical layer of liquid, as in the regular cell model. However, the colloid beyond this spherical layer is modeled as some effective media. The properties of the effective media and the liquid shell thickness depend on the particular application. There are applications for this core-shell model in acoustics, in particular with the theory of the thermodynamic effects associated with sound propagation through the colloid (see Chapter 4).

effective media

particle

FIGURE 2.18 Illustration of the core-shell model.

liquid shell

66

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2

Theoretical developments of the hydrodynamic particle-particle interaction are the backbone of rheological theories that describe reactions of colloids to mechanical stresses. Such reactions might be very complex because colloids possess features of both liquids and solids. They are able to flow and exhibit viscous behavior as liquids. At the same time, they have a certain elasticity as with solids. This is why colloids are referred to as viscoelastic materials. In the simplest linear case, the total stress of the colloid, sshear, in a shear flow of shear rate, Rshear, can be presented as the combination of a Newtonian liquid (stress is proportional to the rate of strain) and a Hookean solid (stress is proportional to the strain) [71]: Ms ðoÞ ð2:63Þ Rshear cosot o where o is the frequency of the shear oscillation, and
s and Ms are, respectively, the macroscopic viscosity and elastic bulk modulus of the colloid. The macroscopic viscosity and bulk modulus depend on the microscopic properties of colloids, such as the PSD and the nature of the bonds that bind the particles together. These links are usually modeled as Hookean springs. There are a number of theories [71, 97–101] that relate the macroscopic viscosity and bulk modulus to the microscopic and structural properties of particular colloids using the Hookean spring model. Later (Chapter 4), we will use this model to describe ultrasound attenuation in structured colloids. This is possible because of the similarities between rheology and acoustics; both deal with responses to mechanical stress: shear stress in rheology and longitudinal stress in acoustics. This parallelism is explored in more detail in the same chapter. sshear ¼
s ðoÞRshear sinot 

2.6.4. Linear Interaction in an Electric Field: Electrokinetics and Dielectric Spectroscopy Applying an electric field to a colloid system generates a variety of effects. One group of such effects is called “electrokinetic phenomena.” Electrophoresis is the best known member of this group, which also includes electro-osmosis, streaming potential, and sedimentation potential [102–106]. These electrokinetic effects depend on the first power of the applied electric field strength, E. There are other effects induced by an electric field which are nonlinear in terms of the electric field strength, but they are usually less well known and less important. They are not well known because the electric field under normal conditions in aqueous systems is rather weak. It becomes especially clear when the electric field is presented in the natural normalized form as follows: EaF E½V=cm E~ ¼ RT 250 a¼1 mm

ð2:64Þ

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Fundamentals of Interface and Colloid Science

In aqueous colloids, the electric field strength is limited by high conductivity and is typically no more than 100 V/cm. A stronger applied electric field creates thermal convection and this masks any electrokinetic effects. Accordingly, the dimensionless electric field strength is small and linear effects are dominant. There are still some peculiar nonlinear effects (described in the next section). The most widely known and used electrokinetic theory was developed a century ago by Smoluchowski [107] to describe the electrophoretic mobility, m, of colloidal particles. It is important because it yields a relationship between the z-potential of particles and the experimentally measured parameter, m: m¼

em e0 z
m

ð2:65Þ

There are two conditions restricting the applicability of Smoluchowski’s equation. The first restriction is that the DL thickness must be much smaller than the particle radius a: ka  1

ð2:66Þ

where k is the reciprocal of the Debye length. In the situation where the DL is not thin compared to the particle radius, Henry (see Ref. [108]) applied a correction to modify Smoluchowski’s equation: m¼

e0 em B fH ðkaÞ
m

ð2:67Þ

where fH ðkaÞ is called the Henry function. A simple, approximate expression has been derived by Ohshima [109]: fH ðkaÞ ¼

2 þ  3 3 1þ

1 2:5 ka½1 þ 2 expðkaÞ

3

ð2:68Þ

Figure 2.19 illustrates the form of the Henry function values as a function of ka. The second restriction on the Smoluchowski equation is that the contribution of surface conductivity, ks, to the tangential component of electric field near the particle surface is negligibly small. This condition is satisfied when the dimensionless Dukhin number, Du, is sufficiently small: Du ¼

ks 1 Km a

ð2:69Þ

where Km is the conductivity of medium outside the double layer. Equation (2.67) is termed the Smoluchowski electrophoresis equation and is usually derived for a single particle in an infinite liquid. In this derivation,

68

CHAPTER

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Henry function, Ohshima approximation

1.0

0.9

0.8

0.7

0.6 0

1

ka

10

100

FIGURE 2.19 Henry function calculated according to the Ohshima expression.

the electrophoretic mobility, m, is defined as the ratio of field-induced particle velocity, V, to the homogeneous field strength, Eext , in the liquid at a large distance from the particle: V ð2:70Þ m¼ Eext Smoluchowski’s equation is applicable to a particle with any geometrical form: a convenient property that makes it applicable to any arbitrary system of particles, including a cloud of particles or a concentrated suspension. This property was experimentally tested by Zukoski and Saville [110]. However, in order to apply Smoluchowski’s equation to a concentrated suspension, one should take into account the difference between the external electric field in the free liquid outside the suspension, Eext, and the averaged electric field inside the dispersion, hEi. The static mobility m in Equation (2.65) is determined with respect to the external electric field, that is, the field strength in the free liquid outside the suspension. For concentrated suspensions, an alternative electrophoretic mobility, hmi, (or electro-osmotic velocity), is defined with respect to the field hEi: hmi ¼

ee0 z Ks n Km

ð2:71Þ

where Ks is the macroscopic conductivity of the dispersed system and hmi ¼

V hEi

ð2:72Þ

An equation similar to Equation (2.71) but with opposite sign is termed Smoluchowski’s electro-osmosis equation, and is usually expressed in this

69

Fundamentals of Interface and Colloid Science

form (Kruyt and Overbeek [47], Dukhin [42], and O’Brien [111]). It is valid under the same conditions as given above for electrophoresis, that is, a thin DL and negligible surface conductivity. The same absolute values and opposite signs of electrophoretic and electro-osmotic velocities reflect the difference in the frames of reference between these two phenomena. In the case of static electrophoresis, the frame of reference is the liquid, whereas in the case of electro-osmosis it is the particle matrix. These two definitions of the electrophoretic mobility are identical if we take into consideration the well-known [23] relationship between Eext and hEi: hEi Km ¼ ð2:73Þ Ks Eext Both expressions (Equations 2.65 and 2.71) specify an electrophoretic mobility (m or hmi) which is independent of both the volume fraction and the system geometry. This happens because the hydrodynamic and electrodynamic interactions of particles have the same geometry [42] when conditions 2.67 and 2.70 are valid. From this viewpoint, Smoluchowski’s equation is unique. Surface conductivity and the associated concentration polarization of the DL destroy this geometric similarity between the electric and hydrodynamic fields and, consequently, the electrophoretic mobility becomes dependent on the particle size and shape; a correction is required and several theories have been proposed. One numerical solution for a single particle in an infinite liquid (for very dilute colloids) was derived by O’Brien and White [112]. However, experimental tests by Midmore and Hunter [113] indicate that the Dukhin-Semenikhin theory [108] is probably the most adequate among the several approximate formulas [48, 114], because it has the advantage of distinguishing between the z-potential and the Stern potential, making it more useful for characterizing surface conductivity. The Dukhin-Semenikhin theory yields the following expression for electrophoretic mobility: " # 4Msh2B þ 2G1 þ lnBchB ð2Msh2B  12mB þ 2G2Þ ee0 z ð2:74Þ 1 m¼
ka þ 8Msh2B  24m lnchB þ 4G1 z2

where

! !

d d   B B D

D 3m c c G1 ¼ s ch  ch ; G2 ¼ s sh  sh ; M ¼1þ 2 ; 2 2 D 2 D 2 z m¼

0:34ðzþ Dþ  z D Þ ; Dþ D ðzþ  z Þ

d

 d ¼ zFc ; B ¼ zFB ; c RT RT

In general, there is a great difference between the theory in dilute systems and that in concentrated systems. Particle interaction makes the theory for concentrated systems much more complicated. Unfortunately, the simplicity and

70

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2

wide applicability of Smoluchowski’s equation is a rather unique exception. The simplest and the most traditional way to resolve the complexity of the electrokinetic theory for concentrated systems is to employ a cell model approach, in a manner similar to that used in the case of hydrodynamics. For instance, Overbeek and coworkers derived an expression for the electrophoretic mobility that takes into account that DLs overlap [49], which later was experimentally verified by Ross and Long [50].   kb  1 a kb  1 expð2kðb  aÞÞ  1þ expðkðb  aÞÞ 1þ 2s kb þ 1 b kb þ 1 m¼ kb  1 3
a ðka þ 1Þ expð2kðb  aÞÞ  ka þ 1 kb þ 1 ð2:75Þ where b is the radius of an imaginary shell around the particles calculated assuming that the volume fraction of solid inside of the shell equals that of the dispersion. This (Overbeek) theory must be used for concentrated dispersions of nanoparticles and even to rather dilute nonpolar dispersions. The first electrodynamic cell model was applied for pure electrodynamic problem by Maxwell, Wagner, and, later, O’Konski [64–66] for calculating conductivity and dielectric permittivity of concentrated colloids comprised of particles having a given surface conductivity. This theory gives the following relationship between complex conductivity and dielectric permittivity: ep  em es  em ¼ ’ es þ 2em ep þ 2em

ð2:76Þ

  Kp  Km Ks  Km ¼  ’    Ks þ 2Km Kp þ 2Km

ð2:77Þ

where the index, s, corresponds to the colloid, m corresponds to the media, and p to the particle. Complex parameters are related to the real parameters as:  ¼ Km  joe0 em Km

ð2:78Þ

  Km em ¼ em 1 þ j oem e0 ep ¼ ep þ j

2Km Du oe0

According to the Maxwell-Wagner-O’Konski theory, there is a dispersion region where the dielectric permittivity and the conductivity of the colloid are frequency dependent. Two simple interpretations exist for this MaxwellWagner frequency, oMW. From DL theory, it is the frequency of the DL

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relaxation to the external field disturbance. From general electrodynamics, it is the frequency at which active and passive currents are equal. Thus, oMW can be defined by two expressions: k2 Deff ¼ oMW ¼

Km e0 em

ð2:79Þ

The Maxwell-Wagner-O’Konski cell model is applicable to the pure electrodynamic problem only. The other cell model mentioned above (Happel or Kuvabara cell model, Section 2.5.3) covers only hydrodynamic effects. An electrokinetic cell model must be valid for both electrodynamic and hydrodynamic effects. It must specify the relationship between macroscopic, experimentally measured electric properties and the local electric properties calculated using the cell concept. There are multiple ways to do this. For instance, the Levine-Neale cell model [115] specifies this relationship using one of the many possible analogies between local and macroscopic properties. The macroscopic properties are current density, hIi, and electric field strength, hEi. According to the Levine-Neale cell model, they are related to the local electric current density, I, and electric field, rf as: hIi ¼

Ir b cosyr¼b

hEi ¼ 

1 @f cosy @rr¼b

ð2:80Þ

ð2:81Þ

Relationships (Equations 2.80 and 2.81) are not unique. There are many other ways to relate macroscopic and local fields [146]. This means that we need a set of criteria to select a proper cell model. One set has been suggested in the electrokinetic cell model created by Shilov and Zharkikh [116]. Their criteria determine a proper choice of expressions for macroscopic “fields” and “fluxes.” The first criterion is the well-known Onsager relationship which constrains the values of the macroscopic velocity of the particles relative to the liquid, hVi, the macroscopic pressure, hPi, the electric current, hIi, and the field hEi: hVi hEi ¼ hIi hrPi¼0 hrPihIi¼0

ð2:82Þ

This relationship requires a certain expression for entropy production, S: 1 ðhIihEi þ hVihrPiÞ ð2:83Þ T It turns out that the expression derived by Shilov and Zharkikh for the macroscopic field strength is different from that of the Levine-Neale model. It is: S¼

hEi ¼

f b cosyr¼b

ð2:84Þ

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However, the expression for the macroscopic current is the same in both models. It is important to mention that the Shilov-Zharkikh electrokinetic cell model condition (Equation 2.84) fully coincides with the relationship between macroscopic and local electric fields of the Maxwell-Wagner-O’Konski electrodynamic cell model. The Shilov-Zharkikh cell model, allowing for some finite surface conductivity but assuming no concentration polarization, yields the following expressions for the electrophoretic mobility and conductivity in a concentrated dispersion [145]: hmi ¼

e0 em B 1’
m 1 þ Du þ 0:5’ð1  2DuÞ

Ks 1 þ Du  ’ð1  2DuÞ ¼ Km 1 þ Du þ 0:5’ð1  2DuÞ

ð2:85Þ

ð2:86Þ

Identical electrodynamic conditions in the Shilov-Zharkikh and MaxwellWagner-O’Konski cell models yields the same low-frequency limit for the conductivity (Equation 2.86). In addition, for the extreme case of Du ! 0, Equations (2.85) and (2.86) lead to Equation (2.71) for electrophoretic mobility. Consequently, the Shilov-Zharkikh cell model reduces to Smoluchowski’s equation in the case of negligible surface conductivity. This is an important test for any electrokinetic theory because Smoluchowski’s law is known to be valid for any geometry and volume fraction. Simultaneous measurement of conductivities Ks and Km makes possible calculation of the Dukhin number. This method was used by Lyklema and Minor [117], but with another conductivity theory [118], which seems to be the linear version of the Maxwell-Wagner theory. In the case of dielectric spectroscopy, there is a method that is similar to the “effective media” approach used by Brinkman for hydrodynamic effects. It was suggested first by Bruggeman [119]. We will use both these approaches for describing particle interaction in acoustic and electroacoustic phenomena.

2.6.5. Nonlinear Interaction in the Electric Field: Nonlinear Electrophoresis, Electrocoagulation, and Electrorheology A strong electric field disturbs DL, which in turn leads to the various deviations in classical linear theories of electrokinetic phenomena. The most obvious is the appearance of a nonlinear component in the electrophoretic mobility. This effect has been known for several decades. A historical overview can be found in a recent article [120]. It turns out that the most pronounced nonlinearity should be observed for conducting particles with a nonconducting shell. Such type of particles is usually referred to as ideally

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polarized particles. A simple theory of this nonlinear electrophoresis has been developed [121]. It yields the following expression for electrophoretic mobility: m¼

em e0 z 9em e0 a2 @Cdl E2  8
Cdl @’ ’¼z


ð2:87Þ

There are several other theories of nonlinear electrophoresis reviewed in the paper mentioned [120]. This effect has attracted serious attention recently because of the development of microfluidics. Induced electric charge can be used for creating wellcontrolled electro-osmotic flow is small channels. Another effect of the strong electric field is associated with the particleparticle interaction. In general, the contribution of the electric field to particle-particle interaction is nonlinear with the electric field strength, E. This follows from simple symmetry considerations. If such a force proportional to E exists, it would depend on the direction of the electric field, or on the sign of E. This implies that, in changing the direction of E, we would convert attraction to repulsion and vice versa. This is clearly impossible. Any interaction force must be independent of the direction of the electric field. This requirement becomes valid if the interaction force is proportional, for example, to the square of the electric field strength, E2. A linear interaction between particles affects their translational motion in an electric field. A mechanism exists for a linear particle interaction with an electrode in a DC field [122], which leads to a particle motion either towards the electrode or away from it [123]. In contrast, a nonlinear interaction induces a mutual or relative motion on the particles. We now give a short description of the theory and experiments related to this nonlinear particleparticle interaction. This nonlinear particle-particle interaction is the cause of two readily observed phenomena that occur under the influence of an electric field. The first phenomenon is the formation of particle chains oriented parallel to the electric field, as shown in Figure 2.20. These chains affect the rheological properties of the colloid [125, 126, 147]. The second effect is less well known but was first reported by Murtsovkin [124]. He observed that under certain conditions the particles build structures perpendicular, or at some other angle, to the electric field, as shown in Figure 2.20. These experiments clearly indicate that the application of an electric field can affect the stability of colloids, and that is why this condition is of general importance for colloid science. We have already mentioned that the volume of the dispersion medium adjacent to a colloid particle in an external electric field differs in its characteristics from those of the bulk dispersion medium. Such intensive

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FIGURE 2.20 Chains of particles in parallel with the electric field strength. Obtained from Murtsovkin [124].

parameters as the electric field, the chemical potential, and the pressure vary from the immediate vicinity of a particle to distances on the order of the dimension of the particle itself. If the volume fraction of the particles is sufficiently large, the particles will frequently fall into these regions of inhomogeneity of the intensive parameters. Naturally, the particles’ behavior in these regions will differ from that in the free bulk. Historically, it happened that the influence of each intensive parameter on the relative particle motion was considered separately from the others. As a result, three different mechanisms of particle interactions in an electric field are known. A short description of each follows. The oldest, and best known, is the mechanism related to the variation of the electric field itself in the vicinity of the particles. The particles gain dipole moments, pind, in the electric field, and these dipole moments change the electric field strength in the vicinity of the particles. This, in turn, creates a particle-particle attraction. This mechanism is usually referred to as a dipole-dipole electrodynamic interaction [127, 128]. This dipole-dipole interaction is responsible for the formation of particle chains and the phenomenon of electrorheology [125]. The dipole-dipole interaction is potential, which means that the energy of interaction, normally characterized with a potential energy, Udd, is independent of the trajectory of the relative motion: Udd ¼

p2ind ð1  3 cos2 yÞ 4pe0 em r 3

ð2:88Þ

where r and y are the spherical coordinates associated with one of the particles. The trajectory of the relative particle motion is shown as (A) in Figure 2.21, where rstart and ystart are the initial particle coordinates.

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FIGURE 2.21 Structure of particles perpendicular to the electric field strength. Obtained from Murtsovkin [124].

Dipole-dipole interaction is the only potential mechanism of particle interaction in an electric field. The two others are not potential, and this means that the energy spent on the relative motion depends on the trajectory of the motion. In order to compare the intensities of these different mechanisms, it is convenient to define them in terms of the radial component of the relative motion velocity, Vr. For the dipole-dipole interaction, Vr is given by: r ¼ Vdd

e0 em a5 E2 ð1  DuÞ2 ð1  3 cos2 yÞ 2
ð1 þ 2DuÞr4

ð2:89Þ

where the value of the dipole moment is substituted according to Equation (2.38). The second, “electrohydrodynamic,” mechanism was suggested by Murtsovkin [124, 125, 129–131]. He observed that an electric field induces an electro-osmotic flow that has quadruple symmetry around the particle. The radial component of the relative (nonconducting) particle motion velocity is: Vhr ¼

3e0 em a3 E2 Dujzjð1  3 cos2 yÞ 2
ð1 þ 2DuÞr2

ð2:90Þ

The trajectory of this relative motion is shown in Figure 2.19B. The third, “concentration” mechanism [132] is related to the concentration polarization of the particles. It is defined as: Vcr ¼

3a3 Cs FE2 Duð1  cos2 yÞ djmj 2ð1 þ 2DuÞRTr 2 dCs

ð2:91Þ

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Dipole-dipole electrodynamic interaction

-

+ r2 r 2start

A

=

-

+

sin2q cosq cosqstart

sin2qstart

Electro-hydrodynamic interaction

1

r(r 2start − a2) r start (r 2 − a2)

B

=

2

sin2q cosq sin2qstart cosq start

Concentration polarization interaction C(r)

ζ+Δζ C0 1

C

2 r sinq = rstart sinq start

FIGURE 2.22 Three mechanisms for the interaction of particles in an electric field.

where Cs is the concentration of the electrolyte in the bulk of the solution. Again, the corresponding trajectory is given in Figure 2.22(C). Comparison of Vr for various mechanisms indicates that the electro2 and hydrodynamic and concentration mechanisms decay with distance as rpp 4 are longer ranged than the dipole-dipole mechanism which decays as rpp . This explains why, in some cases, particles build structures perpendicular to the electric field. These effects are especially pronounced for systems of biological cells and for metal particles [25]. The electrohydrodynamic and concentration mechanisms for nonconducting colloidal particles are only appropriate for static or low-frequency electric fields. They do not operate above the frequency of concentration polarization.

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This means that, at the high frequency of ultrasound, only the dipole-dipole interaction can affect the stability of the colloid.

2.7. TRADITIONAL PARTICLE SIZING There are three groups of traditional particle sizing methods that can be applied to characterize the size of colloidal particles [133, 134]. These groups are (1) counters, (2) fractionation techniques, and (3) macroscopic fitting techniques. Particle counters yield a full PSD weighted by number. This group includes various microscopic image analyzers [135], electrozone counters (Coulter counters), and optical zone counters. Fractionation techniques also yield a full PSD, but weighted by mass. They achieve this by separating particles into fractions with different sizes. This group includes sieving, sedimentation, and centrifugation [148]. The last group, the “macroscopic technique,” includes methods related to the measurement of various macroscopic properties that are particle size dependent. All of these methods require an additional step in order to deduce the PSD from the measured macroscopic data. This is the so-called ill-defined problem [136, 137]. We further discuss this problem later with regard to acoustics in Chapter 7. Each of these three groups has advantages and disadvantages over the others. There is no universal method available that is able to solve all particle sizing applications. For example, the “counters group” has an advantage over “macroscopic techniques” in that it yields full PSD. However, statistically representative analysis requires the counting of an enormous number of particles, especially for polydisperse systems. The technique can be very slow. Sample preparation is rather complicated and might affect the results. Transformation of a number-weighted PSD to a volume-based PSD is not accurate in many practical systems. Fractionation methods are, statistically, more representative than counters, but they are not suitable for small particles. It can take hours to perform a single particle size analysis, particularly of submicrometer particles, even using an ultracentrifuge. In these methods, hydrodynamic irregularities can dramatically disturb the samples and consequently affect the measured PSD. Macroscopic methods are fast, statistically representative, and easy to use. However, the price to be paid is the necessity to solve the ill-defined problem, and that limits particle size information. Fortunately, in many cases there is no need to know the exact details of the full PSD. Median size, PSD width, or bimodality is sufficient for an adequate description of many practical colloids and colloid-related technologies. This is why macroscopic methods are very popular and widely used in many laboratories and plants for routine analysis. Acoustics, as a particle sizing technique, belongs to the third group of macroscopic methods. It competes with the most widely used macroscopic

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method—light scattering [138, 139]. It can be considered also as an alternative to neutron scattering for characterizing microemulsions and polymer solutions. The advantages of acoustics over light scattering were given in the Introduction. There are a number of points of similarity between light and ultrasound that are important in the discussion of the application of acoustics to particle sizing analysis. To this end, a short description of light scattering follows. A full review can be found in the book by Bohren and Huffman [138].

2.7.1. Light Scattering: Extinction = Scattering + Absorption This section is a selection of various important statements and notions presented in the book by Bohren and Huffman [138]. The presence of the particles results in the extinction of the incident beam. The extinction energy rate, Wext, is the sum of the energy scattering rate, Wsca, plus the energy absorption rate, Wabs, through an imaginary sphere around a particle. Wext ¼ Wabs þ Wsca

ð2:92Þ

In addition to reradiating electromagnetic energy (scattering), the excited elementary charges may transform part of the incident electromagnetic energy into other forms (e.g., thermal energy), a phenomenon called absorption. Scattering and absorption are not mutually independent processes, and although, for brevity, we often refer only to scattering, we will always as well imply absorption. Also, we will restrict our treatment to elastic scattering: that is, the frequency of the scattered light is the same as that of the incident light. If the particle is small compared to the wavelength, all the secondary wavelets are approximately in phase; for such a particle, we do not expect much variation of scattering with direction. Single scattering occurs when the number of particles is sufficiently small and their separation sufficiently large so that in the neighborhood of any particle the total field scattered by all particles is small compared to the external field. Incoherent scattering occurs when there is no systematic relation between the phases of the waves scattered by the individual particles. However, even in a collection of randomly separated particles, the scattering is coherent in the forward direction. The extinction cross section Cext may be written as the sum of the absorption cross section, Cabs, and the scattering cross section, Csca: Cext ¼

Wext ¼ Cabs þ Csca Ii

ð2:93Þ

where Ii is the incident irradiance. There is an important optical theorem [138] that defines extinction as depending only on the scattering amplitude in the forward direction.

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This theorem, common to all kinds of seemingly disparate phenomena, involves acoustic waves, electromagnetic waves, and elementary particles. It is anomalous because extinction is the combined effect of absorption and scattering in all directions by the particle. Extinction cross-section is a well-defined, observable quantity. We measure the power incident at the detector with and without a particle interposed between the source and the detector. The effect of the particle is to reduce the detector area by Cext. Hence the description of Cext is as an area. In the language of geometrical optics, we would say that the particle “casts a shadow” of area Cext. However, this “shadow” can be considerably larger or smaller than the particle’s geometric shadow. Importantly, for a particle that is much larger than the wavelength of the incident light, the light scattered tends to be concentrated around the forward direction. Therefore, the larger the particle, the more difficult it is to exclude scattered light from the detector. The optical theorem neglects receiving the scattered light, but it creates a problem for large particles [138]. By observing only the transmitted light, it is not possible to determine the relative contribution of absorption and scattering to extinction; to do so requires an additional, independent observation. Multiple scattering aside, the underlying assumption is that all light scattered by the particles is excluded from the detector. The larger the particle, the more the scattering envelope is peaked in the forward direction and hence the greater the discrepancy between the measured and calculated extinction. Thus, a detection system has to be carefully designed in order to measure an extinction that can be legitimately compared with the theoretical extinction. This is particularly important for large particles. An extinction efficiency, Qext, can be defined as: Qext ¼ Cext = pa2

ð2:94Þ

As the particle radius, a, increases, Qext approaches a limiting value of 2. This is twice as large as that predicted from geometrical optics. This puzzling result is called the extinction paradox, since it seemingly contradicts geometrical optics. Yet geometrical optics is considered to be a good approximation, especially when all particle dimensions are much larger than the incident wavelength. Moreover, the result contradicts “common sense”; we do not expect a large object to dissipate twice the amount of energy that is incident upon it! Although for large objects geometrical optics is a good approximation for the exact wave theory, even for very large objects geometrical optics is still not exact. This arises because all practical materials are anisotropic; they posses “edges.” The edge deflects rays in its neighborhood, rays that, from the viewpoint of geometrical optics, would have passed unimpeded. Irrespective of how small the angle is through which they are deflected, rays are counted as having been removed from the incident beam and this

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contributes, therefore, to the total extinction (roughly speaking, we may say that the incident wave is influenced beyond the physical boundaries of the obstacle). These edge-deflected rays cause diffraction that can be described in terms of Fresnel zones [138, 140]. It is convenient to describe diffraction by considering light transmission through an aperture instead of the scattering by the particle. For instance, we may ask how the intensity changes when the point of observation, Pobs, moves along the axis of an aperture of fixed dimensions which models the particle. Since the radii of the Fresnel zones depend on the position of Pobs, we find that the intensity goes through a series of maxima and minima, occurring, respectively, when the aperture includes an odd or an even number of Fresnel zones. A special case is when the source is very far away, so that the incident wave may be regarded as a plane wave. The radius of the first Fresnel zone, R1, increases indefinitely as the distance from the aperture Robs of Pobs goes to infinity. Thus, if the point of observation is sufficiently far away, the radius of the aperture is certainly smaller than that of the first Fresnel zone. As the distance, Robs, gradually decreases, that is, Pobs approaches the aperture, the radius, R1, decreases correspondingly. Hence an increasingly large fraction of the first Fresnel zone appears through the aperture. Again, as Pobs moves closer and closer to the aperture, maxima and minima follow each other at decreasing distances. When the distance Robs of Pobs from the aperture is not much larger than the radius of the aperture itself, R, then the distances between successive maxima and minima approach the magnitude of the wavelength. Under these conditions, of course, the maxima and minima can no longer be practically observed. From our previous discussion, when the distance of the source and the point of observation from the aperture is large, compared with the square of the radius of the aperture divided by the wavelength (Rayleigh distance, R0), it follows that only a small fraction of a Fresnel zone appears through the aperture. The diffraction phenomenon observed in these circumstances is called Fraunhofer diffraction. When, however, the distance from the aperture of either the point of observation or the source is R0 or smaller, then the aperture uncovers one or more Fresnel zones. The diffraction phenomenon observed is then termed Fresnel diffraction. The exact solutions for scattering and absorption cross-sections of spheres are given in the Mie theory [141]. It might appear that we are faced with the straightforward task of obtaining quantitative results from the Mie theory. However, the number of terms required for convergence can be quite large. For example, if we were interested in investigating the rainbow created by 1-mm water droplets, we would need to sum about 12,000 terms. Even for smaller particles, the number of calculations can be exceedingly large. Computers can greatly reduce this computation time, but problems still remain related to the unavoidable need to represent a number having an infinite

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number of digits by a number with a finite number. The implicit round-off error accumulates as the number of terms required to be counted increases. Unfortunately, there is no unanimity of opinion in defining the condition under which round-off error accumulation becomes a problem. If we were interested only in scattering and absorption by spheres, we would need to go no further than the Mie theory. But physics is, or should be, more than just a semi-infinite strip of computer output. In fact, great realms of calculations often serve only to obscure from view the basic physics that can be quite simple. Therefore, it is worthwhile to consider approximate expressions, valid only in certain limiting cases, that point the way toward approximate methods to be used to tackle problems for which there is no exact theory. If a particle has a nonregular geometrical shape, then it is difficult, if not impossible, to solve the scattering problem in its most general form. There are, however, frequently encountered situations when the particles and the medium have similar optical properties. If the particles, which are sometimes referred to as “soft,” are not too large (but they might exceed the Rayleigh limit), it is possible to obtain relatively simple, approximate expressions for the scattering matrix. This is expressed as the Rayleigh-Gans theory. Geometrical optics is a simple and intuitively appealing approximate theory that need not be abandoned because an exact theory is at hand. In addition to its role in guiding intuition, geometrical optics can often provide quantitative answers to small-particle problems: solutions that are sufficiently accurate for many applications, particularly in light of the precision, accuracy, and reproducibility of actual measurements. Transition from single-particle scattering to scattering by a collection of particles can be simplified using the concept of an effective refractive index. This concept is meaningful for a collection of particles that are small compared with wavelength, at least as far as transmission and reflection are concerned. However, even when the particles are small compared to the wavelength, this effective refractive index should not be interpreted too literally as a true refractive index on the same footing as the refractive index of, say, a homogeneous medium. For example, attenuation, in a strictly homogeneous medium, is the result of absorption and is accounted for quantitatively by the imaginary part of the refractive index. In a particulate medium, however, attenuation may be wholly or in part the result of scattering. Even if the particles are nonabsorbing, the imaginary part of the effective refractive index can be nonzero [138]. When an incident beam traverses a distance, x, through an array of particles, the irradiance is attenuated according to It ¼ Ii expðaxÞ

ð2:95Þ

a ¼ Np Cext ¼ Np Cabs þ Np Csca

ð2:96Þ

where

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Now, the exponential attenuation of any irradiance in particulate media requires that: ax 1

ð2:97Þ

This condition might be relaxed somewhat if the scattering contribution to total attenuation is small: Np Csca x 1

ð2:98Þ

An amount of light, dI, is removed from a beam propagating in the x direction through an infinitesimal distance between x and x þ dx in an array of particles dI ¼ aIdx

ð2:99Þ

where I is the beam irradiance at x. However, light can get back into the beam through multiple scattering; that is, light scattered at any other position in the array may ultimately contribute to the irradiance at x. Scattered light, in contrast to absorbed light, is not irretrievably lost from the system. It merely changes direction, is lost from a beam propagating in a particular direction, but then contributes in other directions. Clearly, the greater the scattering cross-section, the number density of particles, and thickness of the array, the greater will be the multiple scattering contributions to the irradiance at x. Thus, if Np Csca is sufficiently small, we may ignore multiple scattering and can integrate Equation (2.99) to yield the exponential attenuation Equation (2.95). The expression for the field, or envelope, of light scattered by a sphere is normally obtained under the assumption that the beam is infinite in lateral extent; such a beam, however, is difficult to produce practically in a laboratory. Nevertheless, it is physically plausible that scattering and absorption, by any particle, will be independent of the extent of the beam provided that the beam is large compared to the particle size; that is, the particle is completely bathed in the incident light. This is supported by the analysis of Tsai and Pogorzelski [142], who obtained an exact expression for the field scattered by a conducting sphere when the incident beam is cylindrically symmetric with a finite cross-section. Their calculations show no difference in the angular dependency of the light scattered by a conducting sphere between infinite and finite beams provided that the beam radius is about 10 times larger than the sphere radius.

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[3] R. Weast (Eds.), Handbook of Chemistry and Physics, 70th ed., CRC Press, Florida, 1989. [4] S. Temkin, Elements of Acoustics, first ed., Wiley, New York, 1981. [5] P.M. Morse, K. Uno Ingard, Theoretical Acoustics, McGraw-Hill, New York, NY, 1968. Princeton University Press, New Jersey, 1986, p. 925. [6] T.A. Litovitz, Lyon, Ultrasonic hysteresis in viscous liquids, J. Acoust. Soc. Am. 26 (1954) 577–580. [7] Stokes, On a difficulty in the Theory of Sound, Philos. Mag. Nov. (1848). [8] L.W. Anson, R.C. Chivers, Thermal effects in the attenuation of ultrasound in dilute suspensions for low values of acoustic radius, Ultrasonic 28 (1990) 16–25. [9] D.W. Cannon, New developments in electroacoustic method and instrumentation, in: S.B. Malghan (Ed.), Electroacoustics for Characterization of Particulates and Suspensions, NIST, 1993, pp. 40–66. [10] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65. [11] A. Kitahara, Nonaqueous systems, in: A. Kitahara, A. Watanabe (Eds.), Electrical Phenomena at Interfaces, Marcel Decker, New York, 1984. [12] Ph.C. van der Hoeven, J. Lyklems, Electrostatic stabilization of suspensions in non-aqueous media, Adv. Colloid Interface Sci. 42 (1992) 205–277. [13] I.D. Morrison, Electrical charges in non-aqueous media, Colloids Surf. A 71 (1993) 1–37. [14] N. Bjerrum, A new form for the electrolyte dissociation theory, Proceedings of the 7th International Congress of Applied Chemistry, Section X, London, 1909, pp. 55–60. [15] R.M. Fuoss, Ionic association. III. The equilibrium between ion pairs and free ions, J. Am. Chem. Soc. 80 (1958) 5059–5060. [16] J. Bockris, A.K. Reddy, Modern Electrochemistry, vol. 1, A Plenum/Rosetta ed., Plenum Press, New York, 1977. [17] J. Lyklema, Principles of the stability of lyophobic colloidal dispersions in non-aqueous media, Adv. Colloid Interface Sci. 2 (1968) 65–114. [18] D.N.L. McGown, G.D. Parfitt, E. Willis, Stability of non-aqueous dispersions. 1. The relationship between surface potential and stability in hydrocarbon media, J. Colloid Interface Sci. 20 (1965) 650–664. [19] A.S. Dukhin, P.J. Goetz, How non-ionic “electrically neutral” surfactants enhance electrical conductivity and ion stability in non-polar liquids, J. Electroanal. Chem. 588 (2006) 44–50. [20] R. Treffers, M. Cohen, High resolution spectra of cool stars in the 10- and 20-micron region, Astrophys. J. 188 (1974) 545–552. [21] R. Fuchs, Theory of the optical properties of ionic crystal cubes, Phys. Rev. B11 (1975) 1732–1740. [22] H.P. Pendse, T.C. Bliss, W. Han, Particle shape effects and active ultrasound spectroscopy, in: V.A. Hackley, J. Texter (Eds.), Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, Amer. Ceramic Soc., Westerville, OH, 1998. [23] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and the Double Layer in Dispersed Systems and Polyelectrolytes, Wiley, New York, NY, 1974. [24] P.V. Zinin, Theoretical analysis of sound attenuation mechanisms in blood and erythrocyte suspensions, Ultrasonics 30 (1992) 26–32. [25] A.S. Dukhin, Biospecifical mechanism of double layer formation in living biological cells and peculiarities of cell electrophoresis, Colloids Surf. 73 (1993) 29–48. [26] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman Company, New York, 1982.

84 [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

[39] [40] [41] [42] [43] [44] [45] [46]

[47] [48] [49] [50] [51] [52]

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Chapter 3

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

3.1. Longitudinal Waves and the Wave Equation 92 3.2. Acoustic Impedance 95 3.3. Propagation Through Phase Boundaries: Reflection 97 3.4. Longitudinal Rheology and Shear Rheology 99 3.5. Longitudinal Rheology of Newtonian Liquids: Bulk Viscosity 103 3.5.1. In Hydrodynamics 103 3.5.2. In Acoustics 103

3.5.3. In Analytical Chemistry 3.5.4. In Molecular Theory of Liquids 3.5.5. In Rheology 3.6. Attenuation of Ultrasound in Newtonian Liquid: Stokes Law 3.7. Newtonian Liquid Test Using Attenuation Frequency Dependence 3.8. Chemical Composition Influence References

104 104 104

108

110 114 121

This chapter has been significantly modified with a more specific description of the relationship between acoustics and rheology. The purpose of this chapter is to provide a general overview characterizing sound wave propagation in homogeneous liquids. It is important to stress here that practically any liquid system can be modeled as “homogeneous.” For instance, we can treat milk as a homogeneous liquid, forgetting that it is actually a collection of fat droplets in water with sugars and proteins. This approach would provide only limited information about the system. However, it is useful when we deal with very complex systems and some degree of abstraction helps us learn at least something. This approach is essentially identical to a rheological description of the complex liquids. That is why Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23003-X Copyright # 2010, Elsevier B.V. All rights reserved.

91

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rheological aspect of acoustics plays an important role in this chapter. The relationship between acoustics and rheology was recognized long ago by such prominent scientists in the field of acoustics as Rayleigh [1, 2], Morse [3, 4], and Temkin [5]. Unfortunately, there is still very little appreciation of the possibilities that this opens in the rheological field. We will start with introducing some basic information on acoustics in homogeneous media. Then, we will present some general concepts of longitudinal rheology and its relationship with shear rheology. Then, we provide some examples of longitudinal rheology for Newtonian liquids. The last section is dedicated to the influence of the chemical composition on the acoustic properties of Newtonian liquids.

3.1. LONGITUDINAL WAVES AND THE WAVE EQUATION We call traveling compression waves in liquids “longitudinal waves,” in contrast to “transverse waves” typified by a vibrating string. The direction that the material moves, relative to the direction of wave propagation, makes the difference. For a longitudinal wave, the liquid molecules move back and forth in the direction of the wave propagation, whereas for a transverse wave this motion is perpendicular to the direction of propagation. Longitudinal waves, in the most general case, are three dimensional with a potentially very complex geometry. Here, we will consider only two-dimensional plane waves, which have the same direction of propagation everywhere. Longitudinal waves can propagate in a fluid because the fluid has a finite compressibility that allows the energy to be transported across space. In addition to these compression waves, the sound wave itself may induce shear and thermal waves at the boundaries with other surfaces. Compression waves are able to propagate over long distances in the liquid, whereas shear and thermal waves exist only in the close vicinity of phase boundaries. This chapter covers only compression longitudinal waves. Shear and thermal waves play a major role in colloidal systems because of the extended surface area. This aspect is dealt with in detail later, in Chapter 4. We consider the liquid itself to be in equilibrium, and having a certain density, rm, at a given static pressure, P, and temperature, T. The relationship between these three parameters can be described by two partial derivatives, one at constant temperature and one at constant pressure. Together, these equations constitute the so-called equations of state, namely:
 1 @rm p b ¼ ð3:1Þ rm @P T b¼


 1 @rm rm @T P

ð3:2Þ

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

93

We commonly refer to the coefficient bp as the isothermal compressibility and the coefficient b as the thermal expansion coefficient. The presence of a sound wave at a particular instant of time, and point in space, causes an incremental variation of the density, pressure, and temperature about their equilibrium values rm, Peq, and T. We designate the instantaneous values of these incremental changes as drm, dP, and dT, respectively. The pressure variation is the easiest to detect and is in fact a typical output of an acoustic experiment. The sound wave pressure is usually expressed in decibels, abbreviated as dB, and defined as 20 times the logarithm to the base 10 of the ratio of the pressure amplitude, in dynes per centimeter squared, and a reference level of 1 dyne/cm2. The variation of the temperature in the liquid, dT, due to the presence of the sound wave is usually very small, because typically the high thermal conductivity of the liquid rapidly smoothes out any local perturbations. Consequently, the density variation, drm, is adequately described considering only the isothermal compressibility bp:
 @rm P ¼ rm þ rm bp P ð3:3Þ rm þ drm ¼ rm þ @P T The variation of the density is independent of the equilibrium pressure. The variation of the pressure in the case of a plane sound wave is described with a wave equation: @2P 1 @2P ¼ 2 2 2 @x cm @t

ð3:4Þ

where the sound speed cm is found from: c2m ¼

1 b p rm

ð3:5Þ

The values for sound speed and density of several liquids are given in Table 3.1 using the data published in Ref. [6]. The sound speed through most solid materials is much higher than that of fluids and is typically between 5000 and 6000 m/s. Compressibility is a measure of the liquid elasticity. Precise sound speed measurement provided by modern acoustic spectrometers allows fast, nondestructive, and precise calculation of both storage modulus and compressibility as presented in Table 3.1 for all 12 liquids. These values agree well with independent data from the Handbook of Chemistry and Physics [7] for some of the liquids. The speed of sound is temperature dependent. The magnitude of this dependence is usually several meters per second per degree, as shown in Table 3.1 according Ref. [6]. Figure 3.1 illustrates this dependence for water.

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TABLE 3.1 Sound Speed and Compressibility of 12 Newtonian Liquids Sound Sound speed T Storage speed at coef. at 25  C modulus G0long  25 C (m/s) (m/s/ C) at 25  C (109 Pa)

Liquid

Compressibility at 25  C (1010 Pa1)

Cyclohexane

1256

6.66

1.23

1.15

Cyclohexanone

1407

4.20

1.88

0.75

1-Pentanol

1273

2.88

1.31

1.07

Toluene

1308

4.70

1.53

0.92

Methanol

1106

3.65

0.97

1.46

Hexane

1078

4.65

0.77

1.84

2-Propanol

1143

4.40

1.03

1.37

Butyl acetate

1193

4.44

1.26

1.12

Acetone

1166

4.97

1.08

1.31

Ethanol

1147

4.26

1.03

1.36

Pyridine

1417

4.57

1.98

0.72

Water

1496

þ2.40

2.23

0.63

1522 1520

Sound speed (m/s)

1518 1516 1514 1512 1510 1508 1506 1504 1502 1500 28.

29.

30.

31.

32.

33.

34.

35.

36.

Temperature (C) FIGURE 3.1 Speed of sound in water measured at different temperatures using DT-600 [6].

3

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

95

The simple wave equation (Equation 3.4) is valid for uniform, homogeneous liquids having negligible attenuation due to viscous and thermal effects and, furthermore, only if the acoustic pressure is small compared to the equilibrium pressure. This simplified model of acoustic phenomena turns out to be quite adequate in practice, since the typical ultrasound pressure levels used for acoustic characterization are thousands of times weaker than the normal equilibrium pressure of 1 atm. We do not consider any nonlinear effects that might occur with the very high intensity ultrasound sometimes used for various modifications of liquids and colloids. The general solution of the above wave equation describes a pressure wave traveling in the positive, x, direction as a function P(xcmt): Pðx, tÞ ¼ P0 expðam xÞexp½ jðot  FÞ

ð3:6Þ

where am is the attenuation of the medium, o is the frequency of ultrasound, and P0 is the initial amplitude of the pressure. A similar equation can be written for the fluid velocity, v. The ratio between the acoustic pressure and the fluid velocity is called the acoustic impedance, which will be discussed now.

3.2. ACOUSTIC IMPEDANCE There is a convenient analogy between electric and acoustic phenomena. It is based on the linear relationship between the driving force and the corresponding flow. In the case of an alternating electric field, it is electric voltage and electric current. According to Ohm’s law, the current is proportional to the voltage; the coefficient of proportionality is called the electric impedance. The acoustic impedance, Z, is introduced in a similar way, as a coefficient of propotionality between pressure, P, and velocity of the particles, v, in the sound wave: P ð3:7Þ v As with the electric impedance, the acoustic impedance is a complex number. It depends on the other acoustic properties of the media. To derive this relationship, we consider the propagation of a plane longitudinal wave through the medium along the x-axis. This medium is characterized by a given density, rm, sound speed, cm, and longitudinal wave attenuation, am. The particle’s displacement, u, for a plane sound wave is a simple harmonic function: Z¼

uðx, tÞ ¼ u0 expðam xÞexp½ jðot  FÞ

ð3:8Þ

where F is the phase, F ¼ kx with k ¼ 2p/l, l is the wavelength in meters; o ¼ 2pf, where f is frequency in hertz. The attenuation coefficient, am, has dimension of neper per meter. Attenuation expressed in decibels (dB) is

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related to the attenuation expressed in nepers (np) through the following: dB/m ¼ 8.686 np/m. These equations allow us to calculate the wavelength for different fluids. For instance, the wavelength in water varies roughly from 15 mm at 100 MHz up to 1.5 mm at 1 MHz. It is important to note that the wavelength of ultrasound in the megahertz range is much larger than the wavelength of light; for example, the wavelength of green light is about 0.5 mm. This difference in wavelength between light and ultrasound results in a substantial variance between these two wave phenomena, which otherwise have many similarities. Combining together the two exponential functions and expressing the frequency and phase through the wavelength, we can rewrite Equation (3.8) as: 

 2pj am l uðx, tÞ ¼ u0 exp cm t  x 1  j ð3:9Þ l 2p To derive an expression for the pressure, we use Hooke’s law, which relates a force exerted on an element of the medium, Fhook, and a strain caused by the sound wave, @u/@x: Fhook ¼ MS

@u @x

ð3:10Þ

where S is the cross-sectional area. The pressure is simply the opposite of the force per unit area of the surface:
 @u am l 2 @u 2 2pj P ¼ M ¼ rm c ¼ rm cm 1j uðx, tÞ ð3:11Þ @x @x l 2p We express the velocity of the particles’ motion as a time derivative of the displacement: @uðx, tÞ 2pjcm uðx, tÞ ð3:12Þ ¼ l @t Replacing displacement with the particle velocity in the expression for the pressure gives:
 am l v ð3:13Þ P ¼ rm cm 1  j 2p vðx, tÞ ¼

It directly follows by analogy with this expression that the acoustic impedance, Z, is:
 am l Zm ¼ rm cm 1  j ð3:14Þ 2p where the attenuation is in (np/m), wavelength in (m), sound speed of the longitudinal wave in (m/s), and density is in (kg/m3).

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The attenuation does not contribute significantly to the acoustic impedance. Even in highly concentrated colloids at high frequency, when the attenuation becomes about 1000 dB/cm, it still contributes only 2% to the acoustic impedance. This allows us to approximate the acoustic impedance as a real number equal to the product of the density and sound speed.

3.3. PROPAGATION THROUGH PHASE BOUNDARIES: REFLECTION Acoustic impedance is a very convenient property for characterizing effects that occur when the sound wave meets the boundary between two phases. There are certain similarities between longitudinal ultrasound and light reflection and transmission through the phase boundaries. For instance, the ultrasound reflection angle from a plane surface is equal to the incident angle which is the same as for light (see Figure 3.2): yi ¼ yr

ð3:15Þ

where the index i corresponds to the incident wave and index r corresponds to the reflected wave. The transmitted wave angle must satisfy wavefront coherence at the border. Again, this yields the same relationship as with light transmission: sinyi c1 ¼ sinyt c2

ð3:16Þ

where indexes 1 and 2 correspond to the different phases.

reflected wave

incident wave

phase 1

qr

qi

qt phase 2

transmitted wave

FIGURE 3.2 Illustration of the sound propagation through a phase boundary.

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Propagation of the sound wave through the phase border should not create any discontinuities in pressure or the particle’s velocity. This condition yields the following relationships for the pressure in the reflected and transmitted waves: Pr Z2 cosyi  Z1 cosyt ¼ Pi Z2 cosyi þ Z1 cosyt

ð3:17Þ

Pt 2Z2 cosyi ¼ Pi Z2 cosyi þ Z1 cosyt

ð3:18Þ

In the case of normal incidence, when yi ¼ yt ¼ 0, these equations simplify to: Pr Z2  Z1 ¼ Pi Z2 þ Z1

ð3:19Þ

Pt 2Z2 ¼ P i Z2 þ Z 1

ð3:20Þ

and

From these equations, an important relationship is derived between the phases of the reflected and incident waves. If Z2 > Z1, then the reflected pressure wave is in phase with the incident wave; otherwise, it is 180 out of phase. The pressure value determines the intensity of the ultrasound, I: I¼

P2 2rc

ð3:21Þ

For normal incidence, we can use Equations (3.19) and (3.20) to obtain the ultrasound intensity of the reflected and transmitted waves: Ir ðZ2  Z1 Þ2 ¼ Ii ðZ2 þ Z1 Þ2

ð3:22Þ

It 4Z2 Z1 ¼ Ii ðZ2 þ Z1 Þ2

ð3:23Þ

and

At normal incidence, the reflected wave interferes with the incident wave. This leads to the buildup of standing waves. For a perfect reflector, the particle displacement in reflected and incident waves compensate each other completely when they are out of phase. They add together when they are in phase. This leads to a repeating pattern of nodes and maxima. Standing waves do not transmit any power, since the power coming back equals the power going out.

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Standing waves can superimpose with the traveling waves when the reflection is not perfect. This effect occurs when ultrasound propagates through a multilayer medium. The case of three phases is important and well characterized. A standing wave appears in the first and the second layers. They superimpose here with traveling waves if reflection at the phase boundaries is not perfect. In the case of normal incidence, it is possible to derive [5] an analytical expression for the intensities of the incident and transmitted waves: It3 ¼ Ii1

4Z3 Z1
2 2pl Z3 Z1 2pl2 2 ðZ3 þ Z1 Þ2 cos2 þ Z2 þ sin2 l2 Z2 l2

ð3:24Þ

where l2 is the thickness of the second layer. One important conclusion follows from Equation (3.24). There are two cases when the second layer becomes transparent for ultrasound propagation. The first one is rather obvious; it happens when the thickness of the second layer is much less than the wavelength ðl2  l2 =4Þ. The second case is related to the standing waves built up in the intermediate layer, when l2 ¼ nl2/2. These conditions are important for designing acoustic and electroacoustic devices.

3.4. LONGITUDINAL RHEOLOGY AND SHEAR RHEOLOGY As noted in rheological handbooks, there are two distinctively different types of rheology—“shear rheology” and “extensional rheology” [8]. Basically, they differ in the geometry of the applied stress. In shear rheology, the stress is tangential to the surface from which the stress is applied. This causes a sliding relative motion of the liquid layers. In contrast, in extensional rheology, the stress is normal to the surface applying the stress. If the liquid is considered incompressible, the volume is unchanged and we say this is an isochoric process. The rheology of oscillating extensional stress differs substantially from traditional steady-state extensional rheology. The isochoric condition might be violated when the extensional stress varies in time. Extensional nonisochoric rheology with oscillating extensional stress is called “longitudinal rheology.” There is an old review by Litovitz and Davis [9] that presents in parallel both shear and longitudinal rheological theories. Both shear and extensional stress cause displacement in the liquid. In the case of a step change in the shear stress, the liquid displacement propagates with a certain speed from the interface where it has been generated into the bulk. Eventually, it reaches some steady-state condition within the entire system. Many modern rheometers, instead of applying a steady-state stress, apply stress that varies in time. It is achieved by moving the interface backward and forward.

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When the frequency of such an oscillation is low, the displacement originated at the interface has enough time to fill the entire system during the first half of the cycle. All parts of the system would then experience displacement in one direction. Increasing the frequency would eventually alter this situation. The direction of the interface motion would change before the displacement can fill the entire system. Consequently, there would be areas in the system with opposite displacements. Such oscillating stress creates a “wave” of displacement. This is the point where rheology comes into contact with acoustics. Acoustics is a science that studies propagation of mechanical waves. There is a spatial characteristic of the waves—“wavelength.” It is the distance of wave propagation during a single period of oscillation. If wavelength is much shorter than size of the system, the subjects of rheology and acoustics would merge. This is situation that we consider in this paper. We will be dealing with waves of oscillating stresses, both shear and extensional. There are several different ways of generating these oscillations: rotating cylinders, shaking plates, vibrating surfaces of piezo-crystals, etc., see Barnes [8], Dukhin and Goetz [10], Portman and Markgaf [11], Soucemarianadin et al. [12], and Williams et al. [13–15]. There is a set of notations introduced in acoustics for characterizing these oscillating stresses. One of these notations is the frequency of oscillation, usually denoted as o. The other one is pressure P(x,t) at given moment t and given point x. In the case of monochromatic oscillation at a single frequency, this pressure is usually expressed in terms of complex numbers, see Equation (3.6). The attenuation coefficient, a, determines how quickly the pressure amplitude decays with distance due to various dissipative effects. An alternative way to characterize the pressure decay is to use the notion of “penetration depth” x0 where: 1 ð3:25Þ a The pressure amplitude decays by a factor ℓ from its initial value P0 when the distance traveled equals the “penetration depth,” where ℓ  2.71, the base of the natural logarithm. The penetration depth differs from the wave length l, which depends mostly on frequency: x0 ¼

l¼ where f is frequency in hertz.

2pV V ¼ o f

ð3:26Þ

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The attenuation coefficient and the penetration depth are phenomenological and experimentally measurable parameters that are used in acoustics for characterizing dissipative processes. The speed of the wave propagation is related to elastic properties of the liquid. Rheology operates with completely different set of notions for characterizing the viscoelastic properties of liquids. In the case of shear stress, it is a “complex shear modulus” G*, which has a real component “storage modulus” G0 and an imaginary component “loss modulus” G00 : G ¼ G0 þ jG00

ð3:27Þ

In a manner similar to shear rheology, Litovitz and Davis [9] introduce a “complex longitudinal modulus” M*, which corresponds to the longitudinal rheology case, where M ¼ M0 þ jM00

ð3:28Þ

However, it turns out that these two modulii are in fact identical if one makes a suitable change in subscript. Therefore we will use the same symbol G* for both of them, with the subindex “shear” for shear stress, and subindex “long” for longitudinal stress. Similarly, we will use the same indices for both the attenuation coefficient and penetration depth for distinguishing between these two types of stress. The above-mentioned similarity appears in the general relationship between acoustic and rheological parameters for both stresses. Functionally, these relationships are the same for both stresses: 0 1 o  2

G0shear ¼ ro2 V 2 2 long

a2 shear long

4o2 þ a2

shear long

V

B4p2 1 C C ro2 B @ l2  x2 A 0shear

2

V25

¼ 2ro V 2 3

2

long

4o2 þ a2

shear long

where r is density.

12

B4p2 1 B @ l2 þ x2

ð3:29Þ

C C A

0shear long

ro2

a shear V G00shear long

long

32 ¼ 0

32 ¼ 0 V25

lx

4p 0shear long

B4p2 1 B @ l2 þ x2

0shear long

12 C C A

ð3:30Þ

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It is possible to introduce shear viscosity and longitudinal viscosity for non-Newtonian liquids using the same expression for both parameters: G ¼ G0 þ iG00 ¼ G0 þ io shear ðoÞ

ð3:31Þ

long

The derivation of Equations (2.29) and (3.30) for the shear rheology case is from the paper by Williams and Williams [13–15], who in turn make reference to a much earlier book by Whorlow [16]. The derivation of the identical equations for the longitudinal rheology case is taken from the review by Litovitz and Davis [9]. They, in turn, make reference to much earlier works by Miexner [16, 17, 99], who apparently derived the general thermodynamic theory of stress-strain relationship in liquids. Together, Equations (3.29) and (3.30) provide general phenomenological link between acoustics and rheology. One can use them for calculating the shear loss modulus G00 shear and shear storage modulus G0 shear from the acoustically measured shear penetration depth x0shear (or shear wave attenuation ashear) and shear wave sound speed. The same can be done for longitudinal parameters. The longitudinal loss modulus G00 long and longitudinal storage modulus G00 long are linked to the sound attenuation coefficient along. The whole difference between shear rheology and longitudinal rheology is comprised of the specific differences in the penetration depth. It is possible to make some estimates of each penetration depth using well-known theories. The penetration depth of the shear stress xoshear can be estimated using following well-known equation for viscous boundary layer, see for instance Dukhin and Goetz [10]: rffiffiffiffiffiffiffi  ð3:32Þ x0 long  ro Most liquids do not support shear waves very well, which means that the penetration depth is rather short. In water, for instance, the shear penetration depth is approximately 0.1 mm at a frequency of 100 Hz. It quickly decreases with frequency. In the case of ultrasound frequency of 1 MHz, it is only about 10 mm and becomes just 1 mm at 100 MHz. On the other hand, longitudinal waves are able to penetrate much further into the liquid. We can use the Stokes’s law, Equation (3.43), for estimating the longitudinal penetration depth, x0long. It yields the following simple expression for estimating x0long: x0 long 

rV 3 o2 

ð3:33Þ

The longitudinal penetration depth is much longer than the shear penetration depth. In water, the longitudinal penetration depth is about 1000 m at

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1 MHz, and 10 cm even at 100 MHz. Longitudinal waves propagate long distances in liquids. It is easy to generate and measure them. One of the peculiarities of the longitudinal viscosity is the small value of the parameterðaV=oÞ2 . As a result, general equations for longitudinal viscoelastic modulus could be simplified, and longitudinal viscosity is: 2along rV 3 ð3:34Þ o2 The possibility of studying a much higher frequency range is a clear advantage of using longitudinal stress waves over shear stress waves. long ¼

3.5. LONGITUDINAL RHEOLOGY OF NEWTONIAN LIQUIDS: BULK VISCOSITY The dynamics of Newtonian liquids is described by the Navier-Stokes equation, which can be found in most books on hydrodynamics, for example, Happel and Brenner [18], Landau and Lifshitz [19], and the fundamental work on theoretical acoustics by Morse and Ingard [3, 4]. For a compressible Newtonian liquid, the Navier-Stokes equation can be written as:  
 @v 4 r þ ðvrÞv ¼ grad P þ Dv þ b þ  grad div v ð3:35Þ @t 3 The last term on the right-hand side of this equation takes into account the compressibility of the liquid and becomes important only for such effects where liquid compressibility cannot be neglected. It contains two viscosity coefficients: one is the familiar shear viscosity ; the other one, b, is much less known. We use term “bulk viscosity” for this parameter throughout this study, but the same property has been referred to by many different names in various scientific disciplines as described below. A rose by any other name. . .

3.5.1. In Hydrodynamics The earliest known mention of this parameter was made by Sir Horace Lamb in his famous work Hydrodynamics first published in 1879 [20, 100]. The sixth edition of this book contains reference to a “second viscosity coefficient.” Other more modern books on hydrodynamics by Happel and Brenner [18], Landau and Lifshitz [19], as well as Potter and Wiggert [21] use the same name.

3.5.2. In Acoustics Theoretical Acoustics by Morse and Ingard [3, 4] refers to this parameter as “bulk viscosity,” and we follow here their lead. This same name is used by

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Kinsler et al. [22], in Fundamentals of Acoustics, Bhatiain Ultrasonic Adsorption [101], Shortley and Willians in Elements of Physics [102]. However, at the same time Temkin [5] in Elements of Acoustics uses instead the term “expansion coefficient of viscosity.” To obfuscate things a bit further, Litovitz and Davis [9] use yet another term “volume viscosity.”

3.5.3. In Analytical Chemistry Kaatze et al. [23] in an extensive review on ultrasonic spectroscopy for characterizing the chemistry of liquids, uses again the term “volume viscosity” as did Carroll and Patterson in their study using Brillouin spectroscopy [24].

3.5.4. In Molecular Theory of Liquids Several authors calculate this parameter using various molecular theories of liquids [21, 25–32], and following the lead of Morse and Ingard universally adopt the term “bulk viscosity.”

3.5.5. In Rheology Hurtado-Laguna et al. [33] in their studies of molten polymers, so far the most important application associated with volume viscosity, again adopt the term of “volume viscosity.” There is also a historical paper by Graves and Argrow [34] that applies the term “coefficient of bulk viscosity.” Bulk viscosity is critical for understanding the longitudinal rheology of Newtonian liquids. At the same time, it is negligible for traditional lowfrequency shear rheology which operates with incompressible liquids where div v ¼ 0

ð3:36Þ

and the Navier-Stokes equation reduces to the much more familiar form:   @v þ ðvrÞv ¼ grad P þ Dv ð3:37Þ r @t If the liquid compressibility is unimportant, we are then, and only then, justified in using the simpler approximation of Equation (3.35). Indeed, many textbooks just assume the liquid to be incompressible and bulk viscosity therefore plays no role. It is not surprising that bulk viscosity is little known when the modern Handbook of Viscosity by C. Yaws [35] and the Handbook of Chemistry and Physics [7] do not mention it at all. Interestingly, it was noted by Stokes, and indeed known as the Stokes’s hypothesis [22, 36], that there are a few compressible fluids, such as monatomic gases, where this bulk viscosity is zero. This hypothesis raised some question about the very existence of bulk viscosity, even in liquids. Several papers [34, 37] discuss this issue and take up the task of proving that bulk

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viscosity does indeed exists for liquids. Experimental justification for the existence of a bulk viscosity for liquids remained elusive with a paucity of meaningful data. Modern measurement techniques now allow precise determination of bulk viscosity as will be shown here. In fact, we have found that the bulk viscosity is not even close to zero for any of the liquids that we have measured and, indeed, the bulk viscosity is often larger than the dynamic viscosity, as we will show later. One might ask why one should bother to learn about this seemingly obscure bulk viscosity. A general answer was given by Temkin [5], who presented a very clear physical interpretation of both dynamic and bulk viscosities in terms of molecular motion. He points out that molecules have “translational,” “rotational,” and “vibrational” degrees of freedom in liquids and gases. The classical dynamic viscosity, , is associated only with the “translational” motion of the molecules. In contrast, the bulk viscosity b reflects the relaxation of both rotational and vibrational degrees of freedom. This leads one to conclude that, if one wants to study rotational and vibrational molecular effects in complex liquids, one must learn how to measure bulk viscosity. A more specific answer is that measuring the bulk viscosity, which is a typical output parameter of molecular theory models, allows one to then test the validity of such models. As one important example, Enskog’s theory [38] yields an analytical expression for both dynamic and bulk viscosities.
 1 þ 0:8 þ 0:761Y ð3:38Þ  ¼ b0 Y b ¼ 1:002b0 Y

ð3:39Þ

where b0 is second virial coefficient, and Y is relative collision frequency. Another classical theory by Kirkwood [25] and Kirkwood, Buff, and Green [26] presents integral expressions for both dynamic and bulk viscosities. However, at the end of the paper they declare that numerical calculation of the integrals is possible only for dynamic viscosity, stating that a numerical procedure for calculating bulk viscosity is not possible due to “. . .extraordinary sensitivity to the equilibrium radial distribution function . . ..” There are several recent theories for calculating bulk viscosity for Lennard-Jones liquids using the Green-Kubo model. They are described in the theoretical papers by Hoheisel, Vogelsang, and Schoen [27], Okumura and Yonezawa [28, 29], Dyer, Pettitt, and Stell [31], Meier, Laesecke, and Kabelac [30], and Tani and Bertolini [32]. Verification of these theories requires experimental data for the bulk viscosity of Newtonian liquids, but strangely enough there are only a few studies known to us that report experimental bulk viscosities for such Newtonian liquids.

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The first is a 50-year-old review by Litovitz and Davis [9] that reports a bulk viscosity for water of 3.09 cP at 15  C and for methanol 2.1 cP at 2  C. This reported value for the bulk viscosity for water can serve as a reference point for new studies on this subject and validate the applied experimental procedure. It is noted that this review also presents bulk viscosity values for molten salts, molten metals, and some other exotic liquids at very low temperatures, but this is not particularly helpful data in the current context. The second, more recent study by Malbrunot, Boyer, Charles, and Abashi [37] presents bulk viscosity data for liquid argon, krypton, and xenon near their triple point. This article includes a very clear account of the experimental technique, and we will follow a similar procedure with some additional steps that provide better precision. There are also some reports of the bulk viscosity in liquid metals, such as by Jarzynski [39]. This paucity of bulk viscosity data is surprising as expressed by Temkin [5] in his book Elements of Acoustics as well as by Graves and Angrow in their paper “Bulk viscosity: Past and Present” [34]. Recently, more data on bulk viscosity became available thanks to the development of a modern acoustic spectrometer DT-600 by Dispersion Technology, Inc. Bulk viscosity data for 12 Newtonian liquids were published in Ref. [6]. Table 3.2 presents these results. The only one independent data, that is, bulk viscosity of water from the Litovitz-Davis review [9], agrees well with this new result. They report a value of 3.09 cP at 15  C, whereas we obtained a somewhat smaller value of 2.43 cP at a somewhat higher temperature of 25  C. An even better verification would be to compare the ratio of bulk to shear viscosity at a given temperature. Litivitz-Davis report a ratio of 2.81 at 15  C, whereas we obtained a value 2.73 at 25  C. This is the only room-temperature-independent verification of our bulk viscosity measurement that we managed to find. It is interesting to note that the bulk viscosity for these 12 liquids varies over a much wider range than the dynamic viscosity, by almost an order of magnitude. This seems to confirm that bulk viscosity is more sensitive to the molecular structure of the liquid. However, there is no correlation established between this parameter and other intensive properties of these liquids, as stated in Ref. [6]. The only known method of characterizing bulk viscosity is measurement of ultrasound attenuation. This is discussed in Section 3.6. There are at least three different methods of measuring attenuation coefficient that have been applied for characterizing liquids: Brillouin spectroscopy [24, 40], laser transient grating spectroscopy [41–43], and acoustic spectroscopy [10]. Each of these has advantages and disadvantages with regard to this particular task. The first two methods are usually performed at gigahertz frequencies, whereas acoustic spectroscopy is typically employed in the megahertz range.

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TABLE 3.2 Attenuation at 100 MHz, Temp. Coefficient, and Dynamic and Bulk Viscosity for 12 Newtonian Liquids Liquid

Attenuation at 100 MHz (dB/cm/MHz)

Attenuation T coefficient aT

Dynamic viscosity (cP)

Bulk viscosity (cP)

Linear variation coefficient

Cyclohexane

2.07

0.05

0.894

“17.4”

“0.082”

Cyclohexanone

0.63

<0.01

2.017

7.0

0.012

1-Pentanol

0.79

0.007

3.619

2.8

0.009

Toluene

0.72

0.009

0.560

7.6

0.014

Methanol

0.26

0.544

0.8

0.024

Hexane

0.60

0.015

0.300

2.4

0.022

2-Propanol

0.80

0.008

2.040

2.7

0.001

Butyl acetate

0.38

0.004

0.685

“2.4”

“0.137”

Acetone

0.25

<0.01

0.306

1.4

0.038

Ethanol

0.42

<0.01

1.074

1.4

0.015

Pyridine

3.90

<0.01

0.879

62.4

0.012

Water

0.186

0.0043

0.890

2.4

0.036

< 0.01

This difference in frequency ranges makes these techniques complementary for studying processes with widely different relaxation times. However, for the purpose of characterizing bulk viscosity measurements at very high frequency, they might present a serious problem. The attenuation increases rapidly with increasing frequency according to Equation (3.42). This means that Brillouin and laser transient grating spectroscopy must deal with much higher attenuations than acoustic spectroscopy. This would present no problem for liquids such as water which have little attenuation, but might be a serious problem for many others liquids. For instance, according to the data presented below, the attenuation rate for pyridine is 39,000 dB/cm at 1 GHz. Measurement of such a high attenuation would require either very high power excitation which might generate nonlinear effects, or it requires measurement over very short distances. Acoustic spectroscopy is clearly preferable for such highly attenuating liquids. Another problem related to Brillouin and laser transient grating spectroscopy is that these techniques usually report variation of attenuation with temperature, but not frequency. Measurement is usually performed at a single

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frequency. We stated above that frequency dependence is very useful for validating that the measured attenuation can in fact be used to calculate a meaningful bulk viscosity. Acoustic spectroscopy, on the other hand, allows simple attenuation measurement at multiple frequencies within the frequency range 1–100 MHz. This is yet another argument for using acoustic spectroscopy for characterizing bulk viscosity. A final argument in favor of acoustic spectroscopy is precision. Malbrunot et al. [37] compared the error of Brillouin and acoustic spectroscopy and concluded that acoustic spectroscopy with a precision of just 2.5% is considerably more precise. Modern acoustic spectrometers can improve this error by almost an order of magnitude, as discussed below. Furthermore, error analysis of Brillouin spectroscopy presented by Papayonou, Hart, and Anderws [40] confirms the potentially high errors due to difficulty in estimating Brillouin linewidth. There is also a problem with regard to laser transient grating spectroscopy. Calculation of the attenuation from the raw data requires fitting with five adjustable parameters, according to Cao, Diebold, and Zimmt [41]. This makes this method substantially more complicated compared to acoustic spectroscopy, where the attenuation coefficient is measured directly. This short comparative analysis leads us to conclude that acoustic spectroscopy is more suitable for measuring bulk viscosity, which agrees with a similar analysis by Malbrunot et al. [37]. This one of the many reasons that make acoustics suitable for rheological characterizations. In general, there are dozens of papers [44–78] and patents [79–83] devoted to the application of acoustics for characterizing the rheological properties of pure liquids, polymer solutions, and colloids. There are also several commercial instruments developed during last decade based on acoustic spectroscopy. These include the Ultrasizer by Malvern Ltd., an instrument from Ultrasonic Scientific, the Acoustosizer by Colloidal Dynamics, Inc., the APS-100 by Matec Applied Sciences, Inc., and the DT-100 and DT-600 by Dispersion Technology, Inc.

3.6. ATTENUATION OF ULTRASOUND IN NEWTONIAN LIQUID: STOKES LAW Viscosity and thermal conduction are the two mechanisms that cause attenuation when ultrasound propagates through a homogeneous medium and modern theory [3–5, 22, 84] takes both of these into account. Thermal conduction contribution is proportional to the deviation of the ratio of specific heats (g) from unity for a particular fluid, (g  1). Value of specific heat ratio for most liquids is very close to 1 due to low compressibility, in contrast with gases where this parameter may substantially exceed 1. Therefore, the thermal conduction contribution to the ultrasound attenuation in liquids is usually negligible. Consequently, ultrasound attenuation in liquids depends mostly on viscosity-related effects; that is to say, it is

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rheological in nature. The same assumption has been used in previous studies on this subject. The solution of the general Navier-Stokes Equation (3.35) with regard to attenuation of a plane, one-dimensional wave while propagating through a viscous, thermally nonconductive liquid is given in many books on acoustics [3–5, 22]. This solution yields the following expression for the ultrasound attenuation coefficient along:
 along V 2 1 1 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð3:40Þ 2 2 o 1 þ o tv 1 þ o2 t2v where V is sound speed, o is ultrasound frequency, tv is viscous relaxation time that takes into account both the bulk viscosity and the shear viscosity as given by:
 1 4 b þ tv ¼ 2 ð3:41Þ rV 3 This viscous relaxation time is very short, on the order of 1012 s. Attenuation as a function of frequency is a bell-shaped curve. It reaches a maximum value at the so-called critical frequency which corresponds to the wave period, which is approximately equal to the viscous relaxation time. For the Newtonian liquids, this critical frequency is about 1000 GHz due to the very short viscous relaxation time. It is extremely hard to achieve such high ultrasound frequencies in real instruments, but, fortunately, the lowfrequency asymptotic function of Equation (3.40) is quite adequate for interpreting the acoustic data. This asymptotic function is given by:   o2 4 b  þ  ð3:42Þ along ðneper=mÞ ¼ 2rV 3 3 This equation presents attenuation in neper per meter, which is different from the typical units of the data presentation with the modern acoustic spectrometers—dB/cm/MHz. These units assume normalization of the attenuation with frequency. Consequently, attenuation should be a linear function of the frequency when expressed in these normalized units. Figure 3.3 confirms this linearity experimentally for water. This frequency dependence is used below for Newtonian test. Asymptotic expression for ultrasound attenuation is closely related to the Stokes law for sound attenuation [36] derived 160 years ago by the same famous Stokes who derived the well-known expression for the frictional drag on a particle moving in a viscous liquid. He used the Navier-Stokes equation for an incompressible liquid (Equation 3.37), which yields the following simplified equation for sound attenuation along for a Newtonian liquid: along ¼

2o2  3rV 3

ð3:43Þ

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Attenuation (dB/cm/MHz)

0.20

3

Y = −0.001119 + 0.001877X

0.16

0.12

0.08

0.04

0.00 0.

10.

20.

30.

40. 50. 60. 70. Frequency (MHz)

80.

90. 100. 110.

FIGURE 3.3 Attenuation of sound in distilled water.

Stokes’ presentation neglects the liquid compressibility and consequently the bulk viscosity term is missing. The attenuation predicted by Stokes’ equation is often referred to as “classical.” Instead of Stokes’ law, the more general Equation (3.42) serves as a theoretical basis for calculating the bulk viscosity from the measured attenuation and sound speed. All other parameters in this equation, including the dynamic viscosity, are assumed to be known from independent measurements, or perhaps from the Handbook on Chemistry and Physics [7]. Measurement of attenuation might be complicated by its temperature dependence. Figure 3.4 illustrates this dependence for distilled water. The extremely high precision required for determination of the bulk viscosity would require maintaining temperature of the sample within 0.1  C. However, the main constraint on using the notion of the bulk viscosity is associated with the frequency dependence of the attenuation.

3.7. NEWTONIAN LIQUID TEST USING ATTENUATION FREQUENCY DEPENDENCE A valid application of Equation (3.42) for calculating the bulk viscosity of a particular liquid requires that this bulk viscosity is independent of the ultrasound frequency, since this independence is assumed in the derivation of Equation (3.40) from the Navier-Stokes equation (Equation 3.35). This independence of the bulk viscosity with frequency is associated with a somewhat analogous requirement that the dynamic viscosity of a Newtonian liquid is independent of shear rate. A bulk viscosity that is dependent on ultrasound

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

111

Attenuation at 100 MHz (dB/cm/MHz)

0.18

0.17

0.16

0.15

0.14

0.13 28.

29.

30.

31. 32. 33. Temperature (C)

34.

35.

36.

FIGURE 3.4 Attenuation of sound in distilled water at 100 MHz at different temperatures.

frequency or a dynamic viscosity that is dependent on shear rate each indicates that a given liquid is non-Newtonian over the given frequency or shear rate range, respectively. There is a general assumption of longitudinal viscosity long, which determines dissipation due to longitudinal stress in non-Newtonian liquids, Equation (3.34). For a Newtonian liquid, the longitudinal viscosity equals sum of bulk viscosity and four-thirds of the dynamic viscosity, both of them presumably frequency independent: 4 long ðNewtonianÞ ¼ b þ  ð3:44Þ 3 Verification of this frequency independence of the calculated bulk viscosity is essential for subsequent testing of various molecular theories of liquids as mentioned above, which all assume that the liquid obeys the Navier-Stokes equation and is Newtonian from the rheological viewpoint. This analysis points out the necessity of making multifrequency measurements of ultrasound attenuation. Such measurements would allow verification of the frequency dependence predicted by Equation (3.42), which leads to independence of longitudinal viscosity of frequency. Only if an experiment can establish that this theoretically predicted frequency dependence and longitudinal viscosity calculated according to Equation (3.34) would be constant, can one claim bulk viscosity for this particular liquid. Deviation of the measured attenuation frequency dependence from the theoretical one would indicate that there is an additional mechanism of

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3

ultrasound attenuation for a particular liquid in addition to the viscous dissipation. For instance, there is a body of literature reviewed by Kaatze, Udo, Hushcha, and Eggers [23] describing the application of this extra attenuation for characterizing chemical reactions in liquids. If we did not have this frequency dependence test, we might attribute attenuation coming from chemical reactions to bulk viscosity, which would lead to quite erroneous values for the bulk viscosity. Unfortunately, verification of the attenuation frequency dependence does not yield additional information for confirmation of the assumption that thermal conductance contribution is negligible. Frequency dependence of the thermal conductance contribution to the attenuation is the same as for the viscous one. It is still necessary to rely on the small value of the parameter (g  1) for liquids, as stated above. Figures 3.5–3.7 illustrate results of such a frequency test performed with 12 presumably Newtonian liquids. These are the same liquids as presented in Table 3.2. All details of these measurements are given in Ref. [6]. Representative attenuation spectra for all 12 liquids are shown on Figure 3.5 on a logarithmic frequency scale. The precision of these attenuation measurements is 0.7% at 35  C and 0.3% at 28  C. These values are much better than the ones in the older experiments by Malbrunot, Boyer, Charles, and Abashi [37]. The precision of the sound speed measurement is about 0.03%.

Attenuation (dB/cm/MHz)

4

acetone methanol cyclohexanone hexane toluene pyridine cyclohexane butyl acetate 1-pentanol ethanol 2-propanol water

3

2

1

0 100

101 Frequency (MHz)

102

FIGURE 3.5 Attenuation spectra of 12 Newtonian liquids on a logarithmic frequency scale.

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

113

Longitudinal viscosity (cP)

6 5

2-propanol

4 water

3

hexane acetone methanol

2 1

10.

Frequency (MHz)

100.

11 Longitudinal viscosity (cP)

10

cycloxehanone

9

toluene

8 1-pentanol

7 6 5 4 3

ethanol

2 10.

Frequency (MHz)

100.

FIGURE 3.6 Longitudinal viscosity for several Newtonian liquids as a function of frequency.

We can convert these attenuation spectra to longitudinal viscosity using Equation (3.34). For a Newtonian liquid, the longitudinal viscosity, as already given by Equation (3.44), equals the sum of the bulk viscosity and four-thirds the dynamic viscosity and must be frequency independent, which is the essence of the test that we suggest for verifying the Newtonian nature of the liquid. The calculated longitudinal viscosity differs significantly for these different liquids according to plots on Figures 3.6 and 3.7. On each graph, the points are the experimental data for a given liquid and the lines correspond to the arithmetic average viscosity for that liquid. Deviation of the experimental data for a particular liquid from its average value is a measure of its Newtonian behavior. It would be convenient to have a measure of how much the longitudinal viscosity deviates from what would be expected for Newtonian fluid. There are multiple ways of introducing such a statistically justified measure. We can select the most appropriate one using

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3

22 cyclohexane

20

Longitudinal viscosity (cP)

18 16 14 12 10 8 6 butyl acetate

4 2 10.

Frequency (MHz)

100.

FIGURE 3.7 Longitudinal viscosity of two presumably Newtonian liquids, which actually do not obey the frequency dependence required for Newtonian liquids.

the calculated values of the bulk viscosity shown on Figures 3.6 and 3.7. The first feature of these figures that catches eye is that two liquids, cyclohexane and butyl acetate, exhibit regular frequency dependence. This is indication that these liquids are non-Newtonian at this frequency range. The second argument that can be used is an obvious fact that water is a Newtonian liquid. It turns out that the linear variation coefficient reflects these peculiarities the best. This parameter is the standard deviation between the linear frequency fit to the experimental data and average attenuation, normalized with the average attenuation. Table 3.2 presents the value of this parameter for all liquids in the last column. It is seen that it is much larger for the above-mentioned two nonNewtonian liquids compared to all others. We can suggest the use of the linear variation coefficient equal to 0.05 as the boundary that separates Newtonian and non-Newtonian liquids, according to this test based on the bulk viscosity. It is interesting to mention that cyclohexane and butyl acetate are examples of liquids that are judged Newtonian from the viewpoint of only shear viscosity, which nevertheless are judged non-Newtonian when viewed more comprehensively in the light of longitudinal rheology at high frequency. It is perhaps also to interject yet a note of caution.

3.8. CHEMICAL COMPOSITION INFLUENCE It has been known for a long time that the acoustic properties of a homogeneous liquid depend on its chemical composition, and there have been successful attempts to use ultrasound for characterizing such chemical

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

115

properties [85–95]. The homogeneous liquid, by itself, plays only a role in the background, a supporting character, nothing more. The background properties that are important in the present context include the intrinsic attenuation and sound speed (and, later when we speak of electroacoustics, an ion vibration current/potential). It is desirable that the intrinsic attenuation of the medium be small in relation to the incremental attenuation produced by adding particles, and that this intrinsic attenuation, as well as the sound speed, be relatively insensitive to the chemistry. In this case, we would be able to use the same value for intrinsic attenuation and sound speed for all dispersions that use this particular liquid as the dispersion medium. It turns out that the attenuation coefficient, as the measured acoustic parameter, satisfies these two requirements in the vast majority of practical dispersions. The effect of the chemical composition on the attenuation is measurable only for high concentrations of a very few chemicals. In contrast, sound speed is very sensitive to the chemical composition. The corresponding variations are easily measurable, even at relatively low concentrations of many ordinary chemicals, and are quite comparable with the contribution made by colloidal particles. The intrinsic attenuation of many pure liquids is, indeed, small compared to the contribution of the colloid particles added to that medium. Figures 3.3 and 3.5 illustrate the attenuation for several liquids. The intrinsic attenuation of water is about 0.2 dB/cm/MHz at 100 MHz. It is the most transparent liquid for ultrasound. The attenuation of alcohols, acetone, toluene, and heptanes is somewhat higher than that of water, but still much less than 1 dB/cm/MHz at 100 MHz. We will show in Chapters 4 and 6 that the incremental attenuation resulting from adding colloidal particles to these media is typically many times higher. However, there is a large body of literature that demonstrates that both the intrinsic attenuation and sound speed of pure liquids can be dramatically changed by slight changes in chemistry, made by either adding a second liquid (“liquid mixture”) or by adding various solid materials that completely dissolve in the pure liquid. In the published literature, typically, a single paper considers either the attenuation or sound speed, but not both. As examples, the Handbook on Molecular Acoustics [95] provides a lot of data on sound speed, whereas Hunter and Darby [83] present only attenuation data for benzene, dioxane, and polysiloxane. There are very few papers showing both attenuation and sound speed for the same mixture. Figure 3.8 shows the attenuation and sound speed of a water-ethanol mixture. Both attenuation and sound speed vary with the water-to-ethanol content. However, the attenuation spectra remain constant within the instrument precision (0.01 dB/cm/MHz), up to 20% of ethanol in water. Small traces of ethanol would not affect the attenuation measurement. At the same time, sound speed exhibits measurable changes exceeding the instrument precision (0.1 m/s),

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1700

1.0 attenuation

Sound speed (m/s)

1600

0.9 0.8

1500

0.7 0.6

1400

0.5 0.4

1300

0.3 0.2

1200 10

20

30

40 50 60 70 Water-ethanol ratio in %

80

90

Attenuation at 100 MHz (dB/cm/MHz)

1.1 sound speed

0 water

3

0.1 100 ethanol

FIGURE 3.8 Attenuation and speed of sound in a water-ethanol mixture.

even at 0.5% of ethanol. This clearly demonstrates that sound speed is more sensitive than attenuation to any chemical variation. It is interesting that the change in both attenuation and sound speed is very nonlinear with respect to chemical composition. Both go through a maximum, but the position of this maximum for sound speed is much different than for attenuation. This means that the initial addition of ethanol to water does not affect the viscosity of water but does affect its elasticity. The viscosity of water starts to change only when the ethanol content exceeds 20%. The elastic component at this concentration already decays to the value for ethanol. It is not clear why all this happens. However, it is obvious that the acoustic properties of liquid mixtures contain much information about their structure. Having briefly addressed liquid mixtures, we now consider the second possibility for changing the chemistry, namely adding a solid material that completely dissociates into ions on dissolving in the water, creating “ion solutions,” often called “electrolytes.” Researchers at the beginning of the twentieth century used ultrasound to study the properties of electrolyte solutions. And in 1967, Manfred Eigen, Ronald Norrish, and George Porter received the Nobel Prize for their “relaxation spectroscopy” investigations of high-speed chemical reactions. Relaxation spectroscopy includes any experimental technique based on the variation of a thermodynamic parameter with time. Acoustic spectroscopy is just another version of relaxation spectroscopy. “Dielectric spectroscopy,” perhaps more familiar in colloid science [96], is yet another. Both acoustic and dielectric spectroscopies were used in the middle of the twentieth century for investigating properties of electrolyte solutions [85–87]. These studies are associated usually with the name of Manfred Eigen.

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Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

It is ironic that relaxation spectroscopy is unknown in colloid science, and yet it is very close in terms of technique and application. We reproduce here some results obtained originally by Eigen and others. We suggest that readers who are interested in this technique and theoretical interpretation read the original papers by Eigen and his colleagues [85–87]. Modern instruments allow investigators to collect data much more rapidly, and with less effort, than was possible in the middle of twentieth century. All the data presented in Figures 3.9–3.11 and Table 3.3 were collected within 1 week. This higher efficiency allowed us to study one more variable—the electrolyte concentration. We used 10 different inorganic electrolytes; all are listed in Table 3.3. Only five of the selected electrolytes exhibited any pronounced effect on the attenuation spectra (see Figure 3.10). Monovalent electrolytes such as KCl and NaCl appear to have little effect on the attenuation spectra in the 1–100 MHz range. Manganese sulfate (MnSO4) shows the most interesting effect on the attenuation spectra especially as a function of concentration (see Figure 3.11). Interestingly, these curves reproduce results obtained 30 years ago by Jacobin and Yeager [91]. According to Eigen [85], MnSO4, CuSO4, and Al2 (SO4)3 cause the strongest known effects, and this view is supported by our measurements. For our further discussion of acoustics in colloids, it is only important to note merely that for electrolyte concentrations below 0.1 M the attenuation changes by less than 0.1 dB/cm/MHz. We will see later that colloidal particles cause much greater effects on the attenuation. This means that these variations in the intrinsic background attenuation of the dispersion medium due to ion

1640

AlCl3 CaCl2 CuSO4 HCl KCl LiCl MnSO4

Sound speed (m/s)

1620 1600 1580 1560 1540 1520 1500 0.0

0.1

0.2

0.3

0.4 0.5 0.6 Concentration (mol/l)

0.7

0.8

FIGURE 3.9 Attenuation frequency spectra for aqueous solutions of CuSO4.

0.9

1.0

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CHAPTER

concentration in mol/l

1.0

1 0.8

0.8

Attenuation (dB/cm/MHz)

3

0.6 0.6

0.4 0.23

0.4

0.14 0.069 0.03 water

0.2

0.0 100

101 Frequency (MHz)

102

FIGURE 3.10 Attenuation spectra for 1 M aqueous solutions of various electrolytes.

Attenuation (dB/cm/MHz)

1.2 Al2(SO4)3 MgSO4 MnSO4 KCl AlCl3 CuSO4

1.0 0.8 0.6 0.4 0.2 0.0 100

101 Frequency (MHz)

102

FIGURE 3.11 Attenuation frequency spectra for aqueous solutions of MnSO4.

composition are usually negligible, and need only be taken into account for very high electrolyte concentrations approaching 1 M. The situation with sound speed is rather different. Table 3.3 shows sound speed of 1 M electrolyte solutions. It is seen that even 1:1 electrolytes affect the sound speed significantly. We will see later that the magnitude of this

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

119

TABLE 3.3 Sound Speed and Intrinsic Attenuation of Various Electrolytes at 1 M Concentration Sound Attenuation Temperature Density Compressibility speed (m/s) at 100 MHz ( C) (g/cm3) (1010 m2 N) (dB/cm/MHz) Water

1498

0.18

25

0.997

4.47

HCl

1509

0.18

25.3

KCl

1547

0.18

23.7

1.04

4.02

LiCl

1552

0.18

25.4

1.02

4.07

NaCl

1559

0.18

25.6

1.04

3.96

CaCl2

1572

0.18

25.8

1.06

3.82

CuSO4

1555

0.94

25.8

1.13

3.66

MgSO4

1623

0.63

26.6

1.1

3.45

MnSO4

1593

0.69

25.7

1.12

3.52

AlCl3

1640

0.22

23.9

1.08

3.44

Al2(SO4)3

1634

1.28

26.1

1.13

3.31

effect is comparable to that produced by adding colloidal particles. They are comparable even at low ion concentrations. Figure 3.12 shows the dependence of sound speed on the ion concentration for all 10 electrolytes. It is seen that it is practically a linear function of the concentration. This effect can be measured even below 0.1 M, taking into account that the precision of the sound speed measurement is about 0.1 m/s. It is interesting that the linear dependence of the sound speed with ion concentration cannot be explained by the increase in density resulting from the added electrolyte. According to Equation (3.5), the sound speed should decrease with increasing density, assuming that the compressibility remains constant. However, the experiment above indicates just the opposite trend. The data given in Table 3.3 show that the sound speed increases with increasing density of the electrolyte solution (measured using a liquid pycnometer). The increase in sound speed with density suggests that the compressibility is not constant, but is also a function of the ion concentration. It follows that the compressibility of a liquid must decrease with increasing electrolyte concentration. It appears as if the ions enforce some structure in the liquid. This effect becomes stronger for the more highly charged ions (Table 3.3).

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Attenuation (dB/cm/MHz)

0.8

3

concentration in mol/l

0.7

1

0.6

0.8

0.5

0.6

0.4

0.4 0.3

0.3

0.17

0.2

0

0.1 0.0 100

101 Frequency (MHz)

102

FIGURE 3.12 Influence of the various ions on the speed of sound in water.

This effect has been known for a long time [97, 98]. It serves as a basis for one of the methods for estimating the solvation number of ions. It turns out that water molecules trapped in the solvating layer by dipole-ion interaction have negligible compressibility compared to the bulk. Increasing concentration of ions leads to an increasing amount of solvating water, which in turn brings compressibility of the electrolyte solution down. Implications of this effect for characterizing properties of ions will be discussed in more detail in Chapter 11. This structural effect might be important in understanding the nature of the internal parts of the double layer, where electrolyte concentrations become very high. Acoustics allows us to characterize the properties of liquids at extreme ionic strengths. It yields information about both the real and imaginary rheological components. Attenuation is related to the viscous component, whereas compressibility calculated from the sound speed is related to elasticity. The experiments described above indicate that both the real and imaginary components depend on the electrolyte concentration. This means that both the elasticity and viscosity of the liquid within the double layer are quite different than in the bulk of the solution. The viscous component of this effect has been known for a long time [96]. The concept of an elastic change within the double layer is rather new. It is not clear yet as to what impact this new rheological understanding will have on double layer theory.

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[69] J. Kondoh, K. Saito, S. Shiokawa, H. Suzuki, Simultaneous measurement of liquid properties using multichannel shear horizontal surface acoustic wave micro-sensor, Jpn J. Appl. Phys. 35 (Part 1) (1996) 3093–3096. [70] E.H. Trinh, K. Ohsaka, Measurement of density, sound velocity, surface tension, and viscosity of freely suspended supercooled liquids, Int. J. Thermophys. 16 (1995) 545–555. [71] S.H. Sheen, K.J. Reimann, W.P. Lawrence, A.C. Raptis, Ultrasonic techniques for measurement of coal slurry viscosity, in: B.R. McAvoy (Ed.), IEEE 1988 Ultrasonics Symposium. Proceedings (IEEE Cat. No.88CH2578-3), vol. 1, 1988, pp. 537–541. [72] I. Alig, F. Stieber, S. Wartewig, A.D. Bakhramov, Yu.S. Manucarov, Ultrasonic absorption and shear viscosity measurements for solutions of polybutadiene and polybutadiene-blockpolystyrene copolymers, Polymer 28 (1987) 1543–1546. [73] G. Gaunaurd, K.P. Scharnhorst, H. Uberall, New method to determine shear absorption using the viscoelastodynamic resonance-scattering formalism, J. Acoust. Soc. Am. 64 (1978) 1211–1212. [74] T.H. Hulusi, Effect of pulse-width on the measurement of the ultrasoinic shear absorption in viscous liquids, Acustica 40 (1978) 269–271. [75] O. Sulun, Determination of elastic constants by measuring ultra-sound velocity in Al/sub 2/(SO/sub 4/)/sub 3/and Ce/sub 2/(SO/sub 4/)/sub 3/solutions, Rev. Facul. Sci. l’Univ. d’Istanbul Ser. C (Astron. Phys. Chim.) 39–41 (1974–1976) 18–31. [76] M. Sokolov, On an acoustic method for complex viscosity measurement, Trans. ASME Ser. E, J. Appl. Mech. 41 (1974) 823–825. [77] J.C. Bacri, Measurement of some viscosity coefficients in the nematic phase of a liquid crystal, J. Phys. Lett. 35 (1974) L141–L142. [78] C.J. Knauss, D. Leppo, R.R. Myers, Viscoelastic measurement of polybutenes and low viscosity liquids using ultrasonic strip delay lines, J. Polym. Sci. Polym. Symp. USA 43 (1973) 179–186. [79] S.-H. Sheen, W.P. Lawrence, H.-T. Chien, A.C. Raptis, Method for measuring liquid viscosity and ultrasonic viscometer, US Patent number 05365778, 1994. [80] A. Soucemarianadin, R. Gaglione, P. Attane, High-frequency acoustic rheometer and device to measure the viscosity of a fluid using this rheometer, US Patent number 05302878, 1994. [81] M.R. Bujard, Method of measuring the dynamic viscosity of a viscous fluid utilizing acoustic transducer, US Patent number 04862384, 1989. [82] S. Sachiko, M. Toyoe, T. Yoshihiko, Measuring instrument for liquid viscosity by surface acoustic wave, Patent Japan 02006728 JP, 1990. [83] J.L. Hunter, H.D. Dardy, Ultrahigh-frequency ultrasonic absorption cell, J. Acoust. Soc. Am. 36 (1964) 1914–1917. [84] J.O. Hirschfelder, C.F. Curtis, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964. [85] M. Eigen, Determination of general and specific ionic interactions in solution, Faraday Soc. Discuss. 24 (1957) 25. [86] M. Eigen, L. deMaeyer, in: A. Weissberger (Ed.), Techniques of Organic Chemistry, vol. VIII, Part 2, Wiley, New York, 1963, p. 895. [87] L. De Maeyer, M. Eigen, J. Suarez, Dielectric dispersion and chemical relaxation, J. Am. Chem. Soc. 90 (1968) 3157–3161. [88] V.A. Solovyev, C.J. Montrose, M.H. Watkins, T.A. Litovitz, Ultrasonic relaxation in etnanol-ethyl halide mixtures, J. Chem. Phys. 48 (1968) 2155–2162.

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[89] G. D’Arrigo, L. Mistura, P. Tartaglia, Sound propagation in the binary system anilinecyclohaxane in the critical region, Phys. Rev. A 1 (1974) 286–295. [90] O. Nomoto, T. Kishimoto, Velocity and dispersion of ultrasonic waves in electrolytic solutions, Bull. Kobayashi Inst. 2 (1952) 58–62. [91] L. Jackopin, E. Yeager, Ultrasonic relaxation in manganese sulfate solutions, J. Phys. Chem. 74 (1970) 3766–3772. [92] E.G. Richardson, Acoustic experiments relating to the coefficients of viscosity of various liquids, Faraday Soc. Discuss. 226 (1954) 16–24. [93] T.A. Litovitz, T. Lyon, Ultrasonic hysteresis in viscous liquids, J. Acoust. Soc. Am. 26 (1954) 577–580. [94] A.G. Chynoweth, W.G. Scheider, Ultrasonic propagation in binary liquid systems near their critical solution temperature, J. Chem. Phys. 19 (1951) 1566–1569. [95] K.H. Hellwege (Ed.), Numerical Data and Functional Relationship in Science and Technology, vol. 5; W. Schaaffs (Ed.), molecular Acoustics, Springer-Verlag, Berlin, 1967. [96] J. Lyklema, Fundamentals of Interface and Colloid Science, vol. 1-3, Academic Press, London, 1995–2000. [97] A. Passynski, Acta Physicochim. 8 (1938) 385. [98] J.O.M. Bockris, P.P.S. Saluja, Ionic salvation numbers from compressibilities and ionic vibration potential measurements, J. Phys. Chem. 76 (1972) 2140–2151. [99] J. Meixner, Kolloid-Zt. 134 (1954) 47. [100] H. Lamb, Hydrodynamics, sixth ed., Dover Publications, New York, 1945 first ed. 1879. [101] A.B. Bhatia, Ultrasonic Absorption, Dover Publications, New York, 1985. [102] G. Shortley, D. Williams, Elements of Physics, Prentice Hall, Englewood Cliffs, NJ, 1963.

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Chapter 4

Acoustic Theory for Particulates 4.1. Extinction = Absorption + Scattering. Superposition Approach 130 4.2. Acoustic Theory for Dilute Systems 140 4.3. Ultrasound Absorption in Concentrated Dispersions 143 4.3.1. Coupled-Phase Model 144 4.3.2. Viscous Loss Theory 149 4.3.3. Thermal Loss Theory 153 4.3.4. Structural Loss Theory 156 4.3.5. Intrinsic Loss Theory 160 4.4. Ultrasound Scattering 161 4.4.1. Rigid Sphere 165 4.4.2. Rigid Cylinder 167 4.4.3. Nonrigid Sphere 168 4.4.4. Porous Sphere 168

4.4.5. Scattering by a Group of Particles 169 4.4.6. Scattering Coefficient 170 4.4.7. Ultrasound Resonance by Air Bubbles 171 4.5. Ultrasound Propagation in Porous Media 172 4.6. Input Parameters 174 4.7. Estimates of the Dense Particle Motion Parameters 179 4.7.1. Particle Velocity 180 4.7.2. Particle Displacement 181 4.7.3. Shear Rate 181 4.7.4. Quantum Limit for Acoustics 181 References 182

There have been very few changes made in this theory since we published the first edition in 2002. At the same time, we have collected substantial new experimental evidence confirming our approach to the acoustic theory for particulates. There is practically no justification for introducing new theoretical models for a simple system. There are, however, some new models introduced for fractal and porous submicrometer particles. There is also a new adjustable parameter for scattering induced by large particles. There is some new material on this “scattering coefficient.” We have also added one section on the estimatation of parameters that characterize particle motion in an ultrasound field. It yields new insights into these phenomena. Any acoustic theory for particulates should yield a relationship between some measured macroscopic acoustic properties, such as sound speed, Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23004-1 Copyright # 2010, Elsevier B.V. All rights reserved.

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attenuation, acoustic impedance, angular dependence of the scattered sound, etc., and some microscopic characteristics of the heterogeneous system, such as its composition, structure, electric surface properties, particle size distribution (PSD), etc. It is also desirable that this theory satisfies a set of requirements, summarized as follows: (1) The theory must be valid for a wide particle size range, from 10 nm to 1 mm, and for ultrasonic frequencies from 1 to 100 MHz. This is roughly equivalent in magnitude to a ka range of 0.00001-100 (k is the compression wavenumber). (2) The theory should be valid for concentrated suspensions and therefore must take into account particle-particle interactions. (3) The theory should reflect the multiple mechanisms of ultrasound interaction with colloids, namely, viscous, thermal, scattering, intrinsic, structural, and electrokinetic. The heuristic description of these mechanisms is given below. Despite 100 years of almost continuous effort by many distinguished scientists, there is still no single theory that meets all of these requirements. For example, the best known theory, abbreviated as ECAH, following the names of its creators Epstein, Carhart, Allegra, and Hawley [1, 2], meets the first requirement, completely fails the second, and takes into account only four of the six mechanisms mentioned in the third. The ECAH theory is constructed in two stages. We call the first stage the “single-particle theory,” since it attempts to account for all of the ultrasound disturbances surrounding just a single particle. This stage relates the microscopic properties of both the fluid and particle to the system properties at the single-particle level. The second stage, which we refer to as the “macroscopic theory,” then relates this single-particle level to the macroscopic level at which we actually obtain our experimental raw data. Both parts of the ECAH theory (single particle and macroscopic) neglect any particle-particle interactions and are therefore valid only for dilute systems. For instance, in the ECAH macroscopic theory, the total attenuation is regarded as simply a superposition of the contributions from each particle and is determined only by the reflected compression wave. This part, derived by Epstein and Carhart [1], is similar to the well-known optical theorem that states that the extinction cross-section depends only on the scattering amplitude in the forward direction [3]. There have been several attempts to extend this general, two-stage acoustic theory to concentrated systems by incorporating particle-particle interactions. The following are the examples of three different approaches for each stage: For the Single Particle theory: (1) Isolated particle in ECAH theory [1, 2] (2) Cell model [4] (3) Core-shell model [5, 6]

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For the Macroscopic theory: (1) Scattering and multiple-scattering theories [1, 7–11] (2) Coupled-phase model [12–14] (3) Phenomenological theory by Temkin [15, 16] Obviously, these extensions increase the complexity of the ECAH theory. However, the original ECAH theory, even without any modifications, is already very complex, as a result of the authors’ attempt to construct a universal theory that is valid not only for any ka value but also for all interaction mechanisms between sound and the colloidal system. Modifications to implement these particle interactions, such as outlined briefly above, are therefore practically impossible. One might ask: is it necessary to develop such a general theory? Is it possible to introduce some simplifications while at the same time providing room for more readily implementing these particle interactions? Such simplifications are indeed possible. It turns out that there are some very general peculiarities of ultrasound propagation through colloids that allow us to simplify the theoretical process. Historically, these peculiarities prompted the introduction of six different mechanisms of sound interaction with colloids. We give here a short heuristic description of them all. (1) The “viscous” mechanism is hydrodynamic in nature. It is related to the shear waves generated by the particle oscillating in the acoustic pressure field. These shear waves appear because of the difference in density of the particles and the medium. The density contrast causes particle motion with respect to the medium. As a result, the liquid layers in the particles’ vicinity slide relative to each other. The nonstationary sliding motion of the liquid near the particle is referred to as the “shear wave.” This mechanism is important for acoustics. It causes losses of acoustic energy due to shear friction. Viscous dissipative losses are dominant for small, rigid particles with sizes less than 3 mm, such as oxides, pigments, paints, ceramics, cement, and graphite. The viscous mechanisms are closely related to the electrokinetic mechanism, which is also associated with the shear waves. (2) The “thermal” mechanism is thermodynamic in nature and is related to the temperature gradients generated near the particle surface. These temperature gradients are due to the thermodynamic coupling between pressure and temperature. Dissipation of acoustic energy caused by thermal losses is the dominant attenuation effect for soft particles, including emulsion droplets and latex droplets. (3) The “scattering” mechanism is essentially the same as in the case of light scattering. Acoustic scattering does not produce dissipation of acoustic energy. Particles simply redirect a part of the acoustic energy flow, and as a result this portion of the sound energy does not reach the receiving

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sound transducer. The scattering mechanism contributes to the overall attenuation, and this contribution is significant for larger particles with a diameter exceeding roughly 3 mm. (4) The “intrinsic” mechanism refers to losses of acoustic energy due to the interaction of the sound wave with the material of the particles and the medium, considered as homogeneous phases on a molecular level, and unrelated to the state of division of the colloidal dispersion. It must be taken into account when the overall attenuation is low, which might happen for small particles or low volume fractions. (5) The “structural” mechanism links acoustics with rheology. It occurs when particles are joined together in some network. Oscillation of these interparticle bonds can cause additional energy dissipation. (6) The “electrokinetic” mechanism describes the interaction of ultrasound with the double layer of the particles. Oscillation of the charged particles in the acoustic field leads to the generation of an alternating electrical field, and consequently to an alternating electric current. This mechanism is the basis for electroacoustic measurements. However, it turns out that its contribution to acoustic attenuation is negligible [17], which makes acoustic measurements completely independent of the electrical properties of the dispersion, including the properties of the double layer. We can divide all of the mechanisms of ultrasound attenuation into two groups depending on the way the acoustic energy is transformed in the colloid. The ultrasound attenuation in a colloid arises either from (1) absorption (the conversion of acoustic energy into thermal energy) or (2) from scattering (the redirection of incident acoustic energy from the incident beam). The combined effects of scattering and absorption can be described as the extinction cross-section, S, which is an effective area whose product with the incident intensity is equal to the power lost from the sound beam [18–20]. In this respect, ultrasound is similar to light. There is a well-known formula for light, extinction ¼ absorption þ scattering (Chapter 2), which is also applicable to sound. This formula is the basis for the acoustic theory, and therefore we consider it in some detail in the following section.

4.1. EXTINCTION = ABSORPTION + SCATTERING. SUPERPOSITION APPROACH There is a recent trend to snub this extinction equation, to consider viscous and thermal dissipations as additional forms of “scattering,” and to picture all particle-particle interactions at high concentration as simply additional aspects of “multiple scattering.” According to this nouveau view, the very term absorption is absorbed into a “Unified Scattering Theory,” and the term extinction becomes extinct.

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We think that crossing this Rubicon is misleading at best, hides a general understanding of the underlying mechanisms, and complicates rather than simplifies any extensions of the theory. Advocates of this new philosophy offer several arguments for this new perspective. First, they claim that it is simply a semantic change. They argue that within the framework of the ECAH formulation it is impossible, in any case, to clearly separate the various mechanisms. The various components become hidden behind a veil of mathematical formalism, and this complexity defies intuitive understanding in terms of the underlying physical phenomena. This lack of clarity is then used as a reason to discard terminology and philosophical underpinnings that have a long and honorable history. In rebuttal to this first claim, we take strong exception to this semantic argument. It not only completely alters the historical perspective, but the very language is a contradiction in terms. According to Webster, we find the following definitions: Scatter: to separate and go in several directions; to reflect and refract in an irregular, diffuse manner. Absorption: taking in and not reflecting; partial loss of power of light or radio waves passing through a medium.

It is obvious that Webster’s definition stresses “absorption” as being quite different from “scattering,” emphasizing that it is “not reflecting.” This difference was obvious and seemed important to the founder of the scattering theory, Lord Rayleigh. He wrote more than 100 years ago in his book The Theory of Sound, Vol. 2 the following: . . .When we inquire into the mechanical question, it is evident that sound is not destroyed by obstacles as such. In the absence of dissipative forces, what is not transmitted must be reflected. Destruction depends upon viscosity and upon conduction of heat; but the influence of these agencies is enormously augmented by the contact of solid matter exposing a large surfaces. At such a surface the tangential as well as the normal motion is hindered, and a passage of heat to and fro takes place, as the neighboring air is heated and cooled during its condensations and rarefactions. With such rapidity of alternation as we are concerned with in the case of audible sounds, these influences extend to only a very thin layer of the air and of the solid, and are thus greatly favored by a fine state of division. . .

It is clear that Lord Rayleigh considered sound absorption as especially peculiar for colloids, which are systems with the finite state of division and extended surfaces. Secondly, the advocates appeal to the light propagation through the colloids. The general formula extinction ¼ scattering þ absorption, since it expresses only a basic conservation of energy, must be valid for light and sound. However, in the case of light the “scattering” term certainly

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dominates discussion in scientific talks and publications. The terms “extinction” and “absorption” are rarely mentioned, and that too only in special handbooks [3]. This neglect of absorption phenomena in optics might be often justified, but carrying over this bias in case of sound can hardly be justified. The role of absorption is very different in acoustics as compared to optics. In optics, absorption is an electrodynamic effect, whereas in acoustics it is either hydrodynamic or thermodynamic. For optics, absorption presents a difficult problem because it requires information about the imaginary component of the refractive index and this information is generally not available. For optics, this normally leaves only one choice—to ignore the absorption of light. In this case, the description of light propagation thorough a colloid is reduced to a consideration of scattering alone. In many real colloids, this simplification works well enough, because the absorption is often negligible. In acoustics, the situation is dramatically different. The absorption of ultrasound is easy to calculate. No special properties of the particles are required. As we will see later, the absorption of ultrasound by solid, rigid particles depends only on their density, which is readily available or can be easily measured. In the case of ultrasound, absorption is not a complicating factor. Rather, it is a very important source of information about the particles, especially submicrometer particles. Ignoring this term means ignoring the major advantage of ultrasound over light as the characterization technique. Finally, we offer several arguments in support of our view of retaining the historical viewpoint of Rayleigh and others. (1) For acoustics, submicrometer particles do not scatter ultrasound at all in the frequency range under 100 MHz. They only absorb ultrasound. There is no need to develop a general complex scattering-absorption theory for such submicrometer particles. (2) For acoustics, there is a very simple way to eliminate the nonlinear effects of multiple scattering. (We define scattering here in the classical sense, as the interaction of the compression waves scattered by particles with other particles.) The effects of multiple scattering are completely eliminated by following Morse’s suggestion to measure the intensity of the incident ultrasound after transmission. The intensity of this ultrasound is not affected by multiple scattering, but it may be affected by particleparticle interactions through viscous or thermal absorption mechanisms. We prove this statement experimentally later. (3) The ultrasound attenuation in pure liquids and gels is via absorption only. There is no scattering there. This allows us to interpret acoustic spectra in rheological terms and to use an acoustic spectrometer as a high-frequency rheometer. (4) If we combine scattering and absorption in a single mathematical model, such as the ECAH or modification to it, we are forced to use at all times

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an extended set of input parameters, many of which may be unknown in a given case. Even in the case where a given mechanism may be unimportant, the relevant physical properties may still be required because of the complicated, perhaps very nonlinear, characteristics of a “unified” approach. Separating scattering and absorption opens the way to minimize the number of required input parameters. This issue is discussed in detail at the end of this chapter. (5) Separation of scattering and absorption phenomena provides more insight into the nature of the attenuation phenomena. The unified approach is like a “black box.” We input information and get an answer without any understanding of the processes going on. (6) Calculation of the PSD from attenuation spectra is a classical ill-defined problem. The likelihood of multiple solutions can be minimized by carefully using all a priori and independent information. Such information can be more readily employed in helping to solve the inverse problem if the mechanisms concerning the sound attenuation can be linked to all available a priori independent information. The unified or “black box” approach does not easily provide a format for this purpose. Taking into account all these argument, we have decided to proceed in this book with the classical meaning of the terms “scattering” and “absorption,” following Rayleigh, Morse, Sewell, and other distinguished scientists. There is one more important argument supporting our decision. This last argument requires more detailed consideration and justification. (7) In the case of light, it is practically impossible to separate absorption and scattering [3] in measurement, whereas in the case of ultrasound it is easily achievable. For ultrasound, absorption and scattering are distinctly separated in the frequency domain. As a general rule, for a given particle size, the absorption of ultrasound is dominant at low frequencies, whereas scattering is dominant at high frequencies, with an overlap region existing within a narrow intermediate frequency range. This is illustrated in Figure 4.1 (where the sum of the viscous absorption and scattering is plotted as a function of ka) for particles of three different sizes. The low-frequency peaks correspond to the viscous absorption for different size particles, and the high-frequency peak corresponds to the scattering losses. Figure 4.2 shows the measured attenuation spectrum for silica BCR-70. This material is a certified standard with a median size of 2.9 mm [21]. This attenuation spectrum demonstrates very clearly the transition from viscous absorption losses at low frequency, to scattering losses at high frequency. In this case, the transition range occurs in the vicinity of 20 MHz. As with light, ultrasound scattering decreases rapidly with decreasing particle size. For ultrasound frequencies below 100 MHz, scattering becomes unimportant for particles below 1 mm. Thus for samples containing such small

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0.3 micron (ka 100 MHz = 0.06) 1 micron (ka 100 MHz = 0.2) 5 micron (ka 100 MHz = 1)

8

attenuation [dB/cm/MHz]

7 6 5 4 3 2 1 0

10-4

10-3

10-2

10-1

ka

100

101

102

103

FIGURE 4.1 Attenuation spectra calculated for monodisperse colloids with three different sizes showing superposition of viscous absorption and rigid particle scattering. The arrows show the point corresponding to a frequency of 100 MHz.

Attenuation [dB/cm/MHz]

3

silica BCR in ethanol, 11%wt, d50=3 micron

viscous losses

2

scattering losses

1

0 100

101 Frequency [MHz]

102

FIGURE 4.2 Measured attenuation spectra for the particle size standard, silica BCR-70.

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particles, it makes no sense to employ a complicated theory that combines both absorption and scattering; an absorption theory would be sufficient. In the opposite case of large particles with ka > 1, ultrasound absorption is negligible and any theory needs take into account only scattering. Although Figure 4.1 considered only viscous type absorption, this separation between absorption and scattering is also observed for the thermal loss mechanism. Anson and Chivers [22] studied 72 liquids and showed that thermal losses are important up to ka ¼ 0.5. Other mechanisms dominate for higher ka, except in those emulsions where only the thermal properties differ substantially between the two phases. This separation of the loss mechanisms in the frequency domain allows us to express the total attenuation, aT, as the sum of the attenuation due to the absorption, aab, and the attenuation due to the scattering, asc. In addition, we should add a background intrinsic attenuation aint because the dispersion medium attenuates ultrasound as any other liquid (see Chapter 3). Thus: aT ¼ aab þ asc þ aint

ð4:1Þ

A further simplification can be considered from the fact that ultrasound absorption is important for small ka, that is, for wavelengths larger than the particle size. It is the so-called Rayleigh region [23, 24], or long-wavelength limit. At this limit, the two general components of ultrasound absorption (viscous and thermal) are additive. This then allows us to consider these two effects separately, and to neglect their possible coupling. The total attenuation now becomes: aT ¼ avis þ ath þ asc þ aint

ð4:2Þ

The potential contribution of any structural losses in a structured colloid would lead to the modification of the viscous losses, and to additional energy dissipation as a result of oscillation of the interparticle bonds. This last contribution can be accounted for by adding a further term to the total attenuation: aT ¼ avis þ ath þ ast þ asc þ aint

ð4:3Þ

The simple superposition expression for the measured attenuation is not universally applicable. There are some exceptions [25]. However, Equation (4.3) is valid for a very large number of practical concentrated colloids. The most important advantage of this superposition approach is that, compared to the ECAH theory, it allows us to take particle-particle interactions into account more easily. Several independent groups have determined the range of volume fractions for which such particle-particle interactions become important. The conclusion from these studies is that the viscous loss mechanism is the most sensitive to such interactions. For example, Uric [26, 27] showed that the attenuation of rigid, heavy particles becomes a nonlinear function of volume fraction above 10 vol%, reaching a maximum attenuation at 15 vol%. This was confirmed later by Hampton [28], Blue and McLeroy [29], Marlow et al. [30], Dukhin and

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Goetz [4], and Hipp, Stori, and Morbidelli [31]. It is interesting that we find this nonlinear behavior to be independent of the particle shape, and that it occurs at the same critical concentration, even for profoundly nonspherical particles [29]. Perhaps this happens because, for long wavelengths, the particles behave essentially as point sources; shape effects are therefore not pronounced. This same phenomenon is found in light scattering [3]. The thermal loss mechanism, which is thermodynamic in nature, is less dependent on particle-particle interactions. This was shown to be true for several different polymer latices by Dukhin and Goetz [32, 33], and also for several emulsions by McClements and Hemar et al. [5, 6, 34]. The importance of particle-particle interactions, as it relates to the scattering mechanism, depends very much on the method of measurement. In principle, we can measure the sound scattered at some angle to the incident beam, or, instead, consider only the decrease in the intensity of the incident wave as it travels through the colloid. It is not widely understood that this choice of measurement technique plays a very important role in defining the necessary theory. In fact, the effect of “multiple scattering” can be minimized by measuring the attenuation of the incident wave as was clearly pointed out by Morse [18]: . . .whether multiple scattering is important or not, the attenuation of the incident wave in traversing a distance z of region R is given by factor Exp(-Np Pz), where P is the total scattering cross-section, Np is the number concentration of particles. When multiple scattering is important, some energy is absorbed by the scattered wave before it emerges from R, and some of the scattered wave is rescattered, but the formulas for the S and attenuation still tells us what happens to the incident wave, the part of the wave which has not been affected by scattering. . .

Hence, by choosing to measure the attenuation of the incident beam, the scattering mechanism becomes much less dependent on particle-particle interactions. The result is that the variation of the transmitted wave intensity remains a linear function of the volume fraction up to much higher volume fractions than would be possible if directly monitoring the off-axis scattered sound. For example, Busby and Richardson [35] showed that the attenuation for large 95-mm glass spheres remains a linear function of the ultrasound path length up to 18 vol%. Atkinson and Kytomaa [36, 37] concluded that the attenuation of 1-mm particles remains linear to 30 vol%. We have also collected data that confirm these observations by using a variety of disperse systems, each of which exhibits only one of the several possible mechanisms of ultrasound attenuation. Accordingly, Figure 4.3 shows the attenuation spectra for dispersion of 0.3-mm rutile particles, for which we would expect mainly viscous losses. For comparison, Figure 4.4 presents the attenuation spectra for dispersion of 0.16-mm latex particles, for which the thermal losses are dominant. Finally, Figure 4.5 illustrates the attenuation for dispersion of 120-mm quartz particles, for which scattering would be dominant.

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volume fraction

Attenuation [dB/cm/MHz]

14

40.8% 36.2% 32% 25.8% 21.4%

12 10 8 6

vfr=16.6% vfr=11.8%

vfr=45.9%

4

vfr=5.1% vfr=3.2% vfr=1.1%

2 0

100

101 Frequency [MHz]

102

FIGURE 4.3 Measured attenuation spectra for 0.3-mm-diameter rutile in water.

7

vfr=37.4%

Attenuation [dB/cm/MHz]

6 vfr=32.4%

5

vfr=27%

4

vfr=21.5% 3

vfr=16.1%

2 vfr=10.7% vfr=5.4% vfr=3.2% vfr=1.1%

1 0 100

101 Frequency [MHz]

102

FIGURE 4.4 Measured attenuation spectra for 0.1-mm-diameter neoprene latex in water.

The viscous attenuation in Figure 4.3 is strongly volume fraction dependent, leading also to a shift in the critical frequency from about 25 to 60 MHz. This frequency shift is much less pronounced for the thermal attenuation shown in Figure 4.4. Furthermore, we see that the critical frequency for the scattering attenuation in Figure 4.5 is virtually unchanged with volume fraction.

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vfr

40

Attenuation [dB/cm/MHz]

4

15% 10% 5% 3% 1% 20% 25% 30% 35% 40% 46%

30

20

10

0 100

101 Frequency [MHz]

102

FIGURE 4.5 Measured attenuation spectra for 120-mm quartz particles in water.

To better illustrate the difference in dependence on the volume fraction, we can extract the viscous, thermal, and scattering attenuation at a single frequency and plot this versus the volume fraction (Figure 4.6). The viscous attenuation is a linear function of the volume fraction up to only about 10 vol%. In contrast, the thermal attenuation remains linear up to about 30 vol%. Finally, the scattering attenuation remains linear up to 46 vol%. Linear dependence with volume fraction is a strong indication of the absence of particle-particle interactions. Conversely, deviation from linearity implies the presence of such interactions. The boundary between the linear and nonlinear regions determines the volume fraction limit for any dilute case theory. In summary, particle-particle interactions become an important consideration for viscous losses at fairly low volume fraction, whereas such interactions are relatively unimportant for scattering losses even at very high volume fractions. Thus, we can conclude that ultrasound absorption is the most important mechanism to address when developing a more general theory that takes into account hydrodynamic, thermodynamic, and specific particle interactions. Development of an extended scattering theory to account for such particle interactions, the so-called “multiple scattering” problem, is of much less importance for acoustics. However, multiple scattering is indeed a major concern when attempting to analyze similar concentrated systems using optical systems.

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attenuation in [dB/cm/MHz]

25

scattering losses, quartz 120 micron at 5.5 MHz

20

15

10

viscous losses, rutile 0.3 micron at 10.3 MHz

5 thermal losses, latex 0.1 micron at 8.7 MHz

0 0

5

10

15

20 25 30 volume fraction in %

35

40

45

50

FIGURE 4.6 Measured attenuation at one particular frequency as a function of volume fraction for viscous, thermal, and scattering losses.

This major difference between acoustics and optics of heterogeneous systems is often overlooked. For instance, Hipp and coworkers [31] discuss multiple-scattering effects in “lossless scattering,” forgetting that the method of measurement automatically excludes this contribution. The general structure of this expanded ultrasound absorption theory remains the same as in the ECAH theory. Again there are two stages: “single-particle” and “macroscopic.” There are currently several versions of the single-particle theory that addresses particle-particle interactions. They are based on the “cell model,” or the “effective media model,” or their combination in the “core-shell model.” For both absorption mechanisms (viscous and thermal), the “coupled-phase model” can be used as the macroscopic theory in place of the Epstein version of the optical theorem. There is a substantial difference between the ways absorption and scattering are measured. Absorption irreversibly converts acoustic energy to heat. The measured intensity of the ultrasound after propagation through colloid, It, is what remains of the incident power, Ii, after subtraction of the energy absorbed by the particles and liquid. For scattering, the situation is more complicated because the energy disturbed by the particles is simply redirected, although remaining as acoustic energy. This difference provides many options when performing a scattering-type measurement. For instance, it is possible to measure the angular dependence of scattered sound, a possibility that simply does not exist for absorbed sound. Such angular dependence measurements were made by Kol’tsova and Mikhailov [38]

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for very large graphite particles in gelatin media, and by Faran [39] for metal cylinders. However, the measurement of such angular dependence of the scattered sound is not common. The more typical way to measure the scattered sound is to determine the attenuation and/or sound speed. Interpretation of a scattering measurement made in this mode is somewhat more complicated than characterizing simple absorption, because part of the scattered power may still reach the detector together with the remainder of the incident beam. For this reason, we speak of three components of scattered sound: coherent, incoherent, and multiple scattered. There are different ways to separate these components, either in theory or in the experimental design of an instrument. What we want to stress here, however, is that absorption and scattering are very different, not only in their nature and frequency dependencies, but also in the ways the measurement can be performed. In summary, we conclude that there is a strategic approach for deriving acoustic theory which is an alternative to the ECAH theory. We give up the idea of considering simultaneously all the mechanisms of ultrasound attenuation for all ka, but are rewarded with the ability to incorporate particle-particle interactions into the theory of absorption. As a result, acoustics can be more easily applied to the characterization of real, concentrated colloidal systems. Consequently, in the following section we will describe this new theory, which we call the “superposition theory.” The term superposition reflects superposition of the various mechanisms of the ultrasound extinction on an additive basis. As mentioned previously, the single-particle theory might be derived on the basis of either a cell model or a core-shell model. We prefer to use a cell model for the viscous losses since this mechanism is hydrodynamic in nature and the use of a cell model is well established for such effects. At present there is no corresponding cell model theory for thermal losses, but in principle the core-shell model theory can be used. It is not yet clear which approach is more advantageous for characterizing thermal losses. The macroscopic part of the superposition acoustic theory is a combination of the scattering theory and the coupled-phase model. We suggest that the scattering theory be used only for characterizing ultrasound scattering, whereas the coupled-phase model provides a better framework to describe ultrasound absorption.

4.2. ACOUSTIC THEORY FOR DILUTE SYSTEMS The fundamental basis of the ECAH theory was originally developed by Epstein and Carhart [1]. Allegra and Hawley [2] later generalized it for a more representative particle rheology. They also performed the first rigorous experimental tests of the theory, which were more adequate than some earlier

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studies made by Stakutis, Morse, Dill, Beyer, Hartmann, and Focke [40, 41, 75, 76, 77]. Following Allegra and Hawley, we now present a short overview of the ECAH theory. The description of the propagation of a sound wave through a heterogeneous medium requires three potentials: compression fc; thermal fT; and viscous f. A system of wave equations determines the space-time dependencies of these potentials: ðr2 þ k2 Þfc ¼ 0

ð4:4Þ

ðr2 þ kT2 ÞfT ¼ 0

ð4:5Þ

ðr2 þ ks2 Þf ¼ 0

ð4:6Þ

where k¼

o þ ja c

rffiffiffiffiffiffiffiffiffiffiffiffi orCp kT ¼ ð1 þ jÞ 2t sffiffiffiffiffiffiffiffiffi o2 r ks ¼ mL

ð4:7Þ

ð4:8Þ

ð4:9Þ

These equations are applied to both dispersed and continuous phases. The wave equation for a viscous fluid can be derived by replacing the Lame constant, mL, with jo. The general solution of this system of equations can be presented, in the case of spherical symmetry, as an expansion of Legendre polynomials. For the reflected compression wave, for example, this expansion is: frc ¼

1 X

jn ð2n þ 1ÞAn hn ðkc rÞPn ð cos yÞ

ð4:10Þ

n¼0

Altogether, there are six sets of constants similar to An that describe the general solution for the three potentials within the medium plus three potentials inside the spherical particle. There are also a set of six boundary conditions necessary to calculate these constants: vrm ðr ¼ aÞ ¼ vrp ðr ¼ aÞ

ð4:11Þ

vym ðr ¼ aÞ ¼ vyp ðr ¼ aÞ

ð4:12Þ

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CHAPTER

Tm ðr ¼ aÞ ¼ Tp ðr ¼ aÞ tm

4

ð4:13Þ

@Tp @Tm ¼ tp @r r¼a @r r¼a

ð4:14Þ

Prrm ðr ¼ aÞ ¼ Prrp ðr ¼ aÞ

ð4:15Þ

ry Pry m ðr ¼ aÞ ¼ Pp ðr ¼ aÞ

ð4:16Þ

The final system of algebraic equations used to determine these constants is very complex. We refer those interested to the original paper by Allegra and Hawley [2]. Equations (4.4)–(4.16) represent the single-particle portion of the ECAH theory. The macroscopic portion of this theory was derived by Epstein [1]. He showed that the total attenuation is determined only by the constants, An, of the reflected compression wave. Unfortunately, the equations for the system of boundary conditions are interlinked, and it is impossible to calculate each of the individual values of An. Nevertheless, the following expression allows one to calculate the total attenuation, if the constants An are known: a¼

1 3’ X ð2n þ 1Þ ReAn 2kc2 a3 n¼0

ð4:17Þ

Allegra and Hawley stressed that this expression is only valid for dilute suspensions, where the effect of the individual particles is additive. Also, particle-particle interactions were also neglected when the general solution was constructed from Legendre polynomials, automatically eliminating terms that would become zero at an infinite distance from the particle. However, these are the very terms that become important in concentrated colloidal systems where long-distance fields are indeed affected by the presence of other particles. Although the ECAH acoustic theory is the most widely known, it is not the only theory. Temkin [15, 16] formulated an alternative theory in which he suggested that the attenuation and sound speed be defined in terms of the particle-to-fluid velocity and temperature ratios. Consequently, these two parameters can be considered as local driving forces for viscous and thermal losses. Accordingly, he derived the following equations for sound speed and attenuation:

2a

Lw j Imup j

cs ðo ¼ 0Þ ¼ 1 þ Lw cs ðoÞ



 Cp  ghm  1 Lw mp  Cp  Tp  1  þ  Im  Cp  Cp   p 1 þ Lw mp T 1 þ L w p m Cp C p

ð4:18Þ

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Acoustic Theory for Particulates

h

cs ðo ¼ 0Þ  a2 ¼ 1 þ cs ðoÞ

i

Lw Reup  1 1 þ Lw



 Cpp ghm  1 Lw m Cp Tp  1 þ Re ð4:19Þ p Cp Cpp 1 þ Lw m 1 þ Lw Tp m Cp Cp

where up is the particle-to-fluid complex velocity normalized by the colloid velocity, gh is the specific heat ratio, Tp is the particle temperature normalized by the liquid heat transfer rate, and Lw is the mass load related to the weight fraction of the solid ’w simply by: Lw ¼

’w 1  ’w

ð4:20Þ

Temkin’s theory has one significant advantage over the ECAH theory in that it yields an expression for the sound speed. In particular, the low-frequency sound speed limit, cs(o ¼ 0), is important. There are several other derivations for this low-frequency sound speed. The earliest is Wood’s equation [42]: ! c2m rs rm c2m ¼ ð4:21Þ 1’þ’ c2s ðo ¼ 0Þ rm rp c2p where the density of the colloid rs ¼ ’rpþ(1’)rm. Temkin generalized this expression by including thermodynamic effects for the elastic particles, assuming no mass transfer or electric or chemical effects. Temkin’s final expression is:   bp 2 ! 1’þ’  h  c2m rs rm c2m ghp bm ! ð4:22Þ  g ¼ g 1  ’ þ ’  1 m m c2s ðo ¼ 0Þ rm rp c2p ghm Cpp 1 þ ’w 1 Cm p The thermodynamic corrections to sound speed are usually negligible because the specific heat ratio, gm, is very close to unity for almost all fluids. As a result, the simple Wood equation is sufficient for many practical colloidal systems. Temkin, however, stressed that his theory, like the ECAH theory, is valid only for dilute systems.

4.3. ULTRASOUND ABSORPTION IN CONCENTRATED DISPERSIONS Consideration of ultrasound absorption is simplified by the fact that, for most colloidal systems, it occurs at ultrasound frequencies with a wavelength that is much longer than the particle size. This was justified earlier with some

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theoretical and experimental data (Figures 4.1–4.3). The long-wavelength restriction has been used in many studies, as early as the Rayleigh scattering theory [23, 24]. This very convenient restriction allows us to consider the hydrodynamic and thermodynamic effects as both independent and additive. In turn, this allows us to now introduce both hydrodynamic and thermodynamic particle interactions into the theory. Inclusion of hydrodynamic effects is essential above 10 vol%, as was shown first by Urick [26, 27] and later confirmed by Hampton [28], Blue and McLeroy [29], and Dukhin and Goetz [4]. Thermodynamic effects are less sensitive to these particle-particle interactions, but also need to be included above 30 vol%, as demonstrated originally by Dukhin and Goetz [32] and more recently by McClements et al. [5, 6, 34]. The superposition theory is built in two stages, similar to the ECAH theory: a single-particle theory and the macroscopic theory. The longwave restriction allows us to include particle-particle interactions on both levels. The cell model is used as a single-particle theory for the viscous losses. There is a version of the cell model [4] that is valid for the nonstationary mode given a polydisperse system. The single-particle theory for thermal losses was formulated by Hemar et al. [5, 6] using the core-shell model, which is a combination of the cell model and effective medium model (Chapter 2). The coupled-phase model is the logical candidate for the macroscopic theory. We present here the version valid for a colloidal system that is polydisperse and structured. For structured colloidal systems, there is one other dissipative mechanism for ultrasound absorption. This has only recently been incorporated into the absorption theory. It is based on the transient network model that is widely used in rheology. The theory of ultrasound absorption is derived only for spherical particles. There are some experimental indications, however, (Blue and McLeroy [29]) that the equivalent sphere model is reasonably adequate even for particles with pronounced shape.

4.3.1. Coupled-Phase Model Let us consider an infinitesimal volume element in the dispersed system. There is a differential force acting on this element that is proportional to the pressure gradient of the sound wave, rP. This external force is applied to both the particles and the liquid, and is distributed between the particles and liquid according to the volume fraction, ’. Both the particles and the liquid move with an acceleration created by the sound wave pressure gradient. In addition, because of inertia effects, the particles move relative to the liquid, and this causes viscous friction between the particles and the liquid.

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Acoustic Theory for Particulates

The balance of these forces can be represented by the following two equations, written for particles and liquid, respectively: ’rP ¼ ’rp

@up þ gðup  u0 Þ @t

ð4:23Þ

@u0 ð4:24Þ  gðup  u0 Þ @t where um and up are the velocities of the medium and particles in a laboratory frame of reference, t is time, and g is a friction coefficient that is proportional to the volume fraction ’ and the particle hydrodynamic drag coefficient O: ð1  fÞrP ¼ ð1  fÞr0

g¼

9’O 2a2

ð4:25Þ

Fh ¼ 6paOðup  um Þ

ð4:26Þ

where  is the dynamic viscosity and a is the particle radius. Equations (4.23) and (4.24) are well known in the field of acoustics. They have been used in several papers to calculate sound speed and acoustic attenuation [4, 12, 13]. They are valid without any restriction as to the volume fraction. Importantly, it is known that they yield a correct solution, as the concentration is reduced to the dilute case. The two equations together are normally referred to as the coupled-phase model. The word “model” usually suggests the existence of some alternative formulation, but it is hard to imagine what one can change since these equations essentially express Newton’s Second Law. Perhaps, the word “model” is not strong enough in this case. The equations can be solved for the speed of the particle relative to the liquid. The time and space dependence of the unknowns, um and up, are presented as a monochromatic wave AejðotkxÞ , where j is a complex unit and k is a compression complex wavenumber. As a result, the relationship between the gradient of pressure and the speed of the particle relative to the fluid can be expressed as: gðup  um Þ ¼

’ðrp  rs Þ rs þ io’ð1  ’Þ

rp rm rP g

ð4:27Þ

where rs ¼ ’rpþ(1’)rm, and ’ is the volume fraction of the solid particles. Dukhin and Goetz [4] have suggested a generalization of the coupledphase model for polydisperse colloids. Assume that we have a polydisperse system having N size fractions. Each fraction can be described by a certain particle diameter, di, volume fraction, ’i, drag coefficient, gi, and particle velocity, ui (in a laboratory frame of reference). Assume also that the density

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4

of the particles, rp, is the same for all fractions and that the total volume fraction of the dispersed phase is ’. The liquid is characterized by a dynamic viscosity, , a density, rm, and a velocity, um (again in a laboratory frame of reference). The coupled-phase model suggests that we apply the force balance equation to each fraction of the dispersed system, including the dispersion medium. We did this earlier for just one fraction, but now we apply the same principle to the N fractions and obtain the following system of Nþ1 equations: ’1 rP ¼ ’1 rp jou1 þ g1 ðu1  um Þ  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  N equations for particles ’i rP ¼ ’i rp joui þ gi ðui  um Þ ð1’ÞrP ¼ ð1’Þrm joum 

X

ð4:28Þ

gi ðui um Þ an equation for liquid ð4:29Þ

i

where j is a unity imaginary number, P and o are the pressure and frequency of the ultrasound, and gi ¼

18’i Oi di2

ð4:30Þ

Fhi ¼ 3pdi Oi ðui  um Þ

ð4:31Þ

We can solve this system of Nþ1 equations [4]. But to do this, we must first reformulate all equations to introduce the desirable quantities, xi ¼ ui  um, and eliminate the parameter, um, using Equation (4.31) which specifies the liquid velocity in a form X gi xi rP i þ ð4:32Þ um ¼  jorm ð1  ’Þjorm This reformulated system of N equations is now:     X rp rp gi xi þ  1 rP ¼ jorp þ g xi rm ’i ð1  ’Þrm i i

ð4:33Þ

This system can be solved using the principle of mathematical induction. We guess the solution for N fractions, and then prove that the same solution works for N þ 1 fractions. As a result, we obtain the following expression for velocity of the ith fraction particle relative to the liquid:   rp  1 rP r 0 m 1 ð4:34Þ ui  um ¼   N X rp gi @ gi A 1þ jorp þ ’i ð1  ’Þrm i¼1 jorp þ gi ’i

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Acoustic Theory for Particulates

This expression indicates that the coupled-phase model will allow us to calculate the relative particle speed, ui  um, for each fraction, without using the superposition assumption. This is important for the electroacoustic theory (Chapter 5). For the acoustic theory, it is enough that we now have the ability to calculate the complex compression wavenumber, k, of the polydisperse colloidal system, without using any superposition assumption. But to derive k, we must first add the following equation to the coupled-phase model [4, 12, 13]: 

N X @xip @P ¼ K  ð1  ’Þru0 þ K  ’i r @t @t i¼1

ð4:35Þ

Equations (4.28) and (4.29) yield then a solution for the complex wavenumber of the polydisperse colloid:

k2 Ms ¼ o 2 rs

N rp X ’i rs i¼1 1  ð9jrm Oi =4s2i rp Þ ! N P rs ’i 2 2 rp i¼1 1  ð9jrm Oi =4si rp Þ

1 1þ

ð4:36Þ

where s2i ¼

a2i orm 2m

ð4:37Þ

and rs ¼ rp ’ þ rm ð1  ’Þ

ð4:38Þ

At this point, we believe it would be helpful to create a heuristic picture of the physical phenomena that occur when an ultrasound pulses passes through a system of dispersed particles. Such a description, given below, will help provide answers to some general questions. Let us consider an element of the concentrated dispersed system within the sound wave. The size of this element is selected such that it is much larger than the particle size, and also larger than the average distance between the particles. As a result, this element contains many particles. At the same time, this element is much smaller than the wavelength of sound. This dispersion element moves with a certain velocity and acceleration in response to the gradient of acoustic pressure. As a result, an inertial force is applied to this element. At this point we can introduce the principle of equivalence between inertia and gravity. The effect of the inertial force created by the sound wave is equivalent to the effect of the gravitational force. This gravitational force is exerted on both the particles and the liquid in the dispersion element. The densities of the particles and the liquid are different, as are the forces. The force acting on the particles depends on the ratio of the densities.

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4

The question arises as to what density we should consider. To answer this, let us consider the forces acting on a given particle in the gravitational field. The first force is the weight of the particle, and is proportional to its density, rp. This force will be partially balanced by the pressure of the other particles and the surrounding liquid. This pressure is equivalent to the pressure generated by an effective medium having a density equal to the density of the dispersed system. It is clear that this force is proportional to the difference in density between that of the particles, rp, and that of the dispersed system, rs. This statement can be illustrated by the sedimentation in a peat bog, consisting of mainly cellulose-based material dispersed in water. The effective density of this effective medium is much less than that of water, hence a person would drown in such a material if not pulled out. This happens because there are two different scenarios for sedimentation, as shown in Figure 4.7. Case 1 corresponds to the situation where the group settles as a single entity, and the liquid envelopes this settling group from the outside. Case 2 corresponds to a different scenario, in which the liquid is forced to move through the group of particles. A difference exists for the two cases between the forces exerted on the particles within the group. In the first case, the friction occurs only between the settling group and the liquid. In the second case, however, there is an additional frictional force resulting from the liquid pushing through the group of particles. Ultrasound propagation through colloidal suspensions corresponds to the second case, because the motion of the particles and the liquid on the edges of the sound pulse is negligible when the width, w, of the sound pulse is much larger than wavelength, l, at high ultrasound frequencies: wl

ð4:39Þ

It is in the center section of the sound pulse that the liquid is forced to move through the group of particles.

Case 1

Um

Case 2

Um

Up

Um Um

Up

Um

Up

FIGURE 4.7 Two possible scenarios for sedimentation of particles in a liquid.

149

Acoustic Theory for Particulates

Thus, the balance of forces exerted on the particles in a given element of the dispersion consists of the effective gravitational force, the buoyancy force, and the frictional force related to the movement of the liquid through the array of particles.

4.3.2. Viscous Loss Theory The coupled-phase model provides a link between macroscopically measured parameters such as attenuation, as, and sound speed, cs, and the properties of a single particle. The particles are considered to be rigid, and consequently characterized only by a density, rp, and a drag coefficient, Oi, for each particle size fraction. This drag coefficient can be calculated using the Happel cell model (Section 2.5.2). The cell model theory relates the particle size to the drag coefficient, which in turn is linked, using the coupled-phase model, to the experimentally measured attenuation and sound speed. All required mathematical expressions are given in Sections 2.5.2 and 4.2.2. It is instructive to see how the theoretical attenuation changes with respect to various aspects of the colloidal suspension. For example, Figure 4.8 illustrates a decrease in the critical frequency (the frequency that corresponds to the maximum attenuation) as the particle size is increased. According to theory, this critical frequency is inversely proportional to the square of the particle size. In this example, the two monodisperse systems have a size ratio of 2:1 and the corresponding ratio of critical frequencies is seen to be 1:4 (i.e., 4–16 MHz). Figure 4.9 illustrates the effects of polydispersity. Broadening of the PSD leads to a flattening of the attenuation curve.

9

Attenuation [dB/cm/MHz]

8 7 6 5 4 3 2

diameter 1 micron diameter 0.5 micron

1 0 100

101 Frequency [MHz]

102

FIGURE 4.8 Effect of particle size on the theoretical viscous attenuation for a 20 vol% slurry of monodisperse particles having a density contrast of 2.

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4

7

Attenuation [dB/cm/MHz]

6 5 4 3 standard deviation 2

.01 .1 .25

1 0 100

101 Frequency [MHz]

102

FIGURE 4.9 Effect of the width of the particle size distribution on the theoretical viscous attenuation for a 15 vol% slurry of particles having a mean size of 0.5 mm and a density contrast of 2.

Together, Figures 4.10 and 4.11 illustrate the influence of modality on the attenuation spectra. The two PSDs shown in Figure 4.11 have the same median size and the same polydispersity. However, their third and fourth moments are obviously quite different, as is reflected in the shape of the PSD. The corresponding theoretical attenuation curves for these two size distributions are shown in Figure 4.10. Clearly, there is a pronounced difference

Attenuation [dB/cm/MHz]

5

4

3

2

lognormal bimodal

1

0

FIGURE 4.10 Figure 4.11.

100

101 Frequency [MHz]

102

Attenuation spectra corresponding to two particle size distributions from

151

Acoustic Theory for Particulates

1.6 1.4

PSD, weight basis

1.2 1.0 0.8 0.6 0.4 0.2 0.0

10-2

10-1

100

101

Diameter [um] FIGURE 4.11 Particle size distributions with the same median size and width but different higher order moments.

between the two attenuation spectra. This strongly suggests that acoustic spectroscopy can be used to study the differences in the shape of the size distribution of practical dispersions. Figure 4.12 illustrates the effect of particle-particle interactions on the attenuation spectra. It is seen that an increase in the volume fraction increases 10

Attenuation [dB/cm/MHz]

9 8 7 6 5

volume fraction, %

4

1 15 40

3 2 1 0 100

101 Frequency [MHz]

102

FIGURE 4.12 Effect of particle volume fraction on the attenuation spectra for a colloidal suspension of monodisperse 0.5 mm particles.

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the critical frequency. In the least concentrated dispersion, there are two geometric parameters that determine the critical frequency value: the particle radius and the viscous layer thickness, d . The viscous layer thickness is defined as the distance from the particle surface over which the shear wave, generated by the oscillating particle, decays by a factor 1/e as it passes into the bulk liquid. It is thus useful to think of the particle as being surrounded by a viscous layer with a thickness, d , given by [1, 2, 4]: sffiffiffiffiffiffiffiffiffi 2 d ¼ ð4:40Þ rm o The viscous layer thickness decays with frequency. There is a certain frequency at which the viscous layer thickness becomes equal to the particle radius. It turns out that this frequency is approximately equal to the critical acoustic frequency for a dilute suspension. This gives the following approximate expression for the critical frequency: ocr 

2 a2

ð4:41Þ

For the case of a concentrated suspension with hydrodynamically interacting particles, there is one additional important geometrical parameter: the distance between particles. The distance between particles becomes smaller with increasing volume fraction. At some point it becomes less than the particle radius. For the volume fractions exceeding this critical value, the interparticle distance overtakes the particle radius as the parameter which determines the critical frequency. This new condition for calculating the critical frequency can be formulated as an equality between the viscous layer thickness and the interparticle distance for the given volume fraction. The interparticle distance is smaller than the particle radius at this high volume fraction, and correspondingly the critical frequency is higher, as is illustrated in Figure 4.12. Without an adequate theory, this volume fraction effect might be mistakenly attributed to a variation in the particle size, since a decrease in particle size would cause a similar shift in the critical frequency. Ignoring particleparticle interactions would therefore lead to an underestimation of the particle size in concentrated colloidal dispersions. In the example presented in Figure 4.12, the critical frequency shifts almost 10-fold because of volume fraction induced particle-particle interactions. Simply applying the ECAH theory to this attenuation spectrum in order to calculate the particle size of the 40 vol% dispersion would result in a particle size of only one-third the actual size. Hence, to obtain correct particle sizing results using acoustics it is of critical importance to take hydrodynamic particle-particle interactions into account through a suitable theory.

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Acoustic Theory for Particulates

4.3.3. Thermal Loss Theory Isakovich [43] initially developed a theory for thermal losses, again only for dilute colloidal suspensions. Thermal losses dominate any ultrasound propagation in emulsions, polymer latices, or other systems with small (<1 mm) soft particles having low density contrast, such as wax suspensions. Isakovich’s theory yields the following expression for thermal loss attenuation: " #2 bp 3’Tcm rm tm T bm ath ¼  p ReH ð4:42Þ Cm 2a2 C p rp p rm where the function, H, equals: H 1 ¼

zm;p

tm tanh zp 1  1  jzm tp tanh zp  zp

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi orp ,m Cp;m p ¼ að1 þ jÞ 2tp ,m

ð4:43Þ

ð4:44Þ

This original Isakovich theory is valid only for dilute dispersions, because it does not take into account particle-particle interactions. The term “dilute” is somewhat misleading in this instance, since experimental studies have shown that there are often no thermodynamic particle-particle interactions up to 30 vol%. This means that Isakovich’s theory can be applied, without any additional complexity, to a wide variety of practical systems like emulsions, polymer latices, and soft particles. Additional corrections would be required only at exceptionally high volume loading. This surprising difference between “viscous” and “thermal” losses is related to the difference between the “viscous depth” and the “thermal depth.” These two parameters are a measure of the decay of the shear and thermal waves, respectively, in the liquid. Particles oscillating in the sound wave generate these shear and thermal waves that are then damped as they move away from the particle. The characteristic distance for the shear wave amplitude to decay is the “viscous depth,” d (see Section 4.2.2). The corresponding distance for the thermal wave to decay is the “thermal depth,” dT, given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2tm ð4:45Þ dT ¼ rm oCm p The relationship between dT and dT has been considered by McClements, who calculated the thermal depth and viscous depth versus frequency [44]. For aqueous suspensions, it is relatively easy to show that the viscous depth is 2.6 times larger than the thermal depth [33]. As a result, the viscous layers surrounding the particles overlap each other at a much lower volume fraction

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4

than the corresponding thermal layers. The overlap of these boundary layers is a measure of the corresponding particle-particle interactions. Obviously, there are no particle-particle interactions when the corresponding boundary layers are sufficiently separated. Thus, for a given frequency, an increase in the dispersed volume fraction leads first to an overlap of the viscous layers because they extend further into the liquid. Only then, at higher volume fractions, will the thermal layers overlap. In general, we can qualitatively conclude that at lower volume fractions, the hydrodynamic interactions between particles are more important than any thermodynamic interactions. Quantitatively, this factor of 2.6 for the ration of d and dT implies that there is a large difference between these two mechanisms insofar as the critical volume fraction at which the boundary layers begin to overlap (with resultant particle-particle interactions). The dilute case theory is valid for any volume fraction smaller than this critical volume fraction. The ratio of these critical volume fractions for the viscous and thermal losses can be estimated from:  pffiffiffiffiffi 3 a pf þ 1=2:6 pffiffiffiffiffi ð4:46Þ ’vis-th ¼ a pf þ 1 where a is particle radius in micrometers and f is the frequency in megahertz. It is interesting that this important feature of the thermal losses works for virtually all liquids. In order to illustrate this, it is convenient to introduce a parameter referred to as the “depth ratio,” which is completely independent of the properties of the particle and is uniquely defined by the medium alone: depth ratio ¼

d dT

ð4:47Þ

Figure 4.13 shows the values of the depth ratio for many liquids taken from Anson and Chivers [22] and the “Materials Database” of Dispersion Technology, Inc. Without exception, this depth ratio is larger than that of water for all fluids studied. Therefore thermal losses are much less sensitive to particleparticle interactions than viscous losses for virtually all known liquids. For emulsions, etc. we conclude that Isakovich’s theory is valid for a much wider volume fraction range than one might otherwise expect. However, there is still a definite need for a thermal loss theory that would be valid for very high volume concentrations, up to 50 vol%. To extend acoustic theory to these highly concentrated colloidal suspensions, Hemar et al. [5, 6] modified Isakovich’s theory to take into account the overlap of thermodynamic layers. They used the core-shell model for the single-particle theory and a scattering model for the macroscopic theory. A more detailed description of the core-shell model can be found in Section 2.5.2, but here we repeat some essential features. The core-shell model replaces a real suspension with a model in which a single particle is placed inside of a spherical shell of the dispersion medium.

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Acoustic Theory for Particulates

6

depth ratio

5

4

3

2

1 0

10

20

30

40

50 60 liquid index

70

80

90

100

110

FIGURE 4.13 Depth ratio parameter for various liquids. The liquid index determines the liquid in the DTI Material Database.

The radius of this shell, bsh, differs from the radius of the dispersion medium cell, b, in the cell model: a bsh ¼ pffiffiffiffiffiffi , 3’

a b¼p ffiffiffiffi 3 ’

ð4:48Þ

There is another difference between this core-shell model and the straightforward cell model; one applies an effective medium approach to describe the dispersion beyond the liquid core. This effective medium is characterized with “effective” values for the density, ref, the compressibility, bef, the heat conductance, tef, and the heat capacity, Cef p . In turn, these effective values are related to the properties of the dispersion medium and the dispersed phase using the following equations: ref ¼ ð1  ’Þrm þ ’rp

ð4:49Þ

bef ¼ ’bp þ ð1  ’Þbm

ð4:50Þ

Cef p ¼ ð1  ’Þ

rp rm m Cp þ ’ Cpp ref rm

ð4:51Þ

  tp  tm tp  t m 2 2 1 þ 2’  2ð1  ’Þð0:21’  0:047’ Þ tp þ 2tm tp þ 2tm tef ¼ tm   ð4:52Þ tp  tm tp  t m 2 2 1’  2ð1  ’Þð0:21’  0:047’ Þ tp þ 2tm tp þ 2tm

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4

This core-shell model yields a new expression for the function, H: (Equation 4.43)    tm bm zp  tanhðnp aÞ gm  gef bsh bsh H¼ 2bef zm zef þ1 bs a a EC þ FD gp  gm ð4:53Þ þ Cð1 þ zm Þ þ Dð1  zm Þ where gm ¼

bm , rm C m p

gp ¼

bp , rp Cpp

gef ¼

bef ref Cef p

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oref Cef p zef ¼ að1 þ jÞ 2tef

ð4:54Þ

ð4:55Þ

        bsh bsh bsh C ¼ exp zm 1 tm zm  1 þ tef zef þ1 a a a 

       bsh bsh bsh 1 tm zm þ 1  tef zef þ1 D ¼ exp zm a a a E ¼ tp zp þ ½tm ðzm þ 1Þ  tp  tanh zp F ¼ tp zp  ½tm ðzm  1Þ þ tp  tanh zp The introduction of the thermal layer overlap through this core-shell model substantially improves the ability of acoustics to adequately characterize the properties of soft particles such as latices and emulsions.

4.3.4. Structural Loss Theory In addition to hydrodynamic and thermodynamic interaction, particles might experience specific colloid-chemical interactions, especially in concentrated dispersions. These specific interactions can affect the propagation of ultrasound through the colloidal suspension, and provide an additional contribution to the ultrasound attenuation. We refer to this contribution as “structural loss.” The name reflects the fact that this specific interaction usually implies some underlying macroscopic structure in the concentrated suspension. Structural losses are closely related to the rheology of concentrated colloids. According to Voigt’s rheological model [45], a structured suspension exhibits both dissipative and elastic behavior. The main difference between the structural losses in classical rheology and those in acoustics is the

157

Acoustic Theory for Particulates

frequency range over which the loss is measured. Although acoustics utilizes much higher frequencies than classical rheology, the main principles and models are the same. On a macroscopic level, Voigt’s model suggests that we regard stress as a function of the strain and the strain rate. The Rouse-Bueche-Zimm [45] model translates this idea to a microscopic level, by representing the colloidal suspension as Nþ1 beads joined together by N springs. The viscous resistance of the beads moving through the liquid is inversely proportional to the Stokes’ drag coefficient, and causes viscous dissipation. The springs between the beads represent the stress-strain relationship. This model is successfully used to describe the classical rheological behavior. We apply it to acoustics here, following McCann [46]. He added a Hookean factor to the force balance, but only used the first term, in which force is proportional to displacement. We take the next logical step, following Voigt’s model, which states that the stress is proportional not only to displacement but also to the rate of displacement. We can reformulate the coupled-phase model, including additional forces acting on the particles due to the specific interactions. The resulting coupledphase theory reflects both viscous and structural losses. It is based on balancing the forces on both the particles and the liquid, as well as applying the law of conservation of mass. According to this generalized coupled-phase model, expressions for the force balance of the ith fraction of the particles moving with a speed, upi , are: @ 2 xpi þ Fhi þ Fhook i @t2 and for the liquid moving with a speed, u0: ’i rP ¼ ’i rp

ð1  ’ÞrP ¼ ð1  ’Þr0

N @ 2 x0 X  Fhi @t2 i¼1

ð4:56Þ

ð:4:57Þ

These force balance equations assume that the total volume force caused by a sound pressure wave, P, is redistributed between the particles and the liquid according to their volume fractions (’i, for the ith fraction and ’ for the total solid content). This volume force is compensated by an inertia force, a hydrodynamic friction, Fh, and a “specific” Hookean force, Fhook. Fhi ¼ 6pai Oi

@xpi @t

ð4:58Þ

@xpi ð4:59Þ @t Two terms are then necessary to adequately describe the contribution of the non-Hookean springs to the force balance. The first term is an elastic force Fhook ¼ H1 Dxpi þ H2 i

158

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4

proportional to the displacement of the particle with a coefficient H1. The second term is a dissipative force proportional to the particle velocity with a coefficient H2. The coefficients, H1 and H2, are assumed to be the same for all particles. The resulting expression for the complex compression wavenumber of the dispersion, k, is [33]: ! N X gi ðDi  jogi Þ ð1  ’Þ rm þ joDi k2 Ms i¼1 ¼" #2 " # 2 o N N X N o2 ’2 X P jo’i gi gi ðDi  jogi Þ i  ð1  ’Þrm þ 1’þ Di joDi i¼1 Di i¼1 i¼1 ð4:60Þ where Di ¼ o2 rip ’i þ jogi þ joH1 þ H2 gi ¼

9’i Oi 2a2i

ð4:61Þ ð4:62Þ

The attenuation of the ultrasound, a, is related to the complex wavenumber through: as ¼  ImðkÞ

ð4:63Þ

Expression (4.60) is more general than the original coupled-phase theory (Equation 4.36). First, it is valid for polydisperse systems. This is important because practical systems might be polydisperse not only in size but also with respect to density, or perhaps other physical properties. This more general theory allows us to study systems containing mixed dispersed phases, that is, containing particles of different chemical nature and different internal structure. Second, this theory yields an expression for the complex wavenumber in pastes and gels. Attenuation in gels is caused just by the oscillation of the polymer network, or in our terminology it is purely structural loss. By assuming that the drag coefficient, g, in Equation (4.60) is equal to zero, we obtain the following expression for gels: rm ðH1  o2 rp ’Þ þ jorm H2 k2 Ms ¼ o2 ð1  ’ÞðH1  o2 rp ’Þ  o2 ’2 rm þ jð1  ’ÞoH2

ð4:64Þ

This equation can be used to calculate the attenuation and sound speed in pastes and gels, but as of yet has not been tested experimentally. It is seen that attenuation occurs only when the second coefficient, H2, is nonzero. The situation here is analogous to a damped spring; the dissipation of energy depends on the rate at which the spring is compressed, but not on the displacement per se.

159

Acoustic Theory for Particulates

16

no structure, vfr 40%vl, size 1 micron elastic coefficient = 5 elastic coefficient = 15 elastic coefficient = 30

15

attenuation [dB/cm/MHz]

14 13 12 11 10 9 8 7 6 5 4 1

10 frequency [MHz]

100

FIGURE 4.14 Influence of the elastic Hookean coefficient, H1, on the attenuation of sound.

60 initial alumina, vfr 40%vl, size 1 micron loss coefficient = 0.5 loss coefficient = 1 loss coefficient = 2 loss coefficient = 5

attenuation [dB/cm/MHz]

50

40

30

20

10

0 100

101 frequency [MHz]

102

FIGURE 4.15 Influence of the loss Hookean coefficient, H2, on the attenuation of sound.

Figures 4.14 and 4.15 illustrate the effect of structure on the theoretical attenuation spectra of a 40 vol% alumina dispersion having a median size of 1 mm. It is seen that the first Hookean coefficient simply shifts the critical frequency from about 10 to 20 MHz, keeping the shape of the curve more or less intact, and the peak attenuation constant. Since the particle size is inversily proportional to the square root of this critical frequency, the influence of the structure would have to be very substantial in order to create large errors in the calculated particle size. The influence of the elastic coefficient cannot affect the quality of the fit between theory and experiment. For instance, it cannot explain any excess

160

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4

in experimental attenuation over that predicted by the theory. The elastic structure does not change the average amplitude of the attenuation spectra. The excess attenuation sometimes observed in experiments can be explained only with the second coefficient, as follows from Figure 4.15. In principle, this second coefficient can be extracted from the attenuation spectra as an adjustable parameter, as will be shown in Chapter 8.

4.3.5. Intrinsic Loss Theory “Intrinsic loss” reflects an acoustic energy dissipation process that occurs at a molecular level. We can easily speak of the intrinsic loss for a homogeneous material, but it is meaningful in a heterogeneous system as well. It is important, however, to understand that the intrinsic loss in a heterogeneous system is not related to how finely divided one phase is within the other. Roughly speaking, the intrinsic attenuation of a heterogeneous system is a volume-weighted average of the intrinsic attenuation of each component. More precisely, we can compute the intrinsic loss in a heterogeneous colloid system [33] as follows:

aint

r ap am ð1  fÞ þ f m rffiffiffiffiffiffi cm r p a m rs sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rm frm 1f 2 þ 2 cm rp cp

ð4:65Þ

where am and ap are intrinsic attenuations of the liquid medium and the particle material, respectively. The attenuation of any liquid (medium) is easily measured with an acoustic spectrometer. Usually, the attenuation of the internal phase material is negligible for solid particles. For soft particles, such as emulsions, it is easy to measure if the internal phase is available as a pure liquid. If not, we need to use this as an additional adjustable parameter and determine the value from the acoustic data. The intrinsic attenuation of the colloid, aint, can be considered as a background, independent of the PSD, against which any additional attenuation that results from the state of division of the two phases can be judged. Typically, the medium makes the largest contribution to the overall intrinsic attenuation of the system and we can sometimes neglect entirely the contribution of the particulates to the total intrinsic loss in the colloidal suspension. However, there are some interesting exceptions. Figure 4.16 shows the attenuation spectra of a 30% water-in-oil emulsion together with the attenuation for each component of this emulsion, namely, the oil and the water. At first glance it seems strange that the attenuation of the emulsion is actually less than that of the base oil, which comprises the continuous phase of the emulsion. However, this makes perfect sense when we realize that the intrinsic attenuation of the colloid as a whole is much less than that of the oil medium because it is reduced significantly by the relatively low intrinsic attenuation of the water particulates, in accordance with

161

Acoustic Theory for Particulates

12

Attenuation [dB/cm/MHz]

10 base oil

8 6

emulsion

4 2 water

0

100

101 Frequency [MHz]

102

FIGURE 4.16 Attenuation spectra measured for water, oil, and a water-in-oil emulsion.

Equation (4.65). Although not immediately obvious, the division of the water phase into small droplets did cause the attenuation of emulsion to exceed the intrinsic attenuation of this water-in-oil system.

4.4. ULTRASOUND SCATTERING Rayleigh initially developed the theory of ultrasound scattering for both the long-wavelength condition and solid particles. Anderson generalized this for spherical fluid droplets, also within the long-wavelength limit. Atkinson and Kytomaa [36, 37], Hay and Mercer [47], and Faran [39] made attempts to extend the scattering theory to larger ka (shorter wavelength), and to combine it with the asymptotic value of the viscous loss at high frequency. McClements provides a comparison of several attempts to create a multiplescattering theory as developed by Waterman and Truel [7], Lloyd and Berry [8], Ma et al. [9], and Twersky and Tsang [10]. Morse [18] gives the most comprehensive review of single scattering, and we chose his review as the basis for this section because it provides a single scattering theory applicable for any ka. Importantly, Morse points out that there is a simple way to eliminate multiple-scattering effects. Multiple scattering is not important if the experiment measures just the changes in the intensity of the incident beam as it passes through the colloidal suspension. We can demonstrate that this is true by measuring the attenuation of 120-mm quartz particles. As seen in Figure 4.6, the attenuation remains a linear function of volume fraction up to 46 vol%, a good indication of the absence of multiple scattering.

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4

When sound traverses a region filled with small obstacles, each obstacle produces a scattered wave, and these wavelets reinforce in some directions and interfere in others. The wave incident on each scatterer is affected by the presence of the others, which in turn affects the shape of the scattered wave. This gives rise to coherent, incoherent, and multiple scattering and, in the case where the scatterers are regularly spaced, to diffraction. The scattered wave is the sum of all the wavelets arising from the various small obstacles, each having a unique phase and amplitude, determined by the particular obstacle’s size, shape, and position. If the obstacles scattering sound are randomly distributed, no mutual reinforcement occurs, except in the direction of the incident wave. In this forward direction, the reinforced scattering modifies the incident wave. It then behaves as though it were traveling through a region having a different index of refraction. Thus, as far as average terms are concerned, the scattering region behaves as though it has a uniform density and compressibility, which differs from that of the external medium. This difference will produce scattering, or refraction, of the sound incident on the region. In the long-wavelength limit, the scattered intensity depends on the squares of the density and compressibility—in other words, on the square of the number of scattering particles, N, per unit volume in Robs. This cooperative portion of the scattered wave, in which each scatterer adds its contribution to the wave amplitude, is called the coherent scattered wave. Thus the intensity is proportional to N2. The rest of the scattering, the intensity of which is proportional to N rather to N2, is called the incoherent scattered wave. One can calculate various scattering terms if appropriate expressions for scattering cross-sections of the obstacles in the scattering region are available. The most extensive study of this subject appears to have been performed by Morse et al. [18]. Here we provide an overview of this work, with some additions from more recent developments. Consideration of the scattering problem usually starts with a theoretical modeling of the sound radiation by an oscillating object. The simplest model is the radiation from a circular piston which is mounted in a rigid baffle. The general model for the pressure field produced by the vibration of such a piston is a solution called the Rayleigh integral. For the full circular piston it is: Z Z a jkP0 jot p ℓjkR ℓ dc s ds ð4:66Þ Pðr, y, tÞ ¼ p Robs 0 0 where Robs ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ s2  2rs siny cosc

ð4:67Þ

and furthermore, where u0 is the amplitude of the normal velocity of the piston oscillation; a is the piston radius; r, y, and c are the polar coordinates associated with the center of the piston; Robs is the distance from the center

163

Acoustic Theory for Particulates

of the piston to the observation point; and P0 is pressure amplitude that would exist at the face of the piston if the piston diameter were infinite. When the observation point is far away from the piston face ðr  aÞ, the expression for Robs can be simplified to: Robs ¼ r  a siny cosc þ Oða=rÞ2

ð4:68Þ

Including only the first two terms for Robs, we obtain the following expression for the pressure far away from the piston face: jaP0 J1 ðka sinyÞ jðotkrÞ jR0 P0 ¼ ð4:69Þ DðyÞℓjðotkrÞ ℓ r r siny where R0 is the Rayleigh distance. It is defined as a ratio of the piston surface area, S, to the wavelength, l, defined by: Pðr, y, tÞ ¼

R0 ¼

S l

ð4:70Þ

Thus, for a spherical piston: pa2 ka2 ¼ ð4:71Þ l 2 There is an interesting interpretation of the Rayleigh distance that comes from a comparison of the current reciprocating piston to the radiation from a pulsating sphere of radius, R0, and pressure amplitude, P0, at its surface: R0 ¼

R0 P0 jðotkrÞ ℓ ð4:72Þ r A comparison of the two radiation pressure fields (Equations 4.69 and 4.72) shows that, on axis, the piston signal appears to come from a spherical source of radius R0. This interpretation mildly suggests that the piston radiation starts out as a collimated plane-wave beam, where the amplitude is P0 from the face out to r ¼ R0, beyond which point the beam spreads spherically. Thus, R0 roughly marks the end of the near field and the beginning of the far field. This model is a gross oversimplification and must not be taken too literally, but sometimes it is a useful interpretation of the radiation field. A more detailed analysis of the near field [48] concluded that, in a sense, the far field begins much closer to the surface, at about R0/4. At the same time, the far-field formula 1/r for pressure dependence with distance from the piston face is well established at about R ¼ 2 R0. The total on-axis pressure in the far-field region seems to be the sum of two plane-wave signals of the same amplitude, one coming directly from the center of the piston, the other from the edge of the piston. The latter is often called the “edge wave.” Interpreting the axial fields as being made of two plane waves makes some physical sense; one can eliminate the edge wave by making the piston radius large enough. In the limit of an infinitely large value for a, the edge wave Pðr, y, tÞ ¼

164

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4

would never arrive at the axis. The solution is then just an ordinary plane wave, as one would expect, from the vibration of the infinite plane. Far-field and near-field radiation regions determine the long-wavelength and short-wavelength frequency ranges. At low frequencies (long wavelength), the near-field region extends only a very short distance from the sound source. Observation is usually performed in the far field, where sound radiates out with equal intensity in all directions and the amount radiated is small. As the wavelength decreases, more power is radiated, and the intensity has more directionality; the piston begins to cast a “shadow” and a smaller proportion of the energy is sent out on the side of the piston opposite the radiating element. For very short waves, the intensity is large and uniform in the forward direction, and is zero in the shadow behind the cylinder. Figure 4.17 illustrates the dependence of the Rayleigh distance on frequency within the typical ultrasound range of 1–100 MHz. It is seen that even for large particles of several hundred micrometers, the Rayleigh distance does not exceed 1 mm. Normally, the ultrasound propagation path through the colloidal suspension is much longer and, consequently, any observed scattering will correspond to the far field. This general description is characteristic of all wave motion when it strikes an obstacle. When the wavelength is large compared to the size of the obstacle, the presence of the obstacle is of no consequence and can, effectively, be

300 200 micron

Rayleigh distance [micron]

250

100 micron 200 30 micron 150 100 50 10 micron 0

100

101 frequency [MHz]

102

FIGURE 4.17 Rayleigh distance as a function of ultrasound frequency for particles of different sizes.

Acoustic Theory for Particulates

165

neglected. On the other hand, when the wavelength is very short compared to the size of the obstacle, the motion resembles the motion of particles, with the waves traveling in straight lines and the obstacle casting sharp-edged shadows. When the wavelength is about the same size as the obstacle, complicated interference effects can sometimes occur, and the analysis of the behavior of the waves becomes quite involved. When a sound wave encounters an obstacle, some of the wave is deflected from its original course. It is usual to define the difference between the actual wave and the undisturbed wave that would exist if no obstacle were present as the scattered wave. If the obstacle is very large compared to the wavelength, half of this scattered wave spreads out more or less uniformly in all directions from the particle, and the other half is concentrated behind the obstacle in such a manner as to interfere, destructively, with the unchanged wave behind the obstacle, thus creating a sharp-edged shadow. This is the situation in geometrical optics (as well as acoustics); the half of the scattered wave spreading out uniformly is called the reflected wave, and the half responsible for the shadow is called the interfering wave. If the obstacle is very small compared to the wavelength, the scattered wave propagates out in all directions, and no sharp-edged shadow exists. Notice that, in spite of the various peculiarities of the angular distribution of the intensity, the total scattered intensity turns out to be a fairly smooth function of ka.

4.4.1. Rigid Sphere The intensity of the scattered wave, Is, and the total power scattered, Ps, are: 1 a2 I 1 X Is¼ 2 2 2 ð2mþ1Þð2nþ1Þsindmsindncosðdmdn ÞPm ðcosyÞPn ðcosyÞ ð4:73Þ r k a m;n¼0 1 2 X Ps¼2pa2 I 2 2 ð2mþ1Þsin2 dm k a m¼0

ð4:74Þ

where the phase angles, dm, and amplitudes, Bm, are the same as for a radiating sphere and are determined from: mnm1 ðkaÞ  ðm þ 1Þnmþ1 ðkaÞ ¼ ð2m þ 1ÞBm cosdm

ð4:75Þ

ðm þ 1Þjmþ1 ðkaÞ  mjm1 ðkaÞ ¼ ð2m þ 1ÞBm sindm

ð4:76Þ

where jm (ka) is the spherical Bessel function, and nm (ka) is the spherical Neumann function. Phase angles dm are tabulated in Morse [18].

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4

In the extreme cases of either long wavelength or short wavelength, Equations (4.73) and (4.74) can be simplified to: Is ðka  1Þ ¼

16p4 a6 I ð1  3 cosyÞ2 9l4 r 2

  a2 a2 2 y J 2 ðka sinyÞ Is ðka  1Þ ¼ I 2 þ 2 cot 4r 4r 2 1 Ps ðka  1Þ ¼

112p5 a6 I 9l4

ð4:77Þ

ð4:78Þ

ð4:79Þ

Ps ðka  1Þ ¼ 2pa2 I

ð4:80Þ

scattering cross section

where a is a sphere radius; r and y are polar coordinates; and k ¼ o/c ¼ 2p/l. In the short-wave limit, the total power scattered is that contained in an area of primary beam, equal to twice the cross-section of the sphere. This result is the same as that for the light scattering “extinction paradox” [3]. Half of this sound wave is reflected equally in all directions from the sphere, and the other half is concentrated into a narrow beam, which tends to interfere with the primary beam and cause a shadow. The total scattering cross-section versus frequency for the full ka range is shown in Figure 4.18. It looks like a bell-shaped curve, similar to the absorption losses. Figure 4.19 compares the general theory with the long-wavelength asymptote that is reciprocally proportional to the fourth power of the 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

1

10 ka

FIGURE 4.18 Scattering cross-section for a rigid sphere.

100

167

Acoustic Theory for Particulates

2.6 2.4

long wave limit

2.2 2.0

cross section

1.8 1.6 1.4 1.2 1.0 0.8 general

0.6 0.4 0.2 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ka FIGURE 4.19 Scattering cross-section calculated using the general theory and the long-wave limit.

frequency (Rayleigh solution). It is clear that the long-wavelength solution is valid only up to ka ¼ 0.5.

4.4.2. Rigid Cylinder Consider a plane wave traveling in a direction perpendicular to the axis of an infinitely long cylinder having a radius, a. The first-order approximations for the scattered intensity at long and short wavelengths are, respectively: Is ðka  1Þ ¼

p4 a 4 I ð1  2 coscÞ2 l3 r

  a c 1 2 c Is ðka  1Þ ¼ I sin þ sin2 ðka sincÞ cot r 2 pkr 2

ð4:81Þ

ð4:82Þ

The total power scattered by the cylinder per unit length is obtained by multiplying Is by r, and integrating over C, from 0 to 2p. Ps ¼ 4aI

1 1X em sin2 dm ka m¼0

ð4:83Þ

where the phase angles dm have been defined in connection with the radiation of the cylinder [18].

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4

Thus, the total scattered power for these two extreme cases is given by the following simple expressions: Ps ðka  1Þ ¼

6p5 a4 I l3

Ps ðka  1Þ ¼ 4aI

ð4:84Þ ð4:85Þ

These approximate formulas can be used for illustrating a particle shape effect.

4.4.3. Nonrigid Sphere Consider now the case of a scattering sphere composed of a material that allows the sound wave to propagate through it. To describe this motion, one needs the sphere’s density, rp, and the compressibility, bpp . In this nonrigid case, the sphere may now dissipate the sound by absorption, in addition to scattering it. The total extinction is now not just equal to the scattering alone; there will be a contribution from absorption. The scattering and the absorption can be described separately by using two different cross-sections; Ps for the scattering and Pa for the absorption. For scattering the cross-section is given by: 2

3 2 p p 2 5 6 b  b ðrp  rm Þ 7 m 64p a 6 p I4 þ3 Ps ðka  1Þ ¼ ð4:86Þ 5 4 p2 9l bm ð2rp þ rm Þ2 and for absorption the cross-section is: " p # bp  bpm rp  rm 8p2 a3 Pa ðka  1Þ ¼ I Im þ3 3l bpm 2rp þ rm

ð4:87Þ

It follows, from the Equation (4.87), that the absorption exists only if the compressibility and/or the density of the sphere material is complex. This is another way to characterize intrinsic attenuation by the material of the particle, which had been adopted earlier by Morse [18]. For larger spheres or shorter wavelengths, these approximate formulas are not valid, and the exact formulas from Ref. [18] must be used.

4.4.4. Porous Sphere Finally, consider that the sphere material itself is not homogeneous but porous and can be characterized with a porosity, Oe, and flow resistance, Fe. The material of the pore walls is assumed to be stationary. The effective compressibility and density of the fluid in the pores are given by the approximate expressions:

169

Acoustic Theory for Particulates

bpef ¼ bpp Oe ref ¼ rp þ j

Fe o

ð4:88Þ

In this approximation, the compressibility has no imaginary part, although the inclusion of thermal transfer would indeed produce a small imaginary term. Substitution of the effective compressibility and the effective density in the expressions for scattering and absorption cross-section [18] would then yield corresponding values for this system of porous spheres. When the framework of the pore walls is not held at rest, another correction must be added because of the motion of the solid material. If the pore wall material is effectively incompressible, then the radially symmetric term (m ¼ 0) in the series for the pressure inside the sphere is unaffected by this motion; the term ðbpe  bpm Þ=bpm remains unchanged. However, the (m ¼ 1) term, representing linear motion along the z-axis, is changed because the solid part of the sphere as a whole will move back and forth. The mass of a unit volume of the sphere, exclusive of the fluid contained in pores, is rp (1  Oe). In this case, for the (m ¼ 1) term, the impedance corresponding to the fluid flow can be calculated as for a parallel circuit, with the resistance Fe of the porous material being shunted by the reactance, jorm (1Oe), of the solid material that is forced into motion by the drag of the fluid moving through the pores. In other words, the Fe term must be replaced by  1 1 j þ ð4:89Þ Fe orm ð1  Oe Þ The spherically symmetric term, (m ¼ 0), is unaffected by this approximation, because the velocity is out of phase with the pressure at r ¼ a. The axial component, (m ¼ 1), on the other hand, is changed by the motion of the solid material, both in regard to the phase relationship and the magnitude ratios. As for the rigid and nonrigid sphere cases, these approximate formulas are again not valid for larger spheres or shorter wavelengths.

4.4.5. Scattering by a Group of Particles The solution to the problem of sound scattered by a group of particles becomes much simpler if we address just the attenuation of the incident sound pulse as it propagates over a distance through the colloidal suspension. Whether multiple scattering is important or not, the intensity of the ultrasound pulse that has traversed a distance x decays as an exponential function of x; the attenuation coefficient becomes a scale factor. The total amount of the scattered power is, simply, the product of Ps and the number of particles that the pulse encounters while propagating through this distance x. The resulting attenuation coefficient is:

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4

hneperi 3’P s ð4:90Þ ¼ 4pIa3 cm This simple expression allows us to use expressions for Ps derived by Morse [18] for different particles. The attenuation for nonrigid particles at the longwavelength limit, for instance, is given by: 2 !2 !2 3 rp  r m ’o4 a3 41 rm c2m 5 1 asc ¼ ð4:91Þ þ 2c4m 3 rp c2p 2rp þ rm asc

This is a well-known expression for single scattering, which is identical to the explicit result of the ECAH theory. This expression is valid up to approximately kd  1, or, in terms of the wavelength, for diameters which are less than 1/6 the wavelength (i.e., 1/2p). For large particles, the more general expression for Ps [18] must be used. There is yet one more advantage to the elimination of multiple scattering by measuring the attenuation of the incident beam intensity: the ability to treat polydisperse colloidal suspensions using a superposition assumption. The total attenuation can then be presented as a simple sum of the contributions from each of the different size fractions.

4.4.6. Scattering Coefficient There have been number of experiments indicating that theory of elastic scattering proposed by Morse [18] yields correct frequency dependence of real dispersions and emulsions even when scattering is not pure elastic. We have observed this for glass spheres with variable sizes, see Section 6.4 in the first edition. A similar observation was made by Richter and coworkers [49] for emulsions and suspensions [50] with large particle exceeding 10 mm in size. At the same time, the magnitude of attenuation deviates from elastic theory significantly. According to Richter and others [49, 50], it looks like volume fraction is off. We had similar experience with glass spheres and many other dispersions of large solid particles. One example is shown in Figure 4.20 for glass spheres produced by S. Lindner GmbH with sizes in the range from 40 to 70 mm. These empirical observations suggest that effects associated with the particle’s nonrigidity have the same frequency dependence as for the rigid sphere. Accepting this assumption, we can introduce just a single adjustable parameter that would match theory to the experimental attenuation curve by simple multiplication. We suggest calling this empirical adjustable parameter a “scattering coefficient.” It is possible to calculate the value of scattering coefficient by fitting theory to experimental data, if particle size is known. For instance, in the case

171

Acoustic Theory for Particulates

1.6

experiment

attenuation [dB/cm/MHz]

1.4 1.2 1.0 0.8 0.6 Morse rigid sphere theory

0.4 0.2 0.0 1

10 frequency [MHz]

100

FIGURE 4.20 Experimental and theoretical attenuation for glass spheres produced by S. Lindner GmbH with sizes in range from 40 to 70 mm. Theoretical attenuation is calculated using Morse theory for rigid sphere assuming median particle size 55 mm.

presented on Figure 2.40, the best fit can be achieved if we multiply all theoretical attenuation by a coefficient 1.6. Our experience indicates that “scattering coefficient” is a property of the material of the particle. It can be treated as additional input parameter for a large particle. In the case of attenuation measured from 1 to 100 MHz, this parameter is important for particles with sizes above 3 mm.

4.4.7. Ultrasound Resonance by Air Bubbles Acoustic waves propagating through a fluid that contains air bubbles cause the bubbles to resonate. A resonating air bubble vibrates and then reradiates acoustic energy out of the sound beam. In addition, it creates temperature oscillations which then result in thermal absorption. Viscous absorption by air bubbles is relatively much less important. Scattering by these pulsating bubbles occurs within a frequency range that is defined by a resonance frequency, ocr, defined by: o2cr ¼

3gPst a2 r m

ð4:92Þ

where Pst is the total hydrostatic pressure inside the bubble. For instance, the resonance frequency for bubbles with a diameter of 100 mm is about 60 KHz. In addition to the scattering loss by radiation, there is also absorption of energy because of the high thermal conductivity of the fluid.

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The total mechanical impedance Zbub of such a bubble is: Zbub ¼ ðRsc þ Rab Þ þ jðomr  stif=oÞ

ð4:93Þ

where stif is effective stiffness of the bubble given by: stif ¼ 12pagPst

ð4:94Þ

The scattering resistance, Rsc, and absorption resistance, Rab, are given by: Rsc ¼ 4pa2 rm cm ðkaÞ2

ð4:95Þ

Rab ¼ 6:4  104 pa3 rm o3=2

ð4:96Þ

The corresponding scattering cross-section, Ssc, and extinction cross-section, S, are given by: Ssc ¼

64p3 a6 o2 r2m Z2

ð4:97Þ

16p2 a4 rm cm ðRsc þ Rab Þ ð4:98Þ Z2 For very large bubbles, with a radius of say 650 mm, the extinction crosssection is about 7000 times greater than the geometrical cross-section at the 5 KHz frequency. Despite this enormous extinction, such a large bubble would not affect the attenuation in the ultrasound range because the extinction cross-section decreases rapidly above the resonance frequency, and very quickly approaches the geometrical cross-section. S¼

4.5. ULTRASOUND PROPAGATION IN POROUS MEDIA Ultrasound provides an opportunity to characterize porous solids that are saturated with liquid. Both acoustics and electroacoustics are suitable for this purpose. We dedicate a special chapter in this edition for this task, Chapter 13. Here we just briefly describe theoretical basis of acoustics in porous bodies. The first article on this subject was presented 70 years ago by Frenkel [51], who formulated the basic theoretrical principles, which were later developed by Biot [52, 53]. Biot mentions Frenkel’s name as the first founder of this field in his work. There is a large body of subsequent theoretical and experimental [54–65] literature on ultrasound propagation through various porous systems. The simplest approach uses empirical relationships to define certain acoustic properties. Investigators found that the acoustic impedance, Zemp, of many soils and sands can be described empirically according to the following expressions by a single parameter, namely flow resistivity, Fe:

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Acoustic Theory for Particulates

Zemp ¼ Remp þ jXemp

ð4:99Þ

0:754 Þ Remp ¼ rm cm ð1 þ 0:057Cemp

ð4:100Þ

0:732 Xemp ¼ rm cm 0:057Cemp

ð4:101Þ

Cemp ¼

orm 2pFe

ð4:102Þ

These empirical relationships work satisfactorily for many systems as long as the flow resistivity is greater that 0.01, and less than 1.0. This markedly limits the range of applicability. There are also many instances when these relationships do not work at all. A more elaborate theoretical analysis is possible. There are two different approaches to such a theory: phenomenological and microstructural [55]. The phenomenological approach, first developed by Morse and Ingard [7], treats a porous medium saturated with a liquid as a new “effective medium” (We will use a similar idea later for characterizing complex dispersions). They assumed that the solid part of the porous body is completely rigid and incompressible. This system of interconnected pores is a random network having a total volume fraction, Oe, called the porosity. The propagation of sound causes a motion of the liquid with a mean velocity, v(x,t). The equation of continuity for v is: @drm  rm div v ¼ 0 ð4:103Þ @t This equation is still valid for the fluid flow for sufficiently large volume when the irregularities of the pores average out. The density, rm, is the mass of the fluid occupying the fraction Oe of the space available to the fluid. The variation of the density, drm, is related to the acoustic pressure, P, and the effective compressibility, bpef ðwÞ (which is generally frequency dependent): Oe

drm ¼ bpef ðoÞrm P

ð4:104Þ

The equation for dynamic force balance must include both inertia and drag effects arising from fluid motion in the pores. The effective density, rm, of the moving mass might be larger than the actual density of the liquid because part of the structure might be involved in the motion. The frictional retardation of the flow is expressed in terms of a flow resistivity, Fe (o), which might also be frequency dependent. The force balance equation is thus: @v þ Fe ðoÞv þ gradP ¼ 0 ð4:105Þ @t The equation of state contains the ratio of specific heats gh (o), which also might be frequency dependent. rm

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We stress here the frequency dependencies of these three parameters, (bpef ðwÞ, Fe (o), and gh (o)), because this represents the weakest point of the phenomenological approach. There is no independent way to measure these parameters. They must be determined from the acoustic measurements, and this takes away much of the predictive power of the phenomenological approach. In contrast, the microstructural approach allows one to eliminate illdefined parameters, while paying the high price of losing a certain level of abstraction. This approach requires a specifically defined shape and structure for the pores. It goes back to Lord Rayleigh [1, 2], who suggested a model consisting of a rigid matrix of identical, parallel cylindrical pores. The most widely known theory of this kind is associated with Biot [52, 53], who considered a flexible porous system. These theories introduce a total of five parameters specific for the porous body: porosity, flow resistivity, dynamic and static shape factors, and tortuosity. There is discussion in Chapter 13 regarding the methods of measuring these parameters. There is one interesting version of these theories that links acoustics to dielectric spectroscopy. For this reason, it might be especially interesting for people involved in colloid science. This theory was presented by Johnson, Koplik, and Dashen [54]. They operate with two frequency-dependent parameters: tortuosity ator(o), and permeability kper(o). These parameters are related to the each other by the following equation: ator ðoÞ ¼

jOe kper ðoÞorm

ð4:106Þ

They consider Equation (4.106) as an analog of the relationship between complex dielectric permittivity, es ðwÞ, and the conductivity, Ks ðwÞ, of an effective continuous media: es ¼ j

Ks oe0

ð4:107Þ

This analogy allows them to apply the methods developed for dielectric spectroscopy and to obtain rather general and interesting results for acoustics in porous media. The similarity between acoustics and dielectric spectroscopy is very marked. It even justified combining them together under the one name “relaxation spectroscopy,” following Eigen’s Nobel Prize work for ions in liquids [66–68], described in the following section.

4.6. INPUT PARAMETERS To calculate the particle size from an acoustic spectrum requires information about the materials of which the particle and the dispersion medium are composed. For the most general case, the list of the required parameters is quite extensive (Table 4.1). Fortunately, it is not necessary in each case to know

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Acoustic Theory for Particulates

TABLE 4.1 The Complete Set of Properties for Both Particle and Medium that Affect the Ultrasound Propagation Through a Colloidal Suspension Dispersion medium

Dispersed particle

Units

Density

Density

Kg/m3

Shear viscosity (microscopic)

Pa/s

Sound speed (or compressibility)

Sound speed (or compressibility)

M/s

Intrinsic attenuation

Intrinsic attenuation

dB/cm/MHz

Heat capacity at constant pressure

Heat capacity at constant pressure

J/(kg K)

Thermal conductivity

Thermal conductivity

W/(m K)

Thermal Expansion

Thermal expansion

1/K

Scattering coefficient

dimensionless

all of these parameters. The peculiarity of ultrasound interaction with colloidal particles allows one to quite dramatically reduce this list for a specific colloidal dispersion. For example, submicrometer rigid particles cause only viscous absorption of ultrasound. Thus, there is no need for any thermodynamic parameters of either the particles or the medium; nor is sound speed data required. However, sound speed information does become important for larger, rigid particles that generate ultrasound scattering. The sound speed of most solid materials is usually in the range of 5000–6000 m/s and can be found in the literature, or can be obtained from the Material Database of Dispersion Technology. Submicrometer-sized soft particles, such as latices and microemulsions, require information about the thermodynamic properties: the thermal conductivity, t, the heat capacity, Cp, and the thermal expansion, b. Fortunately, t and Cp are almost identical for all liquids except water. The variation of these parameters for some 100 liquids from Anson and Chivers [22] and the Dispersion Technology Material Database has been given in Chapter 2 (Figures 2.1 and 2.2). Thus, in practical terms, the number of required parameters is reduced to just one—thermal expansion. This parameter plays the same role with regard to “thermal losses” as density plays for “viscous losses.” The thermal expansion is known for many liquids, but it presents a problem for polymer latices because the chemical composition of latex particles, and the exact process of their preparation can dramatically affect the thermal expansion coefficient. Initially, this was a serious obstacle to the use of acoustics to characterize such systems. This is no longer a problem because the thermal expansion coefficient can now be determined from the attenuation

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spectrum. Hence, instead of using it as a given input parameter, it can be treated as another adjustable output parameter. To demonstrate this, we measured a set of five polymer latices that had been intentionally synthesized to have different rheological properties, as characterized by a “hard-to-soft ratio” parameter. The thermal expansion coefficient for these latices, calculated from the measured attenuation spectra, correlates with the measured particle rheology (Figure 4.21). Thus the attenuation spectra of polymer latices contain sufficient information to characterize not only the particle size but also the thermal expansion coefficient. In some cases, particularly for dilute measurements or in viscous media, the intrinsic attenuation of the medium and/or the particles may also be required to obtain precise results. This can easily be determined for the medium by measuring it directly using an acoustic spectrometer. For liquid droplets (emulsions), the intrinsic attenuation is also simple to determine as long as a homogeneous sample of the droplet material is available, as is most often the case. For solid particles, the intrinsic attenuation can almost always be ignored. The sound speed in soft particles becomes important for large sizes when scattering is dominant. Scattering coefficient is important for particles with sizes exceeding 3 mm assuming that the frequency is limited to 100 MHz. This parameter would be important for both soft and rigid large particles. It can be calculated directly from the measured attenuation spectra as an adjustable parameter, either with known particle size or with unknown particle size. The first approach is more reliable. Then, it is saved as a material property and used for studying dispersions or emulsions with this material. 7

Thermal Expansion [10E-4, 1/C]

XU 31183.5

6

5

2765

4

2766

3

2770 PP 755

2 0

1

2

3

4 5 6 Hard to Soft Ratio

7

8

9

10

FIGURE 4.21 Thermal expansion coefficient calculated from the attenuation spectra for a series of latices.

177

Acoustic Theory for Particulates

TABLE 4.2 Input Parameters Required to Interpret Acoustic Data for Particular Colloid Classes Colloid

Properties of the particle

Properties of the liquid

Rigid submicron particles

Density

Density, shear viscosity

Soft submicrometer particles

Thermal expansion (might be calculated from attenuation)

Thermal expansion

Large soft particles

Density, sound speed, scattering coefficient

Density, sound speed

Large rigid particles

Scattering coefficient

None Intrinsic attenuation for all colloids (might be easily measured)

The properties necessary to interpret acoustic data are summarized in Table 4.2. They are necessarily simplifications that are not completely general or universal; they are the results of our experience in dealing with a large number of practical colloidal suspensions over many years. They are empirical, not theoretical, and some exceptions are to be expected. In some cases, more input parameters may be required than indicated here, see for instance Babick et al. [25]. In addition to knowing a specific set of material properties for the particles and the suspending medium, it is also necessary to know the relative amount of both materials. This is usually characterized by a weight fraction of the disperse phase and is normally considered as an input parameter. It is usually quite simple to measure, even if unknown, by using a pycnometer, or by weighing a sample of the suspension before and after drying in an oven. Determination of the weight fraction is even possible when the sample has two separate disperse phases (Chapter 8). There is, however, another approach to determining the weight fraction of a suspension. It is very tempting to try to extract the weight fraction from the acoustic data. For example, Alba has suggested using the weight fraction as an additional adjustable parameter when fitting the attenuation spectra [69]. However, based on our experience, such an approach can be misleading, because there is usually insufficient information in the attenuation spectra to extract data for both the PSD and weight fraction. There are certainly exceptions, but they are very rare.

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The attenuation is not the only source of acoustic data that can be collected with ultrasound-based instruments. Sound speed and acoustic impedance are two other independent experimental parameters that can also be used as a source of weight-fraction information. There are four factors, however, that complicate the calculation of weight fraction, from either the sound speed or the acoustic impedance. First, these two parameters are much more sensitive to variations in temperature than attenuation [70]. For instance, a 1 C temperature change causes a 2.4 m/s variation in the speed of sound in water. This can introduce a large uncertainty into the calculated weight fraction. It is possible, of course, to eliminate this factor by either measuring or stabilizing the temperature during the measurement. The second factor is more difficult to tackle. Sound speed might be very sensitive to small variations of the chemical composition of the liquid, as is shown in Chapter 3. It would be a mistake to assign the variation in sound speed caused by the chemical composition of the sample to the weight fraction of the dispersed phase. The third problem is related to the low sensitivity of sound speed to the solid content at certain volume fraction ranges. For instance, silica (Ludox) exhibits very little variation of the sound speed over the concentration range of 7–15 vol%, as is shown in Figure 4.22. In this case, acoustic impedance is a much better candidate for determining the weight fraction. Acoustic impedance is the product of the sound speed and density. So, even when the sound speed remains practically constant, the acoustic impedance will still change as a function of weight fraction, due to variation of the density. The fourth problem is related to the particle size. Sound speed and acoustic impedance are independent of particle size only for very small or very large particles. In general, both of them exhibit a dependence on particle size 1540 experiment theory

sound speed [m/sec]

1530 1520 1510 1500 1490 1480 0

10

20 volume fraction in %

30

FIGURE 4.22 Measured sound speed for silica (Ludox) at different volume fractions.

40

179

Acoustic Theory for Particulates

at the same frequencies as attenuation. This dependence is much weaker than in the case of attenuation, but still might be sufficient to produce a substantial error in the weight fraction. Overall, the most reliable way to measure particle size using acoustics is to assume that the weight fraction will be used as an input parameter that has been determined independently. Acoustics does provide some means to determine the weight fraction and eliminate it from the list of the required input parameters, but this approach must be used with caution. Weight fraction is not the only one input parameter that can be extracted from the acoustic data under certain conditions. For instance, one can use raw data inversely. Instead of calculating particle size with known input parameters, one can calculate input parameters using known particle size. This method is patented for soft particles and structured dispersions by Dukhin and Goetz [71, 72]. It was independently suggested for submicrometer dense particles by Hipp and others [73]. This inverse method can be especially useful for structured dispersions. Characterization of such systems requires at least one additional parameter for describing rheological properties of interparticle bonds. It is not a property of the material but is rather a property of the sample. This parameter was introduced in the Section 4.3.4—Hookean coefficient. It should be introduced if the default theory of separate particles fails to fit experimental data at high frequency. Excess experimental attenuation at high frequency is an indication that the system is structured. In such a case, the Hookean coefficient is calculated from fitting this excess attenuation with structural losses.

4.7. ESTIMATES OF THE DENSE PARTICLE MOTION PARAMETERS Ultrasound moves dense particles relative to the dispersion medium. It is interesting to know the parameters of this motion, at least some estimate of the orders of magnitude. Such parameters include the particle relative velocity up  um, the amplitude of displacement Dx, and the shear rate g. There is an expression for the relative particle velocity, Equation (4.27): gðup  um Þ ¼

’ðrp  rs Þ rs þ io’ð1  ’Þ

rp rm rP g

where rs ¼ ’rpþ(1’)rm, and ’ is the volume fraction of the solid particles; g ¼ 9’O=2a2 and Fh ¼ 6paOðup  um Þ. This equation can serve a starting point for estimating parameters of the particles viscous motion. First of all, we can neglect complex number for the purpose of simple estimate. Second, we neglect particles interaction by assuming O ¼ 1.

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Third, we assume density contrast equals simply 1. The final simplified equation for estimating particle velocity is following: up  um ¼

d2 rP 18

ð4:108Þ

Gradient of pressure equals approximately pressure amplitude P over onequarter of wavelength l: rP ¼ 4P=l Amplitude of pressure can be calculated from ultrasound intensity I: I¼

P2 2rC

where C is sound speed, assumed to be about 1000 m/s. Acoustic sensor of the Dispersion Technology, Inc. (DTI) instruments launches about 10 mW of ultrasound energy into the sample with a beam size of around 1 cm2. We assume the transducer function at best efficiency at the resonance. This yields the value of the pressure amplitude as 10,000 N/m2. Viscosity of the fluid is approximately 0.001 kg/s m. These numbers and expressions allow us to estimate the main parameters of the dense particles motion in the samples being studied with DTI sensors.

4.7.1. Particle Velocity Taking into account approximate values of all parameters mentioned above, the final equation for estimating particle velocity becomes: d2 6 10 l where size and wavelength must be in SI units, as well as the particle velocity. Range of the particle size is 108–106 m, from 10 nm up to 1 mm. Wavelength changes from roughly 105 to 103 m when frequency changes from 100 to 1 MHz. The low limit of particle velocity corresponds to the smallest particles and largest wavelength. It is 107 m/s, or 0.1 mm/s. This is for a particle 10 nm in size. The highest speed limit is for the largest particles in specified range and shortest wavelength. It is 0.1 m/s. Relative particles velocity varies by 6 orders of magnitude from 107 m/s up to 101 m/s. up  um ¼

181

Acoustic Theory for Particulates

4.7.2. Particle Displacement Amplitude of the particle displacement is particle velocity multiplied by a half of the wave period. This leads to the following simple equation: d2 6 T d2 p d2 10 ¼ 106 ¼ 300d 2 106 ¼ 2p l 2 o 2  1500 C o where the particle size is in meters. This determines the range of the particle displacement as from 1013 m for 10 nm particles up to roughly 1 nm for 1 mm particles. DX ¼ ðup  um ÞT=2 ¼

4.7.3. Shear Rate Shear rate is the derivative of the velocity over the characteristic distance. We can assume that characteristic distance is the particle size. This yields immediately expression for shear rate: d ‵g ¼ ðup  um Þ=d ¼ 106 l where particle size and wavelength are in meters. The low limit of the shear rate corresponds to the smallest particle’s size, 108 m, and largest wavelength 103 m. It is 10 s1. This is for a particle 10 nm in size and frequency 1 MHz. The highest speed limit is for largest particles in specified range 106 m and shortest wavelength 105 m. It is 105 s1. Shear rate range is from 10 to 105 s1.

4.7.4. Quantum Limit for Acoustics Very small displacements, especially for nanoparticles, give rise to the question of justifying the use of classical physics at this seemingly quantum range of distances. It turns out that a similar problem has been discussed with regard to cantilever displacements for the atomic force microscope. Details of the analysis are given in the paper by Milburn and others [74]. They concluded that even cantilever based on optical trap scheme is still well within classical physics. They suggested the expression for estimating the “minimum detectable displacement” that we can apply for estimating the transition to quantum physics in acoustics. This is their expression: ħ d 2min ¼ pffiffiffi 2m p o 2 T

ð4:109Þ

where ħ is Plank constant, mp is the mass of the particles, and o is the frequency of oscillation. For particle with size 10 nm and oscillation frequency 1 MHz, this minimum displacement is about 1014 m. This is at least an order of magnitude

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smaller than the estimated actual displacement of the dense particles of the same size under influence of ultrasound generated by DTI sensors, as shown in Section 4.7.2. These displacements are much smaller than for the atomic force microscope due to very small particle size and rather high frequency. This short analysis indicates that there is no problem using classical physics for describing sound propagation through heterogeneous systems, even with nanosize heterogeneity.

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[45] W.P. Mason, Dispersion and absorption of sound in high polymers, Handbuch der Physik, vol. 2, Acoustica part 1, 1961. [46] C. McCann, Compressional wave attenuation in concentrated clay suspensions, Acustica 22 (1970) 352–356. [47] A.E. Hay, D.G. Mercer, On the theory of sound scattering and viscous absorption in aqueous suspensions at medium and short wavelength, J. Acoust. Soc. Am. 78 (1985) 1761–1771. [48] P.H. Rogers, A.O. Williams, Acoustic field of circular plane piston in limits of short wavelength or large radius, J. Acoust. Soc. Am. 52 (1972) 865–870. [49] A. Richter, T. Voight, S. Ripperger, Ultrasonic attenuation spectroscopy of emulsions with droplet size greater than 10 microns, J. Colloid Interface Sci. 315 (2007) 482–492. [50] A. Richter, F. Babick, S. Ripperger, Polydisperse particles characterization by Ultrasonic attenuation spectroscopy for systems of diverse acoustic contrast in the large particle limit, J. Acoust. Soc. Am. 118 (2005) 1394–1405. [51] J. Frenkel, On the theory of seismic and seismoelectric phenomena in a moist soil, J. Phys. USSR 3 (1944) 230–241, Republished J. Eng. Mech. (2005). [52] M.A. Biot, Theory of propagation of elastic waves in a fluid saturated porous solid. I. Low frequency range, J. Appl. Phys. 26 (1955) 168–179. [53] M.A. Biot, Theory of propagation of elastic waves in a fluid saturated porous solid. I. High frequency range, J. Appl. Phys. 26 (1955) 179–191. [54] D. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluid saturated porous media, J. Fluid Mech. 176 (1987) 379–402. [55] K. Attenborough, Acoustical characterization of rigid fibrous absorbents and granular materials, J. Acoust. Soc. Am. 73 (1983) 785–799. [56] G. Shumway, Sound speed and absorption studies of marine sediments by a resonance method, Geophysics 25 (1960) 659–682. [57] A.B. Wood, D.E. Weston, The propagation of sound in mud, Acustica 14 (1964) 156–162. [58] J. Hoverm, Viscous attenuation of sound in suspensions and high porosity marine sediments, J. Acoust. Soc. Am. 67 (1980) 1559–1563. [59] P. Vidmar, T. Foreman, A plane-wave reflection loss model including sediment rigidity, J. Acoust. Soc. Am. 66 (1979) 1830–1835. [60] R. Stoll, Theoretical aspects of sound transmission in sediments, J. Acoust. Soc. Am. 68 (1980) 1341–1350. [61] J. Hoverm, G. Ingram, Viscous attenuation of sound in saturated sand, J. Acoust. Soc. Am. 66 (1979) 1807–1812. [62] L. Hampton, Acoustic properties of sediments, J. Acoust. Soc. Am. 42 (1967) 882–890. [63] E.G. McLeroy, A. DeLoach, Sound speed and attenuation from 15 to 1500 KHz measured in natural sea-floor sediments, J. Acoust. Soc. Am. 44 (1968) 1148–1150. [64] K.V. Mackenzie, Reflection of sound from coastal bottoms, J. Acoust. Soc. Am. 32 (1960) 221–231. [65] S.R. Murpy, G.R. Garrison, D.S. Potter, Sound absorption at 50 to 500 KHz from transmission measurements in the sea, J. Acoust. Soc. Am. 30 (1958) 871–875. [66] M. Eigen, Determination of general and specific ionic interactions in solution, Faraday Soc. Discuss. 24 (1957) 25. [67] M. Eigen, L. deMaeyer, in: A. Weissberger (Ed.), Techniques of Organic Chemistry, vol. VIII, Part 2, Wiley, New York, 1963, p. 895. [68] L. De Maeyer, M. Eigen, J. Suarez, Dielectric dispersion and chemical relaxation, J. Am. Chem. Soc. 90 (1968) 3157–3161.

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[69] F. Alba, Method and apparatus for determining particle size distribution and concentration in a suspension using ultrasonics, US Patent No. 5121629 (1992). [70] R. Chanamai, J.N. Coupland, D.J. McClements, Effect of temperature on the ultrasonic properties of oil-in-water emulsions, Colloids Surf. 139 (1998) 241–250. [71] A.S. Dukhin, P.J. Goetz, Method for determining particle size distribution and structural properties of concentrated dispersions, patent USA, 6,910,367, B1 (2005). [72] A.S. Dukhin, P.J. Goetz, Method for determining particle size distribution and mechanical properties of soft particles in liquids, patent USA, 6,487,894 (2002). [73] A.K. Hipp, G. Storti, M. Morbidelli, Particle sizing in colloidal dispersions by ultrasound. Model calibration and sensitivity analysis, Langmuir 15 (1999) 2338–2345. [74] G.J. Milburn, K. Jacobs, D.F. Walls, Quantum-limited measurements with the atomic force microscope, Phys. Rev. A 50 (1994) 5256–5263. [75] J.C.F. Chow, Attenuation of acoustic waves in dilute emulsions and suspensions, J. Acoust. Soc. Am. 36 (12) (1964) 2395–2401. [76] P.A. Allison, E.G. Richardson, The propagation of ultrasonics in suspensions of liquid globules in another liquid, Proc. Phys. Soc. 72 (1958) 833–840. [77] V.C. Anderson, Sound attenuation from a fluid sphere, J. Acoust. Soc. Am. 22 (1950) 426–431.

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Chapter 5

Electroacoustic Theory 5.1. The Theory of IVP 191 5.2. The Low-Frequency Electroacoustic Limit: Smoluchowski Limit (SDEL) 193 5.3. The O’Brien Theory 195 5.4. The Colloid Vibration Current in Concentrated Systems 198 5.4.1. CVI and Sedimentation Current 200 5.4.2. CVI for Polydisperse Systems 204 5.4.3. Surface Conductivity 206 5.4.4. Maxwell-Wagner Relaxation: Extended Frequency Range 207 5.4.5. Water-in-Oil Emulsions, Conducting Particles 208

5.5. Qualitative Analysis of Colloid Vibration Current 5.6. Electroacoustic Theory for Concentrated Colloids with Overlapped DLs at Arbitrary ka: Application to Nanocolloids and Nonaqueous Colloids 5.6.1. The Homogeneous Model 5.6.2. High-Frequency Model for Overlapped DLs 5.6.3. Theoretical Predictions of Both Models 5.7. Electroacoustics in Porous Body References

210

213 217

220

222 228 234

There are two significant changes in this chapter compared to the first edition. First of all we rediscovered obscure old articles on electroacoustic phenomena in a porous body. These articles describe electroacoustic effects that have been completely unknown in colloid and interface science. These are “electroseismic potential/current” and “seismoelectric potential/current.” We describe them briefly below and then in some detail in Section 5.7. Secondly, we have developed a theory of electroacoustics for nonaqueous systems and nanoparticulates with overlapped double layers (DLs), discussed in Section 5.6. A major contribution to both new theoretical developments was made by Professor V.N. Shilov from the Ukrainian Academy of Sciences. Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23005-3 Copyright # 2010, Elsevier B.V. All rights reserved.

187

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Electroacoustic phenomena, first predicted by Debye in 1933 [1] for electrolytes, arise from coupling between acoustic and electric fields. Debye realized that, in the presence of a longitudinal sound wave, any differences in the effective mass or friction coefficient between anions and cations would result in different displacement amplitudes. In turn, this difference in displacement would create an alternating electric potential between points within the solution. Indeed, this phenomenon is measurable and can yield useful information about the properties of ions. It is usually referred to as an ion vibration potential (IVP). However, the first experimentally recorded electroacoustic effect was observed not in an electrolyte but instead in a porous body saturated with electrolyte solution. It occurred in 1940 in (the then) USSR when Ivanov observed the generation of an electric field caused by a sound wave propagating through wet soil after an explosion [2]. The first and so far the most adequate theory of this effect was developed 4 years later by Frenkel [3]. They called this effect seismoelectric current. There is the possibility of a reverse symmetrical effect when an electric field is the driving force generating a sound wave in a porous body. This effect is known as electroseismic current. The same phenomenon was observed independently in 1948 by Williams [4] and then 60 years later by A. Dukhin in dispersion of carbon nanotubes. Williams suggested the term electrokinetic transducer, whereas Dukhin used the term streaming vibration current, stressing the similarity with classical streaming current first recognized by Frenkel. We will use the term seismoelectric current following the original work by Frenkel. The first experimental observations of this ion vibration current (IVC) predicted by Debye were reported by Yeager [5] in 1949 and by Derouet and Denizot [6] in 1951. A thorough theoretical treatment of the IVP phenomenon was given in 1947 by Bugosh et al. [7]. There was much interest in this effect in the 1950s and 1960s, because it was considered to be a very promising tool for characterizing ion solvation [5–15]. Zana and Yeager [15] summarized the results of this two-decade effort in a review, which we follow here, in our presentation of IVP in this book. Sadly, this phenomenon has been largely forgotten, and virtually all of the interest in electroacoustic phenomena has shifted from pure electrolyte solutions to colloids. Hermans [16] and Rutgers [17] in 1938 were the first to report a colloid vibration potential (CVP). Since that time, there have been several hundred experimental and theoretical works published. For brevity, we mention here only a few key papers. Enderby and Booth [18, 19] developed the first theory for CVP in the early 1950s. The first quantitative experiments were made in 1960 by Yeager’s group [15]. In the early 1980s, Cannon and co-authors [20] discovered an inverse electroacoustic effect, which they termed electrosonic amplitude (ESA). The first commercially available electroacoustic instrument was developed by Pen Kem, Inc. [21]. There are now several commercially available instruments based on both electroacoustic effects. These are manufactured by Colloidal Dynamics, Dispersion Technology, and Matec.

189

Electroacoustic Theory

After the basic works by Enderby and Booth [18, 19], the electroacoustic theory for colloids has been developing in two quite different directions. The original Enderby-Booth theory was very complex as a result of considering both surface conductivity and low-frequency effects. In addition, it did not take particle-particle interactions into account, and consequently was valid only for dilute colloids. Hence, the Enderby/Booth theory required modifications and simplifications. The first extension of the Enderby-Booth approach was performed by Marlow, Fairhurst, and Pendse [22]. They attempted to generalize it for concentrated systems using a Levine cell model [23]. This approach leads to somewhat complicated mathematical formulas, and perhaps this was the reason why it was abandoned. An alternative approach to electroacoustic theory was later suggested by O’Brien [24]. He introduced the concept of a dynamic electrophoretic mobility, md, and suggested a simple relationship between this parameter and measured electroacoustic parameters, such as colloid vibration current (CVI) or ESA. Later, O’Brien stated that his relationship is valid for concentrated as well as dilute systems [25]. According to the O’Brien relationship, the average dynamic electrophoretic mobility, md, is defined as: md ¼

ESAðoÞrm AðoÞFðZT , Zs Þ’ðrp  rm Þ

ð5:1Þ

where the ESA is normalized by the applied external electric field, A (o) is an instrument constant found by calibration, and F(ZT, Zs) is a function of the acoustic impedances of the transducer and the dispersion under investigation. A similar expression can be used for the CVI mode: md ¼

CVIðoÞrm AðoÞFðZT , Zs Þ’ðrp  rm Þ

ð5:2Þ

Here the CVI is assumed to be normalized by the gradient of pressure (grad P), which, in the case of CVI, is the external driving force. According to O’Brien, a complete functional dependence of ESA (CVI) on key parameters, such as z-potential, particle size, and frequency, is incorporated into the dynamic electrophoretic mobility. O’Brien stated that for all considered cases the coefficient of proportionality between ESA (CVI) and md is frequency-independent, and, in addition, is independent of particle size and z-potential. This feature made the dynamic electrophoretic mobility an important and central parameter of the electroacoustic theory. The first theory that relates this dynamic electrophoretic mobility to the properties of the dispersion medium and dispersed particles was initially formulated by O’Brien, but it neglected particle-particle interactions and was therefore limited to the dilute case. We shall call this version the dilute O’Brien theory. Later, Rider and O’Brien [26], Ohshima [27], and Ennis,

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Shugai, and Carnie [28, 29] suggested modifications to extend this approach to concentrated systems. Because it appeared to yield a desirable electroacoustic theory for the concentrated case, the O’Brien approach appeared superior to the EnderbyBooth approach. However, one important question remained unanswered. In principle, these two approaches must give the same result, but it had not been clear if this was indeed the case. It was obvious that such a comparison needed to be done. It would provide strong support for both theories if the two approaches merged. The situation is even more complicated because O’Brien’s approach has been developed mostly for ESA, whereas the Enderby-Booth approach was primarily used to explain the (inverse) effect of CVP or CVI. For dilute systems, this difference does not present any problem. However, it does for concentrates, because the inertial frame of references can be different for the ESA and CVI effects. All of these issues have been addressed recently by an international group of scientists: Shilov from Ukraine, Ohshima from Japan, and Dukhin and Goetz from USA [30–32]. This group has generalized the Enderby-Booth approach to the point that it should now be equivalent to the latest version of O’Brien’s approach. What is of utmost importance is that the group has also developed an electroacoustic analog of Smoluchowski’s equation. As described earlier (Chapter 2), Smoluchowski’s equation is known to be valid for any concentration and any particle shape, provided that the DL is thin ðka  1Þ and the surface conductivity is negligible ðDu  1Þ. By making these same two assumptions, this group derived a general theory for electroacoustics that is valid for any concentration or particle shape. The theory requires no other assumptions. We shall call this theory the Smoluchowski dynamic electroacoustic limit (SDEL). The SDEL is a low-frequency asymptotic solution and is a natural test for all possible electroacoustic theories. Hence, any other proposed theory must reduce to the SDEL under the conditions of ka  1 and Du  1, at sufficiently low frequencies. The SDEL serves as a verification criterion for any proposed electroacoustic theory in the same manner that Smoluchowski’s equation serves for electrophoretic theories. Later Shilov and others further generalized the CVI theory to include particle inertia, surface conductivity, Maxwell-Wagner DL relaxation, thermodiffusion, and barodiffusion [30–33]. This modern CVI theory satisfies the SDEL. We are not aware of any direct comparison between this SDEL and any theory based on the O’Brien approach. Experimental tests by Ennis, Shugai, and Carnie [28, 29] of the latest theory for the dynamic mobility indicate a good correlation even at very high volume fractions for a thin DL. This strongly suggests that the theory may be in compliance with the SDEL; otherwise one would not expect such a good correlation with experiment. In summary, the connection between the modern version of the O’Brien approach to ESA theory and modern versions of the Enderby-Booth approach

191

Electroacoustic Theory

to CVI theory remains uncertain. Although both are able to independently fit corresponding experimental data, the path of theoretical affinity between the two approaches has yet to be resolved. We emphasize here the CVI theory for the following reasons: l

l

l

It is known that in the radio frequency domain electric fields might affect the surface properties and cause strange kinetic and thermodynamic effects [34, 35], including oscillation of the z-potential. The use of ultrasound as the driving force is therefore a better choice than an electric field as required in the ESA method. When using ultrasound, there is an opportunity to calibrate absolute power using internal reflection inside the transducer. This calibration procedure will be discussed in detail in Chapter 7. The ESA method does not offer this option. As a result, the ESA calibration is more difficult and less reliable compared to CVI. The frame of references is well defined for CVI, but problematic for ESA.

We suggest that those readers who are interested in the details of the electroacoustic theory of ESA read the original papers [25, 36–42] or a review by Hunter [43].

5.1. THE THEORY OF IVP Bugosh et al. [7] developed the original IVC theory in 1947. The basis of this theory is a system of (2N þ 1) equations used to characterize the electrodiffusion effects that occur when a plane longitudinal ultrasound wave propagates through a solution of N electrolyte species. For a wave traveling in the x direction, this can be described as follows: ezi E  gi ðui  um Þ  electric

friction

ezi qkE pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3em kT 1 þ qð1 þ jo=oMW Þ relaxation

ezi kE kT@ni @r dui  v i rm m ¼ m i   ni @x @t dt 6p electrophoretic

diffusion

pressure

ð5:3Þ

reaction

@ni @ðni ui Þ þ ¼0 @t @x

ð5:4Þ

@E 4pe X  ni zi ¼ 0 @x em

ð5:5Þ

where e is the charge of an electron, k is the Boltzman constant, n is the ion concentration, the index i specifies the ion species, z is the ion valency, u is the ion velocity, E is the electric field strength, v and m are the volume and

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5

mass, respectively, of the solvated ion, and q is a parameter of the relaxation force (calculated by Bugosh for an electrolyte of two ion species). Herman’s [16] analysis of Equation (5.3) showed that the contributions of the electrophoretic, relaxation, and diffusion terms are negligible. The final expression for a simple binary electrolyte normalizes the IVP by dividing it with the oscillation velocity amplitude Um:   IVP cm tþ W þ t W  1 ¼  ð5:6Þ 1=2 NA e zþ z  o2 Um þ1 o2MW where NA is the Avogadro number, z are the valencies of the anion and cation, cm is the sound speed in the medium, t are the transport numbers of the cation and anion, W are the apparent molar mass of the anion and cation, o is the radian frequency of the acoustic field, and oMW is the Maxwell-Wagner frequency given by: Km ð5:7Þ oMW ¼ k2 Deff ¼ e0 em where Km is the conductivity of the medium. The apparent molar mass of an ion is given by: W  ¼ NA ðm  rm v Þ

ð5:8Þ

and t ¼

K0 Kþ0 þ K0

where K0 are the limiting conductances of the cations or anions and v are the effective volumes occupied by cation and anion, correspondingly. The IVI depends on the concentration of electrolyte because of changes in the Maxwell-Wagner frequency, and this is proportional to the medium conductivity Km. The IVI is a linear function of concentration at frequencies that sufficiently exceed the Maxwell-Wagner frequency. At low frequency, the IVI becomes independent of the electrolyte concentration. The new theoretical analysis of IVC has been performed by Ohshima in 2005 [44]. He derived the following equation, which differs somewhat from Equation (5.6): !  N IVI X zi en1 mi  r0 Vi i ¼ ð5:9Þ gj r0 DP i¼1 where g is drag coefficient of ions. He provides some estimates of IVI in comparison with Equation (5.6). Similar aspects of the electroacoustic phenomena in electrolytes were discussed by Weinmann in the late 1950s [45, 46].

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Electroacoustic Theory

Zana and Yeager [15] provide an interesting discussion of the physical meaning of the apparent molar masses W. Only those solvent molecules whose volume, due to the electrostriction effect, differs from that in the bulk contribute to the apparent molar mass. Solvent molecules that are part of the solvation shell, without appreciable electrostriction or difference in packing, contribute equally to the terms mi and vi in Equation (5.6), and their contributions cancel each other. This makes the IVP/IVI method a unique tool for collecting information about the solvation shell of ions. For a more detailed theoretical analysis, and a large volume of experimental data, we suggest that the interested reader go to the original review. Chapter 11 of this book presents the most important aspects of that review.

5.2. THE LOW-FREQUENCY ELECTROACOUSTIC LIMIT: SMOLUCHOWSKI LIMIT (SDEL) There is a way to derive an expression for CVI using a well-known Onsager reciprocal relationship [47–49]. This relationship is certainly valid in the stationary case, but much less is known about its validity for alternating fields. This uncertainty cautions us to use the relationship only in the limiting case of very low frequencies, or at least frequencies much lower than both the characteristic hydrodynamic frequency, ohd, and the electrodynamic MaxwellWagner frequency, oMW. It thus follows that: o  ohd ¼

 r m a2

ð5:10Þ

Km e0 em

ð5:11Þ

o  oMW ¼

where Km is the electric conductivity of the medium, and e0 and em are dielectric permittivities of the vacuum and the medium, respectively. The Onsager relationship provides the following link between the quasistationary streaming potential CVP, the effective pressure gradient, ▽Prel, which moves the liquid relative to the particles, the electroosmotic current, hIi, and the electroosmotic flow, hVi: hVi hCVPo!0 i ¼ hIi hrPi¼0 hrPrel i hIi¼0

ð5:12Þ

Let us use the Onsager relationship in consideration of the CVP effect for a macroscopically small element of the suspension’s volume (i.e., an element that is small with respect to the length of the ultrasonic wave but that contains the majority of particles). To use Equation (5.12), we need to know the effective gradient of pressurehrPrel i, which provides the velocity of liquid passing around the particles in a vibrated element of the suspension’s volume.

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5

This value can be easily obtained following the coupled-phase model (see Section 4.2.1), which characterizes the particle’s motion in a sound field in concentrated colloids. In the quasistationary (low-frequency) case, the effective gradient of pressure is equal to the specific (per unit volume of suspension) friction force exerted on the particles, which is g(up  um). This force is a part of the pressure gradient that moves the particles relative to the liquid. In the extreme case of low frequencies, Equation (4.27) leads to the following expression for this effective pressure gradient: rPo!0 ¼ rel

’ðrp  rs Þ rP rs

ð5:13Þ

In addition, we can use the fact that the expression on the left-hand side of Equation (5.12) is the electrophoretic mobility, m, divided by the conductivity of the system, Ks. As a result, we obtain the following expression for CVI: CVIo!0 ¼ CVP  Ks ¼ m

’ðrp  rs Þ rP rs

ð5:14Þ

This expression specifies the colloid cibration Current at the lowfrequency limit. This means that m is the usual stationary case electrophoretic mobility. We can thus use the Smoluchowski law for electrophoresis [50] in the form that is valid for concentrated systems: m¼

em e0 z Ks  Km

ð5:15Þ

Here we used two conditions, Equations (2.54) and (2.55), that restrict the applicability of Smoluchowski law. As a result, the asymptotic value of the CVI at low frequency is: CVIo!0 em e0 z’Ks ðrp  rs Þ ¼ rs rP Km

ð5:16Þ

This yields the following asymptotic value for the dynamic electrophoretic mobility that is defined according to the O’Brien relationship (Equation 5.2): md ¼

em e0 z Ks ðrp  rs Þrm  Km ðrp  rm Þrs

ð5:17Þ

The density-dependent multiplier appears because, according to O’Brien’s relationship, the dynamic electrophoretic mobility is defined in relationship to the density contrast between the particle and the medium. This multiplier disappears in dilute systems, but is very important in concentrates, since it conveys the additional volume fraction dependence. Equation (5.17) is very important because it provides a test for the electroacoustic theory. The theory is supposed to be valid for small particles with a thin DL ðka  1Þ and negligible surface conductivity ðDu  1Þ. If these

195

Electroacoustic Theory

three conditions are met, Equation (5.17) is then valid for any volume fraction, and any particle shape and particle size. This makes it analogous to the Smoluchowski theory of microelectrophoresis. There should be a similar equation for ESA, perhaps with a somewhat different density and volume fraction dependence, reflecting the difference in the inertial frame of reference between CVI and ESA, but as yet there is none.

5.3. THE O’BRIEN THEORY O’Brien’s theory was a substantial step forward in our understanding of the electroacoustic phenomena. He provided a link between electroacoustics and classical electrokinetics by introducing the concept of the dynamic electrophoretic mobility, md, through a special relationship between this parameter and an experimentally measured electroacoustic signal: ESAðCVIÞ ¼ AðoÞFðZT , Zs Þ’

rp  rm md rm

ð5:18Þ

The electroacoustic signals (ESA or CVI) in this equation are normalized by the corresponding driving force (E or gradP). This first theory for the dynamic electrophoretic mobility was derived assuming a thin DL and employing the long-wave restriction for frequency. The theory does not take into account particle-particle interactions, and is thus valid only in dilute dispersions. md ¼

2e0 em z GðsÞð1 þ FðDu, o0 ÞÞ 3

ð5:19Þ

where GðsÞ ¼

GðsÞ ¼

1 þ ð1 þ jÞs   for thin DL, ka  1 rp  r m 2s2 3þ2 1 þ ð1 þ jÞs þ j 9 rm

ð5:20Þ

1 ez   for thick DL, ka  1 and < 1 ð5:21Þ rp kT 2s2 1 þ ð1  jÞs þ j 1þ 2 9 rm

  ep ð1  2DuÞþ jo0 1  em   for thin DL ka  1 FðDu, o0 Þ ¼ ep 0 2ð1þ DuÞþ jo 2 þ em s2 ¼

a2 orm o , o0 ¼ 2m oMW

ð5:22Þ

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5

One can see that there are two factors that determine the frequency dependence of the dynamic electrophoretic mobility. The factor G reflects the frequency dependence related to the inertia effects, whereas factor F represents the influence of the Maxwell-Wagner polarization of the DL [51]. Neither of these frequency dependencies is important at low frequency: in the limit 1  2Du when the G and F factors equate to 1 and , respectively. The value 2 þ 2Du of these factors deviates from the low-frequency limit when the frequency approaches some critical value. For the factor G, this critical frequency is ohd, and for the factor F, it is oMW. For aqueous colloids, the inertial factor G plays a more important role than the DL polarization factor F. The inertial factor dramatically reduces the magnitude of the dynamic mobility of larger particles and high frequencies (Figure 5.1). In addition to reducing the amplitude, the inertial factor also causes a lag in the particle motion relative to the external driving force, and this interposes a phase shift on the dynamic mobility. This phase shift reaches a maximum value of 45 at the high-frequency limit. This phase dependence on particle size can be used to determine particle size and is the basis of the method used in the AcoustoSizer by Colloidal Dynamics, Inc. The initial dilute case version of the O’Brien theory was expanded later by Loewenberg and O’Brien [52] by considering the effect of particle shape on the dynamic mobility.

2.0

Dynamic Mobility 10E8 [mm/sec V]

1.8 O'Brien, dilute CVI, monodisperse CVI, stdev=0.2 CVI, stdev=0.5 CVI, stdev=0.7

1.6 1.4 1.2 1.0 0.8 0.6

size 1 micron, density medium 1, density particle 2, vfr 20% thin DL, Du<<1, no Maxwell-Vagner polarization

0.4 0.2 10-1

100

101

frequency [MHz]

FIGURE 5.1 Frequency dependence of the dynamic electrophoretic mobility of colloids.

102

197

Electroacoustic Theory

After O’Brien’s dilute case theory, the main effort in the field of theoretical electroacoustics has been concentrated on incorporating particle-particle interactions into the theory and expanding it to high volume fractions. There are four different approaches within the framework of the O’Brien theory that have been tried: the cell model [38], particle pair interaction [41], the Percus-Yevick approximation [53], and effective media [28, 29]. The first approach using the cell model is considered unsuccessful [43]. The second approach taken by Rider and O’Brien [41] allowed them to expand the volume concentration limit up to 10%. Application of the Percus-Yevick approximation allowed O’Brien to develop a theory for emulsions [53] for volume concentrations up to 60%. In this case, the dynamic electrophoretic mobility is now given by:

md ¼

e0 e m z 



1

H

 ep 1  ’ 1 þ 2Du þ jo0 em 2 þ ’

2s2j s2j Þ



ð2 þ ’Þð3 þ 3sj þ rp  rm 1þ’ HF rm

3’ F 2þ’

ð5:23Þ where sj ¼ ð1 þ jÞs H¼

3 þ 3sj þ s2j   s2j r 3 þ 3sj þ 1þ2 s 3 rm

F¼

J¼

1 2 þ 4sj þ 8ℓ2sj s2j J þ 3 3 þ 3sj þ s2j ð1

½gðrÞ  1rℓ2sj r dr

ð5:24Þ

ð5:25Þ

ð5:26Þ

1

where r is the distance between near-neighbors, and the function, g, is the pair distribution function. Finally, the effective media approach was taken by Ennis, Shugai, and Carnie [28, 29]. They managed to fit electroacoustic spectra even at very high solid loadings up to 40 vol% for dense particles. However, the present authors are encouraged to revive the Enderby-Booth theoretical approach based on six unresolved issues, namely: l

All these recent developments of O’Brien’s theory are for the ESA mode, and it is not yet clear if they are valid for CVI;

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CHAPTER

l

l

l

l

l

5

O’Brien’s approach applies the superposition principle for calculating the electroacoustic signal of polydisperse colloids. This creates an internal contradiction in the theory since superposition does not work when particles interact by definition; There is no definition of the frame of reference. This becomes important since ESA and CVI might require a different inertial frame of reference; O’Brien’s theory completely neglects thermodynamic effects, even for emulsions, and it is not yet clear why thermodynamics is the dominant loss mechanism in the attenuation spectra for soft particles, yet negligible in computing the electroacoustic effects, for both ESA and CVI; It is not clear if these theories, as they are, solely based on O’Brien’s theoretical base, satisfy the low-frequency Smoluchowski electroacoustic limit (Equation 5.17); and It is not clear why the cell model appears to fail in electroacoustics, while being very successful when applied in traditional stationary hydrodynamics and electrokinetics.

The following section presents a new electroacoustic theory for CVI that does not use O’Brien’s relationship. Instead it derives expressions for the electroacoustic signal from first principles.

5.4. THE COLLOID VIBRATION CURRENT IN CONCENTRATED SYSTEMS A new theory for the colloid vibration current has been formulated in close collaboration with Professor Vladimir Shilov, which would have been impossible without his major contribution. We retain the O’Brien expression for introducing dynamic mobility as follows: CVI ¼ AðoÞFðZT, Zs Þ’

rp  rm md rP rm

ð5:27Þ

We also retain the same structure for the dynamic electrophoretic mobility expression, presenting as separate multipliers both the inertial effects (function G) and the electrodynamic effects (function 1 þ F). However, in contrast to the dilute case expressed by Equations (5.20) and (5.21), functions G and F for concentrated systems depend on the particle concentration. There is also ðrp  rs Þrm , which is equal to an additional density-dependent multiplier, ðrp  rm Þrs the ratio of the particle velocity relative to the liquid, and to the particle velocity relative to the center of mass of the dispersion. The convenience of the introduction of such a multiplier, which differs from unity only for

199

Electroacoustic Theory

concentrated suspensions, follows from the exact structure of Smoluchowski’s asymptotic solution for md, given by Equation (5.17). The corresponding equation, which in a convenient way reflects simultaneously limiting transformations both to Smoluchowski’s asymptotic solution (Equation 5.17) and to O’Brien’s asymptotic solution (Equation 5.19), is given as follows: md ¼

2e0 em zðrp  rs Þrm Gðs, ’Þð1 þ FðDu, o0 , ’ÞÞ 3ðrp  rm Þrs

ð5:28Þ

The generalization for the case of polydisperse systems is given by: md ¼

N 2e0 em zðrp  rs Þrm X Gi ðsi , ’Þð1 þ Fi ðDui , o0 , ’ÞÞ 3ðrp  rm Þrs i¼1

ð5:29Þ

The new values of the functions G and F are given by the following equations: 9’i hðsi Þrs    3Hi 4j’ð1  ’Þsi Iðsi Þ rp  rm þ1 2Ii   Gi ðs, ’Þ ¼ 3Hi ’i þ1 N rp X 2Ii   1 3Hi 1  ’ i¼1 rp  rm þ1 2Ii

ð5:30Þ

  ep ð1  ’Þ ð1  2Dui Þð1  ’Þ þ jo 1  em    Fi ðDui , o0 , ’Þ ¼ ep ep 2ð1 þ Dui þ ’ð0:5  Dui ÞÞ þ jo0 2 þ þ ’ 1  em em 0

ð5:31Þ o Km , oMW ¼ , ’i and Dui ¼ ks/Km ai are the 2m e0 em oMW volume fraction and Dukhin number for the ith fraction of the polydisperse colloid, correspondingly, and ’ is the total volume fraction of disperse particles. Special functions H and I are given below. These expressions are restricted to the case of a thin DL and are valid for a broad frequency range, including the Maxwell-Wagner relaxation range. They take into account both hydrodynamic and electrodynamic particle interaction, and are valid for polydisperse systems without making any superposition assumption. The remaining text in this section provides the more interested reader with a step-by-step derivation and justification of the above, perhaps otherwise daunting, expressions.

where s2i ¼

a2i orm

; o0 ¼

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5

5.4.1. CVI and Sedimentation Current There is a simple analogy that helps to create a clearer picture for illustrating the physical nature of the CVP or colloid vibration current. This analogy stresses the similarity between electroacoustics and sedimentation. Sedimentation current is well known from classical colloid chemistry textbooks [54]. Simply put, charged particles sediment because of gravity, and will develop a sedimentation potential between two vertically spaced electrodes. If we externally short-circuit these electrodes, the current that flows is referred to as the sedimentation current. We can extend this simple concept to include colloid vibration current by simply replacing the acceleration due to gravity with an analogous acceleration caused by the applied acoustic field. Figure 5.2 illustrates a particle having a DL and moving relative to the liquid. This motion includes the counterions of the DL. In this example of a negative particle, we need consider only the positive counterions opposing the negative charged particle surface. The hydrodynamic surface current, Is, reduces the number of positive ions near the right-side particle pole and enriches the DL with extra ions near the left-side pole. As a result, the DL shifts from the original equilibrium. A negative surface charge dominates at the right pole, whereas an extra positive diffuse charge dominates at the left pole. The net result is that the particle motion has an induced dipole moment. This induced dipole moment generates an electric field usually referred to as the CVP. This CVP is external to the particle DL. It affects ions in the bulk of the electroneutral solution beyond the DL, and generates an electric current, In. This electric current serves a very important purpose. It compensates for the surface current, Is, and makes the whole system self-consistent. The next step is to add a quantitative description to this simple qualitative picture. To do this, we need a relationship between the CVI and (up  um) as

+ −

+ −

+ + excess plus

Is

−

+

−

+ −

− Is

In

− − +

Up-Um

+

excess minus In

dipole electric field E FIGURE 5.2 Polarization of the particle double layer with the relative liquid flow.

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Electroacoustic Theory

defined in Section 4.2.1. This problem is solved using the Shilov-Zharkikh cell model [55]. The advantages of this cell model over the Levine cell model [23] have been given in Refs [30, 32]. According to both the Shilov-Zharkikh and Levine cell models, the macroscopic electric current density hIi, local electric current density I, and the local gradient of electric potential rf are related by the following expression: In ðr ¼ bÞ 1 @f CVI ¼ hIi ¼ ð5:32Þ ¼ Km cosy cosy @r r¼b Under these conditions, the CVI is identical to the macroscopic electric current density hIi. We start our derivation of the CVI using two simplifications. First, we assume a thin DL; thus: ka  1

ð5:33Þ s

Second, we assume a negligible surface conductivity, k . This requires the Dukhin number to be sufficiently small: Du ¼

ks 1 Km a

ð5:34Þ

This initial simplification will be removed later (Section 5.4.3) but is convenient for the present description. The thin DL simplification allows us to describe the distribution of the electric potential f using the Laplace equation: Df ¼ 0

ð5:35Þ

The general spherically symmetric solution of this equation yields: d cosy ð5:36Þ r2 Equation (5.36) contains only two unknown constants E and d. Two boundary conditions are required for their calculation, that of the particle surface and that of the cell. The particle surface boundary condition reflects continuity of the bulk current, I ¼ Km rf and surface current, Iy, and is defined by: ’ ¼ Er cosy þ

Km rn ’ ¼ divs Iy

ð5:37Þ

There is only one essential component of the surface current when the DL is thin and the surface conductivity is low. It is caused by hydrodynamic involvement of the electric charge rdl in the thin diffuse layer, as given by: ð1 rdl ðxÞuy ðxÞdx ð5:38Þ Iy ¼ 0

For a thin DL, it can be considered to be essentially flat.

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5

We can then apply a Taylor series expansion to the tangential speed, uy, near the particle surface. This is described by the following equation, in which x is the distance from the particle surface. @uy ð5:39Þ @x x¼0 The first term is equal to zero because, at the surface, the liquid does not move relative to the particle. As a result, the surface current within the thin DL can be expressed as: ð1 @uy r ðxÞxdx ð5:40Þ Is ¼ @x x¼0 0 dl uy ðxÞ ¼ uy ðx ¼ 0Þ þ x

At this point, we again employ a peculiarity of the thin DL. There is a known relationship between the electric charge density in the DL and the z-potential: ð1 rdl ðxÞxdx ¼ em e0 z ð5:41Þ 0

Since the surface current is given by: @uy Is ¼ em e0 z ð5:42Þ @r r¼a it then follows that substitution of Equation (5.42) into Equation (5.37) yields the first boundary condition. The cell boundary condition, corresponding to the definition of the CVI effect, specifies a zero value for the macroscopic electric current in the suspension, expressed by a zero value for the macroscopic electric field, E, and, hence, in the frame of the Shilov-Zharkich cell model, by a zero value of the variation of the electric potential on the cell surface: ’ðr ¼ bÞ ¼0 ð5:43Þ b cosy In the framework of the Levine cell model, the relationship between the macroscopic and local electric field is given, contrary to Equation (5.43), 1 @f . A comparison of this expression with Equation by: ELevine ¼  cosy @r r¼b (5.32) (which connects the macroscopic electric current with the local electric field) reveals an onerous contradiction in attempting to employ the Levine cell model. It follows that K Levine ¼ I=ELevine ¼ Km . From this we must conclude that the conductivity of the suspension, Ks, must equal the conductivity of the medium, Km, which is obviously false. Clearly, the conductivity of the suspension must be a function of the properties of particles and their DLs. Therefore the Levine cell model is inappropriate for this task. To compute the CVI, we calculate the unknown constants E and d using Equations (5.37) and (5.43), substitute these constants into the general E¼

203

Electroacoustic Theory

solution of Equation (5.36) for f, then calculate a value of f at r ¼ b, and finally substitute this result into Equation (5.32). Following these steps we obtain the following expression for CVI: CVI ¼

3em e0 z a3 1 @uy 3 3 a b þ 0:5a siny @r r¼a

ð5:44Þ

This expression relates CVI with an unknown, albeit derivative of the tangential component of the liquid motion relative to the particle surface. CVI ¼

3em e0 z ’ 1 @uy a 1 þ 0:5’ siny @r r¼a

ð5:45Þ

where em and e0 are the dielectric permittivities of the medium and vacuum, respectively, z is the electrokinetic potential, a is the particle radius, ’ is the volume fraction, r and y are the spherical coordinates associated with the particle center, and ur and uy are the radial and tangential velocities of the liquid motion relative to the particle. The next step in the development of this CVI theory is to calculate the hydrodynamic field, assuming that the speed of the particle with respect to the liquid is given by Equation (4.27) for a monodisperse colloid, or Equation (4.35) for a polydisperse colloid. This has been solved by Dukhin et al. [31] using a Happel cell model, and later using the Kuwabara cell model [56], under the assumption of an incompressible liquid. This condition is valid only when the wavelength, l, is much larger than the particle size (the so-called long-wavelength requirement): la

ð5:46Þ

The final expressions for the drag coefficient and tangential velocity are:     3 dh h j ð5:47Þ  þ g ¼ orm ’ 2I dx x x¼a duy 3ðup  um ÞhðsÞ ð5:48Þ ¼ dr r¼a 2I where s ¼ a√o/2n, n is the kinematic viscosity, and  is the dynamic viscosity. The values of the special functions h(x) and I(x) are presented in Chapter 2, Section 2.5.2. Substituting the drag coefficient into Equation (4.27) and applying the result to Equation (5.45) leads to the following expressions for CVI and dynamic mobility: CVI ¼

9em e0 z ðrp  rs Þ ’ rs 4 1 þ 0:5’

jhðaÞ rP ð1  ’Þrp  rs I 1:5H  rs

ð5:49Þ

204

CHAPTER

md ¼

9em e0 z ðrp  rs Þrm 1 4 ðrp  rm Þrs 1 þ 0:5’

G¼

9j 2s

jhðsÞ rP 2ð1  ’Þrp  rs I H 3rs

5

ð5:50Þ

hðsÞ 1’  F ¼ rp 2þ’ 3H þ 2I 1  ð1  ’Þ rs

These expressions are valid for nonconducting particles with a thin DL and a low surface conductivityðDu  1Þ. The frequency dependence of CVI and dynamic mobility given by Equations (5.49) and (5.50), correspondingly, is caused by the factor G only, which reflects the formation of hydrodynamic fields. Factor F, which reflects the formation of electric fields, in the case of Equation (5.50) is frequency independent. This happens because we did not yet take into account the nonstationary process for the storage of soundinduced ionic charge in the DL when deriving Equations (5.35) and (5.37), which reflect the balance of the electric fields and currents.

5.4.2. CVI for Polydisperse Systems Let us assume now that we have a polydisperse system with N conventional fractions. Each fraction of particles has a certain particle diameter, di, volume fraction, ’i, drag coefficient, gi, and particle velocity, ui, in the typical laboratory frame of reference. We also assume the density of the particles to be the same for all fractions, rp. The total volume fraction of the dispersed phase is ’. The liquid is characterized by a dynamic viscosity, , a density, rm, and a velocity, um. The coupled-phase model allows us to calculate the difference ui  um for each fraction, without the need to invoke the superposition assumption (Equation 4.35), viz:   rp  1 rP rm 0 1 ð5:51Þ ui  um ¼   N X rp g B gi C jorp þ i @1 þ A ’i ð1  ’Þrm i¼1 jor þ gi p ’i The next step is the calculation of the corresponding electric field. We can apply Equation (5.44) to calculate the electric current generated by the ith fraction of the polydisperse system, CVIi, keeping in mind that the number of dipoles in this fraction is ’i/’ times smaller: CVIi ¼

’i 3em e0 z a3i 1 @uiy 3 3 ’ ai bi þ 0:5ai siny @r r¼a

ð5:52Þ

205

Electroacoustic Theory

This equation has been derived using the cell model. Usually, a cell model is formulated only for a monodisperse system. We have proposed [31, 56] and described briefly (Section 2.5.2) a way to generalize the cell model to account for polydispersity. This yields the following simplification: CVIi ¼

3ee0 z’i ’ 1 @uiy ai 1 þ 0:5’ siny @r r¼a

ð5:53Þ

The derivative from the tangential velocity contains a dependence on the speed of the particle motion in the sound wave given by Equation (5.48): 1 @uiy 3ðui  um Þhðsi Þ ¼ siny @r r¼a ai Iðsi Þ

ð5:54Þ

where h and I are special functions (see Section 2.5.2). The total CVP generated by all particles is then calculated as the superposition of all the fractional CVPi. The total electroacoustic current equates to CVP multiplied by the complex conductivity of the dispersion Ks. The final expression for CVI and the dynamic mobility of the polydisperse system is then given by: N P

CVI ¼

9em e0 zðrs  rm Þ ’ 4rs 1 þ 0:5’

i¼1

r ’ hðs Þ s i i   3Hi þ1 jð1  ’Þsi Iðsi Þ rp  rm 2Iðsi Þ   3Hi þ 1 ’i N rp X 2Iðsi Þ   1 3Hi 1  ’ i¼1 þ1 rp  r m 2Iðsi Þ ð5:55Þ

N X i¼1

md ¼



2em e0 zðrs  rm Þrs 3ðrp  rm Þrm

9rs ’i hðsi Þ 0 0

11 3H i 4j’ð1  ’Þsi Iðsi Þ@rp  rm @ þ 1AA 2Iðsi Þ 3 0 1 2þ’ @ 3Hi þ 1A’i 2Iðsi Þ N rp X 0 1 1 1  ’ i¼1 3H i þ 1A rp  r m @ 2Iðsi Þ

N 2em e0 zðrs  rm Þrs X Gðsi :’Þð1 þ Fi Þ 3ðrp  rm Þrm i¼1

ð5:56Þ

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CHAPTER

5

where the special functions h, H, and I are given in Section 2.5.2, Hi ¼ H(si), and Ii ¼ I(si). A comparison of Equation (5.56) with the form (Equation 5.29) gives rise to the conclusion that for the considered case of negligible value for the surface conductivity ðDu  1Þ and a negligible contribution of the P Maxwell-Wagner dispersion ðo0  1Þ, we have Gðs, ’Þ ¼ ni¼1 Gi ðsi , ’Þ and Fi ¼ ð3=ð2 þ ’ÞÞ  1 ¼ ð1  ’Þ=ð2 þ ’Þ. The procedure used for the derivation of Equations (5.55) and (5.56) might be also applied for taking into account the surface conductivity of the particles and (or) the Maxwell-Wagner dispersion. Such a generalization would require modification of the function Fi only, without any revision of Gi(si, ’).

5.4.3. Surface Conductivity So far, the surface current that generates CVP has been attributed only to the hydrodynamic component. Isi ¼ em e0 z

@uy @r r¼ai

ð5:57Þ

This restricts the theory to the situation of negligible surface conductivity and small Dukhin number: Du  1. Now, as stated earlier, we are going to expand the theory to include surface conductivity effects and remove the restriction on the Dukhin number. In the presence of substantial surface conductivity, the expression for the surface current must take into account not only the hydrodynamic component but also the electrodynamic components of the surface current: @uy ks @f þ Isi ¼ em e0 z @r r¼ai ai @r r¼a

ð5:58Þ

The addition of this second component to the surface current modifies the expression for the dynamic mobility: md ¼

N 2e0 em zðrp  rs Þrm X Gðsi , ’i Þð1 þ Fi ðDui , ’ÞÞ 3ðrp  rm Þrs i¼1

ð5:59Þ

The surface conductivity might be assumed to be the same for all fractions, especially in the case of a thin DL. Under such an assumption, the substitution of Equation (5.58) into the surface condition (Equation 5.37), and following the application of Equations (5.37) and (5.43) for the determining the constants in the general solution, Equation (5.35), and substitution of the resulting potential distribution into Equation (5.32), lead to:

207

Electroacoustic Theory N P

9rs ’i hðsi Þ 3    3H 2þ’þ2Du i i ð1’Þ i¼1 4j’ð1’Þsi Iðsi Þ rp rm þ1 2em e0 zðrs rm Þrs 2Iðsi Þ   md ¼ 3Hi 3ðrp rm Þrm þ1 ’i N rp X 2Iðsi Þ   1 3Hi 1’ i¼1 þ1 rp rm 2Iðsi Þ

ð5:60Þ s

where Dui ¼ k /Km ai is the Dukhin number for the ith fraction of the polydisperse colloid.

5.4.4. Maxwell-Wagner Relaxation: Extended Frequency Range So far we have neglected the variation of the surface charge density within the DL. This was justified for low frequencies much smaller than the Maxwell-Wagner relaxation frequency defined by: Km o  oMW ¼ e0 em Consequently, the boundary condition of the continuity bulk and surface currents was defined earlier by Equation (5.37) as: Km rn f ¼ divs Iy To remove the restriction on frequency and to incorporate the MaxwellWagner relaxation into the CVI theory, we need to add a field-induced charge density variation to the electrodynamic boundary condition: @sd ð5:61Þ @t where sd is the surface charge density induced and stored in the DL. The appearance of this additional surface function, namely @sd/@t, itself requires an additional surface boundary condition to complete the mathematical formulation of the problem. The electrostatic condition of the continuation of the normal components of electrostatic displacement and the tangential components of electric field strength serve this purpose: Km rn ’ ¼ divs Iy þ

ep Er ðr ¼ a  0Þ  em Er ðr ¼ a þ 0Þ ¼ sd

ð5:62Þ

Ey ðr ¼ a  0Þ  Ey ðr ¼ a þ 0Þ ¼ 0

ð5:63Þ

Substituting an induced surface charge density into the charge conservation condition (Equation 5.61) yields the following surface condition that expresses the continuity of the complex currents near the particle’s surface: ðKm  joem ÞEr ðr ¼ a þ 0Þ  ðKpef  joep Þ Er ðr ¼ a  0Þ @uy ¼ em e0 zdivs @r r¼a

ð5:64Þ

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CHAPTER

5

where 2ks a The tangential components of the electric fields have the same value on both sides of the particle’s surface: Kpef ¼ DuKm ¼

Ey ðr ¼ a þ 0Þ ¼ Ey ðr ¼ a  0Þ

ð5:65Þ

Solution of the Laplace equations for the potential distributions outside and inside the particle surface together with the surface conditions (Equations 5.43, 5.64, and 5.65), and following substitution of the resulting potential distribution into Equation (5.32) lead to the final expression for the frequency-dependent electrodynamic contribution to CVI:   ep ð1  2DuÞð1  ’Þ þ jo0 1  ð1  ’Þ em 0    FðDu, o , ’Þ ¼ ep ep 0 2ð1 þ Du þ ’ð0:5  DuÞÞ þ jo 2 þ þ ’ 1  em em ð5:66Þ This expression for the function F is only valid for a thin DL but for any surface conductivity, frequency, and volume fraction.

5.4.5. Water-in-Oil Emulsions, Conducting Particles In order to apply theory described in previous sections to water-in-oil emulsions and dispersions of conducting particles, we should simply introduce the conductivity of the particles in the expression for the function F. A water-in-oil emulsion can be considered as conducting particles in a nonconducting media. In order to achieve this, we should simply use more general expression for the Dukhin number: Du ¼

K p ks þ Km Km

ð5:67Þ

where Kp is conductivity of the water droplets. This modified Dukhin number can be used in the general expression for the F function, Equation (5.66). The properties of the water-in-oil emulsion allow us to approximate the value of this function. It is known that the conductivity of water is much higher than that of nonpolar liquids. This means Kp  1. that Du

Km In addition, the dielectric permittivity of water is about 40 times higher ep  1. than that of nonpolar liquids: em

Electroacoustic Theory

209

Using these two strong inequalities, we can neglect 1 in comparison to the relevant parameters in Equation (5.66). As a result, we get the following approximate value for the function F in water-in-oil emulsions: ep 2Du  jo0 1f em 0 FðDu, o , fÞ

¼ 1 1  f 2Du þ jo0 ep em This approximate value is actually quite accurate because of the large conductivity and permittivity of water. If we use this value for the function F in Equation (5.59) for the dynamic mobility, we would come to the interesting result:   Kp ep md  1,  1 ¼ 0 for thin isolated DLs only Km em This means simply that water-in-oil emulsions with isolated and thin DLs should not exhibit any electroacoustic effect. It is possible to give a simple explanation to this unexpected conclusion. In order to do this, we compare the electric field structure induced by the relative particle-liquid motion for nonconducting and conducting particles, as illustrated in Figure 5.3. It is known that ultrasound generates particle motion relative to the liquid because of the density contrast. In drawing Figure 5.3, we assume that the particles move from left to right. This causes a liquid motion relative to the particle surface, which is illustrated with arrows above and below the particle. This motion of the liquid drags ions within the diffuse layer towards the left pole of the particle. This redistribution of the diffuse layer leads to different results for conducting and nonconducting particles. In the case of a nonconducting particle, the surface charge cannot move. It retains its spherical symmetry. As a result, the particle gains an excess negative charge at the right-hand side pole and excessive positive charge at the left-hand side pole. It gains a dipole moment. This dipole moment generates an external electric field, which gives rise to the CVC. In the case of a conducting particle, the charge carriers inside the particle can follow the counterions of the DL that are being dragged to the left-hand side pole. This means that the surface charge, or charge associated with the particle, loses its internal spherical symmetry as well as the charge of the diffuse layer. Most importantly, each element of the surface, as well as the adjacent diffuse layer, remains electroneutral. No external electric field appears and the electroacoustic effect is practically zero. However, this analysis does not mean that electroacoustics is useless for water-in-oil emulsions. It is so only for water-in-oil emulsions with thin and non-overlapping DLs. So far we have considered a water-in-oil emulsion with thin isolated DLs. In water-in-oil emulsions with overlapping DLs, the nature of electroacoustic effect is quite different.

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5

Non-conducting particle + + + − − − − − + − − + − + − − − − + + + + +

Liquid flow

−

+

Colloid Vibration Potential

Conducting particle

+

+

+ + − − −−

+ − − + − + − −− − + + + + +

+

Liquid flow

− −

FIGURE 5.3 Illustration of the exterior double layer polarization by the liquid flow relative to the particle surface induced with ultrasound.

Homogeneous distribution of the counterions eliminates polarization charges induced on the left-hand side pole of the particles by the liquid motion. The electroacoustic effect is related simply to the displacement current of the oscillating motion of the particle surface charges. This conclusion is a strong motivation for developing an electroacoustic theory for overlapped DLs. This was accomplished in Ref. [2] and presented below in Section 5.6.

5.5. QUALITATIVE ANALYSIS OF COLLOID VIBRATION CURRENT It is helpful to create some heuristic understanding of the physical phenomena that take place when an ultrasound pulse passes through a dispersed system. This also provides answers to some general questions. For instance, why do we need a density contrast in the case of ESA when the particles already

211

Electroacoustic Theory

move relative to the liquid under the influence of an electric field? Why do we need a density contrast to generate a CVI at low frequency when the particles already move in phase with the liquid? As far as we can find, there are no simple published answers to these questions. To find them, we utilize again the analogy between the sedimentation potential and electroacoustic phenomena. Marlow has already used this analogy [21, 22], and we will give further justification for his approach. Let us consider an element of the concentrated dispersed system in the sound wave (Figure 5.4). The size of this element is selected such that it is much larger than the particle, and also larger than the average distance between particles. As a result, this element contains many particles. At the same time this element is much smaller than the wavelength. This dispersion element moves with a certain velocity and acceleration in response to the gradient of acoustic pressure. As a result, an inertia force is applied to this element. At this point, we can use the principle of equivalence between inertia and gravity. The effect of the inertia force created by the sound wave is equivalent to the effect of the gravity force. This gravity force is exerted on both the particle and liquid inside of the dispersion element. The densities of the particles and liquid are different, as are the forces. The force acting on the particles depends on the ratio of the densities. element of dispersion

Up - Um

Up Um

acceleration gradient of pressure

FIGURE 5.4 An element of the colloid in the ultrasound pressure field.

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CHAPTER

5

The question then arises as to what density we should take into account. To answer this, let us consider the forces acting on a given particle in the gravity field. The first force is the weight of the particle, which is proportional to its density, rp. This force will be partially balanced by the pressure of the surrounding liquid and other particles. This pressure is equivalent to the pressure generated by an effective medium with a density equal to the density of the dispersed system. This may become clearer when one imagines a large particle surrounded by smaller ones. We are coming to the well-known conclusion that the resulting force which moves the particles with respect to the liquid is proportional to the density difference between particle, rp, and dispersed system, rs. It is important to mention here that this force is not necessarily a buoyant force. A buoyant force would be proportional to the difference between the particle density rp and the dispersion medium density rm, as given by Archimedes’ law. Figure 5.5 illustrates this difference for the example of the sedimentation of a small spherical group of spherical particles in the liquid. Case 1 corresponds to the situation when the array settles as one entity and the liquid envelopes this settling entity from the outside. Case 2 corresponds to the different situation in which the liquid is forced to move through the array. There will be a difference between the forces exerted on the particles within the array. There is an additional force in Case 2 caused by the liquid pushing through the array of particles. Electroacoustic phenomena correspond to Case 2. It happens because the width of the sound pulse w is much larger than the wavelength l at high ultrasound frequencies. wl

Case 1.

ð5:68Þ

Case 2.

Um

Um

Um

Up

Up

FIGURE 5.5 Two possible ways in which particles can sediment.

Electroacoustic Theory

213

The balance of forces exerted on the particles in a given element of the dispersion consists of the effective gravity force, the buoyancy force, and the friction force, related to the filtration of the liquid through the array of particles. As a result, the particles move relative to the liquid with a speed (up  um) that is proportional to the density difference between the particle and the system (rp  rs). The motion of the particles relative to the liquid disturbs their DLs, and as a result generates an electroacoustic signal. This electroacoustic signal is zero when the speed of the particle is equal to the speed of the medium. This occurs when the density of the particle is equal to the density of the dispersed system. This implies that the electroacoustic signal must be proportional to (rp  rs). This conclusion is rather unexpected, because O’Brien’s relationship makes the CVI proportional to (rp  rm). This discrepancy in the density contrast might be related to the initial definition of the inertial frame of reference. When sound is the driving force (for either CVI or CVP), the correct inertial frame is a laboratory frame of reference because the acoustic wavelength is much shorter than the size of the sample chamber. Therefore, particles move with different phases inside the narrow sound beam. The chamber as an entity remains immobile. The question of the frame of reference is more complicated in the ESA case where the electric field is the driving force. The wavelength of the electric field is much longer, and as a result, all the particles move in phase. This motion exerts a given force on the chamber. The motion of the chamber depends both on the mass of the chamber and mass of the sample. Depending on the construction of the instrument, the inertial system might be related to the chamber, to the center of mass, or to some intermediate case (depending on the ratio of the masses of the chamber and the sample). This means that, for the ESA method, the final expression relating any measured ESA signal with the properties of the dispersed system might contain a multiplier that is dependent on the mass of the chamber.

5.6. ELECTROACOUSTIC THEORY FOR CONCENTRATED COLLOIDS WITH OVERLAPPED DLs AT ARBITRARY ka: APPLICATION TO NANOCOLLOIDS AND NONAQUEOUS COLLOIDS This section represents the theory developed by Shilov and others [2]. All theories discussed above employ one important assumption that restricts the range of their validity. They do not take into account overlap of DLs that might occur in concentrated dispersions. In addition, most of them employ the so-called thin DL assumption. The following analysis would demonstrate that these two assumptions are not acceptable for variety of nanocolloids and nonaqueous dispersions.

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CHAPTER

5

The thin DL assumption implies that the thickness of the DL, the so-called Debye screening length k1, is much smaller than particles radius a. The thin DL assumption restricts the particle size. In addition, it limits ionic strength, or alternatively conductivity of the solution, because of the Debye length dependence on these parameters. There is a simple, approximate expression that allows us to scope with a range of particle sizes and conductivity values for valid thin DL assumption, Equation (2.18). The classical electrokinetic theory [47, 57] introduces three approximate separate ranges of ka values: ka > 10; ka < 0.1; 10 > ka > 0.1, illustrated in Figure 5.6. The numbers 10 and 0.1 are approximate borders of these ranges. There are much more sophisticated ways to determine these borders using modern electrophoretic theories [58–63]. However, these approximate numbers allow us to explain the main ideas of this section. The range of Km and a where ka > 10 can be considered as a range of validity of the thin DL theory. In regard to the theory of electrophoresis, it is the range where Smoluchowski theory [61] works. The range of Km and a where ka < 0.1 can be considered as a range of the thick DL is the range of the Huckel theory [62] for electrophoresis. The range of Km and a where 10 > ka > 0.1 can be considered as a transition range between the thin DL and the thick DL is described by the Henry-Ohshima function [63, 64] in the classical electrophoresis theory, if surface conductivity is negligible. 101 Thin isolated DLs

100 10−1

aqueous

conductivity [S/m]

10−2 10−3 10−4

ka=10 permittivity=80

10−5 10−6

ka=1

10−7 10−8 10−9

Thick overlapped DLs

10−10 10−11

10−2

10−1

ka=0.1 permittivity=2 100

101

diameter [microns] FIGURE 5.6 Various ranges of ka depending on the conductivity of dispersion and particle diameter.

Electroacoustic Theory

215

The same conventional rages of ka values, and consequently Km and a values, can be used for the electroacoustic theory development. The main efforts, as we stated before, have been concentrated on the range of ka > 10, which is the range of thin DL. The one exception, the theory developed by Babchin and coauthors [65], unfortunately is not valid for concentrates, as we will show below. As one can see, the thin DL range covers aqueous dispersions with sizes above 1 mm. However, the large portion of aqueous dispersions with submicrometer particles belongs to the transition range of moderate ka values. Practically all nanocolloids with sizes under 100 nm belong to this range. This means that thin DL electroacoustic theories would have problems dealing with nanocolloids. This situation becomes even more pronounced for nonaqueous systems. Usually, conductivity of the nonaqueous systems is orders of magnitude lower than that of aqueous ones. This leads to much smaller ka values. Practically all nonaqueous dispersions belongs to the range of either moderate or low ka. Nanocolloids and nonaqueous dispersions are very important classes of heterogeneous systems. The lack of the appropriate electroacoustic theory certainly justifies efforts for formulating one. This is not an easy task compared to the case of the thin DL. The last one offers several simplifications that cannot be used for the other two ka ranges. One of the main advantages of the thin DL range for theoretical development is the possibility to consider DLs as isolated, not overlapped. This assumption brings enormous simplification for mathematical modeling. Expansion of the electroacoustic theory to the other ka ranges would force us to drop this simplification. It is clear that thick DLs would overlap at much lower volume fractions compared to the case of the thin DLs. It is possible to introduce a critical volume fraction, ’over, above which DLs overlap, as was done in Section 2.4.2 in Equation (2.35). This expression indicates that a strong nonlinear ka dependence leads to the DLs’ overlap even for practically dilute systems if ka < 1. For instance, DLs start to overlap at the volume fraction 0.04% if ka ¼ 0.1. Babchin’s [65] electroacoustic theory for small ka neglects DLs overlap. This makes it of little use for electroacoustics, which targets concentrated dispersions and emulsions. Creation of a new theory that would take DLs overlap into consideration is imperative for further successful development of this characterization technique. As far as we know, there is only one attempt has been made to develop electroacoustic theory for concentrates at the low and moderate ka values with overlapping DLs. It is Ohshima’s electroacoustic theory for “salt-free media” [66]. This is an imaginary case where there are only counterions present in the bulk between particles. This could correspond to the real systems with completely overlapped DLs and extremely high surface charge. Unfortunately, this model is hardly applicable to the real dispersions, but it reveals several peculiar features of the “not-thin DL” theory.

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5

For instance, it predicts a completely different nature of the “surface charge-surface potential” relationship. It predicts the so-called condensation effect which restricts uncontrolled growth of the surface potential. Stern layer follows directly from the mathematical theory, in contrast to the classical DL theory where it must be introduced as an additional entity. We will follow this line and show that surface charge and surface potential (or z-potential) play very different roles in the surface characterization, compared to the “thin DL” case. Ohshima’s theory tells us also that appropriate modeling of the ion distribution in the interparticles space might lead to a significant simplification of the mathematical theory. The question is to find a more suitable distribution. There are two different models described in Ref. [67]. The first one, following Ohshima, considers strongly overlapped DLs, when the Debye screening length exceeds substantially the characteristic distance between the surfaces of neighboring particles in the dispersion. In this so-called homogeneous case, the relative variation of the concentration of every ionic component in the equilibrium DL is negligible. This allows us to model ion concentrations as space-independent, which leads also to the space-independent constant conductivity. This homogeneous model allows us to take into account all ions species, not only counterions as in Ohshima’s theory. Consequently, homogeneous model is much closer to real colloids. The homogeneous model does not apply any restrictions on the frequency. However, it is clear that the model is valid for the restricted range of low and, perhaps, moderate ka values. It is impossible to determine whether the ka validity range is within this model framework. It turns out that there is a possibility for one more model that would allow us to determine the validity range of the “quasihomogeneous model.” This possibility corresponds to the case when the DL is thick enough compared to the ion diffusion drift during the period of reciprocal cyclic frequency of ultrasound. It occurs when the DL relaxation time exceeds substantially the reciprocal cyclic frequency of the ultrasound. In other words, this is the case when the frequency of measurement is much higher than Maxwell-Wagner frequency. It is known that the active conductivity current equals the passive dielectric displacement current at the Maxwell-Wagner frequency. For frequencies much higher than Maxwell-Wagner frequency, the active conductivity current becomes negligible compared to the dielectric displacement current. Dielectric displacement current is independent of the concentration of ions, whereas conductivity current completely depends on it. If the measurement frequency exceeds significantly the Maxwell-Wagner frequency, we can neglect conductivity current and consequently eliminate dependence on the ion concentration in the complex structure of the overlapped DLs from the mathematical electrodynamic equations. This simplification is sufficient

217

Electroacoustic Theory

for creating an analytical electroacoustic theory. We will call this model a “high-frequency model.” Summarizing, we can say that there are two approximate analytical electroacoustic theories for concentrates. The first one, the homogeneous model, is valid for any frequency but restricts the ka from above. The second one, the high-frequency model, restricts frequency but is valid for any ka. The second model is much more complicated mathematically. However, it allows us to determine the range of ka where the first model is valid. Both theories have been developed for the CVC mode of electroacoustics. Expressions for dynamic electrophoretic mobility should be valid for the ESA mode of elctroacoustics as well.

5.6.1. The Homogeneous Model In the previous section we gave a short overview of the role of the parameter ka in the electrokinetic and electroacoustic phenomena. This traditional presentation is based on the comparison of the DL thickness with characteristic dimension of the dispersion. In the case of the traditional dilute systems, there is only one special parameter with which to compare the Debye length: the particle size a. That is why parameter ka is chosen for characterizing many DL-related effects. In concentrated dispersions, the other special parameter appears, which is the average distance between particles, d. In the case of the thick DL, this parameter is of greater importance because it determines the degree of DL overlap. In this section, we consider situation when DLs overlap strongly, which corresponds to the following relationship between the Debye length and the interparticle distance: kd  1

ð5:69Þ

In this case, the characteristic space that is accessible for ions is much narrower than the Debye screening length, k1. There is no noticeable deviation from homogeneous distribution of ions that may arise in the dispersing medium. We may neglect these deviations completely, considering that the ion concentration C 0 is constant and space-independent. This is the essence of the quasihomogeneous model. In the frame of this model, there is one well-known [28–30] feature of the equilibrium electric potential F0 in the dispersion medium: in the depth of the suspension with strongly overlapped DLs, the value of F0 is spacehomogeneous and equal to the potential of the surface (z-potential). F0 ¼ z

ð5:70Þ

The theory of diffuse screening layer connects directly the distribution of screening charge density r0 with equilibrium electric potential F0 and, hence, r0 also is spatially homogeneous in the dispersion medium:

218

CHAPTER

5

RT FF0 RT Fz

em k2 sinh em k2 sinh ð5:71Þ RT F F RT In the frame of the same theoretical approach, the local concentrations of cations and anions (upper index “þ” and “”) may be expressed as:   1 RT Fz  2 ð5:72Þ em k exp  C0 ¼ 2 F2 RT r0 ¼

The space-independence of ion concentration C 0 means space-independence of the local conductivity Km: F2 þ þ ðD C0 þ D C ð5:73Þ 0Þ RT In the frame of the homogeneous model, we can derive a simple relationship between the surface charge density s and z-potential, even without using the cell model (i.e., for an arbitrary geometry of the disperse phase). The condition of the electroneutrality of the dispersion can be written as a following simple equation: Km ¼

sfs ¼ ð1  fÞr0

ð5:74Þ

Where s is specific (for unit volume of disperse phase) surface of disperse phase and r0 is the local bulk charge density given by Equation (5.71). It follows from Equations (5.20) and (5.23) that: s¼

1 RT 1  f em k2 sinh ~z s F f

ð5:75Þ

For monodisperse dispersion of spherical particles, when s ¼ 3/a, Equation (5.75) turns to: s¼

1 RT 1  f em ak2 sinh ~z 3 F f

ð5:76Þ

This equation reflects a very important distinction between thin DL and thick, overlapped DL. In the case of the traditional thin DL theory, both surface charge and z-potential are surface properties, independent of the volume fraction. In the case of thick DL, surface charge is a true parameter of the surface properties. Electrokinetic z-potential might lead to erroneous conclusions because it depends on the volume fraction, not only on the surface charge. For instance, if we have surface charges caused by adsorption of ions, the value of the surface charge characterizes this adsorption. It remains constant and independent of the volume fraction for the same adsorption. At the same time, z might be different due to the volume fraction dependence. This means that the z-potential value might lead to the wrong conclusions about adsorption. This all indicates that both z and surface charge must be reported when dealing with concentrated dispersions with thick DLs.

219

Electroacoustic Theory

Due to the space independence of the local conductivity Km and equilibrium screening charge density r0 within the interparticle electrolyte medium, equation for electric potential distribution reduces to the Laplace equation, Equation (5.35). The problem of the distribution of field-induced electric potential f in the limiting case of strongly overlapped electric DL is completely equivalent to the problem of the potential distribution in the system of dielectric particles in a dielectric medium. It is clear that the condition of zero macroscopic potential gradient (characteristic for CVI mode) means automatically zero ! values of the local electric fields E 1 everywhere in the dispersion. This leads to the zero values of the local electromigration currents everywhere in the dispersion. As a result, formula for CVI reduces to the simple form: RT Fz ! ð5:77Þ em k2 sinh hvi F RT Ð ! ! where hvi V1 V vdV is the macroscopic velocity of the dispersion medium with respect to the dispersed particles. We take also into account that, in the case of strongly overlapped DLs, the density of screening charge r0 is space-independent everywhere in the liquid that flows with respect to particles ! with the velocity v. ! The value hvi represents the drag of the liquid through the system of the particles under the action of inertial forces caused by sound-induced vibrational movement of the considered infinitesimal element V. This drag of the liquid may be expressed by the pressure gradient in the sound wave hrP1i, by means of the well-known coupled-phase model, Equation (4.27). Consequently, we would have the new equation for the CVI of the strongly overlapped diffuse DLs between the particles (for the quasihomogeneous approximation):   rp  rs 2RTem Fz ðkaÞ2 sinh CVI ¼ hrP1 i ð5:78Þ f 9FO RT rs þ ioð1  fÞ rp rm g !

CVI ¼ r0 hvi ¼

One of the most interesting and peculiar features of this expression is the absence of the linear dependence on the volume fraction. This unusual result becomes clearer if we take into account the volume-fraction-dependent coefficient between surface charge and z-potential. We can use this equation in order to express CVI through the surface charge: CVI ¼

2ssa2 f 9O 1  f

rp  r s rs þ ioð1  fÞ

f rr g p m

hrP1 i

ð5:79Þ

or for monodisperse dispersion CVI ¼

2sa f 3O 1  f

rp  rs rs þ ioð1  fÞ

f rr g p m

hrP1 i

ð5:80Þ

220

CHAPTER

5

These expressions for CVI clarify that the homogeneous model is consistent with general notion about electroacoustics as a tool for characterizing surface properties. It shows that CVI is proportional to the degree of the surface ionization indeed. Following usual electroacoustic theory path, we introduce dynamic electrophoretic mobility using Equation (5.2). Dynamic electrophoretic mobility in the case of strongly overlapped DLs equals: md ¼

2sa rm 3O rs þ ioð1  fÞ fg rp rm

ð5:81Þ

The applicability of these equations is restricted only with the condition of the strongly overlapped diffuse layers. There is no restriction applied on the frequency of ultrasound and characteristic frequency of the screening charge relaxation (Maxwell-Wagner relaxation). The second model presented below does just the opposite: it restricts ultrasound frequency keeping ka free.

5.6.2. High-Frequency Model for Overlapped DLs In this version of the electroacoustic theory, we restrict the frequency instead of ka. If the measurement frequency is much higher then frequency of Maxwell-Wagner relaxation oMW, that is, o

Km oMW em

ð5:82Þ

the contribution of the ionic conductivity current to the total complex current is negligible. In this case, the equation for electric potential distribution reduces to the Poisson equation: r2 ’ ¼ 

i ! v grad r0 oem

ð5:83Þ

In addition, we can neglect the contribution of the ionic current to total current density. As a result, the total current density transforms to the following: !

!

i ¼ joeE 1 þ r0n !

ð5:84Þ

Unfortunately, these simplifications are still not sufficient for an analytical solution of the general electroacoustic problem for the complicated geometry of the concentrated dispersed phase. To solve the problem for concentrated suspension with an arbitrary thickness of the DL, we use the electrokinetic cell model developed by Shilov and Zharkikh and presented in Section 2.5.4.

221

Electroacoustic Theory

In the long-wave approximation (wavelength much larger than particle size), the distribution of acceleration of a macroscopically small element of suspension volume vibrating in the sound wave may be considered as a homogeneous vector distribution. Consequently, the components of local sound! induced vector fields, as for example electric current density j and liquid ! velocity v, may be represented as: !

!

!

!

jðr, y, tÞ ¼ ejotjðr, y, oÞ ¼ ejot fi r jr ðr, oÞ cosy þ i r jy ðr, oÞ sinyg !

!

!

!

vðr, y, tÞ ¼ eiotvðr, y, oÞ ¼ eiot fi r vr ðr, oÞ cosy þ i y vy ðr, oÞ sinyg

ð5:85Þ ð5:86Þ

while the distributions of sound-induced scalar fields, as for example electric potential ’1 and pressure P, have the following form: f1 ðr, y, tÞ ¼ eiot f1 ðr, y, oÞ cosy ¼ eiot f1 ðr, oÞ cosy

ð5:87Þ

Pðr, y, tÞ ¼ eiot Pðr, y, oÞ cosy ¼ eiot Pðr, oÞ cosy

ð5:88Þ

In order to find the sound-induced electric potential and current which are in accordance with the above-mentioned basic principles, we have to consider the Laplace equation (Equation 5.12) for the inner space of the particle (0  r  a) and Equation (5.35) for the space of spherical shell (a < r  b) that represents the dispersion medium. The surface condition at the outer boundary of the cell (r ¼ b) in the case of CVI should reflect the condition of the absence of sound-induced macroscopic electric field in the dispersion. It requires that the definition of the cell analogue of the macroscopic electric field strength be expressed in the terms of electric potential distribution at the outer surface of the cell. We use here a definition that yields, according to Ref. [33], the correct form for entropy production through cell: hEi ¼ f1 ðrÞjr¼b

ð5:89Þ

hii ¼ ir ðrÞjr¼b

ð5:90Þ

The electrokinetic cell model formulated by Levine [37] does not satisfy the Onsager symmetry requirements between kinetic coefficients for electrophoresis and streaming potential (See Ref. [22]). This leads to incorrect qualitative results (See Ref. [23]). It is interesting to note that definition (Equation 5.89), in which is embedded specially the Onsager symmetry relations, coincides with the definition of the macroscopic electric field given in the Maxwell-Wagner theory of the dielectric permittivity. It follows from Equation (5.42) that the condition (Equation 5.15) of zero macroscopic field in the dispersion in the frame of the cell model transforms to: f1 ðrÞjr¼b ¼ 0

ð5:91Þ

222

CHAPTER

5

Using the CVI definition in accordance with Equation (5.16) and comparing it with Equations (5.36) and 5.43) and we obtain the following expression for CVI which is a high-frequency asymptotic: dfðrÞ CVI ¼ joem þ r0 ðbÞvr ðbÞ ð5:92Þ dr r¼b The solution of Equation (5.35), corresponding to the form (Equation 5.40) with surface conditions (Equations 5.13–5.15) (surface S is determined by r ¼ a) may be represented as: f1 ðrÞ ¼

j 3r 2 oe

m

  M b3 I1 ða; bÞ  I2 ða; bÞ  r3 I1 ða; rÞ þ I2 ða; rÞ

M¼

ð5:93Þ

a3 ðep  em Þ  r3 ðep þ 2em Þ a3 ðep  em Þ  b3 ðep þ 2em Þ

I1 ða, rÞ ¼

ðr

vr ðr1 Þ

dr0 dr1 dr1

vr ðr1 Þ

dr0 3 0 r dr dr1 i

a

I1 ða, rÞ ¼

ðr a

Substitution of f1(r) into the previous equations leads to the next integral form, which expresses the CVI for our limiting case in which the ratio of ultrasound frequency to Maxwell-Wagner frequency is large: CVI ¼ 

1 I2 ða; bÞð2em þ ep Þ b3 þ vr ðbÞr0 ðbÞ ð5:94Þ ð1  fÞep þ ð2 þ fÞem

f I1 ða, bÞðem  ep Þ þ

Further development of the CVI cell theory requires a definite expression for the distribution of electric charge density r0(r) in the equilibrium DL, as well as ! liquid velocity vðr,yÞ, with respect to the particle within the cell. The hydrodynamic problem of the mutual motion of the components of concentrated suspension induced by its vibration in ultrasonic field derived by Dukhin and Goetz (see Ref. [56]) for the Kuwabara cell model is given in Chapter 2.

5.6.3. Theoretical Predictions of Both Models In this section we analyze the dependencies derived in the previous sections for CVI and dynamic mobility for the two approximate versions of the electroacoustic theory. Comparing these versions between themselves would allow us to determine the ranges of their validity. The first version, the so-called homogeneous model, corresponds to the case when the space density of the screening charge is almost constant in

223

Electroacoustic Theory

the liquid medium between the particles. One of the most peculiar features of this approximate theory is CVI dependence on the volume fraction. It is very different for the case of the constant, volume-fraction-independent surface charge s or z-potential. In the case of strongly overlapped DLs, the vector field of the density of sound-induced electric current is similar to the vector field of the local liquid velocity with the factor of similarity RT Fz em k2 sinh F RT As a result, the homogeneous model with constant z-potential predicts that dependence of CVI on the volume fraction and geometric parameters of the particles is completely presented by the factor: r0 ¼

rp  rs rs þ ioð1  fÞ

f rr g p m

This factor includes, besides the densities of the dispersion medium and the dispersed phase, only one characteristic of dispersion, namely hydrodynamic friction coefficient g. It does not include any electric characteristics. At the same time, the similarity factor, r0, being expressed through the z-potential includes no geometrical parameters. It is illustrated in Figure 5.7, which

CVI 1010 gradP

4

0.4

2 0.3

0 4

j 5

0.2 6 F (H z)

Log

7 8 0.1

FIGURE 5.7 The dependence of CVI/h▽Pi on the frequency and on the volume fraction, calculated with quasihomogeneous model for constant z ¼ 25 mV and the concentration C ¼ 104 mol/l of 1-l electrolyte being in thermodynamic equilibrium with suspension. Parameters used: a ¼ 1 mm, rm ¼ 1 g/cm3, rp ¼ 2 g/cm3,  ¼ 103 kg/m s, em ¼ 30e0.

224

CHAPTER

5

demonstrates that the magnitude of CVI decreases monotonically with increasing volume fraction. This corresponds just to the increase of the hydrodynamic friction coefficient g(f,o), while the bulk charge density r0 remains constant under the condition of ’-independent z-potential. It is important to emphasize, however, that the homogeneous approximation is not valid for small volume fractions ’. The volume fraction and frequency dependence of CVI for the case of the constant surface charge density in the frame of quasihomogeneous approximation is presented in Figure 5.8. Here, contrary to Figure 5.7, CVI increases almost proportionally to volume fraction in the range of small ’. It occurs because the total charge of the particles dispersed in unit volume is almost proportional to the volume fraction of the particles. According to the electroneutrality of dispersion, the space-independent density of the equilibrium bulk charge is proportional to f(r0 / f). It is also evident that for small enough volume fractions the movement of every particle would be independent of the movement of its neighbors and, correspondingly, the macroscopic velocity ! hvi will be independent of the volume fraction ’; hence, under the above! mentioned conditions, CVI ¼ hvir0 . This means that the magnitude of CVI becomes proportional to the volume fraction. Further increase of ’ induces the hydrodynamic particle-particle interactions, and the friction coefficient will increase with ’ very fast. Correspondingly, the magnitude of the macro! scopic velocity hvi would decrease fast. The linear increase of CVI at small ’ will be replaced by the decrease of CVI with ’ when it reaches values that are large enough.

CVI 1010 < gradP >

10 7.5 5 2.5 0 4

0.4 j 5

0.2 6 F( Hz)

Log

7 8

FIGURE 5.8 The dependence of CVI/h▽Pi on the frequency and on the volume fraction, calculated with quasihomogeneous model for constant surface charge density s ¼ 6.64  105 C/m2. Other parameters are the same as in Figure 5.6 with the exception of the z-potential.

225

Electroacoustic Theory

There is only one frequency-dependent coefficient in the expression for CVI in the case of the quasihomogeneous approximation: f gðf, oÞ

rp  rs rs þ ioð1  fÞ

f rr gðf, oÞ p m

It reflects the finite time of establishing the stationary profile of the local velocity in the liquid between the dispersed particles. As pointed in Ref. [56], this time th decreases with increasing volume fraction of the solid phase in the suspension and a corresponding decrease of the effective hydrodynamic   r d2 radius dh of pores in the suspension th / m h . This fact manifests itself  in Figures 5.7 and 5.8 by the extension of the area with relatively weak frequency dependence of CVI with increasing of the volume fraction. Such shift of the CVI relaxation frequency with increasing volume fraction is characteristic for both constant surface charge and constant z-potential. Figures 5.9 and 5.10 correspond to the second version of the theory, the high-frequency model. These figures show the dependencies of CVI on ka at the constant z-potential (Figure 5.9) and constant surface charge density (Figure 5.10). Low-frequency limit corresponds to the frequencies that satisfy nonequality (Equation 5.95) and simultaneously exceed substantially the characteristic frequency of hydrodyamic relaxation Km  o em rm dh2

ð5:95Þ

4 3

CVI 1010

2

1

3

2

4

1 5

1

2

4

ka

6

8

10

FIGURE 5.9 The dependence of high-frequency CVI/h▽Pi on the parameter ka for constant z ¼ 25 mV and for different values of the volume fractions: curve 1  ’ ¼ 0.1; curve 2  ’ ¼ 0.2; curve 3  ’ ¼ 0.3; curve 4  ’ ¼ 0.4; curve 5  ’ ¼ 0.6. Other parameters are the same as in Figure 5.7, and ep ¼ 3e0.

226

CHAPTER

5

10 3 CVI 10 10

8 2

6

4

4 1 2

5 1 2

4

ka

6

8

10

FIGURE 5.10 The dependence of high-frequency CVI/h▽Pi on the parameter ka for constant surface charge density s ¼ 6.64  105 C/m2 for different values of the volume fractions: curve 1  ’ ¼ 0.01; curve 2  ’ ¼ 0.05; curve 3  ’ ¼ 0.1; curve 4  ’ ¼ 0.3; curve 5  ’ ¼ 0.6. Other parameters are the same as in Figure 5.8 with the exception of z-potential.

The CVI(ka) curves in Figure 5.9 are close to the parabolic dependences in the vicinity of the coordinate origin. The slope is steeper for the smaller volume fraction. These features are characteristic of the quasihomogeneous model of overlapped DLs under the condition of ’-independent z-potential. The CVI(ka) dependence slows down in the range of larger ka, with the rate reciprocal to the volume fraction. As a result, if ka is large enough, the partial inversion of the CVI(ka) dependence takes place (e.g., the curve 1 lies below curves 2 and 3). Such the inversion reflects the breakdown of the DL overlap for smaller ka if the volume fraction is small enough. In these conditions, the central zone of the space between the particles becomes filled with electroneutral solution. The expansion of this zone leads to the slowing down of the CVI(ka) dependence. In the case of a constant surface-charge density, the dependencies CVI(ka) are shown in Figure 5.10. They are decreasing functions, just contrary to the case of constant z-potential. It occurs because the total charge of the diffuse layer does not increase (contrary to the case of constant z-potential) with the increasing of ionic strength. Increasing ka or increasing ionic strength causes just one effect in this case—approaching of the moving electric charge to the particle surface due to the decreasing DL thickness. Velocity of the liquid-particle motion is lower close to the surface. This would lead to the reduction of CVI, which is proportional to the product of the moving electric charge, by the velocity of this motion. Such decay of CVI is faster for the smaller volume fraction because of the above-mentioned breaking of the DL overlap.

227

Electroacoustic Theory

It is important to mention that, if information about both parameters k and a is available, interpretation of the CVI measurement would yield data on both the z-potential and the surface charge s. Volume fraction dependence of CVI would reveal whether the constant z-potential model or the constant surface charge model is valid for a particular system. However, it seems to us that the surface charge model is more adequate. This would be justified by the fact that adsorption forces, which could be responsible for the surface charge formation, exceed substantially electrostatic forces induced with volume fraction variation. The validity of the high-frequency model is not restricted with respect to the parameter ka. This makes it possible to apply this equation for finding of the ranges of applicability of the homogeneous model. It is convenient to carry out this analysis using dynamic mobility instead of CVI. Figure 5.11 shows the ratio of the dynamic electrophoretic mobilities calculated using the quasihomogeneous model mdh, and the high-frequency model for several volume fractions. As one could expect, the quasihomogeneous approximation works better for larger volume fractions. As a quantitative measure, we can use 10% deviation between the quasihomogeneous model value and the exact number calculated using the high-frequency model. This degree of deviation occurs at: ka < 0.65 for ’ ¼ 0.3; ka < 1 for ’ ¼ 0.4; ka < 2 for ’ ¼ 0.6.

10 3 CVI 10 10

8 2

6

4

4 2

1 5 1 2

4

ka

6

8

10

FIGURE 5.11 Dependence of the md /mdh ratio on ka (mdh is the dynamic electrophoretic mobility corresponding to quasihomogeneous approximation, md corresponding to high-frequency approximation) for different values of the volume fractions: curve 1  ’ ¼ 0.3; curve 2  ’ ¼ 0.4; curve 3  ’ ¼ 0.6. Other parameters are the same as in Figure 5.9.

228

CHAPTER

5

In general, we may say that the quasihomogeneous model, being independent of the particle’s shape and polydispersity, provides good accuracy for characterizing concentrated dispersions with ka almost up to unity.

5.7. ELECTROACOUSTICS IN POROUS BODY The first theory was developed in 1944 by J. Frenkel [3]. He used the Smoluchwski theory of the streaming surrent as the starting point. That is why his theory is valid only for isolated DLs. He derived the following expression for the electric field strength Esee, induced by ultrasound in the porous body saturated with water:   8ee0 zkD o2 Oe rm Mm b0 Esee ¼  1 usc ð5:96Þ Km r 2 rm b0 c2s where usc is amplitude of displacement, cs is the largest value of the sound speed in the porous body, Mm is compressibility modulus of the liquid, rm is density of liquid, Oe is porosity, and kD is the Darcy constant which is proportional to the square of the pores radius r, b0 ¼

1 ; f ð1 þ av Þ

b0 ¼ 1 þ ðb0  1Þ

Mm Mp

ð5:97Þ

where Mp is compressibility modulus of the solid phase, which differs from the compressibility modulus of the dry porous skeleton Msc, and av is the coefficient proportionality between variations of volume of the solid and liquid phases. Frenkel predicted that electroseismic electric field strength is proportional to porosity and independent of pore size, because of the dependence of the Darcy constant on pore size. There is no conclusive experimental confirmation of this prediction so far. Frenkel’s equation does not present Esee as a function of the pressure gradient, similar to the modern electroacoustic theories. It also ignores phase shifts that occur at high ultrasonic frequencies. Existing theoretical developments in this field, instead of following the lines driven by Frenkel 60 years ago, shifted emphasis to the structure of the acoustic field in the soil. They ignore further development of the electrokinetic theories regarding coupling of rheological and electric fields. We have now such an experience in describing this coupling occurring in concentrated particulates. We can use it for testing Frenkel’s theory both theoretically and experimentally. Such a theoretical test of this theory was performed by Shilov and Dukhin, with results published in Ref. [68]. We reproduce here some derivations and final conclusions from that paper. The starting point is the Smoluchowski expression for streaming potential, Esrt: e0 em z @Peff ð5:98Þ Estr ¼  Km @x

229

Electroacoustic Theory

where Peff is effective pressure that squeezes liquid through the porous skeleton in the direction x. Frenkel calculated the velocity of this relative motion of the liquid and solid phases urel, Equation (5.45), in Ref. [3]: urel ¼

  iokD Oe rm cs Mm b0 @usc  1  rm b0 c2s @x

ð5:99Þ

where kD is filtration Darcy coefficient which is independent of porosity and proportional to r2. The parameter usc is the displacement of the skeleton. The relationship between effective pressure P and the velocity of the relative phase motion is given by the Darcy equation: @Peff  ¼  urel @x kD

ð5:100Þ

k D ¼ O e kD

ð5:101Þ

where

Substituting Equation (5.100) in Equation (5.98), we get the following equation for the electric field strength: Estr ¼ 

e0 em z urel Km kD

ð5:102Þ

Then, we can use Equation (5.99) for expressing the velocity of the relative phase motion: Estr ¼

  e0 em z iokD Oe rm cs Mm b0 @usc 1 kD rm b0 c2s @x Km

ð5:103Þ

One can see that porosity dependence disappears from the equation for the electric field strength because of the relationship between the filtration coefficients (Equation 5.101). A derivative of the displacement in Equation (5.103) can be expressed through frequency and sound speed as follows: @usc o ¼ i usc @x cs

ð5:104Þ

This yields the following equation for the electric field strength: Estr ¼

  e0 em z o2 rm cs Mm b0  1 usc Km  rm b0 c2s

ð5:105Þ

The experimental setup of the modern electroacoustic instruments functions in the current measurement mode. On multiplying the expression for electric

230

CHAPTER

5

field strength by conductivity of the porous body Ks we obtain the following equation for the seismoelectric current Isee:   e0 em zo2 rm cs Ks Mm b0  1 usc ð5:106Þ Isee ¼  Km rm b0 c2s The last simplification step that brings theory closer to the experiment is substitution of the displacement with gradient of pressure in the porous body. This can be done using the equation of general acoustics from Chapter 3: usc ¼

PℓikðxctÞ iocs rs

The final expression for seismoelectric current is:   e0 em z rm Ks Mm b0  1 rP Isee ¼  rs Km rm b0 c2s

ð5:107Þ

One of the most important features of the electroseismic current is its dependence on porosity. In order to specify this dependence, Frenkel first of all assumed that the term in the brackets in Equations (5.105) and (5.107) is independent of porosity. It is not quite obvious. It turns out that we can somewhat rearrange the parameters in this term for verification. As the first step, let us consider the expression for the sound speed in a porous body. Frenkel was the first to dicover that a sound wave splits when propagating through a porous body. Consequently, there are two values of the sound speed for a longitudinal wave. He derived the rather simple expression for the wave that propagates faster and with small attenuation. It is Equation (5.33a) in Ref. [3]:   b0 Msc E þ 0 Mm 1  Mp b ð5:108Þ c2s ¼ g1 þ g 2 Before we substitute the expression for the sound speed into the equation for the seismoelectric current, let us note that the sum of parameters g in the denominator equals simply the density of the wet porous body: g1 þ g2 ¼ rs ¼ rm Oe þ rp ð1  Oe Þ

ð5:109Þ

Substituting the expression for sound speed with simplifications (Equation 5.109) into Equation (5.107) yields the following expression for seismoelectric current: 2 3 Isee ¼

e0 em z Ks 6 1 r 7 6    m7 4 E E E  Msc  Km rs 5 þ  þ1 Oe ð1 þ av Þ Mp Mm Mp

ð5:110Þ

231

Electroacoustic Theory

The parameters av, E, and Msc are still unknown functions of porosity. However, one may state that both elastic characteristics of the dry skeleton, E and Msc, are of lower values in comparison with the corresponding characteristics of solid particles which form the skeleton. Besides, it is clear that both E and Msc decrease with increasing porosity and with decreasing rigidity of the contacts between the particles that compose it. If the compressibility modulus of either phase is large enough, while the skeleton is easily deformable, so that Mp , M0  E, Msc ,

ð5:111Þ

the first term in the denominator with unknown dependency on porosity tends to zero because the compressibility moduli Mp and M0 are in the denominator. The final expression for the seismoelectric current in such a case is:   e0 em z r Ks Isee ¼ rP ð5:112Þ 1 m rs K m  In accordance with the previous equation, Equation (5.112) describes the case of low elasticity of a skeleton compared to the elasticity of the liquid. Note that a suspension of nonbound particles or their aggregatively stable deposit corresponds to the case of zero skeleton elasticity. This opens the possibility for verification of this theory, which will be realized in the last chapter. It is interesting to note that seismoelectric current becomes nonzero even when both compressibilities and densities of the phases are equal. Under such a condition, Isee has a finite value if Young’s modulus and compressibilities of the skeleton are of different magnitudes: Isee ðMp ¼ Mm ; rp ¼ rm Þ ¼

ee0 z Ks E þ Msc rP  Km E  Msc þ Mp

ð5:113Þ

Thus, Frenkel apparently was right in assuming that the term in brackets is practically independent of compressibility. However, his final conclusion was not exactly correct. He stated that the electric field strength is proportional to the porosity, assuming that the terms in brackets are independent of porosity. Our derivations indicate that the electric field strength within the scope of this assumption is completely independent of porosity, Equation (5.105). However, the seismoelectric current is indeed proportional to the porosity. We can show this using the appropriate model for conductivity of the porous body, for instance, the Maxwell-Wagner theory. It yields an expression for conductivity of heterogeneous systems, Equation (2.87). In the case of negligible surface conductivity, when Du  1: Ks 1f Oe ¼ ¼ Km 1 þ 0:5f 1:5  0:5Oe

ð5:114Þ

232

CHAPTER

5

At the same time, we can use the equation for the density of the porous body, Equation (5.109). Substituting this equation in the expression for the current Equation (5.110), we would get the following dependence of porosity: ! e0 em z 1 Oe Isee ¼ rP ð5:115Þ 1 r  Oe þ r p ð1  Oe Þ 1:5  0:5Oe m

seismoelectric current dependence on porosity

Figure 5.12 illustrates this dependence of the seismoelectric current on porosity for the limiting case of a rigid skeleton. It is clear that this dependence is close to linear if the porosity is not very large. In the vicinity of the porosity value around 0.6, seismoelectric current reaches a maximum and then tends to zero as porosity tends to 1. The theory presented here has several serious limitations. First of all, the DL thickness is assumed to be much smaller than the pore size. This is the condition of a thin DL. It is usually assumed for streaming potential/current studies. However, there are some papers regarding streaming potential in nonaqueous media when the DL thickness becomes comparable to the pore size [69, 70]. Second, we neglected surface conductivity. The Dukhin number is assumed to be much less than 1. Third, we ignored all relaxation processes. Frequency is assumed to be low enough that all fields can reach their steady space distributions. There are several critical frequencies that are supposed to be much larger than the measurement frequency: hydrodynamic, Maxwell-Wagner for electric field, electrokinetics for DL, and rheological.

density ratio = 3

0.2

0.15 density ratio = 2 0.1

0.05

0.2

0.4

0.6

0.8

1

porosity

FIGURE 5.12 Dependence of the seismoelectric current on porosity according to Equation (5.115) for the large compressibility case and for two different ratios of the skeleton density to the liquid density.

233

Electroacoustic Theory

Hydrodynamic relaxation restriction on frequency and pore size equals o  oh ¼

2 rm r 2

ð5:116Þ

Maxwell-Wagner relaxation restricts frequency as follows: o

Km oMW em

ð5:117Þ

Simultaneous validity of the conditions (Equation 5.116) and (Equation 5.117) is sufficient for establishing steady-state electrokinetic flow in the pores. Finally, the rheological stress field should reach a steady configuration, which restricts the frequency as follows: o

2pcs r

ð5:118Þ

The last limitation of the presented theory is the assumption that the DLs do not overlap in the pores. It is possible to derive an expression of the seismoelectric current for the opposite case of completely overlapped DLs. This situation is similar to the homogeneous model of the completely overlapped DLs discussed in the previous section for nonaqueous systems. Seismoelectric current in this case is simply the product of the average velocity of the relative phase motion, urel, and electric charge density in the pores, r0. We can use the expression (5.99) for the first parameter and expression (5.71) for the last one. As a result, we come to the following expression of seismoelectric current for the homogeneous DL: Isee ¼

RTk2 em rm sinh~z kD f rP Frs

ð5:119Þ

where ~z ¼ Fz = RT This expression indicates that the seismoelectric current in the case of completely overlapped DLs is also proportional to the porosity if we neglect porosity dependence of the density and the porous body elasticity. In addition, it is proportional to the square of the pore size. The filtration coefficient kD is proportional to r2. For instance, for the system of cylindrical capillaries, kD ¼ r2/8. These theoretical derivations would be experimentally tested in Chapter 13.

234

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5

REFERENCES [1] P. Debye, A method for the determination of the mass of electrolyte ions, J. Chem. Phys. 1 (1933) 13–16. [2] A.G. Ivanov, Bull. Aca. Sci. USSR Ser. Geogr. Geophys. 5 (1940) 699. [3] J. Frenkel, On the theory of seismic and seismoelectric phenomena in a moist soil, J. Phys. USSR 3 (5) (1944) 230–241, Re-published J. Eng. Mech. (2005). [4] M. Williams, An electrokinetic transducer, Rev. Sci. Instrum. 19 (1948) 640–645. [5] E. Yeager, J. Bugosh, F. Hovorka, J. McCarthy, The application of ultrasonics to the study of electrolyte solutions. II. The detection of Debye effect, J. Chem. Phys. 17 (1949) 411–415. [6] B. Derouet, F. Denizot, C. R. Acad. Sci. Paris, 233 (1951) 368. [7] J. Bugosh, E. Yeager, F. Hovorka, The application of ultrasonic waves to the study of electrolytes. I. A modification of Debye’s equation for the determination of the masses of electrolyte ions by means of ultrasonic waves, J. Chem. Phys. 15 (1947) 592–597. [8] E. Yeager, F. Hovorka, The application of ultrasonics to the study of electrolyte solutions. III. The effect of acoustical waves on the hydrogen electrode, J. Chem. Phys. 17 (1949) 416–417. [9] E. Yeager, F. Hovorka, Ultrasonic waves and electrochemistry. I. A survey of the electrochemical applications of ultrasonic waves, J. Acoust. Soc. Am. 25 (1953) 443–455. [10] E. Yeager, H. Dietrich, F. Hovorka, Ultrasonic waves and electrochemistry. II. Colloidal and ionic vibration potentials, J. Acoust. Soc. Am. 25 (1953) 456. [11] R. Zana, E. Yeager, Determination of ionic partial molal volumes from ionic vibration potentials, J. Phys. Chem. 70 (1966) 954–956. [12] R. Zana, E. Yeager, Quantitative studies of ultrasonic vibration potentials in polyelectrolyte solutions, J. Phys. Chem. 71 (1967) 3502–3520. [13] R. Zana, E. Yeager, Ultrasonic vibration potentials in tetraalkylammonium halide solutions, J. Phys. Chem. 71 (1967) 4241–4244. [14] R. Zana, E. Yeager, Ultrasonic vibration potentials and their use in the determination of ionic partial molal volumes, J. Phys. Chem. 71 (1967) 521–535. [15] R. Zana, E. Yeager, Ultrasonic vibration potentials, Mod. Aspects Electrochem. 14 (1982) 3–60. [16] J. Hermans, Philos. Mag. 25 (1938) 426. [17] A.J. Rutgers, W. Rigole, Ultrasonic vibration potentials in colloid solutions, in solutions of electrolytes and pure liquids, Trans. Faraday Soc. 54 (1958) 139–143. [18] J.A. Enderby, On electrical effects due to sound waves in colloidal suspensions, Proc. Roy. Soc. Lond. A207 (1951) 329–342. [19] F. Booth, J. Enderby, On electrical effects due to sound waves in colloidal suspensions, Proc. Am. Phys. Soc. 208A (1952) 32. [20] T. Oja, G. Petersen, D. Cannon, Measurement of electric-kinetic properties of a solution, US Patent 4,497,208, 1985. [21] B.J. Marlow, T. Oja, P.J. Goetz, Colloid analyzer, US Patent 4,907,453, 1990. [22] B.J. Marlow, D. Fairhurst, H.P. Pendse, Colloid vibration potential and the electrokinetic characterization of concentrated colloids, Langmuir 4 (1983) 611–626. [23] S. Levine, G.H. Neale, The prediction of electrokinetic phenomena within multiparticle systems. 1. Electrophoresis and electroosmosis, J. Colloid Interface Sci. 47 (1974) 520–532.

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[24] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190 (1988) 71–86. [25] R.W. O’Brien, B.R. Midmore, A. Lamb, R.J. Hunter, Electroacoustic studies of moderately concentrated colloidal suspensions, Faraday Discuss. Chem. Soc. 90 (1990) 1–11. [26] P.F. Rider, R.W. O’Brien, The dynamic mobility of particles in a non-dilute suspension, J. Fluid Mech. 257 (1993) 607–636. [27] H. Ohshima, Dynamic electrophoretic mobility of spherical colloidal particles in concentrated suspensions, J. Colloid Interface Sci. 195 (1997) 137–148. [28] J.P. Ennis, A.A. Shugai, S.L. Carnie, Dynamic mobility of two spherical particles with thick double layers, J. Colloid Interface Sci. 223 (2000) 21–36. [29] J.P. Ennis, A.A. Shugai, S.L. Carnie, Dynamic mobility of particles with thick double layers in a non-dilute suspension, J. Colloid Interface Sci. 223 (2000) 37–53. [30] A.S. Dukhin, V.N. Shilov, Yu. Borkovskaya, Dynamic electrophoretic mobility in concentrated dispersed systems. Cell model, Langmuir 15 (1999) 3452–3457. [31] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. New theory and CVI experiment, Langmuir 15 (1999) 6692–6706. [32] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. Effect of the surface conductivity, Langmuir 16 (2000) 2615–2620. [33] A.S. Dukhin, P.J. Goetz, Acoustic and electroacoustic spectroscopy for characterizing concentrated dispersions and emulsions, Adv. Colloid Interface Sci. 92 (2001) 73–132. [34] M. Colic, D. Morse, The elusive mechanism of the magnetic memory of water, Colloids Surf. 154 (1999) 167–174. [35] M. Lubomska, E. Chibowski, Effect of radio frequency electric fields on the surface free energy and zeta potential of Al2O3, Langmuir 17 (2001) 4181–4188. [36] R.W. O’Brien, Electro-Acoustic Effects in a Dilute Suspension of Spherical Particles, Preprint, School of Mathematics, The University of New South Wales, Sydney, Australia, 1986. [37] R.W. O’Brien, W.N. Rowlands, R.J. Hunter, Determining charge and size with the acoustosizer, in: S.B. Malghan (Ed.), Electroacoustics for Characterization of Particulates and Suspensions, NIST, 1993, pp. 1–21. [38] R.W. O’Brien, Determination of particle size and electric charge, US Patent 5,059,909, Oct.22, 1991. [39] R.W. O’Brien, Particle size and charge measurement in multi-component colloids, US Patent 5,616,872, 1997. [40] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190 (1988) 71–86. [41] R.W. O’Brien, T.A. Wade, M.L. Carasso, R.J. Hunter, W.N. Rowlands, J.K. Beattie, Electroacoustic determination of droplet size and zeta potential in concentrated emulsions, J. Colloid Interface Sci. (1996). [42] R.W. O’Brien, D.W. Cannon, W.N. Rowlands, Electroacoustic determination of particle size and zeta potential, J. Colloid Interface Sci. 173 (1995) 406–418. [43] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65. [44] H. Ohshima, Colloid vibration potential and ion vibration potential in dilute suspensions of spherical colloidal particles, Langmuir 21 (2005) 12100–12108.

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[45] A. Weinmann, Phenomenological theory of ultrasonic vibration potentials in liquids and electrolytes, Proc. Phys. Soc. LXXIII (3) (1958) 345–353. [46] A. Weinmann, Theory of ultrasonic vibration potentials in pure liquids, Proc. Phys. Soc. LXXIV (1) (1959) 102–108. [47] J. Lyklema, Fundamentals of Interface and Colloid Science, vol. 1–3, Academic Press, London/New York, 1995–2000. [48] M.P. Bondarenko, V.N. Shilov, About some relations between kinetic coefficients and on the modeling of interactions in charged dispersed systems and membranes, J. Colloid Interface Sci. 181 (1996) 370–377. [49] A. Katchalsky, P.F. Curran, Nonequlibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, 1967. [50] H.R. Kruyt, Colloid Science, vol. 1: Irreversible Systems, Elsevier, New York, 1952. [51] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and the Double Layer in Dispersed Systems and Polyelectrolytes, Wiley, New York, 1974. [52] M. Loewenberg, R.W. O’Brien, The dynamic mobility of nonspherical particles, J. Colloid Interface Sci. 150 (1992) 159–168. [53] P.F. Rider, R.W. O’Brien, The dynamic mobility of particles in a non-dilute suspension, J. Fluid. Mech. 257 (1993) 607–636. [54] H. Ohshima, T.W. Healy, L.R. White, R.W. O’Brien, Sedimentation velocity and potential in a dilute suspension of charged spherical colloidal particles, J. Chem. Soc. Faraday Trans. 80 (1984) 1299–1377. [55] V.N. Shilov, N.I. Zharkih, Yu.B. Borkovskaya, Theory of nonequilibrium electrosurface phenomena in concentrated disperse system. 1. Application of nonequilibrium thermodynamics to cell model, Kolloid. Zh. 43 (1981) 434–438. [56] A.S. Dukhin, P.J. Goetz, Acoustic spectroscopy for concentrated polydisperse colloids with high density contrast, Langmuir 12 (1996) 4987–4997. [57] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65. [58] S.S. Dukhin, B.V. Derjaguin, Electrokinetic phenomena, in: E. Matijevic´ (Ed.), Surface and Colloid Science, vol. 7, Wiley, New York, 1974. [59] S.S. Dukhin, N.M. Semenikhin, Theory of double layer polarization and its effect on electrokinetic and electrooptical phenomena and the dielectric constant of dispersed systems, Kolloid. Zh. 32 (1970) 360–368. [60] R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. II 74 (1978) 1607. [61] M. Smoluchowski, Handbuch der Electrizitat und des Magnetismus, vol. 2, Barth, Leipzig, 1921. [62] J. Huckel, Phys. Z. 25 (1924) 204. [63] D.C. Henry, Trans. Faraday Soc. 44 (1948) 1021. [64] H. Ohshima, J. Colloid Interface Sci. 168 (1994) 269–271. [65] A.J. Babchin, R.S. Chow, R.P. Sawatzky, Electrokinetic measurements by electroacoustic methods, Adv. Colloid Interface Sci. 30 (1989) 111. [66] H. Ohshima, Dynamic electrophoretic mobility of spherical colloidal particles in salt-free media, J. Colloid Interface Sci. 265 (2003) 422–427. [67] V.N. Shilov, Y.B. Borkovskaya, A.S. Dukhin, Electroacoustic theory for concentrated colloids with overlapped dls at arbitrary ka. Application to nanocolloids and nonaqueous colloids, J. Colloid Interface Sci. 277 (2004) 347–358.

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[68] V.N. Shilov, A.S. Dukhin, Seismoelectric effect—streaming current at ultrasonic frequencies. Theory, J. Colloid Interface Sci. submitted for publication 2009. [69] P. Chowdian, D.T. Wasan, D. Gidaspow, On the interpretation of streaming potential data in non-aqueous media, Colloids Surf. 7 (1983) 291–299. [70] G.B. Tanny, E. Hoffer, Hyperfiltration by polyelectrolyte membranes. 1. Analysis of the streaming potential, J. Colloid Interface Sci. 44 (1973) 21–36.

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Chapter 6

Experimental Verification of the Acoustic and Electroacoustic Theories

6.1. 6.2. 6.3. 6.4.

Viscous Losses Thermal Losses Structural Losses Scattering Losses

239 243 247 250

6.5. Electroacoustic Phenomena References

254 258

In the previous sections describing acoustic and electroacoustic theory, we tried to justify the superposition approach to the description of the interaction of ultrasound with colloids. Altogether, six interaction mechanisms have been introduced. They are assumed to operate independently, with very little interaction in real-world practical colloids. In this chapter we demonstrate that it is indeed possible to prepare a real disperse system in which only one of the six mechanisms dominates the acoustic and electroacoustic properties of the colloid. This chapter is structured such that each subsection corresponds to a different mechanism of ultrasound interaction. In each of these subsections we present experimental tests that verify the corresponding theory.

6.1. VISCOUS LOSSES According to our theory, it is the viscous loss that is dominant, compared to all other loss mechanisms, for those colloids composed of submicrometersized rigid particles that have a substantial density contrast with respect to their suspension medium. It is obvious that a wide variety of such colloids exist. We will test our viscous loss theory by selecting two examples of stable colloids by performing a dilution experiment and testing whether (1) the measured attenuation conforms to the theory, (2) the calculated size using this theory is correct, and, importantly, (3) the calculated size remains constant Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23006-5 Copyright # 2010, Elsevier B.V. All rights reserved.

239

240

CHAPTER

6

with dilution. If these three criteria are met, then we have strong evidence that the viscous mechanism theory adequately describes the particle-particle interactions. We closely follow here the published works by Dukhin and Goetz [1, 2] using a Dispersion Technology Acoustic Spectrometer DT-100 (Mount Kisco, NY, USA). The protocol for testing volume fraction dependence involves making an equilibrium dilution of an initially concentrated and stable system. The initial concentrate itself must be stable if we are to ensure that the actual particle size remains constant throughout the experiment. The dilution protocol requires that we obtain a suitable diluent that is chemically identical, in all respects, to the original suspending medium, in order to ensure stability as we dilute the sample. The chemical composition of the medium must remain the same for all volume fractions. In summary, if the initial sample is stable, and we retain the same chemical environment, then the particle size should remain constant for any volume fraction. In other words, the measured particle size should be independent of the volume fraction. We performed this dilution test using dispersions of two different materials: colloidal silica (Ludox TM-50) and rutile (R-746), both produced by DuPont. We selected silica because of its small size; the nominal area-based size, reported by DuPont, based on titration measurement, is approximately 22 nm and it is provided as a 50 wt% aqueous slurry. Acoustic measurement of the silica gave a weight-basis particle size of 28 nm. We selected rutile as an example of enhanced density contrast. We took 100 ml of this dispersion and weighed it; it weighed 234 g, which yields a particle density of 3.9 g/cm3. This density was somewhat lower than the density of regular rutile, perhaps because of the presence of stabilizing additives. The rutile sample is also supplied as an aqueous slurry, at a weight fraction of 76.8%; the nominal size is about 300 nm. We used two different techniques to produce the equilibrium dispersion medium needed for dilution: centrifugation and dialysis. For the larger rutile particles, having a significant density contrast, we easily extracted the needed equilibrium solution by centrifugation. However, such gravimetric separation does not work for the very small silica Ludox particles; for this we employed dialysis. The measured attenuation spectra for the silica and rutile samples are shown in Figures 6.1 and 6.2, respectively. It is clear that the attenuation of the silica is much smaller than that of the rutile. This is because of the smaller size and lower density contrast of the silica. For the silica slurry, the attenuation spectra are almost indistinguishable at concentrations above 9 vol%. This reflects a nonlinear dependence of the attenuation with volume fraction. It arises because of particle-particle interactions. Such nonlinearity has been known for quite some time [1] and it is even more pronounced for rutile (Figure 6.2). The interactions also shift the critical frequency to higher values. Hence the attenuation, at low frequency, decreases

241

Experimental Verification of the Acoustic and Electroacoustic Theories

0.6

volume fraction 0.15 0.18 0.21 0.24 0.27 0.312 0.092 0.06 0.03 0.01 0.043 0.09 0.12

Attenuation (dB/cm/MHz)

0.5

0.4

0.3

>9 vol% 6 vol% 4.3 vol% 3 vol%

1 vol%

0.2

0.1

0.0 100

101

102

Frequency (MHz) FIGURE 6.1 Attenuation spectra of 28-nm silica at various volume fractions. 16

volume fraction

Attenuation (dB/cm/MHz)

14

40.8% 36.2% 32% 25.8% 21.4%

12 10 8 6

vfr=45.9%

4

vfr=16.6% vfr=11.8%

vfr=5.1% vfr=3.2% vfr=1.1%

2 0

100

101

102

Frequency (MHz) FIGURE 6.2 Attenuation spectra of 300-nm rutile at various concentrations.

with increasing volume fraction above 16.6%. It is exactly the same effect that makes the attenuation constant with volume fraction for silica. Our new theory takes this nonlinear effect into account. As a result, the values for particle size, calculated from these attenuation spectra, become almost constant irrespective of volume fractions for silica (Figure 6.3). The increase in the

242

CHAPTER

6

0.5

Median size (mm)

0.4 rutile 0.3

0.2

0.1 silica Ludox 0.0 0

5

10

15

20 25 30 Volume fraction (%)

35

40

45

50

FIGURE 6.3 Median particle size of rutile and silica Ludox calculated from the attenuation spectra in Figures 6.1 and 6.2.

particle size of the rutile slurry at high volume fraction cannot be accounted for by particle-particle interaction. It is plausible that some increase might be due to unavoidable aggregation, but it is also possible that the theory may yet need some further refinement. Nevertheless, using the simple dilute case theory would certainly yield particle sizes that would decrease very dramatically with increasing volume fraction. In the case of rutile sample, the dilute case theory would yield a size of about 100 nm at the highest volume fraction, giving an error exceeding 200%. In addition to attenuation, the viscous loss theory also yields an expression for sound speed, as the real part of a complex wavenumber (Equations (2.24) and (4.37)). The dilution experiment allows us to test this part of the theory as well. Figure 6.4 shows the experimental and theoretical sound speed for silica slurries at various volume fractions. The variation in sound speed is relatively small: only about 2% for a change in volume fraction from 1% to 31%. Nevertheless, this small variation is measurable, and is in good agreement with theoretical calculations. Verifying the accuracy of the size measurement at high volume fraction requires standard materials with well-known particle size. Unfortunately, there are no accepted particle size standards for materials in the submicrometer range that have, in addition, a high density contrast. Traditional polymer latex standards do not exhibit the needed viscous loss because of their low density contrast. Accordingly, we show here data that we have

Experimental Verification of the Acoustic and Electroacoustic Theories

1540

243

experiment theory

Sound speed (m/s)

1530 1520 1510 1500 1490 1480 0

10

20 Volume fraction (%)

30

40

FIGURE 6.4 Sound speed at 10 MHz of silica Ludox at various volume fractions.

collected over the last 5 years using 10 commercially available slurries, namely, colloidal silica Ludox TM, silica Geltech 1.5 and 0.5, silica quartz BCR-70, silica Cabot SS25, rutile Dupont R746, alumina Sumitomo AKP 15 and AKP30; zirconia TOSO TZ-3YS, and silicon nitride Ube E-10. The particle sizes of these materials, as determined by their respective manufacturers, are summarized in Table 6.1 and compared with the particle sizes determined using acoustic attenuation. Figures 6.5 and 6.6 show the attenuation spectra for all 10 materials at various volume fractions. All the slurries are considered relatively stable. The solid lines represent the best theoretical fit to the experimental data that can be achieved with the new theory for viscous losses, using the particle size distribution as an adjustable parameter. The median particle size corresponding to this best fit for each slurry is given in Table 6.1. The median size measured using acoustic spectroscopy agrees very well with the independent data from the manufacturer’s literature. Unfortunately, we do not know the measuring technique used by every manufacturer to determine the particle size of their material. Nevertheless, the general agreement for this wide variety of materials is compelling evidence for the accuracy of our viscous loss theory. The precision will be discussed later in Chapter 7.

6.2. THERMAL LOSSES Allegra and Hawley [3] provided an early demonstration of the ability of Isacovich’s theory to characterize moderately concentrated colloids. They observed almost perfect correlation between experiment and their dilute case

244

CHAPTER

6

TABLE 6.1 Median Particle Sizes Reported by Manufacturer and Calculated from the Acoustic Attenuation Spectra for Various Solid Particle Materials Sample material

Median size (manufacturer’s data), µm

Median size (acoustics), µm

Silica Ludox TM

0.022a

0.028

Silica Geltech 1.5

1.5

1.55

Silica Geltech 0.5

0.5

0.53

2.9

b

2.85

Silica Cabot SS25

0.1

c

0.087

Rutile Dupont R746

0.3

0.31

Alumina Sumitiomo AKP-15

0.7

0.67

Alumina Sumitomo AKP-30

0.3

0.33

Zirconia TOSO TZ-3YS

0.3

0.34

Silicon nitride Ube E-10

0.5

0.47

Silica quartz BCR 70

a

Using titration method, thus giving smaller area-weighted particle size. Using sedimentation method, thus giving a weight-basis hydrodynamic size. Using chromatographic hydrodynamic fractionation (CHDF), thus emphasizing the largest dimension for irregularly shaped particles. b c

ECAH theory for several systems: a 20 vol% toluene emulsion, a 10 vol% hexadecane emulsion, and a 10 vol% polystyrene latex dispersion. This fortuitous result can be understood by noting that their ECAH dilute case theory reduces to Isacovich’s theory for the special case of submicrometer, soft particles. Studies by McClements and Povey [4–6], using emulsions, have provided similar results. Recent work by Holmes, Challis, and Wedlock [7–9] also shows good agreement between the ECAH theory and experiments on a 30 vol% polystyrene latex suspension. Dukhin and Goetz [10] have shown that particle-particle interactions can be neglected for small (160 nm) neoprene latex dispersions. In this case, they found that the attenuation was a linear function of volume fraction up to 30 vol% (Figures 4.4 and 4.6). Such linear behavior is a strong indication that each particle fraction contributes to the total attenuation independently of other fractions, that is, the total attenuation is a superposition of the individual contributions. Superposition works only when particle-particle interactions are insignificant.

245

Experimental Verification of the Acoustic and Electroacoustic Theories

2.0

Attenuation (dB/cm/MHz)

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 100

101 Frequency (MHz) silica silica silica silica silica

102

Ludox-TM, 4.8 vol% Cabot, SS25, 14.2 vol% Geltech 1.5, 7 vol% Geltech 0.5, 6.3 vol% quartz BCR in ethanol, 4.3 vol%

FIGURE 6.5 Experimental attenuation spectra compared with theoretical fit calculated for various silica suspensions.

Attenuation (dB/cm/MHz)

5

4

3

2 alumina AKP 15, 5 vol% zirconia TZ-3YS, 3 vol% rutile Dupont R746, 2.7 vol% alumina AKP-30, 4.7 vol% silicon nitride, Ube, 10 vol%

1

0 100

101 Frequency (MHz)

102

FIGURE 6.6 Experimental attenuation spectra and theoretical fit calculated for various solid materials.

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6

Anson and Chivers have also shown [11] that thermal losses are important for soft particles up to ka ¼ 0.5. Above this limit, other mechanisms may dominate the attenuation. However, for those emulsion systems in which, of all the relevant physical properties, only the thermal ones differ substantially between the two phases, the thermal losses may be important for much larger values of ka. Their study involved 72 liquids. Water-based emulsions are somewhat unique because of their unusual thermal properties. Allison and Richardson [12] measured attenuation in several emulsions including benzene in water, water in benzene, and benzene in glycerin-water mixtures. To interpret their experiments, they applied a combination of Urick’s theory for the viscous losses and the Rayleigh scattering theory. They completely neglected “thermal losses” and consequently observed large deviations between experiment and their theory. One of the most convincing confirmations of Isacovich’s theory was obtained by Wines et al. [13] with a classical water-in-heptane microemulsion, stabilized with amount of surfactant (AOT) surfactant. Independent information of the droplet size was obtained using both neutron nd X-ray scattering [14–19]. Figure 6.7 shows the data from these earlier experiments together with new data obtained from attenuation spectroscopy. The droplet size is seen to increase with increasing water content from just a few nanometers to about 20 nm, probably because of a deficit in the AOT necessary to stabilize the additional surface. Notwithstanding this size increase, it is clear that the particle size calculated from the attenuation spectra using Isacovich’s theory correlates well with the particle size from two totally different techniques. 24 22

X-rays scattering (1982) neutron scattering (1979) neutron scattering (1982) acoustics

Mean diameter (nm)

20 18 16 14 12 10 8 6 4 2 0 0

10

20

30 40 50 Water-to-AOT molar ratio

60

70

FIGURE 6.7 Comparison of the median particle size of a water-in-heptane microemulsion measured using several techniques.

247

Experimental Verification of the Acoustic and Electroacoustic Theories

Recently, the focus of research in this area has shifted towards more concentrated systems where Isacovich’s theory certainly fails. In particular are the experimental tests by McClements, and Hemar et al. [20–23] for verifying the core-shell model theory. These experiments were made using hexadecanein-water emulsions and corn oil-in-water emulsions. The results from these studies show that deviation occurs (from Isacovich’s dilute case theory) at volume concentrations exceeding approximately 30%. However, the new core-shell model provides a good fit with experimental data up to 50 vol%. In summary, we can conclude that multiple experiments performed by different independent groups have confirmed the validity both of Isacovich’s theory for dilute soft colloids and the core-shell model theory for more concentrated soft colloids.

6.3. STRUCTURAL LOSSES Here we use data from experiments performed at the National Institute for Resources and Environment, Tsukuba, Japan [24]. The authors used two alumina powders: Showa Denko AL-160SG-4 and Sumitomo Chemical Industry ALM-41-01. The median size of very dilute aqueous samples of each powder was measured using both laser diffraction and photocentrifugation (Table 6.2). Slurries of both alumina powders, stabilized using a sodium polycalboxyl acid surfactant and ball-milled for 3 days before measurement, were prepared at various concentrations from 1 to 40 vol%. The acoustic attenuation spectrum was then measured for both slurries at various concentrations. For concentrations up to 20 vol%, the particle size calculated from these attenuation spectra

TABLE 6.2 Measured Particle Size of Two Alumina Samples, Showing the Importance of Structural Losses Measurement details

Median particle size, µm ALM-41-01

AL-160SG-4

Photocentrifugation, Horiba CAPA-700, very dilute sample

1.47

0.56

Laser diffraction, Sympatec Helos, very dilute sample

1.98

0.71

Acoustic spectroscopy, DT1200, <20 vol%

1.79

0.52

Acoustic spectroscopy, DT1200, 40 vol%. Initial analysis ignoring structural losses

1.07 (fit error 19.2%)

0.8 (fit error 18.4%)

Acoustic spectroscopy, DT1200, 40 vol%. Subsequent analysis including structural losses

1.63 (fit error 6.1%)

0.77 (fit error 2.3%)

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6

agreed quite well with independent data on dilute samples. However, for the highest volume fraction of 40%, the agreement was not as good. The actual attenuation spectrum for both slurries at 40 vol% is shown in Figures 6.8 and 6.9. In both cases, it is clear that the experimental data exceed the theoretical values by a very substantial degree. The theory, however, is

20

experiment intrinsic log, total

Attenuation (dB/cm/MHz)

18 16 14 12 10 8 6 4 2 0 100

101 Frequency (MHz)

102

FIGURE 6.8 Experimental attenuation for alumina AL-160SG-4 and theoretical fit assuming only viscous losses.

16

Attenuation (dB/cm/MHz)

14 12 10 8 6 experiment intrinsic log, total

4 2 0 100

101 Frequency (MHz)

102

FIGURE 6.9 Experimental attenuation for alumina ALM-41-01 and theoretical fit assuming only viscous losses.

249

Experimental Verification of the Acoustic and Electroacoustic Theories

based only on viscous losses. Hence, based on this “excess attenuation,” the authors concluded that there was an as-yet-unknown factor that becomes significant only at high volume fractions. We suggest the possibility of “structural losses” to explain this excess attenuation. We have described such structural loss in some detail previously (Section 4.2.4). Specifically, Equation (4.61) can be included in the analysis to compute any additional structural loss (to account for this excess attenuation). In doing this, we assume that the first coefficient, H1, is zero. The second coefficient, H2, is then used as an adjustable parameter, in addition to the median size and standard deviation that are used to define the lognormal particle size distribution. In essence, the searching routine looks for that particle size distribution and that value of H2 which provide the best fit between theory and experiment. The addition of this new adjustable parameter, H2, allowed us to achieve a much better fit between theory and experiment for both alumina samples, as can be clearly seen in Figures 6.10 and 6.11. The resultant particle size and fitting errors, after allowing for these structural losses, are given in Table 6.2. Since the addition of structural losses leads to a dramatic improvement in the fitting error, this strongly suggests that such a mechanism does explain, at least in part, the observed excess attenuation. The calculated particle size data confirms this conclusion, because the calculated values are then much closer to independent measurements on the dilute samples and to acoustic measurements of the more dilute case.

20

Attenuation (dB/cm/MHz)

18 16 14 12 10 8

experiment log, total

6 4 2 0 100

101 Frequency (MHz)

102

FIGURE 6.10 Experimental attenuation for alumina AL-160SG-4 and theoretical fit including both viscous and structural losses.

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16

Attenuation (dB/cm/MHz)

14 12 10 8 6 4 2 0 100

101 Frequency (MHz) experiment

intrinsic

102 log, total

FIGURE 6.11 Experimental attenuation for alumina ALM-41-01 and theoretical fit including both viscous and structural losses.

It is interesting that the value of the adjustable parameter, coefficient H2, turns out to be the same for both samples, that is, 0.8. It is independent of the particle size, as one might expect, because it is a property of the polymer “springs.” This parameter characterizes the flexibility of the bonds linking the particles together into a structure at high volume fractions. In the case of polymer bridging, it characterizes the rheology of the polymer chains. This rheological property might be very useful in optimizing the chemical formulation of concentrated systems, such as these ceramic slips, to obtain specific structural characteristics. Unfortunately, we do not know of any independent way to calculate or measure this parameter. The addition of these structural losses is only justified, however, when a simpler theory neglecting such losses fails, and the experiment shows such a characteristic excess attenuation. This excess attenuation then becomes a source of additional experimental information about the microrheological properties of concentrated systems.

6.4. SCATTERING LOSSES There appear to be only two experimental studies where the scattering losses are the dominant mechanism and all other losses can be neglected. The first was by Faran [25], who in 1951 measured the angular dependence of scattered ultrasound for large spheres and cylinders. The second was by Busby and

251

Experimental Verification of the Acoustic and Electroacoustic Theories

Richrdson [26], who in 1956 measured the attenuation of large (95 µm) glass spheres. They observed a linear dependence of attenuation with respect to concentration, up to 13.6 vol%. There are many other studies in which the scattering was not dominant, but rather just one component of the total attenuation spectra. One example was shown previously in Figure 4.2, for the silica quartz BCR-70 particle size standard. The scattering losses for this sample dominate only the highfrequency portion of the attenuation spectra, whereas at low-frequency viscous dissipation is predominant. Ideally, to verify the scattering loss theory, the particle size must be sufficiently large to eliminate the contributions by any absorption mechanism to the total attenuation spectrum. We have made such a test using glass beads of different sizes produced by Sigmund Lindner GmbH. These particles are quite monodisperse and almost spherical. The median particle size varies from 25 to 200 µm (Table 6.3). In addition, we tested alumina Sumitomo AA-18 having a particle size reported by the manufacturer of approximately 18 µm. These particles are also sufficiently large to eliminate ultrasound absorption over the frequency range of 3–100 MHz. Both the glass and alumina dispersions were prepared in distilled water; no stabilizer was used. Attenuation spectra were measured using a Dispersion Technology DT-100 acoustic spectrometer. This instrument measures the attenuation of the transmitted sound pulse, as opposed to the sound scattered outside the incident beam path, which eliminates undesirable nonlinear effects that result from multiple scattering [27]. To demonstrate the absence of such multiple scattering, we measured one of the glass samples at different concentrations, TABLE 6.3 Particle Size Calculation Using Simple Algorithm Sample Size range, Median size, Median Critical Wavelength, ID manufacturer’s manufacturer’s size, frequency, µm data, µm data, µm acoustics, MHz µm S 5210

0–50

25

30

30

50

S 5211

40–70

55

54

18

83

S 5212

70–110

90

74

13

115

S 5213

90–150

120

107

9

166

S 5214

100–200

150

137

7

214

18

14

70

22

AKP-18

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6

Attenuation at 5% MHz (dB/cm/MHz)

30 linear regression 25 20 15

statistical confidence range

10 5 0 0

5

10

15

20 25 30 35 Volume fraction (%)

40

45

50

FIGURE 6.12 Attenuation of 120-µm glass spheres at 5 MHz as function of concentration.

from 1 to 46 vol%. The measured attenuation is seen to depend linearly with concentration, within the statistical confidence range (Figure 6.12). Figures 6.13 and 6.14 show the attenuation spectra for various large particle dispersions. The concentration in each case was 4 vol%. The samples were continuously pumped through the chamber using a peristaltic pump to prevent sedimentation. 3

d50=55

d50=25

Attenuation (dB/cm/MHz)

d50=90

400-800 90-150 70-110 0-50 40-70 100-200

d50=150

2

size range [mm]

d50=120

1

d50=600

0 100

101 Frequency (MHz)

FIGURE 6.13 Attenuation spectra for glass spheres having different sizes.

102

253

Experimental Verification of the Acoustic and Electroacoustic Theories

Attenuation per 1 vol% (dB/cm/MHz)

0.7 0.6 0.5

quartz <50 mm

0.4 0.3 alumina 18 mm

0.2 0.1 0.0 100

FIGURE 6.14

101 Frequency (MHz)

102

Attenuation spectra normalized by volume fraction for glass spheres and alumina.

As predicted by theory, the attenuation spectra are bell-shaped. The position of the critical frequency shifts to lower values with increasing particle size, which is in agreement with the theoretical predictions of Morse [27]. According to Morse, there is a simple way to estimate the median particle size from these attenuation spectra. The critical frequency at which attenuation reaches a maximum can be defined with: pd 2 ð6:1Þ l This simple equation allows us to calculate the median particle size as approximately two-thirds of the wavelength at the critical frequency. Table 6.3 lists the particle sizes calculated from the peak frequency from each curve in Figure 6.13. This calculated particle size is very close to the independent particle size data provided by the manufacturer. These results confirm the main conclusions of the Morse theory. There is, however, one unexplained feature of the experimental data: the variation of the amplitude of the attenuation spectra with particle size. It is not completely clear what causes this variation. It is not density variation; alumina and quartz with similar particle size range exhibit the same attenuation per 1 vol% (Figure 6.14). There are two potential factors to explain this phenomenon: polydispersity and particle shape. The latter might be much more important for scattering losses compared to the various mechanisms of ultrasound absorption. We conclude from this test that Morse’s scattering theory allows the particle size range, which can be characterized using acoustics, to be expanded up to millimeters. ka ¼

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6.5. ELECTROACOUSTIC PHENOMENA The first experimental confirmation of the electroacoustic phenomena in colloids was made by Hermans [28] in 1938. Then, in 1958 Rutgers and Rigole published data on the colloid vibration potential (CVP) of silver iodide sols [29]. At about the same time, Yeager’s group made measurements of colloidal silica (Ludox) at various weight fractions [30]. They showed that the measured signal (CVP) increased with weight fraction, and they observed this increase for solutions having different ionic strengths. These first measurements were performed using an instrument employing a continuous wave technique. This creates complications when attempting to separate the desired electroacoustic response from other unwanted signals. Nevertheless, this first work did illustrate the existence of the electroacoustic effect in colloids, but did not yield sufficient precision for quantitative analysis. The next step forward was made by the Yeager-Zana group when they switched to a pulse technique [31] in the early 1960s. Their new measurements with colloidal silica gave the first quantitative confirmation of the main conclusions of the Enderby-Booth theory. Marlow, Fairhurst, and Pendse [32] made CVP measurements on concentrated rutile dispersions and observed a very good correlation with independent microelectrophoretic measurements. Although they observed a nonlinear dependence in the CVP on concentration starting at about 5 vol%, they managed to perform measurements up to 30 vol%. These studies confirmed the volume fraction dependence predicted by the Levine cell model. Marlow and Rowell [33] used CVP measurements to characterize the electric surface properties of coal slurries. They observed a deviation from linear dependence above 5 vol%, but were able to perform measurements even at 40 vol%. They tried to apply a “cell model” concept for describing the nonlinearity of this volume fraction dependence. This initial period of electroacoustic studies dealt only with CVP, using ultrasound as the driving force. At the end of the 1980s, the focus began to shift to the reverse electroacoustic effect, electric sonic amplitude (ESA), using the electric field as the driving force. This new direction was initiated by Cannon, O’Brien, Hunter, and others from the international AustralianUSA group associated with development of the first Matec instruments. One of the first results, presented by Hunter [34] in 1988, concerned a quantitative comparison of mobility (of polymer latex particles) as estimated from the ESA effect (after correction for inertia using O’Brien’s theory [35]) with microelectrophoresis results. Shortly afterwards, Babchin, Chow, and Samatzky [36] compared electroacoustic dynamic mobility and electrophoretic mobility for a silver iodide sol and showed very good correlation. They also measured the CVP in both bitumen-in-water emulsions and water-in-crude oil emulsions, and showed that it is possible to monitor the effect of a surfactant on emulsion stability.

Experimental Verification of the Acoustic and Electroacoustic Theories

255

The most vigorous test of O’Brien’s theory for dilute systems has been carried out by O’Brien’s own group, using monodisperse cobalt phosphate and titanium dioxide dispersions [37]. They observed very good correlation with microelectrophoresis data for different chemical compositions of the suspensions. They also observed a linear dependence of CVP for monodisperse latex of particle size 88 nm. Later ESA investigations of dilute dispersions (below 5 vol%) have been directed towards characterizing polydispersity, aggregation, and the double layer (DL model), for example the work by James et al. [38] on alumina and silicon nitride. They confirmed that it is the mass-averaged particle size that should be used when estimating inertial effects in dilute colloids. Gunnison, Rasmussen, Wall, Dahlberg, and Ennis [39] tested a standard Stern model in electroacoustics using titration of dilute aqueous hematite dispersions. They obtained good agreement at high ionic strengths of 50 and 100 mM NaNO3. They found some deviation at lower ionic strengths, which they explained as being a consequence of aggregation of the hematite particles. However, the most intriguing issue remained—the volume fraction dependence. Indeed, this was the strongest limitation on the early stages of electroacoustic theory. For instance, James, Texter, and Scales [40] showed that the ESA remains a linear function of concentration only up to 5 vol%. Their studies used polymer latices, alumina (AKP), and colloidal silica (Ludox). Using an organic pigment, Texter [41] observed a linear dependence only up to 2 vol%, and then a strongly nonlinear behavior. The overall conclusion arrived at by Hunter is that the dilute case theory is valid only up to 2–5 vol%. Beyond this concentration, particle-particle interactions result in nonlinearity. Marlow suggested using a cell model to incorporate particle-particle interactions, and derived a corrected volume fraction dependence [32]. Despite his initial success with rutile and coal dispersions [33], this path was not followed further until recently. O’Brien and Hunter even reported failure of the Levine cell model when describing ESA effects [42]. The development of the ESA theory for concentrates has proceeded along different lines, and there are experimental data confirming these new developments. For details, we suggest that the interested reader go to the original publications [42, 43]. As an example, O’Brien, Hunter, and others [44, 45] have tested the ESA theory using dairy cream, intravenous fat emulsions, and bitumen emulsions. Both particle size and zeta potential were reported at volume fractions up to 38%. The results suggest that electroacoustics can indeed be used to characterize concentrated emulsions. Now, we would like to present an experimental verification of the new CVI theory that was described in Chapter 5. From our viewpoint, this new theory has several advantages over the latest ESA theories. In particular, it does not require the use of a superposition principle to describe

256

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polydispersity. This and other advantages have already been described. The main goal of the studies described here is to test the validity of this new CVI theory in concentrated systems. There are two aspects that are subjected to experimental verification. The first one is a comparison with independent microelectrophoretic results, and the second is a test of the volume fraction dependence. We have successfully addressed the first aspect of this test many times. However, the most convincing results are from independent tests. As an example, we present results by Van Tassel and Randall [46], who studied the z-potential of both concentrated and dilute dispersions of alumina in ethanol. For the concentrated slurries, they used the CVI measurements from the Dispersion Technology Model DT-1200. For the dilute samples, they used the Coulter Delsa 440 microelectrophoretic measurements. They observed almost perfect correlation between the CVI-based data and microelectrophoretic measurements. This confirms the new CVI theory since it is the basis of the Dispersion Technology DT-1200 software. In regard to the volume fraction dependence test, the equilibrium dilution is again the logical experimental protocol, as with the case of viscous attenuation. Equilibrium dilution maintains the same chemical composition of the dispersion medium for all volume fractions, which means that the z-potential calculated from the CVI should remain constant for all volume fractions. Any variation of the z-potential with volume fraction is an indicator that a particular theory does not correctly reflect the particle-particle interactions that occur at high volume fraction. Colloidal silica (Ludox TM-50) is ideal for these studies, because the small particle size allows us to eliminate the particle size dependence dictated by Equation (5.28) since the factor G is reduced to unity. Using small particles gives another simplifying advantage; it eliminates any contribution to the overall attenuation because small particles do not attenuate sound at low frequency. Thus the choice of small particles allows us to test only the volume fraction dependence. This is critical because volume fraction dependence is what produces the most pronounced difference between the various theories. In addition, the small particle size would then reproduce the lowfrequency Smoluchowski limit, the important reference point in the electroacoustic theories. With a nominal size of less than 30 nm, colloidal silica satisfies all specified conditions of Equations (5.10) and (5.32). The measured sizes using acoustics agrees fairly well with the manufacturer’s specified values (Table 6.1). At the same time, the silica slurry also satisfies the thin DL restriction (Equation (5.32)) because of its relatively high ionic strength, about 0.1 mol/l. Otherwise, we would have to generalize the theory to remove this thin DL restriction, following the Henry-Ohshima correction [47]. The selection of rutile as the second dispersion gives us an opportunity to test the particle size dependence and enhance the density contrast

257

Experimental Verification of the Acoustic and Electroacoustic Theories

contribution. An equilibrium dilution protocol requires a pure solvent that is identical to the medium of the given dispersed system. In principle, one can try to separate the dispersed phase from the dispersion medium using either sedimentation or centrifugation. The dilution protocol used is the same as that described earlier (Section 6.1) for testing the theory for viscous losses; centrifugation for rutile and dialysis for colloidal silica. We carried out our studies using a Dispersion Technology Acoustic and Electroacoustic Spectrometer DT-1200. Details concerning the hardware are described in Chapter 7. Figures 6.15 and 6.16 summarize the results of the dilution experiment for the colloidal silica and rutile slurries, respectively. It is seen that z-potential calculated from the measured CVI using the new theory yields a z-potential that remains almost constant over the volume fraction range studied. Variations do not exceed 10%. These experimental measurements strongly support the new CVI theory for concentrated colloids. Figures 6.15 and 6.16 also show results using the dilute case theory. For rutile, the error exceeds 100%. The volume fraction dependence is, however, less pronounced for the less dense silica particles, in agreement with the results obtained by Rasmussen [48] for latex particles using ESA. This confirms one of the main positions of the new CVI theory: the magnitude of the volume fraction dependence is significantly affected by the choice of inertial frame of reference. For the CVI, this is reflected in the multiplier (rprs)/rs, and this is less volume fraction dependent for the lighter particles. Volume fraction (%) 0

5

10

15

20

25

30

35

z-potential (mV)

0

-10 dilute system theory

-20

-30

new CVI theory

FIGURE 6.15 Comparison of the zeta potential of colloidal silica calculated using the dilute case theory and the new CVI theory.

258

CHAPTER

0

5

10

15

Volume fraction in % 20 25 30 35

40

45

6

50

0 -5 z-potential (mV)

dilute system theory

-10 -15 -20 -25 -30

new CVI theory

-35 FIGURE 6.16 Comparison of the zeta potential of rutile using the dilute case theory and the new CVI theory.

REFERENCES [1] A.S. Dukhin, P.J. Goetz, Acoustic spectroscopy for concentrated polydisperse colloids with high density contrast, Langmuir 12 (1996) 4987–4997. [2] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. New theory and CVI experiment, Langmuir 15 (1999) 6692–6706. [3] J.R. Allegra, S.A. Hawley, Attenuation of sound in suspensions and emulsions: Theory and experiments, J. Acoust. Soc. Am. 51 (1972) 1545–1564. [4] J.D. McClements, M.J. Povey, Ultrasonic velocity as a probe of emulsions and suspensions, Adv. Colloid Interface Sci. 27 (1987) 285–316. [5] D.J. McClements, Ultrasonic characterization of emulsions and suspensions, Adv. Colloid Interface Sci. 37 (1991) 33–72. [6] D.J. McClements, M.J.W. Povey, Scattering of ultrasound by emulsions, J. Phys. D: Appl. Phys. 22 (1989) 38–47. [7] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth study of ultrasound velocity and attenuation in suspensions: Comparison of theory with experimental measurements, J. Colloid Interface Sci. 156 (1993) 261–269. [8] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth ultrasonic study of suspensions: The variation of velocity and attenuation with particle size, J. Colloid Interface Sci. 168 (1994) 339–348. [9] A.K. Holmes, R.E. Challis, Ultrasonic scattering in concentrated colloidal suspensions, Colloids Surf. A 77 (1993) 65–74. [10] A.S. Dukhin, P.J. Goetz, C.W. Hamlet, Acoustic spectroscopy for concentrated polydisperse colloids with low density contrast, Langmuir 12 (1996) 4998–5004. [11] L.W. Anson, R.C. Chivers, Thermal effects in the attenuation of ultrasound in dilute suspensions for low values of acoustic radius, Ultrasonic 28 (1990) 16–25.

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[12] P.A. Allison, E.G. Richardson, The propagation of ultrasonics in suspensions of liquid globules in another liquid, Proc. Phys. Soc. 72 (1958) 833–840. [13] T.H. Wines, A.S. Dukhin, P. Somasundaran, Acoustic spectroscopy for characterizing heptane/water/AOT reverse microemulsion, J. Colloid Interface Sci. 216 (1999) 303–308. [14] P.D.I. Fletcher, B.H. Robinson, F. Bermejo-Barrera, D.G. Oakenfull, J.C. Dore, D.C. Steytler, in: I.D. Rob (Ed.), Microemulsions, Plenum Press, New York, 1982, p. 221. [15] P.C. Cabos, P. Delord, J. Appl. Cryst. 12 (1979) 502. [16] S. Radiman, L.E. Fountain, C. Toprakcioglu, A. de Vallera, P. Chieux, Prog. Colloid Polym. Sci. 81 (1990) 54. [17] J.P. Huruguen, T. Zemb, M.P. Pileni, Prog. Colloid Polym. Sci. 89 (1992) 39. [18] M.P. Pileni, T. Zemb, C. Petit, Chem. Phys. Lett. 118 (4) (1985) 414. [19] B. Bedwell, E. Gulari, in: K.L. Mittal (Ed.), Solution Behavior of Surfactants, vol. 2, Plenum Press, New York, 1982. [20] Y. Hemar, N. Herrmann, P. Lemarechal, R. Hocquart, F. Lequeux, Effect medium model for ultrasonic attenuation due to the thermo-elastic effect in concentrated emulsions, J. Phys. II (7) (1997) 637–647. [21] J.D. McClements, Y. Hermar, N. Herrmann, Incorporation of thermal overlap effects into multiple scattering theory, J. Acoust. Soc. Am. 105 (2) (1999) 915–918. [22] R. Chanamai, J.N. Coupland, D.J. McClements, Effect of temperature on the ultrasonic properties of oil-in-water emulsions, Colloids Surf. 139 (1998) 241–250. [23] R. Chanamai, N. Hermann, D.J. McClements, Influence of thermal overlap effects on the ultrasonic attenuation spectra of polydisperse oil-in-water emulsions, Langmuir 15 (1999) 3418–3423. [24] T. Hayashi, H. Ohya, S. Suzuki, S. Endoh, Errors in size distribution measurement of concentrated alumina slurry by ultrasonic attenuation spectroscopy, J. Soc. Powder Technol. Jpn 37 (2000) 498–504. [25] J.J. Faran, Sound scattering by solid cylinders and spheres, J. Acoust. Soc. Am. 23 (4) (1951) 405–418. [26] J. Busby, E.G. Richrdson, The propagation of ultrasonics in suspensions of particles in liquid, Phys. Soc. Proc. 67B (1956) 193–202. [27] P.M. Morse, K. Uno Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968; Princeton University Press, New Jersey, 1986, 925pp. [28] J. Hermans, Philos. Mag. 25 (1938) 426. [29] A.J. Rutgers, W. Rigole, Ultrasonic vibration potentials in colloid solutions, in solutions of electrolytes and pure liquids, Trans. Faraday Soc. 54 (1958) 139–143. [30] E. Yeager, H. Dietrick, F.J. Hovorka, Ultrasonic waves and electrochemistry. II. Colloidal and ionic vibration potentials, J. Acoust. Soc. Am. 25 (1953) 456. [31] R. Zana, E. Yeager, Ultrasonic vibration potentials and their use in the determination of ionic partial molal volumes, J. Phys. Chem. 71 (13) (1967) 521–535. [32] B.J. Marlow, D. Fairhurst, H.P. Pendse, Colloid vibration potential and the electrokinetic characterization of concentrated colloids, Langmuir 4 (3) (1983) 611–626. [33] B.J. Marlow, R.L. Rowell, Acoustic and Electrophoretic mobilities of coal dispersions, J. Energy Fuel 2 (1988) 125–131. [34] R.J. Hunter, Liversidge lecture, Proc. R. Soc. New South Wales 121 (1988) 165–178. [35] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190 (1988) 71–86.

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[36] A.J. Babchin, R.S. Chow, R.P. Sawatzky, Electrokinetic measurements by electroacoustic methods, Adv. Colloid Interface Sci. 30 (1989) 111. [37] R.W. O’Brien, B.R. Midmore, A. Lamb, R.J. Hunter, Electroacoustic studies of moderately concentrated colloidal suspensions, Faraday Discuss. Chem. Soc. 90 (1990) 1–11. [38] M. James, R.J. Hunter, R.W. O’Brien, Effect of particle size distribution and aggregation on electroacoustic measurements of Zeta potential, Langmuir 8 (1992) 420–423. [39] M. Gunnarsson, M. Rasmusson, S. Wall, E. Ahlberg, J. Ennis, Electroacoustic and potentiometric studies of the hematite/water interface, J. Colloid Interface Sci. 240 (2001) 448–458. [40] R.O. James, J. Texter, P.J. Scales, Frequency dependence of electroacoustic (electrophoretic) mobilities, Langmuir 7 (1991) 1993–1997. [41] J. Texter, Electroacoustic characterization of electrokinetics in concentrated pigment dispersions, Langmuir 8 (1992) 291–305. [42] R.W. O’Brien, W.N. Rowlands, R.J. Hunter, Determining charge and size with the acoustosizer, in: S.B. Malghan (Ed.), Electroacoustics for Characterization of Particulates and Suspensions, NIST, 1993, pp. 1–21. [43] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65. [44] R.W. O’Brien, T.A. Wade, M.L. Carasso, R.J. Hunter, W.N. Rowlands, J.K. Beattie, Electroacoustic determination of droplet size and zeta potential in concentrated emulsions, ACS Symposium Series, 693 (1998) 311–319. [45] L. Kong, J.K. Beattie, R.J. Hunter, Electroacoustic study of concentrated oil-in-water emulsions, J. Colloid Interface Sci. 238 (2001) 70–79. [46] J. Van Tassel, C.A. Randall, Surface chemistry and surface charge formation for an alumina powder in ethanol with the addition HCl and KOH, J. Colloid Interface Sci. 241 (2001) 302–316. [47] H. Ohshima, A simple expression for Henry’s function for retardation effect in electrophoresis of spherical colloidal particles, J. Colloid Interface Sci. 168 (1994) 269–271. [48] M. Rasmusson, Volume fraction effects in electroacoustic measurements, J. Colloid Interface Sci. 240 (2001) 432–447.

Chapter 7

Acoustic and Electroacoustic Measurement Techniques 7.1. Historical Perspective 261 7.2. Difference Between Measurement and Analysis 263 7.3. Measurement of Attenuation and Sound Speed Using Interferometry 264 7.4. Measurement of Attenuation and Sound Speed Using the Transmission Technique 264 7.4.1. Historical Development of the Transmission Technique 264 7.4.2. Detailed Description of the Dispersion Technology DT100 Acoustic Spectrometer 267 7.5. Precision, Accuracy, and Dynamic Range for Transmission Measurements 278

7.6. Analysis of Attenuation and Sound Speed to Yield Desired Outputs 283 7.6.1. The Ill-Defined Problem 283 7.6.2. Precision, Accuracy, and Resolution of the Analysis 288 7.7. Measurement of Electroacoustic Properties 291 7.7.1. Electroacoustic Measurement of CVI 291 7.7.2. CVI Measurement Using Energy Loss Approach 294 7.8. Zeta Potential Calculation from the Analysis of CVI 296 7.9. Measurement of Acoustic Impedance 297 References 300

Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23007-7 Copyright # 2010, Elsevier B.V. All rights reserved.

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7.1. HISTORICAL PERSPECTIVE Ultrasound is a versatile tool for characterizing materials. The most widely used application is the nondestructive characterization (NDC) of defects in solid bodies; a review has been given by Bhardwaj [1]. Ultrasound can also yield vital information on texture, microstructure, density, porosity, elastic constants, mechanical properties, corrosion, and residual stresses. However, we do not cover all these applications concerning solids. Instead, we will describe the use of acoustic and electroacoustic measurements to characterize only liquid-based systems. Over the years, many useful fluid properties have been computed from acoustic measurements. For example, Hamstead [34] measured sound speed using the time-of-flight (TOF) method to determine the density of fluid in a pipe. Dann and Halse [2] used acoustic impedance measurements to determine not only the density of slurries but also their rheological properties. Kikuta [3] used attenuation measurements to characterize the composition of electrodeposition coating materials containing resin and pigment. Sowerby [4] used a combination of gamma-radiation and acoustic measurements to determine the volume fraction of solid particles in suspensions. Although a wide range of techniques have been used to measure fluids, many of these methods have not worked well for studying colloidal dispersions. For example, one technique is based on the interaction between light and ultrasound. Variations in the density of the liquid induced by the pressure variation within the sound wave cause a periodic pattern in the refracted light wave. Although this interaction allows one to measure the sound speed of pure liquids with a precision of 0.01% [5], the method is not suitable for colloids because of the additional scattering of the sound waves caused by the colloidal particles. Another technique is related to the use of surface acoustic waves (SAWs). Although Ricco and Martin [6] describe construction of a simple acoustic viscosity meter, and Kostial [7] used this to monitor the concentration of ions in water, the method has not been employed to characterize colloidal dispersions. Finally, Apfel [8] suggested an exotic droplet levitation technique for the measurement of sound speed, density, and compressibility. None of the above methods has been employed successfully to characterize colloidal dispersions. However, there are a number of other techniques that have indeed been found useful for characterizing not only pure liquids but colloidal systems as well. For attenuation and sound speed, these techniques can be divided into two basic categories: interferometric and transmission. The interferometric technique uses standing waves, whereas transmission utilizes transverse waves. This classification was initially suggested by Sette [5] and later adopted by McClements [9–11].

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7.2. DIFFERENCE BETWEEN MEASUREMENT AND ANALYSIS At this point, it is important to stress the difference between the measured data and the desired output parameters. For acoustics, the measured data can include the attenuation coefficient, the sound speed, and the acoustic impedance. For electroacoustics, the measured data might include the colloid vibration current (CVI), colloid vibration potential (CVP), or electrosonic amplitude (ESA). However, an investigator is typically not very much interested in these measured properties; more commonly, the particle size distribution, z-potential, or rheological properties of the colloidal dispersion are of prime importance. Obtaining these desired output parameters involves two steps. The first is to perform experiments on the disperse system to obtain a set of measured values for macroscopic properties such as temperature, pH, acoustic attenuation, CVI, etc. The second step is to then analyze these measured data and compute the desired microscopic properties such as particle size or z-potential. Such an analysis requires three tools: a model dispersion, a prediction theory, and an analysis engine. A model dispersion is an attempt to describe a real dispersion in terms of a set of model parameters including, of course, the desired microscopic characteristics (Chapter 2). The model, in effect, makes a set of assumptions about the real world in order to simplify the complexity of the dispersion and thereby also simplify the task of developing a suitable prediction theory. For example, most particle size measuring instruments make the assumption that the particles are spherical, which allows a complete geometrical description of the particle to be given by a single parameter, namely, its diameter. Obviously such a model would not adequately describe a dispersion of high-aspect-ratio carpet fibers, and any theory based on this oversimplified model might well give incorrect results. The model dispersion may also attempt to limit the complexity of a particle size distribution by assuming that it can be described by certain conventional distribution functions, for example, a lognormal distribution. A prediction theory consists of a set of equations that describe some of the measurable macroscopic properties in terms of these microscopic properties of the model dispersion. For example, a prediction theory for electroacoustics would attempt to describe a macroscopic property such as CVI in terms of microscopic properties such as the particle size distribution, zeta potential, viscosity, and dielectric permittivity. An analysis engine is essentially a set of algorithms implemented in a computer program that calculate the desired microscopic properties from the measured macroscopic data, using the knowledge contained in the prediction theory. The analysis can be thought of as the opposite, or inverse, of prediction. Prediction describes some of the measurable macroscopic properties in terms of the model dispersion. Analysis, on the other hand, given perhaps only a few values for the model parameters, attempts to calculate the

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remaining microscopic properties by an evaluation of the measured data. There are many well-documented approaches to this analysis task. In the following sections we shall first present details concerning the measuring technique for obtaining the required set of macroscopic properties, and then discuss the specific analysis methods that can be used to extract the desired microscopic properties.

7.3. MEASUREMENT OF ATTENUATION AND SOUND SPEED USING INTERFEROMETRY We will consider interferometry briefly before turning our full attention to the transmission method. Interferometry requires generating standing waves inside the sample chamber. This can be achieved by placing an acoustic reflector in front of the ultrasound transducer at some distance across the chamber. The incident wave generated by the transducer interferes with the reflected wave. This interference creates a standing wave with a structure dependent on both the wavelength and the distance between the transducer and the reflector. In this setup, the transmitting transducer works as the receiver as well. It monitors the intensity of the ultrasound at its surface. It is possible to change this amplitude by varying either the wavelength (swept-frequency interferometry, Sihna [12]) or the distance (swept-distance interferometry, McClements [9–11]). The measured intensity of the ultrasound goes through a series of minima and maxima when either the wavelength or the distance changes. The distance between successive maxima is equal to half of the ultrasound wavelength in the colloid. This allows a simple calculation of the sound speed using the following equation: cs ¼ f ls

ð7:1Þ

where cs is the sound speed, f is frequency, and ls is the measured wavelength in the colloid. This is a very accurate and precise method for sound speed measurement. Attenuation measurement using interferometry is also possible. The amplitude of the maxima depends on the distance between the transducer and the receiver. The amplitude decays with increasing distance. It is possible to measure this maximum amplitude-distance dependence and to thereby derive an attenuation coefficient. However, Sette [5] suggested that the transmission technique is better suited for attenuation measurements because of its ability to handle a much larger dynamic range.

7.4. MEASUREMENT OF ATTENUATION AND SOUND SPEED USING THE TRANSMISSION TECHNIQUE 7.4.1. Historical Development of the Transmission Technique The first transmission measurements of ultrasound attenuation had nothing to do with colloids or particle sizing. The goal was only to measure the ultrasound attenuation of pure liquids. The general principles of this technique

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were initially formulated in the early 1940s by Pellam and Galt [13], and independently by Pinkerton [14]. Further developments continued through the mid-1960s. These early investigators were perhaps the first to apply two basic principles: the use of a pulse technique and the use of a sensor with a variable acoustic path length (variable gap). These early instruments worked only at a single frequency, using one transducer in a pulse-echo mode. However, they both managed to measure sound velocity with an accuracy of 0.05% and sound attenuation with an accuracy of 5%. In each case, the velocity measurement was made by determining the distance the transducer must be moved to delay the received echo by a specified increment [5, 9–11, 15–19]. Similarly, the attenuation was measured by determining the attenuation that needed to be added or subtracted in order to keep the received signal constant as the transducer was moved. The velocity was obtained directly from the slope of the measured pulse delay versus the distance traveled. The attenuation was obtained from the slope of the compensating attenuation versus distance. In both cases, there was no need to know the exact distance traveled by the sound because the measured parameters depend only on the difference in the acoustic path. This approach was significantly improved by Andrea et al. from 1958 to 1962 [20–22]. They expanded the Pellam-Galt approach to include measurements at multiple frequencies. Instead of using a single transducer as in a pulse-echo mode, they used separate transducers for transmitting and receiving. Like Galt, they also used a variable gap and measured the change in the pulse amplitude due to variation of the gap. Again, the final attenuation and sound speed are computed from the slope of the measured attenuation or time delay versus gap. These researchers made four key observations concerning this measurement technique that are still important today: 1. At any one frequency, there is a wide variation in the attenuation coefficient of different liquids, and this variation is even more pronounced for colloids. Different samples require different path lengths. 2. The attenuation is strongly frequency dependent. For pure liquids the attenuation (in dB/cm) typically increases with the square of frequency. Again this makes different path lengths necessary for different frequencies. 3. At low frequencies, diffraction effects caused by the transmitting transducer having a finite radius must be taken into account. The sound field can approximately be divided into the Fresnel region (near the transducer) and the more distant Fraunhofer region; the boundary between two is called the Rayleigh distance (Chapters 2 and 4). In the Fresnel region, the total ultrasonic energy is still confined to a narrow beam, but the pressure amplitude fluctuates rapidly across the beam and along its axis. If, however, the receiving transducer is made somewhat larger than the transmitting transducer, and thus includes the whole cross section of the beam, a constant signal is obtained throughout the Fresnel region (in the absence of attenuation). This is not exactly true for

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transmitting and receiving transducers of equal radius, but even then the diffraction effects are, in practice, usually negligible at frequencies above 5 MHz and with ultrasonic path lengths not exceeding the Rayleigh distance of aT2/2l, where aT is transducer radius. On the other hand, in the Fraunhofer region a correction for beam divergence is required. 4. When the time taken for the ultrasonic pulse to traverse the liquid path is less than the pulse duration, an additional time delay must be provided in the ultrasonic system. The necessary time delay may be provided conveniently by introducing a length of fused quartz rod into the ultrasonic path. Fused quartz possesses a low attenuation and gives a delay of about 1.7 ms/cm. There are eight US patents dedicated to transmission attenuation measurements for the purpose of obtaining raw data from which particle size might be derived: Gushman et al., 1973 [23]; Uusitalo et al., 1983 [24]; Riebel, 1987 [25]; Alba, 1992 [26]; and Dukhin and Goetz, 2000 [15, 16, 27, 28]. However, from the standpoint of the actual measuring technique, these patents offer only a few improvements over the basic ideas suggested originally by Andrea. The Uusitalo et al. patent [24] suggests using the measurement of the incident beam attenuation in combination with a measurement of the sound scattered off-axis. The scattered part of the measurement provides complementary information only for large particles (>3 mm) because smaller particles do not scatter sufficient ultrasound. This might be useful for monitoring aggregation, the presence of a few large particles on a background of a large number of small particles. Erwin and Dohner [29] similarly proposed monitoring colloid aggregation by using a single transducer equipped with a spherical quartz lens to monitor agglomerates passing the focal point. To our knowledge, this interesting idea has not been implemented in any commercial instrument. Neither Riebel [25] nor Alba [26] brings new ideas to the acoustic measurement per se, but rather address methods of calculating the desired results. Riebel assumes that the particle size range is known in advance and selects a correspondingly appropriate frequency range. This makes it difficult to use as a general research tool but offers utility for dedicated online applications. Alba repeats some ideas from Andrea’s work and then concentrates on the conversion of these attenuation data into particle size information using the ECAH theory (see Chapter 4). Perhaps the most significant modification to the original transmission method is called the wide-bandwidth method. It is closely associated with the names of Holmes, Challis, and others from the United Kingdom [17, 18]. Whereas Andrea used an ultrasonic tone burst containing a finite number of ultrasound cycles at a selected set of frequencies, the widebandwidth method instead applies a single sharp video pulse to the transmitting

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transducer. Such a sharp pulse contains a wide range of frequency components. After the pulse has traveled through a known path length of the sample, the frequency components of the received signal are extracted using a Fourier transformation of the received pulse. Like all techniques, this wide-bandwidth technique has some advantages and disadvantages. The hardware to generate the video pulse is somewhat simpler than in the tone-burst method, but the (computer) computations are more difficult because of the necessity to maintain an extreme degree of precision. It does have the advantage that the entire spectrum is obtained during the same time interval, which may be important in colloidal systems that are very unstable and varying with time. Among the commercial instruments, Colloidal Dynamics and Dispersion Technology both use a tone-burst technique. There is no commercially available wide-bandwidth instrument.

7.4.2. Detailed Description of the Dispersion Technology DT100 Acoustic Spectrometer The Dispersion Technology DT100 instrument is a modern example of an acoustic spectrometer that, in its basic design, follows the transmission tone burst variable gap technique pioneered by Andrea [20–22]. A photograph of a typical system is shown in Figure 7.1. The unit consists of two parts, the measuring unit and the associated electronics. The description here follows various patent disclosures by Dukhin and Goetz [15, 16, 27, 28].

7.4.2.1. Acoustic Sensor The acoustic sensor is the portion of the measuring unit that contains the sample being measured and positions the piezoelectric transducers appropriately

FIGURE 7.1 The Dispersion Technology DT-100 acoustic spectrometer.

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FIGURE 7.2 (A) The acoustic sensor. (B) Top view of the acoustic sensor.

for the attenuation measurement (Figure 7.2). The sensor utilizes two identical piezoelectric transducers separated by an adjustable gap, which is controlled by a stepping motor. Each transducer incorporates a quartz delay rod that provides a delay of 2.5 ms. The transmitting transducer (on the right side of photo) is fixed, but has an adjustment to accurately set and maintain the parallelism between the two transducers. The receiving transducer (on the left side) is mounted in a piston assembly whose position can be adjusted. The gap between the face of the transmitting and receiving transducers can be set from close to zero up to 20 mm. The moving piston is sealed against the sample by a wiper seal and supplemental O rings. The space between the seals is filled with a fluid to keep the withdrawn portion of the piston wetted at all times. This prevents buildup of any dried deposit. Ports associated with this space between the seals can be used to periodically inspect the intraseal fluid for any contamination. The presence of any contamination serves as advance warning of possible seal degradation when making continuous measurements of very aggressive samples such as concentrated Portland cement. The position of the receiving transducer piston is controlled by a stepping motor linear actuator, which, in turn, is controlled by the electronics and software, with a precision of a few micrometers. The sample is contained in the space between the transducers. Depending on the installation, the sample might be stirred with a magnetic stir bar, pumped through the sensor with a peristaltic pump, or in some cases not stirred at all. When the piston is fully withdrawn, the transducers are essentially flush with the walls of the sample compartment to facilitate cleaning. During operation, tone bursts at selected frequencies are applied to the transmit transducer. The amplitude of the received pulses, received with some delay, is then measured.

7.4.2.2. Electronics A block diagram of the electronics unit is shown in Figure 7.3. A frequency synthesizer 30 generates a continuous wave (CW) RF signal 31. A fieldprogrammable gate array (FPGA) 32 controlled by the host computer via

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50

Quad Mixer

46

52

53

A/D

DSP

51 34

35

37

49

60

32 FPGA Field programmable Gate Array

30 Frequency Synthesizer

31

54

33

Computer bus

38

39

44

58

36

RF Input Signal

61

43

41

Receive Switch

Transmit Switch

A

A

C

B

40

C

B

42 Reference Attenuator

Transmit RF Pulse Received RF pulse

Movable Receive Transducer

Fixed transmit transducer

Colloid under test

FIGURE 7.3 Block diagram of electronics.

digital interface 60 generates the transmit gate 33, attenuator control 34, receive gate 35, switch control 36, and A/D strobe command 37. The transmit gate 33 is used by the gated amplifier 38 to form a pulsed RF signal from the synthesizer CW signal 31. A power amplifier 39 increases the peak power of this RF pulse to approximately 1 W. A directional coupler 40 provides a

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low-loss path for the forward signal from the power amplifier to the transmit switch 41, which then routes this 1-W RF pulse 61 to any of three channels, labeled A, B, and C according to the FPGA switch control 36. Channel A is called the reference channel and in this case the output pulse is routed to a precision, fixed 40-dB reference attenuator 42. The output of this fixed attenuator is connected to channel A of the receive switch 43, where it is then routed to the RF input signal 44. The RF input signal is connected to a wide-band RF amplifier 45 having a programmable gain set by the FPGA attenuator control 34; thence to a gated amplifier 46 controlled by the FPGA receiver gate 35 which allows only pulses arriving within a predetermined time interval to pass; thence to a quadrature mixer 47 keyed by the synthesizer output 31 which demodulates the received signal providing analog pulses 48 and 49 proportional to the in-phase and quadrature components of the received RF pulse, respectively; thence to separate matched filters 50, 51 for the in-phase and quadrature components; thence to a dual A/D converter 52 which digitizes the peak amplitude of these quadrature signals upon an A/D strobe command 37; and thence to a digital signal processor (DSP) 53 chip that accumulates data from a large number of such pulses and generates statistical data on a given pulse set. Finally, these statistical data are routed to the computer bus 54 so that the data can be analyzed to determine the amplitude and phase of the input signal level 44 for this reference condition. This reference signal level is computed in the following way. The statistical data from the two-channel A/D provide an accurate measure of the in-phase and quadrature components of the received signal at the A/D input. The magnitude is computed by calculating the square root of the sum of the squares of these two signals. The phase is computed from the inverse tangent of the ratio of the in-phase and quadrature components. The magnitude of the RF input signal 44 is then computed by taking into account the overall gain of the RF signal processor as modified by the gain control signal 34. The use of this reference channel with a known attenuation allows the attenuation to be measured precisely when using the remaining channels B and C. Since we know the input signal level for the 40-dB reference attenuator, any increase or decrease in this signal when using either of the other channels will be proportional to any increase or decrease with respect to this 40-dB reference value. When the transmit switch and receive switch are set to position C, the pulsed RF signal is routed to the transmit transducer in the acoustic sensor where it is converted to an acoustic pulse and launched down the quartz delay rod. A portion of the pulse is transmitted into the colloidal dispersion, traverses this sample, is partially transmitted into the quartz delay rod of the receiving transducer, and, after passing through this delay rod, is converted back into an electrical pulse. The received pulse is then routed to port C of the receive switch for processing by the receiver.

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Measurements can be performed for just one frequency or for a chosen set of frequencies from 1 to 100 MHz. Typically, the system averages at 800 pulses but may accumulate as many as several million pulses if necessary to achieve a satisfactory signal-to-noise ratio. The number of pulses measured will depend on the properties of the colloid. For example, concentrated slurries at high frequency and large gaps may require a large number of pulses.

7.4.2.3. Measurement Procedure Since the attenuation is determined by measuring the relative change in signal versus relative change in gap, it is not necessary to know precisely the absolute value of the gap. The precision with which the relative gap can be measured is determined by the accuracy of the lead screw used in the linear displacement stepping motor. Nevertheless, the absolute position is calibrated to an accuracy of just a few micrometers the first time a measurement is made after starting the program. The calibration is performed by simply moving the receiving transducer towards the transmitting transducer for a sufficient time so that it is certain that the two have contacted one another and the motor has stalled in that position. The gap position is then initialized and further measurements are then relative to this known initial position. Measurements are made using a defined grid consisting of a certain number of frequencies and gaps. In default, the system selects 18 frequencies between 1 and 100 MHz in logarithmic steps, and 21 gaps between 0.15 and 20 mm, also in logarithmic steps. These grid values may be modified automatically by the program, depending on a priori knowledge of the sample, or manually by an experienced user. The signal processor generates a transmit gate which defines the 1-W pulse as well as the necessary signals to set the frequency for each point on the grid. At the beginning of each measurement, the signal is routed through the reference channel attenuator as previously described to calibrate the attenuation measurement at each frequency. The next step in the measurement is to determine the losses in the acoustic sensor for each point on the measurement grid. The signal processor now substitutes the acoustic sensor for the reference channel attenuator. The 1-W pulses are now sent to the transmitting transducer which converts these electric pulses into sound pulses. Each sound pulse propagates through the transducer’s quartz delay rod, passes through the gap between the transducers (which is filled with the dispersion under test), enters an identical quartz rod associated with the receiving transducer, is converted back to an electrical signal in the receiving transducer, and finally is routed through the interface to the input signal port on the signal processor, where the signal level of the acoustic sensor output is measured. For each point on the grid, the signal processor collects data for a minimum of 800 pulses. The number of pulses that are collected will automatically increase, as necessary, to obtain a target signal-to-noise ratio. Typically, the target is set for a signal-to-noise ratio of 40 dB, which means that the

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10

Sensor Level, dbm

0

7

100 Ref .325 .399 .491 .603 .741 .91 1.118 1.374 1.689 2.075 2.55 3.133 3.849 4.73 5.811 7.141 8.774 10.781 13.247 16.277 20.

-50

-100

-150 Frequency, MHz FIGURE 7.4 Sensor level for 10 wt% silica slurry.

received signal power will be 10,000 times that of the noise. This is usually more than sufficient to ensure an acoustic spectrum of very high quality. However, an experienced user can tailor the measurement parameters to provide trade-offs between measuring speed and precision. A typical plot of the signal level for both the reference and acoustic sensor channel for all points on the grid for a 10 wt% dispersion of silica (Minusil 15) is shown in Figure 7.4. Comparison of the amplitude and phase of the acoustic sensor output pulse with that of the reference channel output pulse allows the software to calculate precisely the overall loss in the acoustic sensor at each frequency and gap. The overall sensor loss for this same sample is shown in Figure 7.5. The overall sensor loss consists of the sum of two components: the “colloid loss” in passing through the gap between the transmitting and receiving transducers, and a “transducer loss.” This transducer loss includes the loss in converting the electrical pulses to acoustic energy in the transmitting transducer, the reflection losses at the two interfaces between the transducers and the sample, and the loss in converting the acoustic energy received by the receiving transducer back to an electrical signal. Regression of the sensor loss with the gap yields two useful results. Since the loss in the sample, by definition, is zero for a gap of zero, the intercept of this regression analysis provides a measure of all other losses in the acoustic sensor, which we have defined as the transducer loss (Figure 7.6). Moreover, the slope of this regression analysis yields the attenuation per unit distance of the dispersion, expressed in

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.325 .399 .491 .603 .741 .91 1.118 1.374 1.689 2.075 2.55 3.133 3.849 4.73 5.811 7.141 8.774 10.781 13.247 16.277 20.

Sensor Loss, db

150

100

50

0 1

10 Frequency, MHz

100

FIGURE 7.5 Sensor loss for 10 wt% silica slurry.

100 Measured Model Estimated

Transducer Loss, db

80

60

40

20

0 1

10 Frequency, MHz

100

FIGURE 7.6 Transducer loss computed while measuring a 10 wt% silica slurry.

dB/cm. Since, in general, the attenuation for most materials increases strongly with frequency, it is conventional to divide the above attenuation by the frequency in megahertz so that the final output is expressed in [dB/cm/MHz] (Figure 7.7). Subtracting the transducer loss from the sensor loss now enables us to plot the colloid loss (Figure 7.8).

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Attenuation db/cm/MHz

4

3 Measured Measured Error Polynomial fit

2

Polnomial fit Error 1

0 1

10 Frequency, MHz

100

FIGURE 7.7 Attenuation of 10 wt% silica slurry.

80

3. 3.686 4.529 5.565 6.838 8.402 10.324 12.685 15.586 19.151 23.532 28.914 35.527 43.653 53.637 65.905 80.979 99.5

Colloid Loss, db

60

40

20

0 0

5

10 Gap, mm

15

20

FIGURE 7.8 Colloid loss for 10 wt% silica slurry.

Although the transducer loss is itself not necessary for the calculation of the attenuation, it does provide a good integrity check on the system performance and allows any degradation in the sensors to be detected long before any loss in system performance would be observable by the user. Inspection of Figure 7.6 shows a typical transducer loss. Both transducers are resonant

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at approximately 30 MHz and therefore have the maximum conversion efficiency (i.e., minimum loss) at 30 MHz. Inherently, such resonant transducers have a loss maximum at even multiples of their resonant frequency and a minimum in their loss at odd multiples. Following this, we see a maximum in the transducer loss occurring at about 60 MHz and a subsequent minimum at about 90 MHz. Accurate attenuation measurements require a reasonably precise knowledge of the sound speed in the sample. The transmitted pulse arrives at the receiving transducer with a time delay dependent on the width of the gap, as well as the sound speed in the sample. At the beginning of the measurement, the expected sound speed computation is based on the recipe used to make the sample. However, this estimated velocity may not be exact, and the error becomes progressively more important as the gap is increased. For this reason, the sound speed is automatically determined during each measurement, at least at one frequency, typically 10 MHz. The instrument employs a certain synergism between the attenuation and sound speed measurements, using each to improve the performance of the other. Errors in sound speed lead to an apparent decrease in the signal level because the received pulse is sampled at the wrong time. At the same time, the attenuation data is used to improve the accuracy of the sound speed measurements by accepting only data at those gaps for which the attenuation is not so great as to preclude an adequate signal-to-noise ratio. Although the transmitted tone-burst pulse has a rectangular envelope, the received RF pulse processed by the signal processor exhibits a bell-shaped response, primarily as a result of the matched filter used to optimize the received signal-to-noise ratio. For maximum accuracy, the amplitude of this received pulse must be measured at the exact peak of this bell-shaped curve. For small gaps, the pulses are sampled by the receiver at a time delay calculated using only the estimated sound speed and the present gap. For larger gaps, a more precise pulse arrival time is determined, for at least one selected frequency, by making a few additional measurements at time intervals slightly shorter and longer than the expected nominal value. The optimum arrival time, referred to as the time of flight (TOF), is then computed by fitting a polynomial curve to these few points measured on this bell-shaped response. This procedure typically provides a precision of a few nanoseconds in the determination of the optimum arrival time. The measured error in the arrival time for this single frequency is accumulated with each increasing gap. For each subsequent point on the grid, the signal is sampled at a time interval which is the sum of the time calculated from the theoretical sound speed and the last updated value for the cumulative error in the arrival time. When the measurement for the entire grid is complete, a regression analysis of the optimum arrival time with gap provides an accurate measure of the group sound speed. (Figure 7.9) This procedure of TOF adjustment

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Time of flight, usec

25

20 10.3

15

10 0

5

10 Gap, mm

15

20

FIGURE 7.9 Time-of-flight data for 10 wt% silica slurry.

achieves two goals. First, it eliminates possible artifacts in the attenuation measurement. Second, it provides the group sound speed measurement, which is of interest in its own right. This final sound speed measurement is much more reliable than that which might be obtained by measuring the TOF for just two gaps, because it reflects a regression analysis over data for many gaps. The attenuation of the sound pulse as it passes through the sample makes measurement of the TOF possibly less accurate at larger gaps where the intensity of the arriving signal may be low. To ensure the integrity of the sound speed measurement, the software selects only those gaps where the signal-to-noise ratio is sufficient to ensure reliable data. This method makes it possible to measure the group sound speed over a much wider range of attenuation than would be possible with any fixed-gap technique. It is clear that taking data at all points on the grid provides some redundant information. In principle, we need only two gaps for each frequency to determine the attenuation. The extra data provide a very robust system that can operate despite many difficult conditions. For example, when monitoring a flowing stream in a process control application, there may be occasional clumps of extraneous material, or perhaps a large air bubble, or maybe even a momentary loss of sample material altogether. In performing the regression at each frequency, the software automatically rejects any data points where that residual (to the best fit linear regression) differs by more than three sigma from the average. After any “outlaw” points are removed, it uses the remaining residuals to estimate the overall experimental error in the measured

Acoustic and Electroacoustic Measurement Techniques

277

Status Magnitude (Gap, Freq.)

Good data Lo Signal Lo S/N Lo colloid loss Local outlaw Other problem

Not measured Not computed Hi sensor loss Hi colloid loss Global outlaw

FIGURE 7.10 Status array for 10 wt% silica slurry.

attenuation at each frequency. The final status of each point in the grid is then reflected in a status array as shown in Figure 7.10. The legend gives the reason why any given point was rejected. In this example, valid data for low frequency were obtained only for larger gaps, whereas for high frequency valid data were obtained only for smaller gaps. Three points were discarded as being “outlaws.” Often it is desirable to make several measurements on the same sample. For example, if performing a titration experiment to test the effect of pH on particle size distribution, it will be necessary to measure the same sample after reagent addition. After the first measurement, the software constructs a “Next Mask” (Figure 7.11) that notes all of the points on the grid that produced useful results on the first measurement. Typically, the number of such “good” points is only half those measured the first time the sample was measured. Deleting these nonproductive points for the succeeding measurements typically doubles the measurement speed after the first measurement. However, since the sample may change slowly from one measurement to the next, Next Mask always includes additional points at the boundary of the acceptable points from the previous measurement. Next Mask allows the system to adapt to samples that are changing from measurement to measurement, yet does not waste time collecting useless data.

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Next Mask (Gap, Freq)

FIGURE 7.11 Next Mask for 10 wt% silica slurry.

7.5. PRECISION, ACCURACY, AND DYNAMIC RANGE FOR TRANSMISSION MEASUREMENTS Precision is a measure of the reproducibility, whereas accuracy is a measure of agreement with a given standard. Quite often, these two metrics of system performance are confused, or not clearly stated. For example, Andrea et al. [20–22] reported that the accuracy of their attenuation measurement was 2% over the frequency range of 0.5–200 MHz, but they did not give any measure of the precision. Holmes and Challis [17–19] simply reported an error in sound speed measurement, using their wide-bandwidth technique, of 0.2 m/s. They characterized the error in the attenuation measurement for water by specifying only a signal-to-noise ratio of 66 dB at 5 MHz, after averaging 4000 pulses. We demonstrate the precision of our technique using two test samples: a 10 wt% colloidal silica sample (Ludox TM-50) and a 20 wt% slurry of much larger silica particles (Minusil 5). The colloidal silica sample provides a difficult test, because the attenuation of the slurry is only slightly greater than the intrinsic attenuation of the aqueous medium. Figure 7.12 shows several attenuation spectra for two Ludox samples having different weight fractions, with a

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Acoustic and Electroacoustic Measurement Techniques

10/17/98 10/17/98 10/17/98 10/17/98 10/17/98 10/17/98 10/17/98 10/17/98 10/17/98 10/17/98

0.5 25%wt ludox Attenuation (dB/cm/MHz)

0.4

3:15:57 PM 12:57:22 PM 10:40:35 AM 10:18:21 AM 10:12:01 AM 10:05:19 AM 9:26:53 AM 9:20:07 AM 9:13:25 AM 8:43:36 AM

0.3 12%wt ludox 0.2

0.1

water

0.0 100

101 Frequency (MHz)

102

FIGURE 7.12 Several attenuation spectra for colloidal silica slurries at two concentrations, with comparison to that of water.

typical water spectrum included for comparison. Table 7.1 summarizes this numerical data. It is seen that the precision is better than 0.01 dB/cm/MHz for all frequencies. The precision of the attenuation measurement over a longer period (30 h) is shown in Figure 7.13. The standard deviation at 100 MHz over this 30-h period was approximately 0.002 dB/cm/MHz. Figure 7.14 shows the measured sound speed for the same colloidal silica. The standard deviation for these 43 measurements was 0.28 m/s. For comparison, Holmes and Challis [17–19] reported an error in sound speed measurement using their wide-bandwidth technique in transmission mode as 0.2 m/s. Figure 7.15 shows the attenuation spectra for eight repeat measurements of a 10 wt% silica (Minusil 5) sample. The precision of the attenuation measurement at mid-band (10 MHz) was approximately 0.01 dB/cm/MHz. To be useful as a general-purpose tool, an acoustic spectrometer must be able to accommodate a large dynamic range, that is, work over a large range of attenuation. The instrument must be able to work for low attenuation fluids such as the colloidal silica Ludox TM-50 already described, as well as for high attenuation slurries such as concentrated rutile slurry. The Dispersion Technology database contains over 60,000 measurements accumulated over

280

TABLE 7.1 Attenuation Spectra for 12 wt% Silica, Measured Nine Times with Five Different Loads Frequency in MHz 3

3.7

4.5

5.6

6.8

8.4

1

0.03

0.03

0.02

0.04

0.04

0.04

0.05

0.07

0.08

0.09

0.1

0.13

0.15

0.19

0.23

0.27

0.33

0.4

2

0.04

0.02

0.03

0.04

0.04

0.04

0.05

0.06

0.07

0.09

0.1

0.13

0.15

0.19

0.23

0.28

0.34

0.4

3

0.05

0.05

0.04

0.05

0.05

0.06

0.05

0.05

0.07

0.09

0.1

0.12

0.15

0.19

0.22

0.25

0.33

0.39

4

0

0.03

0.01

0.04

0.04

0.04

0.05

0.06

0.07

0.09

0.1

0.12

0.15

0.19

0.22

0.24

0.33

0.39

5

0.03

0.04

0.02

0.05

0.04

0.04

0.05

0.06

0.07

0.09

0.1

0.12

0.15

0.19

0.22

0.27

0.33

0.4

6

0.03

0.03

0.02

0.03

0.04

0.05

0.04

0.06

0.07

0.08

0.1

0.12

0.15

0.18

0.22

0.24

0.32

0.39

7

0.03

0.04

0.03

0.04

0.04

0.05

0.04

0.06

0.07

0.08

0.1

0.12

0.15

0.18

0.23

0.24

0.33

0.39

8

0.01

0.02

0.02

0.04

0.04

0.04

0.04

0.06

0.07

0.08

0.1

0.12

0.15

0.19

0.22

0.26

0.32

0.39

9

0.02

0.03

0

0.04

0.04

0.04

0.05

0.06

0.07

0.09

0.1

0.12

0.15

0.19

0.22

0.27

0.33

0.39

0.007 0.008 0.003 0.001 0.004

0.003

0.002

0.001

0.003

0

0.002

0

0.002

0.003

0.013

0.003

0.003

Average 0.027 0.032 0.021 0.041 0.041 0.044

0.047

0.06

0.071

0.087

0.1

0.122

0.15

0.188

0.223

0.258

0.329

0.393

Std dev. 0.01

10.3

12.7

15.6

19.1

24

29

35.5

43.6

53.6

65.9

81

100

CHAPTER

Run#

7

281

Acoustic and Electroacoustic Measurement Techniques

db/cm/MHz at 100 MHz

0.40

0.38

0.36

0.34

0.32

0.30 0.

50.

100. 150. 200. Measurement number

250.

300.

FIGURE 7.13 Precision of attenuation at 100 MHz for 10 wt% silica Ludox over 30 h.

1505 1504

Sound speed, m/s

1503 1502 1501 1500 1499 1498 1497 1496 1495 0.

10.

20. 30. Measurement number

40.

50.

FIGURE 7.14 Precision of sound speed measurement for 10 wt% silica (Ludox TM-50).

the past 8 years from many instruments around the world for hundreds of different applications. This database was polled to find the sample that had the maximum attenuation at each of three frequencies, namely 3, 10, and 100 MHz. Several spectra for each of these three samples are shown in Figure 7.16. If we take the envelope of the spectra for these three samples, they provide a

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Attenuation (dB/cm/MHz)

2.4 2.0 1.6 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02

1.2 0.8 0.4

6:33:26 6:21:50 6:02:21 5:51:08 5:36:33 5:22:48 5:09:59 4:57:01

PM, PM, PM, PM, PM, PM, PM, PM,

1.644 1.73 1.628 1.657 1.652 1.69 1.705 1.628

0.0 100

101 Frequency (MHz)

102

FIGURE 7.15 Several spectra for 20 wt% silica.

140

40% hematite in solvent 11.2% carbon black in water 11.2% carbon black in water 11.2% carbon black in water 70% rutile in water 70% rutile in water 70% rutile in water

Attenuation (dB/cm/MHz)

120 carbon black 100 80 60 40 hematite 20

rutile

0 100

101 Frequency (MHz)

102

FIGURE 7.16 Spectra for several very highly attenuating slurries.

conservative estimate of the maximum attenuation that can be reliably measured with this technique. Considering these samples, the maximum attenuations at 3, 10, and 100 MHz is approximately 130, 60, and 18 dB/cm/MHz, respectively. Describing the attenuation in terms of [dB/cm] (by multiplying by frequency), we obtain values of 390, 600, and 1800 dB/cm.

Acoustic and Electroacoustic Measurement Techniques

283

7.6. ANALYSIS OF ATTENUATION AND SOUND SPEED TO YIELD DESIRED OUTPUTS 7.6.1. The Ill-Defined Problem It has been known for a long time that different particle sizing techniques can yield quite different particle size distributions. Professor Leschonski wrote in 1984: Each group of measurement techniques based on the same physical principle provides, in general, different distribution curves. The reason for this must partly be attributed to the fact that, in particle size analysis, every possible physical principle has been used to determine these distributions. Due to many reasons, the results will, in general, not be the same. . . . It means that the instruments based on different physical principles yield different PSD. . . . However, apart from this, the distributions differ due to two reasons. Firstly, different physical properties are used in characterizing the “size” of individual particles and, secondly, different types and different measures of quantity are chosen [30].

We suggest that particle sizing methods should be broadly classified into three general categories: (1) counting methods, (2) fractionation techniques, and (3) macroscopic fitting techniques. Each category is closely linked to a most appropriate way to describe the particle size distribution. Particle counting yields a “full PSD” on a “number basis.” This category includes various microscopic image analyzers, electro-zone methods such as the Coulter Counter, and light obscuration-based counters. Fractionation techniques also yield a full PSD, but usually on “volume basis.” They operate by physically separating the sample into fractions that contain different sized particles. This category includes sieving, sedimentation, and centrifugation methods. Macroscopic techniques include methods that are related to the measurement of macroscopic properties that are particle size-dependent. All macroscopic methods, in addition to a measurement step, require the step of calculating the particle size distribution from this measured macroscopic data. For attenuation spectroscopy, for example, measuring the spectra is only the first step in characterizing the particle size. The second step is to use a set of algorithms, implemented in a computer program, to calculate the desired microscopic properties from the measured macroscopic data, using the knowledge contained in the prediction theory. The analysis can be thought of as the opposite, or inverse, of prediction. Prediction describes some of the measurable macroscopic properties in terms of the model dispersion. Analysis, on the other hand, given perhaps only a few values for the model parameters, attempts to calculate the remaining microscopic properties by an evaluation of the measured data. However, in attempting the analysis we often run into a problem. There may be more than one solution. Perhaps more than one size distribution yields

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the same attenuation spectrum, at least within the experimental uncertainties. This problem of “multiple solutions” is often termed the ill-defined problem. It is not unique to acoustics, but is common to all macroscopic methods, particularly light scattering methods. Each category of measuring techniques has some advantages and disadvantages. No one method is able to address all particle sizing applications. The counting method, for instance, has an advantage over macroscopic techniques in that it yields a full PSD. Obtaining a statistically representative analysis, however, requires counting an enormous number of particles, especially for a polydisperse system, and this can be very time consuming. Also, sample preparation is rather complicated and may affect the results. Additionally, transformation of the number-based PSD into a volume-based PSD is not accurate in many practical systems. Fractionation methods allow one to collect data on a very large numbers of particles and may more easily provide statistically significant results. However, fractionation methods are not very suitable for small particles. For example, when using an ultracentrifuge it may take hours to perform a particle size analysis of a sample containing submicrometer particles. These methods also dramatically disturb the sample and consequently might affect the measured particle size distribution. Macroscopic methods, in general, are fast, statistically representative, and easy to use. However, there is a price to pay. We must solve the ill-defined problem. Fortunately, in many cases there is no need to know all of the details of the particle size distribution. A value for the median size, a measure of the width of the distribution, and perhaps some measure of bimodality may provide sufficient description for many practical colloidal dispersions. For this reason, macroscopic methods are very widely used in many research laboratories and manufacturing plants. One of the first steps in selecting a particle sizing technique should be to decide how much detail one needs concerning the PSD. A full PSD is only needed in rare cases. Do you need this degree of detail? Ask yourself what you would actually do with such detail. If this elaborate PSD information is indeed necessary, you should consider some counting or fractionation techniques. If not, you can use macroscopic techniques, with the understanding you will not obtain the exposition of the PSD. The nature of this ill-defined problem is well understood in the field of light scattering. For instance, Bohren and Huffman, in their well-known handbook on light scattering [31], wrote: The Direct Problem. Given a particle of specified shape, size and composition, which is illuminated by a beam of specified irradiance, polarization, and frequency, determines the field everywhere. This is the “easy” problem. The Inverse Problem. By suitable analysis of the scattered field, describe the particle or particles that are responsible for the scattering. This is the “hard” problem . . . the ill-defined problem, . . .

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Acoustic and Electroacoustic Measurement Techniques

To understand why we have labeled this the “hard” problem, consider that information necessary to specify a particle uniquely is (1) the vector amplitude and phase of the field scattered in all directions, and (2) the field inside the particle. The field inside the particle is not usually accessible to direct measurements. The total amplitude and phase of the scattered light are rarely achieved in practice. The measurement usually available for analysis is the irradiance of the scattered light for a set of directions. We are therefore almost always faced with the task of trying to describe a particle (or worse yet, a collection of particles) with a less than theoretically ideal set of data in hand. . . .Geometrical optics can often provide quantitative answers to small-particle problem which are sufficiently accurate for many purposes, particularly when one soberly considers the accuracy with which many measurements can be made. . . .

This quote from a respected expert in the field of particle characterization suggests that we should be cautious in our expectations when any macroscopic method is employed. In general, we are able to measure some macroscopic property (scattered light, neutrons, X-rays, or sound attenuation) with some experimental error, Dexp. In turn, this experimental error limits our ability to calculate an exact PSD from the measured experimental data. Figure 7.17, for instance, illustrates a hypothetical “full” size distribution as a histogram. One can see that there is a certain irregular variation in the height of the columns. Is it possible to characterize this detailed variation with any macroscopic technique? Is it important to characterize this fine detail in order to formulate, manufacture, or use a colloidal product? The answer to both questions in many cases is “No.” These small variations of the PSD would create corresponding variations in the measured parameter, DM. If this variation is less than the experimental error (i.e., DM < Dexp), then the given macroscopic method will not be able to qr(X)

modified log-normal distribution

histogram

Xmin

Xmax

FIGURE 7.17 Example of the “full PSD” and fitting modified lognormal PSD.

size

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resolve this variation in PSD. The problem of “multiple solutions” prevents us from finding a unique PSD using the measured parameters. There may be several PSDs that would each yield the same measured parameter within the limit of experimental errors. This rather simple discussion shows that all macroscopic techniques have some limits in the detail with which they can describe the actual PSD of a given colloidal dispersion. There are many instances, however, when this limitation is simply ignored, and a “full PSD” is claimed as a valid output of some macroscopic measurement and analysis. For instance, US patent 6,119,510 by Carasso et al. [32] suggests an inversion procedure for calculating a “full PSD” from either acoustic or optical attenuation spectra. This invention presents the PSD as an abstract form, based on a selected number of parameters—perhaps 50 or even more. No means is suggested for determining this important number, which can be considered as a measure of the amount of information contained in the experimental data. Is it possible to determine the limitations of a given “macroscopic technique” for characterizing the finer details of a PSD? We suggest a simple procedure that follows the general scientific method. Let us assume that we have two PSDs, each proposed as fitting the same set of experimental data. In the case when both PSDs yield the same quality of fit, we suggest that the solution employing the fewer number of adjustable parameters should be selected. This common sense idea was formulated in the fourteenth century by William of Occam [33] as a general philosophical principle (Occam’s razor), which briefly states that “given a set of otherwise equivalent explanations, the simplest is most likely to be correct.” In the case of fitting theory to experimental data, two solutions producing the same fitting error are deemed equivalent, whereas the number of adjustable parameters is inversely related to simplicity. If we decide to stay with general scientific principles, we should start with the minimum number of adjustable parameters, and increase it in steps till the fitting error no longer improves significantly relative to the experimental errors. This procedure then determines the maximum number of adjustable parameters that one can extract from the particular set of experimental data. This is just a general principle. There are many ways to implement it in calculating the PSD from the attenuation spectra. The first step is obvious: we assume a monodisperse PSD with the size as the only adjustable parameter. It is possible to run a global search for the best monodisperse solution (the one that yields the best fit between the theoretical attenuation and the experimental data). In the case of very small particles, this first step will often be the last because the attenuation of the dispersion may be very low, and only slightly greater than the intrinsic attenuation of the sample. The added attenuation from the presence of the particles may be comparable to the experimental error, which then limits the amount of particle size information that one can extract from the measured data. Our experience with DT1200 indicates that this often occurs when the particle size is 10 nm or less.

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Acoustic and Electroacoustic Measurement Techniques

For particles larger than 10 nm, there is usually sufficient information in the attenuation spectra for calculating additional adjustable parameters. The next step is to use two adjustable parameters, resulting in a “unimodal PSD.” There are several ways to select these two parameters. By default we use a lognormal PSD, which is of course defined by two adjustable parameters: a median and standard deviation. It is also possible to use a modified lognormal PSD (see Chapter 2) when the particle size range is known. This is helpful for characterizing nonsymmetrical PSDs, as was illustrated in Figure 7.17. The next level of complexity brings us to a “bimodal PSD,” which can be modeled as the sum of two lognormal PSDs. This increases the number of adjustable parameters to five (a median and standard deviation for each mode and the relative concentration of these two modes) or perhaps only four if we assume that the two modes have similar standard deviation. Our experience with many thousands of samples is that it is seldom necessary, or allowable, to increase the complexity much beyond a bimodal PSD. The analysis of an attenuation spectrum proceeds by increasing the complexity (the number of adjustable parameters) in steps, and at each step computing the best fit PSD and the corresponding fitting error. We can illustrate how this process works by examining the results of a pH titration experiment on a 7 vol% rutile dispersion. The rutile particles aggregate somewhat in the vicinity of the isoelectric pH, about 4.0. Thus, we would expect that the PSD becomes bimodal close to the isoelectric point (IEP), with one peak representing the primary particles and the second peak comprising the aggregates. Figure 7.18 shows fitting errors for the lognormal and bimodal cases, as compared with the experimental errors. For pH values larger than 6,

12

error, %

10

experment bimodal lognormal

8 6 4 2 0 3

4

5

6

7

8

9

10

11

12

pH FIGURE 7.18 Experimental error and errors of theoretical fit based on lognormal and bimodal PSD for 7 vol% rutile dispersion.

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the lognormal and bimodal fitting errors are identical. Following Occam’s principle, we should select then the simpler lognormal distribution. However, in the vicinity of the IEP, that is, for pH < 6, the fitting error of the lognormal solution dramatically increases, an indication that the lognormal PSD fails to fit the experimental data in this pH range. In contrast, the fitting error for the bimodal solution remains small, indeed close to the experimental error. In light of this information, following Occam’s principle allows us to claim that the PSD is bimodal in this pH range.

7.6.2. Precision, Accuracy, and Resolution of the Analysis The precision (not accuracy) of the PSD measurement is determined almost exclusively by the error in the attenuation measurement since the computations can be done with high precision. Figure 7.19 shows the mean size for a colloid of silica (Cabot SS25) measured over 18 h. The standard deviation was less than 1 nm. The absolute accuracy (as opposed to precision) depends on several factors, including the following: 1. 2. 3. 4. 5.

The accuracy of the required physical properties of the sample How well the sample conforms to the limitation of the model The accuracy of the attenuation measurement The accuracy of the prediction theory The performance of the analysis engine in finding the best fit between theory and experiment

The accuracy of attenuation spectroscopy for determining the PSD was briefly demonstrated in Chapter 6 using a glass (BCR-70) slurry. Here we show results for another sample, a 20 wt% silica (Minusil-5) slurry. The attenuation spectra for this sample were shown previously in Figure 7.15. The analysis of these

Series1

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Mean size, um

100 measurements on Cabot SS25 on system 187 0.090 0.088 0.086 0.084 0.082 0.080 0.078 0.076 0.074 0.072 0.070

Measurement number FIGURE 7.19 Precision of mean particle size for 100 measurements of 10 wt% silica slurry (Cabot SS25).

289

Acoustic and Electroacoustic Measurement Techniques

1.0 0.9

Cumulative, weight basis

0.8 0.7 0.6

1/7/02 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02 1/7/02

6:33:26 6:21:50 6:02:21 5:51:08 5:36:33 5:22:48 5:09:59 4:57:01

PM, PM, PM, PM, PM, PM, PM, PM,

1.644 1.730 1.628 1.657 1.652 1.690 1.705 1.628

0.5 0.4 0.3 0.2 0.1

0.0 FIGURE 7.20

0.1

1.0 Diameter, um

10.0

0.0 100.0

Precision of cumulative distribution for a 20 wt% silica (Minusil 5) dispersion.

TABLE 7.2 Accuracy of PSD Data for 20 wt% Silica (Minusil 5) Cumulative

Acoustic measurement (mm)

Manufacturer’s data (mm)

D10

0.45

0.6

D50

1.67

1.6

D90

6.0

3.5

spectra yields the corresponding cumulative size distributions shown in Figure 7.20. The eight separate measurements are virtually indistinguishable. Table 7.2 compares key points on this cumulative distribution with values provided by the manufacturer (US Silica). The term “resolution” is meant to characterize the ability of the acoustic technique to resolve subpopulations of particles having different sizes. An interesting resolution parameter is the amount of large particles that can be distinguished against a background of much smaller particles. Resolution defined in this manner is important for characterizing the aggregation in materials such as chemical-mechanical polishing (CMP) slurries. One can estimate this resolution parameter by comparing the theoretical predictions for the attenuation spectra of a series of bimodal slurries in which small amounts of large particles (representing the aggregates) are sequentially added to a base slurry of much smaller particles. Clearly, the precision with which one can measure the attenuation spectra will determine the sensitivity

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with which one can detect any small changes due to the introduction of small amounts of larger particles. The precision in determining the attenuation of the base slurry was depicted earlier in Table 7.1, and was on the order of 0.01 dB/cm/MHz at low frequency. It follows that we should be able to detect a change from the addition of the larger particles if this addition results in an increase in attenuation that is larger than this basic precision. Figure 7.21 shows the increment in the theoretical attenuation of a bimodal slurry having 1% large particles with respect to a unimodal distribution having no large particles. This increment is shown for three different sizes for the larger particles: 0.5, 1, and 3 mm. The precision of the attenuation measurement (from Table 7.1) is also shown for comparison. Agglomerates having a particle size of 0.5 mm would clearly be detectable, as the increment in attenuation would be considerably larger than the instrument error. On the other hand, we see that the incremental attenuation for 3-mm particles is somewhat less, which means that such particles would be more difficult to detect at this 1% level. As a result, a 1% addition of 3-mm particles would be detected only in the high-frequency portion of the spectra. Another conclusion follows from the theoretical curves of Figure 7.6, which is that the resolution depends on the particle size. We note that the addition of either the 0.5- or 1-mm particles contributes additional viscous loss and results primarily in an added attenuation at low frequencies, whereas the addition of the 3-mm particles contributes additional scattering losses, resulting primarily in added attenuation at higher frequencies. It is a fortunate coincidence that this acoustic technique is particularly sensitive for distinguishing between these “large” and “larger” particles. Qualitatively, we can say that an increase in the attenuation at

attenuation resolution (MHz)

0.04

0.5 micron 1 micron 3 micron experiment

0.03

0.02

0.01

0.00 100

101 frequency (MHz)

102

FIGURE 7.21 Increment in the predicted attenuation caused by agglomeration of 1% of the total particulates in a 12 wt% slurry of colloidal silica, for three sizes of agglomerates.

291

Acoustic and Electroacoustic Measurement Techniques

Attenuation (dB/cm/MHz)

0.5 9% 9% 7.4% 7.4% 5.7% 5.7% 3.8% 3.8% 2% 2% ludox only

0.4

0.3

0.2

0.1

0.0

100

101 Frequency (MHz)

102

FIGURE 7.22 Attenuation spectra for a 12 wt% slurry of colloidal silica (Ludox TM-50), with added amounts of larger silica (Geltech 0.5).

lower frequencies indicates that we are dealing with “large” particles of about 1 mm, whereas an increase in the attenuation spectra at higher frequencies is an indication that we have even “larger” particles exceeding 3 mm. Quantitatively, of course, these calculations are made more precise by the analysis software. Figure 7.22 illustrates how large particles affect the total attenuation spectra. It is seen that 2% of larger particles induce visible changes in the attenuation spectra. In conclusion, we can describe this level of resolution in the following terms. Acoustic spectroscopy is able to resolve a single large particle of 1 mm size against a background of 100,000 small 100-nm particles. This conclusion is rather approximate; it should be considered as reference point but not as absolute number. It is obvious that the resolution might vary with particle size, weight fraction, and the properties of the particle and the liquid medium.

7.7. MEASUREMENT OF ELECTROACOUSTIC PROPERTIES 7.7.1. Electroacoustic Measurement of CVI The Dispersion Technology DT300 is an example of a modern instrument for determining the zeta potential of colloidal dispersions by the measurement of electroacoustic properties and the subsequent analysis of these data. This instrument measures the CVI (as opposed to ESA). Similar to the DT100, it consists of two parts: a CVI probe and an electronics unit. A photo of a typical system is shown in Figure 7.23. The electronics for the electroacoustic measurement is identical to that used for acoustic measurements. In fact, Dispersion Technology offers a

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7

FIGURE 7.23 The probe for measuring colloid vibration current.

single instrument that combines the two sensors (acoustic and CVI) in a single instrument, the DT1200, for characterizing both particle size and zeta potential. The CVI probe consists of two parts: a transmitting transducer and a receiving transducer. The transmitting transducer consists of a piezoelectric transducer to convert a radio frequency (RF) pulse into acoustic energy and a delay rod that launches this acoustic pulse into the slurry after a suitable delay. This acoustic excitation of the slurry causes the particles to gain induced dipole moments. The dipole moments from the particles add up to create an electric field, which can be sensed by a receiver. This receiver, in essence, consists of a two-element antenna immersed in the sample. The electric field changes the potential of one element with respect to the other. If the impedance of the measuring circuit associated with the antenna is relatively high with respect to the colloid, then the antenna senses an open circuit voltage, which is typically referred to as the CVP. Alternatively, in the preferred configuration, the impedance of the circuitry associated with the antenna circuit is low compared to that of the colloid, and therefore the electric field causes a current to flow in the antenna. This short circuit current is referred to as the CVI. The probe can be dipped into a beaker for laboratory use, immersed in a storage tank, or inserted into a pipe for online monitoring of a flowing process stream. Figure 7.24 shows a cross section of the probe. The transmitting portion utilizes an off-the-shelf transducer 3 in conjunction with a mating standard UHF cable connector 1 and input cable 2. This transducer contains a cylindrical piezoelectric device 4 having a front electrode 5 and a back electrode 6 across which an RF pulse can be applied to generate an acoustic pulse. The front and

293

Acoustic and Electroacoustic Measurement Techniques 2 RF pulse input

15 CVI Output 23

23

17

Figure 6 1 13

13 7

6 5

4 3 off-the-shelf piezoelectric transducer 10 Ceramic spacer

Quartz Delay Rod 61 9 Buffer Rod

11 21 22

16 Stainless steel shell

8

12

14

18

19 20 16 10

Figure 6A

14

FIGURE 7.24 Diagram of the probe for measuring colloid vibration current.

back electrodes of the piezoelectric device are connected to corresponding terminals of the UHF connector by means of internal jumper wires 7 in a manner well known to those skilled in the art of such transducer design. The piezoelectric device in turn is bonded to a quartz delay rod 8 by means of a suitable adhesive. The resonant frequency of the piezoelectric device is selected depending on the frequency range for which electroacoustic data are required. Typically, the resonant frequency is selected in the range of 2–10 MHz and the quartz delay rod has an acoustic delay of about 2.5 ms. The quartz delay rod is extended by an additional buffer rod 9 having an acoustic impedance that is more closely matched to that of the slurry than the quartz material. The plastic known commercially as Rexolite has been found to be quite suitable for this purpose. The length of this buffer rod is chosen to provide an additional time delay at least as long as the pulse length, typically

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2.5 ms. The interface between the delay rod and the buffer rod 62 provides a well-characterized reflection and transmission coefficient for the incident acoustic pulse since the acoustic impedance of both materials is known precisely. The acoustic transducer and the additional buffer rod are cemented together and inserted fully into a ceramic spacer 10 till the recess 11 on the buffer rod mates with the shoulder 12 on the spacer, whereupon it is fastened securely by means of the brass retaining ring 13. The end of the buffer rod is coated with gold 14 in order to provide an electrode for measuring the electrical response of the colloid when excited acoustically. The gold coating wraps around the end of the buffer rod for approximately 2 mm in order to provide a means for making an electrical connection to the gold electrode. In Figure 7.24, part Figure 6A provides an end view of the probe showing the relative location of the gold electrode 14, the spacer 10, and the stainless steel shell 16. A coaxial cable 15 carries the CVI signal, connecting to the two electrodes used to receive the colloid vibration signal. The center conductor of the coax connects to the gold electrode and the shield of the coax makes connection to the outer stainless steel shell 16 as will be explained shortly. A groove 17 is provided along the length of the spacer to provide a path for the coaxial cable. The outer insulation of the coax is removed from the cable over the length of this groove exposing the outer braid such that it will contact the stainless steel shell. The shield is removed from the final portion 18 of the center conductor of the coaxial cable, which is inserted into an access hole 19 in the spacer such that it makes contact with the part of the gold that wraps around the end of the buffer rod. The center conductor of the coaxial cable is permanently fastened to the gold by means of silver-filled epoxy 20. The spacer, with its internal parts, is inserted into the stainless steel shell until a recess 21 at the periphery of the spacer mates with a flange 22 on the stainless steel shell, whereupon it is fastened securely by means of a retainer ring 23. When the probe is immersed in a test sample, the colloid vibration signal between the central gold electrode and the surrounding stainless steel outer housing is thus available at the coaxial cable output.

7.7.2. CVI Measurement Using Energy Loss Approach The measured CVI signal reflects a series of energy losses that are experienced between the point the RF pulse is sent to the probe and the point at which it later returns. These losses include l l

the loss in converting the RF pulse to an acoustic pulse the reflection losses at the quartz/buffer rod interface

Acoustic and Electroacoustic Measurement Techniques

l l

295

the reflection loss at the buffer rod/colloid interface; and, importantly the conversion from an acoustic pulse to a CVI pulse by virtue of the electroacoustic effect, which we are trying to characterize

The following paragraphs describe how one calculates the CVI in terms of these effects. The experimental output of the electroacoustic sensor, Sexp, is the ratio of the output electric pulse from the receiving antenna, Iout, to the intensity of the input RF pulse to the acoustic transmitter, Iin. It is different from the acoustic impedance measurement (Equation 7.11) due to the different nature of the output signal. The intensity of the sound pulse in the delay rod, Irod, is related to the intensity of this input electric pulse through some constant, Ctr, which is a measure of the transducer efficiency, and any other energy losses to this point. This relationship is given as Equation (7.12). In the case of the electroacoustic measurement, we are more interested in the sound pressure than the sound intensity. The sound pressure in the rod, according to Equations (3.33) and (3.34) in Chapter 3, is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7:2Þ Prod ¼ 2Irod Zr ¼ 2Ctr Zr Iin At the receiving transducer, we can define the electric pulse intensity as being proportional to the square of the CVI received at the antenna: Iout ¼ CVI2 Cant

ð7:3Þ

where the constant Cant depends on some geometric factor of the CVI space distribution in the vicinity of the antenna, as well as the electric properties of the antenna. Substituting Equation (7.2) into Equation (7.3), we obtain the following expression which relates CVI to the measured parameter Sexp: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sexp CVI ¼ ð7:4Þ 2Zr Ctr Cant Prod The magnitude of the measured CVI is proportional to the pressure near the antenna surface, Pant, not in the rod itself. The pressure at the antenna is lower than the pressure in the rod, Prod, because of a reflection loss at the rod-colloid interface, which gives us the following expression for Pant : Pant ¼ Prod

2Zs Zs þ Zr

This correction leads to the following expression for CVI: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sexp Zs þ Zr CVI ¼ Pant 2Zr Ctr Cant 2Zs

ð7:5Þ

ð7:6Þ

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The gradient of pressure rP in Equation (5.27) for CVI is equal to the gradient of pressure Pant. Using this fact and substituting CVI from Equation (5.27), we obtain the following equation relating the properties of the dispersion with the measured parameter Sexp: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sexp Zs þ Zr CVI ð7:7Þ ¼ 2Zr Ctr Cant 2Zs cs rP where cs is the sound speed in the dispersion. For calculating the z-potential or particle size from the measured CVI, one should use Equations (5.27)–(5.30). One can calculate the z-potential for a known particle size or both z-potential and particle size. In order to use this relationship, we should define a calibration procedure for resolving the two calibration constants in Equation (7.7). There are two ways to eliminate them. It is possible to combine them in one unknown calibration constant, Ccal, which is independent of the properties of the dispersion. This calibration constant can be determined using a colloid with a known zeta potential. For this purpose, we use Ludox TM-50, a commercially available colloidal silica, which is diluted to 10 wt% with 102 mol/l KCl. From independent measurements, we have determined that this silica preparation has a z-potential of 38 mV at pH 9.3. However, there is a more efficient way. The conversion efficiency of the transducer, Ctr, is frequency dependent, whereas the geometric constant for the CVI field distribution, Cant, is not. Furthermore, the transducer efficiency may vary with the age of transducer, as well as the ambient temperature. It would be more accurate if we apply a calibration procedure for eliminating the calibration constant Ctr. As a result, we obtain the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CVI Sexp jZb  Zr j Zs þ Z r ¼ ð7:8Þ rP 2Cant Scal Zr ðZb þ Zr Þ 2Zs cs The calibration constant, Cant, requires calibration with a known colloid, as described above. The unknown impedance of the colloid can be either measured using acoustic impedance measurement (Section 7.8) or calculated if colloid density and sound speed are known.

7.8. ZETA POTENTIAL CALCULATION FROM THE ANALYSIS OF CVI The analysis of the CVI signal to determine z-potential is considerably simpler than the analysis of an attenuation spectrum to determine the PSD. Unlike the PSD calculation, which requires no calibration, the zeta potential is calibrated using a colloid of known electrokinetic properties as described above. Figure 7.25 shows the precision that is normally obtained following calibration. The standard deviation for the zeta potential measurement is typically 0.3 mV.

Acoustic and Electroacoustic Measurement Techniques

297

-40

zeta pitential, mV

-39

-38

-37

-36 0.

10. 20. 30. 40. 50. 60. 70. 80. 90. 100. measurement number

FIGURE 7.25 Zeta potential for 10 wt% colloidal Silica (Ludox TM-50).

7.9. MEASUREMENT OF ACOUSTIC IMPEDANCE Sound speed measurements, in principle, allow us to calculate the acoustic impedance of a sample if the density of the slurry is known. However, there is an independent way to measure acoustic impedance and thereby obtain additional data for the colloidal dispersion. The acoustic impedance, Z, is introduced as a coefficient of proportionality between pressure, P, and the velocity of the particles, v, in the sound wave. It directly follows from this (see Chapter 3) that the acoustic impedance equals:   al ð7:9Þ Z ¼ rc 1  j 2p where the attenuation Z is in [nepers/m], the wavelength l is in [m], the sound speed of the longitudinal wave c is in [m/s], and the density r is in [kg/m3]. For most colloids, the contribution of attenuation to acoustic impedance in the ultrasound range of 1–100 MHz is negligible. Thus we can simplify Equation (7.9) to obtain: Zs ¼ rs cs

ð7:10Þ

If sound speed, cs, is known, a measurement of the acoustic impedance allows us to calculate an effective density of the slurry, which in turn can be used for monitoring the colloid composition. According to Equations (3.33) and (3.34), the acoustic impedance controls the propagation of the ultrasound wave through phase boundaries. The

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7

intensity of the reflected and transmitted waves depends on the relationship between the acoustic impedances of the various phases. These intensities are measurable, which opens up the way to characterize the otherwise unknown acoustic impedance of the colloid. The design of the acoustic impedance probe is somewhat similar in design to that of the CVI probe previously described (Figure 7.24). The difference between this device and the CVI probe is the nature of the sound path and the reflections at the various interfaces. These reflections and transmissions of the sound pulse are illustrated in Figure 7.26. An electric pulse of intensity, Iin, is introduced to the piezoelectric crystal which converts it to an ultrasound pulse of intensity, Irod, This pulse propagates through the delay rod, and eventually meets the boundary between r the delay rod and the buffer rod. Part of the pulse, Irb , reflects back to the t piezoelectric crystal, and the other part, Irb , transmits further to the boundary between the buffer rod and the colloid. The last part of the pulse splits at this r boundary, again into a reflected Ibc pulse and a transmitted pulse. The reflected part comes back towards the piezoelectric crystal, experiencing one more reflection-transmission split at the delay rod-buffer rod boundary. t2 The last part of the initial acoustic pulse, Irb , reaches the piezoelectric crystal, which converts it back to the electric pulse with intensity Iout. The experimental output of the impedance probe, Sexp, is the ratio of intensities of the input and output pulses: Sexp ¼

Iout Iin

ð7:11Þ

We can express Iout through Iin by following the pulse path, and applying reflection-transmission laws (Equations (3.33) and (3.34)) at each border using the acoustic impedances of the delay rod, Zr, of the buffer rod, Zb, and of the colloid, Zs. This gives us the following set of equations: Irod ¼ Ctr Iin:conversion of electric pulse into ultrasound Quartz Delay Rod

Buffer Rod

Piezo Crystal

Iout cal I out

Iin

ð7:12Þ

Colloid t2 I r−b

r I r−b

t2 I r−b

I br−c r I r−b

t I b−c

I rt1−b

I rod

FIGURE 7.26 Transmission of an ultrasound pulse through the acoustic impedance probe.

299

Acoustic and Electroacoustic Measurement Techniques

t1 Irb ¼ Irod

4Zb Zr

¼ Iin Ctr

4Zb Zr

¼ Iin Ctr

4Zb Zr

ðZb þ Zr Þ ðZb þ Zr Þ2 :transmission delay rod-buffer

r t1 ¼ Irb Ibc

2

ðZb  Zc Þ2

ð7:13Þ

ðZb  Zs Þ2

ðZb þ Zc Þ2 ðZb þ Zr Þ2 ðZb þ Zs Þ2 :reflection buffer rod-colloid

t2 r Ibr ¼ Ibc

4Zb Zr

¼ Iin Ctr

16Zb2 Z2r ðZb  Zs Þ2

ðZb þ Zr Þ2 ðZb þ Zr Þ4 ðZb þ Zs Þ2 :transmission buffer-delay rod

t2 ¼ Iin C2tr Iout ¼ Ctr Ibr

ð7:14Þ

ð7:15Þ

16Zb2 Zr2 ðZb  Zs Þ2

ðZb þ Zr Þ4 ðZb þ Zs Þ2 :conversion of ultrasound into electric pulse

ð7:16Þ

Substituting the final equation for the intensity of the output electric pulse into the definition of the experimental output of the impedance probe (Equation 7.11), we get the following result: Sexp ¼

Iout 16Zb2 Zr2 ðZb  Zs Þ2 ¼ C2tr Iin ðZb þ Zr Þ4 ðZb þ Zs Þ2

ð7:17Þ

There are two unknown parameters in this equation: the acoustic impedance of the colloid, Zc, and the efficiency of the piezoelectric crystal for conversion of electric energy into ultrasound, Ctr. It turns out that we can eliminate the last parameter using the pulse reflected from the delay rod-buffer interface during forward transmission r [16]. The intensity of this pulse is marked as Irb . This pulse returns back to the piezo crystal before the pulse reflected from the colloid. The piezoeleccal . We can tric crystal converts it back into the electric pulse with energy Iout measure the ratio of this intensity to the intensity of the initial pulse: Scal ¼

2 cal r Iout Ctr Irb Ctr Irod ðZb  Zr Þ2 2 ðZb  Zr Þ ¼ ¼ ¼ C tr Iin Iin Iin ðZb þ Zr Þ2 ðZb þ Zr Þ2

ð7:18Þ

Substituting this equation into Equation (7.17), we obtain the following simple expression for calculating the acoustic impedance of the colloid: Zs ¼ Zb

1  SNexp 1 þ SNexp

ð7:19Þ

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where SNexp

rffiffiffiffiffiffiffiffi 2 Sexp jZb  Zr2 j ¼ Scal 4Zb Zr

This calibration procedure minimizes the potential problem related to variations in the transducer conversion efficiency.

REFERENCES [1] M.C. Bhardwaj, Advances in ultrasound for materials characterization, Adv. Ceramic Mater. 2 (1987) 198–203. [2] M.S. Dann, N.D. Hulse, Method and apparatus for measuring or monitoring density or rheological properties of liquids or slurries, UK Patent GB, 2 181 243 A, 1987. [3] M. Kikuta, Method and apparatus for analyzing the composition of an electro-deposition coating material and method and apparatus for controlling said composition, US Patent 5,368,716, 1994. [4] B.D. Sowerby, Method and apparatus for determining the particle size distribution, the solids content and the solute concentration of a suspension of solids in a solution bearing a solute, US Patent 5,569,844, 1996. [5] D. Sette, Ultrasonic studies, in: Physics of Simple Liquids, North-Holland, Amsterdam, 1968. [6] A.J. Ricco, S.J. Martin, Acoustic wave viscosity sensor, Appl. Phys. Lett. 50 (1987) 1474–1476. [7] P. Kostial, Surface acoustic wave control of the ion concentration in water, Appl. Acoust. 41 (1994) 187–193. [8] R.E. Apfel, Technique for measuring the adiabatic compressibility, density and sound speed of sub-microliter liquid samples, J. Acoust. Soc. Am. 59 (1976) 339–343. [9] J.D. McClements, M.J.W. Powey, Scattering of ultrasound by emulsions, J. Phys. D Appl. Phys. 22 (1989) 38–47. [10] D.J. McClements, Ultrasonic characterization of food emulsions, in: V.A. Hackley, J. Texter (Eds.), Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, Am. Ceramic Soc., 1998, pp. 305–317. [11] J.D. McClements, Principles of ultrasonic droplet size determination in emulsions, Langmuir 12 (1996) 3454–3461. [12] D.N. Sinha, Non-invasive identification of fluids by swept-frequency acoustic interferometry, US Patent 5,767,407, 1998. [13] J.R. Pellam, J.K. Galt, Ultrasonic propagation in liquids: application of pulse technique to velocity and absorption measurement at 15 megacycles, J. Chem. Phys. 14 (1946) 608–613. [14] J.M.M. Pinkerton, A pulse method for measurement of ultrasonic absorption in liquids, Nature 160 (1947) 128–129. [15] A.S. Dukhin, P.J. Goetz, Method and device for characterizing particle size distribution and zeta potential in concentrated system by means of acoustic and electroacoustic spectroscopy, Patent USA, 09/108,072, 2000. [16] A.S. Dukhin, P.J. Goetz, Method and device for determining particle size distribution and zeta potential in concentrated dispersions, USA Patent 6,449,563, 2002.

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[17] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth study of ultrasound velocity and attenuation in suspensions: comparison of theory with experimental measurements, J. Colloid Interface Sci. 156 (1993) 261–269. [18] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth ultrasonic study of suspensions: the variation of velocity and attenuation with particle size, J. Colloid Interface Sci. 168 (1994) 339–348. [19] A.K. Holmes, R.E. Challis, Ultrasonic scattering in concentrated colloidal suspensions, Colloids Surf. A 77 (1993) 65–74. [20] J. Andreae, R. Bass, E. Heasell, J. Lamb, Pulse technique for measuring ultrasonic absorption in liquids, Acustica 8 (1958) 131–142. [21] J. Andreae, P. Joyce, 30 to 230 Megacycle pulse technique for ultrasonic absorption measurements in liquids, Br. J. Appl. Phys. 13 (1962) 462–467. [22] P.D. Edmonds, V.F. Pearce, J.H. Andreae, 1.5 to 28.5 Mc/s pulse apparatus for automatic measurement of sound absorption in liquids and some results for aqueous and other solutions, Br. J. Appl. Phys. 13 (1962) 551–560. [23] C.R. Cushman et al., Particle size and percent solids monitor, US Patent 3779070, 1973. [24] S.J. Uusitalo, G.C. von Alfthan, T.S. Andersson, V.A. Paukku, L.S. Kahara, E.S. Kiuru, Method and apparatus for determination of the average particle size in slurry, US Patent 4,412,451, 1983. [25] U. Riebel, Method of and an apparatus for ultrasonic measuring of the solids concentration and particle size distribution in a suspension, US patent 4,706,509, 1987. [26] F. Alba, Method and apparatus for determining particle size distribution and concentration in a suspension using ultrasonics, US Patent 5,121,629, 1992. [27] A.S. Dukhin, P.J. Goetz, Method for determining particle size distribution and structural properties of concentrated dispersions, USA Patent 6,910,367, 2005. [28] A.S. Dukhin, P.J. Goetz, Method for determining particle size distribution and mechanical properties of soft particles in liquids, USA Patent 6,487,894, 2002. [29] L. Erwin, J.L. Dohner, On-line measurement of fluid mixtures, US Patent 4,509,360, 1985. [30] K. Leschonski, Representation and evaluation of particle size analysis data, Part. Char. 1 (1984) 89–95. [31] C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983, 530pp. [32] M. Carasso, S. Patel, J. Valdes, C.A. White, Process for determining characteristics of suspended particles, US Patent 6,119,510, 2000. [33] A. Hyman, J.J. Walsh, Philosophy in the Middle Ages, Hackett, Indianapolis, 1973. [34] P.J. Hamstead, Apparatus for determining the time taken for sound energy to cross a body of fluid in a pipe, US Patent 5,359,897, 1994.

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Chapter 8

Applications for Dispersions 8.1. Characterization of Aggregation and Flocculation 8.2. Principles of Particle Sizing in Mixed Colloids with Several Dispersed Phases 8.3. Mixtures with High Density Contrast: Ceramics, Oxides, Minerals, Pigments

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312

8.4. Composition of Mixtures with High Density Contrast 8.5. Cosmetics: Mixtures of Solids in Emulsions 8.6. Milling 8.7. Monitoring Dissolution 8.8. Effect of Air Bubbles References

327 332 336 338 339 340

314

It is practically impossible to present all applications of our 350 ultrasoundbased instruments and hundreds more installed by other companies. We will give here only the most important ones that are based on unique features offered by ultrasound instruments. This chapter is dedicated solely to dispersions. The following chapters present applications to other classes of heterogeneous systems. First of all we will describe in some detail the characterization of dispersion stability and aggregation state using a combination of acoustics and electroacoustics. The next important application is dispersions with several dispersed phases—mixtures. Ultrasound makes possible characterization of possible interaction between particles of different phases. This section will also include ceramics—a very important field for these methods. Then we will describe briefly application of these instruments for controlling milling, dissolution, and precipitation.

8.1. CHARACTERIZATION OF AGGREGATION AND FLOCCULATION Determining the stability of dispersions is one of the central problems in colloid science. Accurate and nondestructive stability characterization is the key Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23008-9 Copyright # 2010, Elsevier B.V. All rights reserved.

303

304

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8

to the successful formulation and use of many colloids. Ultrasound-based techniques offer several advantages over traditional means for stability characterization. Eliminating the dilution protocol is one obvious advantage. Another important advantage is the ability to distinguish between aggregation and flocculation. Aggregation is a process that results in the buildup of rigid agglomerates of the initial primary particles. These agglomerates are essentially new, inherently larger particles, which move as rigid entities in the liquid under stress. Flocculation, on the other hand, is a process in which the original particles retain, to some degree, their independent motion. The particle size distribution remains the same as in the original stable system, but the particles are now weakly linked together by specific colloidal-chemical forces. The nature of these forces might arise from various sources, including perhaps the creation of polymer chains that form bridges between the particles, or electrodynamic forces corresponding to particles being held in the secondary minima. Figure 8.1 illustrates these conceptual differences between aggregation and flocculation. One might expect that the measured acoustic properties somehow reflect this difference between aggregation and flocculation, and in fact we have been able to observe such differences for several colloids. Here we will show results for rutile and alumina that can be interpreted as aggregation, and data for anatase that can be seen as flocculation. The aggregation data were previously published in Langmuir [1], whereas the flocculation data for anatase are presented here for the first time. Stable colloid Primary particles

Aggregation New moving entities

Flocculation Particles retain their motion Specific forces link particles in floc

FIGURE 8.1 Illustration of the difference between aggregation and flocculation.

305

Applications for Dispersions

We present here data for alumina and rutile slurries that illustrate how attenuation spectra can be utilized to characterize an agglomeration process. We purposely chose two materials having quite different isoelectric points (IEPs); pH 4 for rutile, and pH 9 for alumina. The rutile sample was prepared from commercially available aqueous rutile slurry, R746 from DuPont. This slurry is supplied at 76.5 wt% (44.5 vol%), but was diluted to 7 vol% using distilled water. The particle density was 4.06 g/cm3, which is slightly less than that for some other rutile because of various surface modifiers used to stabilize the slurry. The manufacturer reported a median particle size of about 0.3 mm. The alumina slurry was prepared at a concentration of 11.6 wt% (4 vol%) from dry powder. The initial pH was 4. The manufacturer reported a median particle size of 100 nm; acoustically we measured 85 nm for the initial stable sample. Titration of these alumina and rutile slurries and the corresponding measurements of the z-potential confirmed the expected IEPs for each material, as shown in Figure 8.2. According to general colloid chemistry principles, a dispersed system typically loses stability when the magnitude of the z-potential decreases to less than approximately 25 mV. As a result, there will be some region surrounding a given isoelectric pH for which the system is not particularly stable. Within this unstable region, the particles may perhaps agglomerate, thereby increasing the particle size and resulting in a corresponding change in the attenuation spectra. Indeed, this titration experiment showed that the attenuation spectra of both the alumina and rutile slurry do change with pH (Figure 8.3). For the rutile slurry, the attenuation spectra remained unchanged as the pH was decreased until the z-potential reached approximately
30 mV, at which point the spectra rapidly changed to a new state entirely. Similarly, for the alumina slurry, the attenuation spectra remained unchanged as the pH was increased 50 40

zeta potential (mV)

30

alumina

20 10 0 -10

2

3

4

5

6

7 pH

8

9

10

11

-20 -30 -40

rutile

-50 FIGURE 8.2 Electroacoustic pH titrations of rutile (7 vol%) and alumina (4 vol%).

12

306

CHAPTER

8

8

Attenuation (dB/cm/MHz)

7 pH > 6

6

rutile 25 wt% pH aroud 5

5 4

pH < 4 3 pH < 7.5 2

pH=7.9

alumina 12 wt%

pH > 8

1 0 100

101

102

Frequency (MHz) FIGURE 8.3 Attenuation spectra for 7 vol% rutile (300 nm) and 4 vol% alumina (85 nm) at different pH values.

until the z-potential reached þ30 mV, at which point the spectra also changed dramatically. In both cases, that pH corresponding to the point where the magnitude of the z-potential decreased to 30 mV became a critical pH, beyond which the pH could not be changed without degrading the stability of the slurry. In both cases, the spectra assumed a quite different, although more or less constant, shape within this unstable region: one spectrum characteristic of the stable system, and the second spectrum characteristic of the unstable system. The particle size distributions for the stable and unstable conditions of the rutile and alumina slurries are shown in Figures 8.4 and 8.5, respectively. An error analysis presented in the following section proves that the distribution becomes bimodal beyond the critical pH. It is interesting that the size of the aggregates does not exceed several micrometers. Apparently, there is some factor that prevents particles from collapsing into very large aggregates. We believe that these bimodal particle size distributions are stable with time because of stirring. The titration experiment requires the sample to be stirred to provide a rapid and homogeneous mixing of the added reagents. The shear stress caused by this stirring might break the larger aggregates. This shear can be a factor that restricts the maximum size that any aggregate might achieve. This idea can be tested in the future by making measurements at different stirring rates. There remains another unresolved question. We do not know whether the aggregation that we observe is reversible. Neither the alumina nor the rutile sample allows us to reach a z-potential above this critical 30 mV value after

307

Applications for Dispersions

pH=3.9

pH=8.9

1.0

1.6

0.6 0.4 0.2 0.0 10-2

10-1

100

percent diameter

percent

diameter

10%

0.146

10%

0.138

20%

0.218

20%

0.181

30%

0.332

30%

0.219

40%

0.642

40%

0.258

50%

1.017

50%

0.302

60%

1.345

60%

0.352

70%

1.704

70%

0.415

80%

2.169

80%

0.504

90%

2.939

90%

0.660

PSD, weight basis

PSD, weight basis

0.8

1.2

0.8

0.4

0.0 10-2

101

Diameter (mm)

10-1

100

101

Diameter (mm)

FIGURE 8.4 Particle size distribution of stable and unstable rutile at 7 vol%.

pH=4

pH=9 0.9

0.8 0.6 0.4 0.2 0.0 10-3

0.8 percent diameter

percent diameter 10%

0.026

20%

0.039

30%

0.053

40%

0.067

50%

0.085

60%

0.107

70%

0.137

80%

0.183

90%

0.274

10-2 10-1 100 Diameter (mm)

101

PSD, weight basis

PSD, weight basis

1.0

0.7

10%

0.020

0.6

20%

0.028

30%

0.037

40%

0.048

50%

0.066

60%

0.109

70%

0.257

0.2

80%

0.424

0.1

90%

0.678

0.5 0.4 0.3

0.0 10-3

10-2 10-1 100 Diameter (mm)

101

FIGURE 8.5 Particle size distribution of stable and unstable alumina at 4 vol%.

crossing the IEP. For both alumina and rutile samples, the magnitude of the z-potential value is limited by the increased ionic strength at the extreme pH values. We will need another material with an IEP at an intermediate pH in order to answer this question. We can summarize this experiment on rutile and alumina slurries with the conclusion that the combination of acoustics and electroacoustics allows us to characterize the aggregation of concentrated colloids. Electroacoustics yields information about electric surface properties such as the z-potential, while acoustic attenuation spectra provide data for particle size characterization.

308

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8

In Chapter 3 we noted that the acoustic attenuation can be interpreted in terms of longitudinal rheology. We can consider the rutile slurry in this light as a homogeneous non-Newtonian liquid. The attenuation decreases near the IEP. If we follow the longitudinal rheology interpretation, this would suggest that the longitudinal viscosity of the colloid decreases near the IEP, yet this contradicts our experience. Of course, the rutile slurry is not a pure liquid, and the attenuation changes can readily be explained by changes in particle size resulting from aggregation of the primary particles. The rheological interpretation fails in this case because aggregation is essentially a heterogeneous effect. It needs to be interpreted using the notion of a “particle size distribution,” which itself is heterogeneous in nature. A rheological interpretation assumes a homogeneous model for the colloid, which is apparently not valid for describing aggregation. As for flocculation, it turns out that the relationship between these two possible interpretations of the acoustic attenuation spectra is reversed. We will demonstrate that such an aggregation-based interpretation of the variation in the attenuation spectra of an anatase dispersion due to flocculation leads to meaningless results. At the same time, a rheological interpretation makes perfect sense, and suggests an additional structural loss contribution to the total attenuation. The rutile and alumina slurries exemplified an agglomeration process, and we next present data for an anatase slurry, which illustrates how the attenuation spectra can be used to characterize flocculation as well. For this purpose, we selected a pure anatase supplied by Aldrich, following the suggestion of Rosenholm and Kosmulski et al. [2–6], who published several papers about the electrokinetic properties of this material at high ionic strength. We describe such an application later in this chapter. The anatase is easily dispersed at 2 vol% in a 0.01 M KCl solution, resulting in an initial pH value between 2 and 3. Figure 8.6 shows the

60

start

zeta potential (mV)

40 20 0

pH 2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

-20 -40 -60

FIGURE 8.6 Three sweep pH titration of anatase at 2 vol% in 0.01 M KCl solution.

309

Applications for Dispersions

z-potential of this anatase during three back-and-forth titration sweeps over a wide pH range. During each pH sweep, the measured isoelectric pH was about 6.3. Following our earlier results on rutile and alumina, we might again expect a significant loss in stability when the magnitude of the z-potential is reduced to less than 30 mV, corresponding in this case to an unstable pH range of 5–8. This loss in stability might then be expected to produce some change in the particle size distribution, along with a concomitant change in the attenuation spectra, in much the same way as we observed with the rutile and alumina slurries. Indeed, measurements of the attenuation versus pH during these three pH sweeps did confirm that the attenuation varied with pH. However, this variation was quite different from that observed during the aggregation of the rutile or alumina. Figure 8.7 shows selected attenuation curves illustrating the changes in the attenuation spectra resulting from changes in pH. After an initial adjustment caused by pumping, the attenuation reached the lowest level. This happens at pH 4.6, which is still in the stability region. Approaching the IEP more closely caused an increase in the attenuation, not a decrease as was the case for the rutile and alumina aggregation. In order to illustrate this behavior more clearly, we present in Figure 8.8 a plot of the attenuation, at only a single frequency (35 MHz), versus pH. It can be seen that this attenuation reaches a maximum value at the IEP. There are two ways to evaluate this anatase experiment. The first approach is to assume that this sample is aggregating in the same manner as the rutile and alumina samples described above. We can calculate a new PSD from the attenuation spectra measured at each pH. The corresponding mean particle

1.8

IEP

Attenuation (dB/cm/MHz)

1.6 start 1.4 1.2 40 min pumping 1.0

pH 10.57 6.32 4.64 2.34

0.8 0.6 100

101 Frequency (MHz)

102

FIGURE 8.7 Attenuation spectra of 2 vol% anatase in 0.01 M KCl for different pH values.

310

CHAPTER

starting points

1.7

Attenuation at 35 MHz (dB/cm/MHz)

8

1.6

1.5

1.4

1.3 2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

pH FIGURE 8.8 Attenuation of 2 vol% anatase at 35 MHz versus pH.

size and z-potential computed in such a way are shown in Figure 8.9. We see that there is a dramatic correlation between the mean particle size and z-potential—but this correlation makes no sense. If we assume that an aggregation mechanism explains the changes in the colloid behavior, then the mean particle size should increase with decreasing z-potential, as we observed for the rutile and alumina slurries. For the anatase slurry, however, the size

0.40

mean size

zeta potential

0.38

50

0.36 40

0.34 0.32

30

0.30 20

0.28

Mean size (mm)

Zeta potential, absolute value (mV)

60

0.26

10

0.24 0

0.22 0

5

10

15

20

25

30

35

pH titration measurement index FIGURE 8.9 Mean particle size and z-potential of anatase assuming aggregation at the isoelectric point.

311

Applications for Dispersions

decreases with decreasing z-potential. Clearly, our assumption of an aggregation mechanism must be incorrect. We need to look for some other mechanism to explain the attenuation behavior near the isoelectric pH. For this anatase sample, let us consider replacing the initially proposed aggregation mechanism with an alternative flocculation mechanism. Rather than the creation of new large aggregates in the unstable pH region, instead we postulate some flocculation structure developing as was described in Figure 8.1. In Chapter 4 we introduced the concept of structural losses that can be described by two Hookean coefficients H1 and H2. We can now include the possibility of such structural losses, adding them to the viscous and other losses when calculating the particle size from the attenuation spectra for this anatase sample. If we do this, we find that the PSD remains practically constant over the whole pH range, whereas the second Hookean coefficient, H2, changes with pH as shown in Figure 8.10. H2 reflects the variation with pH of the strength of the interparticle bonds. These bonds become stronger closer to the IEP, as one might expect. This might happen because of the weakening electrostatic repulsion due to the decrease in z-potential decay in the vicinity of the IEP. In conclusion, we can formulate several general statements as follows:

l l

Electroacoustics provides a means to determine the IEP and the expected stability range of concentrated colloids. Acoustics allows us to distinguish between aggregation and flocculation. In the case of aggregation, acoustic attenuation measurements yield information about variations in particle size.

Zeta potential, absolute value (mV)

60

Hookean coefficient

zeta

0.8 0.7

50

0.6 40

0.5

30

0.4 0.3

20

0.2 10

Second Hookean coefficient

l

0.1

0

0.0 2

3

4

5

6

7

8

9

10

11

pH FIGURE 8.10 z-potential and Hookean coefficient H2 for anatase at 2 vol% versus pH, assuming flocculation at the isoelectric point.

312 l

l

CHAPTER

8

In the case of flocculation, acoustic measurements allow us to characterize not only the particle size distribution but also the strength of the interparticle bounds and the micro-viscosity of the colloid. To do this, the structural losses must be taken into account. Interpretation of the attenuation spectra changes due to aggregation in terms of longitudinal rheology fails, whereas is succeeds if these changes are related to flocculation.

8.2. PRINCIPLES OF PARTICLE SIZING IN MIXED COLLOIDS WITH SEVERAL DISPERSED PHASES There are many important natural and man-made dispersed systems containing a high concentration of more than just one dispersed phase. For instance, whole blood contains many different types of cells; paint usually consists of latex with added pigment to provide color; and sunscreen preparations include an emulsion as well as ultraviolet-light-absorbing particles. In many such systems there is a practical need to determine the particle size distribution of more than just one ingredient. In general, light-based techniques are not well suited for these applications, because most optical methods require the sample to be diluted prior to measurement, thereby distorting, or destroying altogether, the particle size information being sought. Furthermore, most light-based systems cannot handle multiple disperse phases, even in the most dilute case. In contrast, acoustic attenuation spectroscopy enables us to eliminate this undesirable step of sample dilution. It is now well known that acoustic spectroscopy can characterize particles size at concentrations up to 50% by volume. Furthermore, acoustic attenuation spectroscopy can characterize the particle size distribution of concentrated dispersions having more than one dispersed phase. These two unique features make acoustic spectroscopy very attractive for characterizing the particle size distribution of many real-world dispersions. There are at least three quite different philosophical approaches for interpreting the acoustic spectra in complex mixed colloids. In the simplest empirical approach, we forego any size analysis per se and simply observe the measured acoustic attenuation spectra to learn whether, for example, the sample changes with time or if “good” or “bad” samples differ in some significant respect. Importantly, this empirical approach provides useful engineering solutions even in cases where nothing is known about the physical properties of the sample or whether, indeed, the sample is adequately described by our theoretical model. In order to make this approach more familiar, we can convert attenuation spectra into longitudinal viscosity as it shown in Chapter 3. Basically this empirical approach is based on the same routes as traditional rheology. In a more subtle validation approach, we assume in advance that we know the correct particle size distribution, and furthermore assume that the real dispersion conforms to some well-defined model. We then use some predictive

Applications for Dispersions

313

theory based on this model, as well as the assumed size distribution, to test whether the predicted attenuation matches that actually measured. If the validation fails, it is a very strong indication that the model is inadequate to describe the system at hand. As an example of this validation approach, consider the case where we construct a mixed system by simply blending two single-component slurries. The PSD of each single-component slurry can be measured prior to blending the mixed system. Since we have control over the blending operation, we know precisely how much of each component is added. If we claim that the combined PSD is simply a weighted average of the individual PSD for each component, we are in effect assuming that there is no interaction between these components. In this case, the prediction theory allows us to compute the theoretical attenuation for this mixed system. If the experimental attenuation spectrum matches the predicted spectrum, then the assumption that the particles did not interact is confirmed. However, if the match is poor, it is likely that the mixing of the two components caused some changes in the aggregation behavior of the system. Perhaps new composite particles were formed by the interaction of the two species, or some chemical component in one sample interacted with the surface of another. Many interaction possibilities exist. Nevertheless, it seems appropriate to conclude that a necessary condition for ruling out aggregation in mixed systems is that the experimental and predicted attenuation curves match. In addition, we can probably also conclude that an error between theory and experiment is sufficient to imply that some form of aggregation or disaggregation has occurred within the mixed system. We will show that such prediction arguments are indeed able to monitor such aggregation phenomena. Finally, we can take the ultimate leap of faith and use an analysis algorithm to search for that particle size distribution which, in accordance with the model and the predictive theory, best matches the experimental data. Whereas there are several papers [7–10] which demonstrate that acoustic spectroscopy is able to characterize bimodal distributions in dispersions when both modes are chemically identical, it is less well known that acoustic spectroscopy is also suitable for characterizing mixed dispersions when each mode is chemically quite different. Here, we will discuss two models that can be particularly helpful for describing such mixed dispersions. The first, “multiphase,” model assumes that we can represent the PSD of a real-world dispersion as a sum of separate lognormal distributions, one for each component in the mixed system. We assume that there are only two components, which then reduces the overall PSD to a simple bimodal distribution. When we calculate the attenuation of such a multiphase system, we take into account the individual density, among other particle properties, for each component in the mixture. For a bimodal case, the multiphase approach would typically require the analysis algorithm to fit five adjustable parameters: the

314

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8

median size and standard deviation of both modes, and the relative mass fraction of each mode. In this work, we will assume that the weight fraction of each mode is known in advance. Furthermore, in an effort to avoid the well-known problem of multiple solutions, we will further assume that both modes have the same standard deviation. Altogether, these simplifications reduce the number of adjustable parameters to just three: the median size of each mode, and the shared standard deviation. The implications of these simplifications will be discussed later. The second, “effective medium,” model further assumes that one needs to determine the PSD of just one component in a multicomponent mixed system. All other disperse phases are lumped together into an effective homogeneous medium, characterized by some composite density, viscosity, and acoustic parameters. By adopting this viewpoint, we significantly reduce a complex real-world mixture to a simpler dispersion of a single preselected dispersed phase, in a newly defined “effective medium.” We need not even define the exact nature and composition of this new medium since we can simply measure, or perhaps calculate, the required composite density, viscosity, attenuation, and sound speed. If we assume that the key disperse phase can be described by a lognormal distribution, then we will have reduced the degree of freedom to just two adjustable parameters, a median size and standard deviation. Here we initially evaluate the effectiveness of both the multiphase and the effective medium models for rigid particles with high density contrast. These might be ceramic slurries, paints, and other oxides and minerals. Then we illustrate how this method works with the example of real cosmetic products, such as sunscreens. Then we will present results for cosmetics, when one of the phases is an emulsion with low density contrast.

8.3. MIXTURES WITH HIGH DENSITY CONTRAST: CERAMICS, OXIDES, MINERALS, PIGMENTS There are several published papers describing the application of ultrasoundbased instruments for ceramics [11–15]. We present here results of these tests presented in Ref. [16]. We used three pigments from Sumitomo Corporation: AKP-30 alumina (nominal size 0.3 mm), AA-2 alumina (2 mm), and TZ-3YS zirconia (0.3 mm). A mixture of such oxides plays an important role in many ceramic composites [17]. In addition, we used precipitated calcium carbonate (PCC) (0.7 mm) and Geltech silica (1 mm). Slurries of the AA-2 alumina and the zirconia were prepared in such a manner as to have quite good aggregative stability. Each slurry was prepared at 3 vol% by adding the powder to a 10
2 mol/l KCl solution, adjusted initially to pH 4 in order to provide a significant z-potential. Although the alumina showed very quick equilibration,

Applications for Dispersions

315

the zirconia required about 2 h for the z-potential and pH to equilibrate. Electroacoustics allows us to establish this equilibration time by making continuous z-potential measurements over an extended time. This will be discussed later in Section 8.5. Preparation of a 3 vol% PCC slurry was more problematic, since the z-potential right after dispergating was very low (1.3 mV). Control of pH alone was insufficient, and we therefore used sodium hexametaphosphate in order to increase the surface charge and thereby improve the aggregative stability of this slurry. In order to determine the optimum dose, we ran a z-potential titration as described later in Chapter 11. The z-potential reaches a maximum at a hexametaphosphate concentration of about 0.5% by weight, relative to the weight of the PCC solid phase. The Geltech silica and the AKP-30 alumina were used only as dry powders, being added to the PCC slurry as needed. The goals of the experiment were met in the following steps. Step 1. A single-component slurry was prepared for the alumina, zirconia, and PCC materials as described above. Step 2. The attenuation spectra of these single-component slurries were measured, and the particle size distribution for each was calculated. Step 3. Three mixed alumina/zirconia slurries were prepared by blending the above slurries in different proportions, and the attenuation spectrum for each mixture was measured. Step 4. Geltech silica powder was added to the initial PCC slurry, and the attenuation spectrum was measured for this mixed system. Step 5. AKP-30 alumina powder was added to the initial PCC slurry, and the attenuation spectrum for this mixed system was measured. Step 6. The particle size distribution was calculated for all of the mixed systems using the multiphase model. Step 7. The properties of the effective medium were calculated for all mixtures. Step 8. The particle size distribution for each of these mixed systems was calculated using the effective medium model. Step 9. The results of the particle size calculation using two different approaches were compared. Step 10. The validation approach was used to test for possible particle interactions in the mixed systems. The experimental attenuation spectra for the three single-component slurries and five mixtures are shown in Figures 8.11 and 8.12. In order to demonstrate reproducibility, each sample shown in Figure 8.11 was measured at least three times. Mixture 1, in fact, was measured yet a fourth time after a fresh sample was loaded just to show that sample handling was not a detrimental factor. It is clear that the reproducibility is sufficient for resolving the relatively large differences in attenuation between the different samples.

316

CHAPTER

8

Attenuation (dB/cm/MHz)

5 mixture 1, load 2 mixture 1 mixture 1 mixture 1 mixture 2 mixture 2 mixture 2 mixture 3 mixture 3 mixture 3 AA-2 AA-2 AA-2 TZ-3YS TZ-3YS TZ-3YS

4 zirconia

3 mixture 1 mixture 2 mixture 3

2

alumina

1 0 100

101 Frequency (MHz)

102

FIGURE 8.11 Experimental attenuation spectra for initial alumina AA-2 and zirconia TZ-3YS from Sumitomo and their mixtures with weight fractions given in Table 8.1. This figure illustrates reproducibility by showing repeat measurements, including filling mixture #1 a second time.

5

Attenuation (dB/cm/MHz)

4

3

2 PCC PCC PCC PCC

1

and alumina AKP-30 and silica Geltec only only

0 100

101 Frequency (MHz)

102

FIGURE 8.12 Experimental attenuation spectra for initial PCC slurry and its mixture with the added silica and alumina powders. Weight fractions are given in Table 8.2.

The attenuation spectra for the single-component slurries of the alumina, the zirconia, and the PCC allow us to calculate the particle size distribution for each material. The calculated sizes are given in the Tables 8.1 and 8.2, and it can be seen that these acoustically defined sizes agree quite well with the nominal sizes given by the manufacturers of these materials.

Initial Alumina Volume fraction, % Weight fraction, %

3 10.96

Eff. Viscosity, cp 3

Eff. Density, g/cm

Mixture 1 Zirconia

3 15.91

Alumina

Zirconia

Mixture 2 Alumina

Zirconia

Mixture 3 Alumina

1.55

1.45

1.85

1.15

2.28

0.72

5.5

7.9

6.6

6.3

8.2

4

0.92

0.93

0.94

1.04

1.05

1.06

Att. M0

1.593

1.21

0.982

0.823

Att. M1

0.0845

0.0642

0.0521

0.0437

Att. M2


1.251

Att. M3

0.528

Zirconia


0.95 0.401


0.771


0.646

0.326

0.273

Applications for Dispersions

TABLE 8.1 Characteristics of Alumina AA-2 and Zirconia TZ-3YS Slurries and Their Mixtures

Continued

317

318

TABLE 8.1 Characteristics of Alumina AA-2 and Zirconia TZ-3YS Slurries and Their Mixtures—Cont’d Initial Alumina

Mixture 1 Zirconia

Alumina

Zirconia

Mixture 2 Alumina

Zirconia

Mixture 3 Alumina

Zirconia

Parameters of the particle size distributions, effective medium approach Median lognormal, mm

2.15  0.02

0.33  0.006

0.293  0.006

0.303  0.005

0.317  0.003

Std deviation

0.26

0.43

0.38

0.378

0.372

Fitting error, %

6.6

1.9

1.4

1.2

0.95

Parameters of the particle size distributions, two dispersed phases approach Median size, mm

0.565  0.002

0.558  0.001 2.922  0.088

0.352  0.005

3.582  0.182

Std deviation

0.53

0.3

0.21

Fitting error, %

5

7.6

4.4

0.303  0.003

CHAPTER 8

319

Applications for Dispersions

TABLE 8.2 Characteristics of PCC Slurry and Its Mixtures with Alumina AKP-30 and Silica Geltech Initial PCC

Volume fraction, %

10.55

Weight fraction, %

23.53

Initial silica

PCC and silica

PCC and alumina

Powder

PCC

PCC

9.19 19.6

Silica 6.29 11.3

10.27

2.52

21.6

8.1

Eff. viscosity, cp

1.125

1.094

1.118

Eff. density, g/cm3

1.17

1.13

1.15

Att. M0

1.053

Att. M1

4.431

Att. M2


3.648

Att. M3

Alumina

0.9296

Parameters of the particle size distributions, effective medium approach Median lognormal, mm

0.684

1.26

0.454

0.325

Std deviation

0.31

0.35

0.015

0.015

Fitting error, %

1.1

1.3

7.5

2.4

Parameters of the particle size distributions, two dispersed phases approach Median size, mm

0.449

Std deviation

0.16

0.19

Fitting error, %

8

1.9

0.681

0.798

0.2715

As shown in Figures 8.11 and 8.12, the attenuation spectra of the mixtures differ significantly from those of the single-component slurries. This difference in the attenuation spectra reflects the differences in both the particle size distributions and the density of the constituent components in the mixtures. We want to compare the effectiveness of the multiphase and the effective medium approaches in calculating the PSD of these five different mixed systems.

320

CHAPTER

8

First, let us consider the more or less straightforward multiphase model. To use this approach, we need only know the weight fraction and density of both disperse materials. The present DT-1200 software implementation assumes that the total particle size distribution is bimodal, and that each mode corresponds to one disperse phase material. In the alumina/zirconia mixture, for example, the smaller mode corresponds to the zirconia, and the larger mode corresponds to the alumina. The software takes into account the difference in densities between materials of the first and the second modes. The PSD of each mode is itself assumed to be lognormal. In order to reduce the number of adjustable parameters, and in an effort to reduce the likelihood of multiple solutions, the present software implementation assumes that both modes have the same standard deviation. The software searches for some combination of the three adjustable parameters (two median sizes and their common standard deviation) that provide the best fit to the experimental attenuation spectra. It assumes the relative content of the modes to be known. The corresponding PSD for these five mixed systems are shown in Figures 8.13 and 8.14. The parameters of these PSDs are given in Tables 8.1 and 8.2. It is seen that in some cases this multiphase approach yields more or less the correct size. For instance, the two zirconia/alumina mixtures with a lower zirconia content (mixtures 2 and 3) have almost the correct size combination. The size of the alumina particles is somewhat higher than expected

1.4 1.2

PSD, weight basis

1.0

mixture 1 mixture 2 mixture 3

0.8 0.6 0.4 0.2 0.0 10-2

10-1

100 Diameter (mm)

101

102

FIGURE 8.13 Particle size distributions calculated for alumina-zirconia mixtures using the multiphase model. The smaller size mode corresponds to zirconia, and the larger size mode to alumina AA-2. The weight fraction and PSD parameters are given in Table 8.1.

321

Applications for Dispersions

PSD, weight basis

2.0 alumina in PCC dispersion silica in PCC dispersion

1.6 1.2 0.8 0.4 0.0

10-2

10-1

100

101

Diameter (mm) FIGURE 8.14 Particle size distributions calculated for PCC-alumina and PCC-silica mixtures using the multiphase model. Weight fraction and PSD parameters are given in Table 8.2.

(2.15 mm), but is still fairly acceptable. We can say the same about the PCC/ alumina mixture from Table 8.2. The difference of the sizes relative to the nominal values does not exceed 10%. The multiphase model, on the other hand, appears to be a complete failure for the alumina/zirconia mixture #1 as well as the PCC/silica mixture. It is not clear yet why this model works for some systems and not for others. We think that it is probably related to the fact that the present software assumes that both particle size modes have the same width. It is seen that the singlecomponent zirconia slurry has a PSD that is much broader (std dev. ¼ 0.43) than the PSD of the AA-2 alumina (std dev. ¼ 0.26). The bimodal searching routine finds the correct intermediate value for the standard deviation (0.3) only for mixture #2. It is interesting that this PSD solution is the closest match to the superposition of the initial PSD. The standard deviations for the other two mixtures are out of range completely, and the corresponding PSDs also deviate from the expected superposition. This observation allows us to conclude that our restriction that the standard deviation be the same for both modes might itself create a wrong solution. It is easy to eliminate this restriction, but, in general, as one adds additional degrees of freedom, it is not uncommon to be faced with the problem of multiple solutions. This multiple solution problem appears when the error function (difference between experimental and theoretical attenuations) has several local minimums resulting from different combinations of the adjustable parameters. In general, the problem of multiple solutions increases as the number of adjustable parameters increases. It seems clear that the maximum number of

322

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8

adjustable parameters that will avoid multiple solutions is not a fixed number, but rather depends on a combination of factors: the accuracy and amount of experimental data points; the degree to which the real-world sample is described by the model; and how accurately the key parameters of the colloid such as weight fraction, density, etc. are known. Our experience is that even bimodal PSDs with only four adjustable parameters sometimes exhibit multiple solutions. We have found ways to resolve these multiple solutions in the case of single-component dispersions; however, the situation is more complicated for mixed dispersions with two or more chemically different components. For this reason, we restricted the number of the adjustable parameters to only three for this work. These results indicate that the multiphase model might sometimes lead to wrong solutions and it is unclear at this point how to completely eliminate the problem. In contrast, the effective medium approach circumvents this problem by addressing only the question of determining a simple lognormal distribution, which describes only one disperse phase in an otherwise complex mixture. Since we are then dealing only with two adjustable parameters (median size and standard deviation), the possibility for multiple solutions is diminished. On the downside, when using the effective medium approach, we need to perform an additional experiment to measure the properties of this effective medium, and this may not always be possible, or without other difficulties. In the case of the PCC mixtures with the added alumina or silica, the original PCC slurry itself serves as the effective medium. We need just three parameters to characterize this effective medium, namely, density, viscosity, and attenuation. Importantly, all three parameters can be measured directly if we have access to this medium. The attenuation is the most important of these three required parameters. It is also the most challenging to characterize, because we need the attenuation of this medium as a function of frequency from 3 to 100 MHz. The current version of the DT-1200 software allows us to define the attenuation of the effective medium much the same way we would normally define the intrinsic attenuation of a pure liquid medium. This intrinsic attenuation as measured in [dB/cm/MHz] can be described in terms of a polynomial function: að f Þ ¼ M0 þ fM1 þ f 2 M2 þ f 3 M3

ð8:1Þ

where f is frequency in MHz, and M0, M1, M2, and M3 are the polynomial coefficients. For example, in the simplest case we can say that our effective medium is just water. Water has an attenuation that for practical purposes can be said to simply increase as a linear function of frequency, if attenuation is expressed in [dB/cm/MHz]. Thus M0, M2, and M3 are zero and M1 represents this linear dependence.

323

Applications for Dispersions

To use the effective medium approach for mixed systems, we simply need to define new coefficients that describe the intrinsic attenuation of this new medium. In the case of the alumina/zirconia mixtures, we use the alumina slurry as the effective medium. The coefficients for the alumina slurry can be calculated by fitting a polynomial to the attenuation data, as shown in Figure 8.15. These coefficients are also given in Table 8.1. Similarly, the coefficients for the PCC effective medium can be calculated from a polynomial fit of the attenuation data for that material, as shown in Figure 8.15. Likewise, these coefficients are given in Table 8.2. We should keep in mind that the initial alumina slurry gets diluted when we mix it with increasing amounts of the zirconia slurry. As a result, we need to recalculate the attenuation coefficients for each mixture, taking into account the reduced volume fraction of the alumina in each mixture. The suitably modified values for the attenuation coefficients of the effective medium for all three alumina/zirconia slurries are also given in Table 8.1. We avoided the need for making these additional calculations in the case of the PCC mixtures by simply adding dry silica or alumina powder to the PCC effective medium, and therefore the coefficients for the PCC effective medium are the same for both mixtures. For an aqueous medium, the software automatically calculates the intrinsic attenuation of water and subtracts this from the measured attenuation in order to deduce the attenuation caused solely by the presence of the disperse particles. When using the effective medium model, the software actually works in the same way, except that the intrinsic attenuation of the water medium Alumina AA-2

PCC slurry

1.6 Y = 1.593 + 0.08454X - 1.251X2 + 0.5281X3

Attenuation (dB/cm/MHz)

Attenuation (dB/cm/MHz)

1.4

5

1.2 1.0 0.8 0.6 0.4

4 Y = 1.053 + 4.431X - 3.648X2 + 0.9296X3

3

2

1

0.2 0.0 100

101 102 Frequency (MHz)

0 100

101 Frequency (MHz)

102

FIGURE 8.15 Experimental attenuation spectra measured for individual alumina AA-2 slurry and PCC slurry with polynomial fits.

324

CHAPTER

8

is replaced by the attenuation of this new effective medium. For instance, in the case of the PCC/alumina mixture, the software calculates the attenuation due to the PCC contribution and subtracts it from the total attenuation of the mixture. The residual part corresponds to the attenuation due to the alumina particles, and is the source of the particle size information for the alumina component. The software assumes a lognormal PSD and fits this residual attenuation using the median size and standard deviation as adjustable parameters. This effective medium approach allows us to calculate the particle size distribution of the zirconia in the alumina/zirconia mixtures and of the silica or the alumina in the case of PCC mixtures. The corresponding values are shown in Tables 8.1 and 8.2. Figure 8.16 illustrates the corresponding PSD for each case. In the case of zirconia, we have almost the same PSD for all three mixtures. This PSD agrees well with the initial slurry. The fitting error is much smaller than in the multiphase model, which is an additional indication of the consistency of this method. In the case of PCC mixtures, the situation is more complicated. We have a very good correlation with the nominal size of the AKP-30 alumina in the case of the PCC-alumina mixture and a good fitting error. However, the other PCC-based mixture gives a particle size which is half that expected. Table 8.2 shows that the calculated size of the silica Geltech in Zirconia 30

Alumina AKP-30 and silica in PCC slurry

mixture 1, 1.45% vl mixture 2, 1.15% vl mixture 3, 0.72%vl TZ-3YS initial, 3%vl

0.8 0.6 0.4

PSD, weight basis

PSD, weight basis

1.0

20 alumina silica 10

0.2 0.0 10−2

10−1 100 Diameter (µm)

101

0 10−1

100 Diameter (µm)

FIGURE 8.16 Particle size distribution calculated using the effective medium model. In the case of zirconia, the alumina AA-2 dispersion is the effective medium. Attenuation of the alumina is reduced according to volume fractions from Table 8.1. The density and viscosity are adjusted for the effective medium. In the case of alumina AKP-30 and silica, the PCC dispersion acts as the effective medium.

325

Applications for Dispersions

this mixture is only 0.454 mm, whereas the nominal size is at least 1 mm. Acoustically, we measured an even larger 1.26 mm for this silica: perhaps the size was slightly larger than reported by the manufacturer because of the problems in dispersing this sample. We have found that this silica is difficult to disperse even at high pH and at high z-potential. For instance, we measured a z-potential of
66 mV for this silica at pH 11, but even this was apparently not sufficient to disperse it completely. Summarizing the analysis results for these five mixtures, we conclude that in the case of the three mixed dispersions (alumina-zirconia mixtures #2 and #3, and the PCC-alumina mixture), the multiphase model and the effective medium model gave similar results and reasonable PSD values. For the other two mixtures, the results are more confusing. We suspect that the failure of the multiphase model for the alumina-zirconia mixture #1 is related to the restriction on the PSD width, but particle aggregation is still a candidate as well. In the case of the PCC-silica mixture, the failure of both models certainly points towards particle aggregation. We can evaluate these ideas about aggregation of the two difficult mixtures using the validation approach. To do this, we must first compute the total PSD using the known PSD of the individual single-component dispersions. Next, we calculate the predicted attenuation for this combined PSD. This predicted attenuation should agree with the experimental spectrum for the mixed system if there is no particle interaction between the species. Figure 8.17 illustrates the predicted and experimental attenuation spectrum for the Zirconia-alumina mixture #1 and the PCC-silica Geltech mixture. For

Zirconia-alumina mixture 1

PCC-silica Geltech mixture 4

2 size1 size 2

1

experiment analysis 0.56 0.57 prediction 0.33 2.15

0 100 101 102 Frequency (MHz)

stdev1 stdev2 0.54 0.43

0.54 0.26

Attenuation (dB/cm/MHz)

Attenuation (dB/cm/MHz)

3

3

2

size 1

size2

experiment analysis 0.449 prediction 0.684

0.681 1.26

stdev1 0.16 0.31

stdev2 0.16 0.35

1

0 100

2

10 101 Frequency (MHz)

FIGURE 8.17 Experimental and theoretical attenuation for zirconia-alumina mixture #1 and the PCC-silica mixture. The theoretical attenuations are calculated for the best analysis result, and for the combined PSD built from the individual distributions, assuming no particle aggregation.

326

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8

both mixtures, we have also added the predicted attenuation corresponding to the best PSD calculated using the multiphase-model analysis. It is seen that, in the case of the zirconia-alumina mixture, a superposition PSD generates an attenuation spectrum that fits the experimental spectra much better than the best multiphase model PSD. The fitting error has improved from 5% to 2.3%, and becomes comparable with the best fitting errors of the effective medium model. This correlation between prediction and experiment proves that our concern about using a common standard deviation for both modes was well founded. The prediction program allows us to apply independent standard deviation for each mode of the PSD, and as a result we achieve much better fitting than in the case of the analysis multiphase model that uses the same standard deviation for both modes. In addition, we conclude that there is no aggregation between the alumina and zirconia particles in this mixed dispersion. Otherwise, the theoretical attenuation based on the superposition assumption would not fit the experimental data. The situation with the second mixture (PCC-silica) is very different. In this case, the predicted attenuation provides a much worse fit than the best multiphase model analysis. The fitting error degrades from 8% to 17.2%. This means that the superposition assumption is not valid. In this case, there is apparently some aggregation between the PCC and silica particles. We can conclude from these tests that acoustic spectroscopy is able to characterize the particle size distribution of dispersions with more than one dispersed phases. There are two models that we can use to describe the dispersion: a multiphase model and an effective medium model. The multiphase model describes the total distribution as a number of separate lognormal distributions, one for each disperse phase. Although this provides a complete description of the system, it also entails certain risks of multiple solutions because of the large number of adjustable parameters. For this reason, we assumed here that the width for each mode of our binary mixtures was the same. This assumption might lead in some cases to an incorrect solution, especially when dealing with mixture of dispersed phases with widely different standard deviations. The effective medium model is not complicated by multiple solution problems. It requires only two adjustable parameters for characterizing the lognormal distribution of the selected dispersed phase. All other components of the mixed dispersion are considered as a new, homogeneous effective medium. We assume that the attenuation of this effective medium is the same as measured for the mixed materials separately. In those cases where this assumption is valid, the effective medium model yields robust and reliable PSD values for the selected dispersed phase. In the opposite case, when the assumption is not valid, we obtain a PSD that differs from the expected particle size of this dispersed phase. Observation of this difference can be used as an indication of particle-particle interactions between the particles from the effective medium and the selected dispersed phase.

327

Applications for Dispersions

In addition to the particle size distribution, acoustic attenuation spectroscopy is able to indicate the presence of particle aggregation. To perform this test, we need to calculate the attenuation spectra for the PSD that is built from the individual particle size distributions using the superposition assumption. Failure of this theoretical attenuation to fit experimental data is an indication of the particle aggregation.

8.4. COMPOSITION OF MIXTURES WITH HIGH DENSITY CONTRAST Acoustic spectroscopy provides a means to determine the relative amounts of a substance in a mixed dispersion comprised of several different materials. In the case of rigid particles, this opportunity is related primarily to the density differences between the various dispersed phase materials. A variation in the relative amount of each material causes a variation in the total volume fraction, and consequently the attenuation spectra. Experimental monitoring of these variations in the attenuation spectra allows one to characterize the corresponding variation in the composition of the dispersed phases. This idea was suggested by Kikuta in a US patent [18]. We illustrate this application with an example of a mixture of rutile and aluminum hydroxide. Figure 8.18 shows the measured attenuation spectra for the initial single-phase dispersions of rutile and aluminum hydroxide (ATH) as well as their mixtures. The content of ATH in these mixtures varies from 23% to 31%. Each sample was measured three times, in order to demonstrate reproducibility of the measured attenuation spectra, which in turn indicates the reliability of this method for characterizing the ATH content.

Attenuation (dB/cm/MHz)

7

rutile to ATH ratio

rutile only

6 5

ATH 23%

4

ATH 31%

3

69:31 69:31 69:31 73:27 73:27 73:27 74:26 74:26 74:26 75:25 75:25 75:25 76:24 76:24 76:24 77:23 77:23 77:23

aluminum hydroxide only

2 100

101 Frequency (MHz)

102

FIGURE 8.18 Attenuation spectra measured for rutile, aluminum hydroxide, and their mixtures at 25 wt% total solid.

328

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8

The variation in the measured attenuation spectra reflects not only changes in the volume fraction and density but the particle size as well. The attenuation of the mixture depends on the particle size distributions of the components. This fact is used in previous sections for characterizing particle size distributions of mixed dispersions. Particle size dependence is a potential obstacle for characterizing mixture compositions. Fortunately, it is possible to minimize this factor’s influence by proper selection of the frequency range. Figures 8.19 and 8.20 illustrate this procedure. Figure 8.19 shows particle size distributions 1.8 1.6

aluminum hydroxide

rutile

PSD, weight basis

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

10-2

10-1

100

101

Diameter (mm) FIGURE 8.19 Particle size distributions of rutile and ATH calculated from the attenuation spectra in Figure 8.18.

8

Attenuation (dB/cm/MHz)

rutile

7 0% ATH 10% ATH 20% ATH 30% ATH 40% ATH 50% ATH 60% ATH 70% ATH

6 5 4

80% ATH

3

90% ATH

2

aluminum hydroxide

100% ATH

1 100

101 Frequency (MHz)

102

FIGURE 8.20 Attenuation spectra calculated for rutile-ATH mixture at 25 wt%, assuming particle size distributions from Figure 8.19.

329

Applications for Dispersions

calculated from the attenuation spectra measured for rutile only and ATH only. Note that the rutile particles are much smaller than the ATH particles. We can assume that the particle size distribution of each material remains the same in the mixture, because the stability ranges for these two materials are about the same. Using this assumption, we can calculate theoretical attenuation spectra for mixtures with different ATH contents. These theoretical attenuation spectra are shown in the Figure 8.20. It can be seen that the level of ultrasound attenuation is almost proportional to the ATH content. The variation is relatively small at low frequency, and reaches a maximum at the mid-frequency range. This means that the mid-frequency range (from 20 to 40 MHz) is the most desirable for characterizing the ATH content. Attenuation of ultrasound is most sensitive to the ATH particles in this frequency range. It turns out that the second factor determining ultrasound attenuation, that is, particle size distribution, is also pointing to the mid-frequency range as the most suitable for calculating the ATH content. Ultrasound attenuation is least sensitive to particle size within the mid-frequency range. Figure 8.21 illustrates this statement. The attenuation spectra shown in this figure are calculated for various particle size distributions. The particle size of rutile varies from 0.25 to 0.35 mm, and the particle size of ATH varies from 0.7 to 2 mm. There are two conclusions that follow from the attenuation spectra in Figure 8.21. First of all, variation of the ATH particle size affects the attenuation spectra much less than the variation of the rutile particle size. These variations become minimal in the mid-frequency range between 20 and 40 MHz. The values of ultrasound attenuation given in the Table 8.3 indicate

5.8 5.6

Attenuation (dB/cm/MHz)

5.4 5.2 5.0 4.8 4.6 4.4

experiment; 25% of ATH, solid wfr 25% d1=0.298; d2=1.46; stdev=0.32 d2=1 d2=0.7 d2=2 d1=0.255 d1=0.275 d1=0.325 d1=0.35

4.2 4.0 3.8 3.6 3.4 3.2 100

101 Frequency (MHz)

102

FIGURE 8.21 Theoretical attenuation spectra for rutile-ATH mixtures with different individual sizes.

330

TABLE 8.3 Attenuation of 75:25 Rutile-Aluminum Hydroxide Mixture at 25% Total Weight Fraction for Selected Frequencies and Particle Size Distributions Frequency [MHz]

Experiment

d1 ¼ 0.3

d1 ¼ 0.3

d1 ¼ 0.3

d1 ¼ 0.3

d1 ¼ 0.25

d1 ¼ 0.28

d1 ¼ 0.33

d1 ¼ 0.35

d2 ¼ 1.46

d2 ¼ 1.0

d2 ¼ 0.7

d2 ¼ 2

d2 ¼ 1

d2 ¼ 1

d2 ¼ 1

d2 ¼ 1

19.2

5.718398

5.600715

5.645591

5.679868

5.56728

5.541092

5.604283

5.66003

5.647861

23.5

5.651757

5.612663

5.658526

5.697022

5.582969

5.619873

5.651227

5.636786

5.594701

28.9

5.656034

5.587781

5.631235

5.67256

5.565599

5.659421

5.658074

5.574058

5.503365

35.5

5.506359

5.527928

5.568101

5.608497

5.517046

5.658066

5.624988

5.478223

5.381812

Standard deviation of each fraction is assumed to be 0.32

CHAPTER 8

331

Applications for Dispersions

that the attenuation varies as little as 0.1 dB/cm/MHz for the rather wide variation of the particle sizes mentioned above. This short analysis indicates that the frequency range between 20 and 40 MHz is optimal for characterizing the ATH content in rutile-ATH mixtures. The attenuation spectra are most sensitive to the ATH particles in this frequency range and the least affected by the possible variation of the particle size distribution of the components. The precision of the measurement of the ATH content is related to the precision of the attenuation measurement. The total difference in attenuation between pure rutile and pure ATH is about 5 dB/cm/MHz in this frequency range, as follows from the curves in Figure 8.18. Attenuation varies with the ATH content almost as a linear function, according to Figure 8.20. This means that a variation of the ATH content of 1% causes a variation of attenuation of about 0.05 dB/cm/MHz. It turns out that we can measure attenuation within this frequency range with such a high precision. Figure 8.22 presents results of multiple measurements performed with rutile-ATH mixtures with different component ratios. The frequency of the measurement is 40 MHz. The step size of the component ratio is about 1.1% of ATH. Eleven measurements were made for the each value of the component ratio. There is a clearly reproducible difference between the attenuations corresponding to the different component ratios. This means that this instrument allows us to characterize the content of ATH in the rutile-ATH mixtures with a precision of less than 1%.

Attenuation at 40 MHz in (dB/cm/MHz)

5.6 25% ATH

5.5 26.2% ATH 27.3% ATH

5.4 28.4% ATH

5.3 29.5% ATH

5.2 0

10

20

30

40

50

60

Measurement number FIGURE 8.22 Attenuation of rutile-ATH mixture at 40 MHz for different values of ATH content.

332

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8

This high level of precision is achievable in the particular case of the rutile-ATH mixtures. However, we cannot generalize this precision claim for every mixed dispersion. A similar analysis would be required in each particular case. The suggested acoustic method of determining component ratios requires information about the total solid content. There are two ways to gain this information. The first one is a simple drying procedure, which is widely used in many industrial applications. The other method is to use density measurements. The density of the dispersion, rs, depends on the total weight fraction of solids, ws, as well as on the ratio of the partial weight fraction of the components, wp1 and wp2. The following formula relates these parameters: wp1 ¼

rp1 rp2 ðrm
rs Þ þ ws rs rp1 ðrp2
rm Þ ðrp2
rp1 Þrs rm

ð8:2Þ

wp2 ¼

rp1 rp2 ðrm
rs Þ þ ws rs rp2 ðrp1
rm Þ ðrp1
rp2 Þrs rm

ð8:3Þ

where the indices 1 and 2 correspond to the first and the second solid phases. In principle, density measurements combined with a drying method for determining the total solid content can provide information about the ratio of component weight fractions.

8.5. COSMETICS: MIXTURES OF SOLIDS IN EMULSIONS Sunscreen cosmetics are a very interesting and complex application. They have attracted much attention lately due to nanotoxicology concerns. Some aspects of this problem can be found in Refs [19–26]. Ultrasound characterization can be very useful for resolving this problem. We present in the next chapter some arguments illustrating advantages of ultrasound particle sizing for monitoring the nanoparticle content in broad size distributions. Here we discuss a completely different approach, which treats sunscreens as mixtures of an oil-in-water emulsion with solid light-absorbing particles mixed in the oil phase. In some cases, this approach allows us to estimate particle size of the intact system, approaching the goal formulated in Ref. [26]. A detail description is presented in Ref. [27]. The most widely used solid particles are rutile and zinc oxide. For this application, we can speak of two different continuous phases: a water phase for the oil droplets, and an oil phase as the dispersion medium for the solid particles. There are also two dispersed phases: the oil droplets in the water and the solid particles in the oil. Correct characterization of the size distribution of the solid particles size is critical to ensure the performance of sunscreen preparations. The efficiency of

333

Applications for Dispersions

these sunscreens in adsorbing ultraviolet radiation, thereby preventing it from penetrating the skin, depends strongly on the particle size. This particle size characterization must be done in the final sunscreen preparation. Any dilution of the final product might well affect the particle size distribution and yield misleading and possibly dangerous results. The fact that the attenuation spectra of the complete sunscreen moisturizer is sensitive to particle size follows from the data presented in Figure 8.23 for moisturizers with two different ZnO particles: microfine ZnO with a particle size of 165 nm, and USP ZnO with a size of about 500 nm. There are two ways to extract this information. First of all, we can measure the attenuation spectra of the oil phase containing the solid active ingredient. This is a simple two-phase system. In order to calculate the particle size from the total attenuation curve, the intrinsic attenuation of the pure oil phase is required. The attenuation spectra corresponding to the pure oil, and oil with added 14 wt% USP ZnO are shown in Figure 8.24. The attenuation of the pure oil allows us to estimate the viscosity of the oil phase. This is important because this phase often includes various additives. We can make an estimate of the viscosity using Stokes law, which relates the viscosity to the measured attenuation (see Chapter 3). The attenuation of the oil is about 35 times higher than that of water, which means that we might estimate the viscosity of the oil phase as about 35 cp. The particle size calculated from the attenuation spectra using this viscosity and intrinsic attenuation is shown in Figure 8.25. It is very close to the expected value provided by the manufacturer.

Attenuation (dB/cm/MHz)

5

4 USP ZnO, size = 0.55 mm

3

2

Microfine ZnO, size = 0.165 mm

1

0 100

101 Frequency (MHz)

102

FIGURE 8.23 Attenuation of two sunscreens with identical chemical composition but having ZnO particles of different sizes.

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CHAPTER

8

14

Attenuation (dB/cm/MHz)

12 oil + 14 wt% USP ZnO

10 8

oil phase only

6 4 2 Sun screen, 23 wt% oil + 3.5 wt% USP ZnO

0 100

101 Frequency (MHz)

102

FIGURE 8.24 Attenuation spectra of the complete sunscreen and its components.

1.4

PSD, weight basis

1.2 1.0 0.8 0.6

size 0.0804 0.0975 0.1184 0.1436 0.1743 0.2114 0.2566 0.3113 0.3777 0.4583 0.5561 0.6748 0.8188 0.9935 1.2056 1.4628 1.7750 2.1537 2.6133 3.1709 3.8476

1.0

percent 0.0004 0.0017 0.0049 0.0122 0.0269 0.0540 0.0991 0.1667 0.2587 0.3715 0.4966 0.6220 0.7357 0.8288 0.8977 0.9437 0.9716 0.9868 0.9943 0.9977 0.9990

0.9 0.8 0.7 0.6 0.5 0.4 0.3

0.4

0.2

0.2 0.0 10−2

Cumulative

1.6

0.1 0.0 10−1

100

101

Diameter (µm) FIGURE 8.25

Particle size distribution of the ZnO particles.

A second way to characterize the particle size using the attenuation spectra can be accomplished by measuring the oil-in-water emulsion alone, without the particles. This emulsion would represent the background for the particles in the attenuation spectra of the complete sunscreen moisturizer. It would allow us to calculate the intrinsic attenuation and extract the contribution of

335

Applications for Dispersions

5

Attenuation (dB/cm/MHz)

4

3

2

sun screen, 3.5% ZnO

emulsion only

1

0 100

101 Frequency (MHz)

102

FIGURE 8.26 Attenuation spectra of the complete sunscreen, and of the basic emulsion.

the particles to the total attenuation spectra. Figure 8.26 illustrates this approach, providing attenuation curves for a complete moisturizer prepared with microfine ZnO and the emulsion only. The corresponding particle size is shown in Figure 8.27. It is very close to the value reported by the manufacturer for this ZnO.

ZnO, median size=165 nm 2.0

PSD, weight basis

1.6

1.2

oil droplet, median size=1.92 mm

0.8

0.4

0.0

10-2

10-1

100 Diameter (mm)

101

102

FIGURE 8.27 Particle size distribution of the ZnO particles and oil droplets.

336

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8

It is important to point out that in this second approach we can calculate the size of the oil droplets. The attenuation for the emulsion consists of the intrinsic attenuation of water and oil plus the thermal losses caused by the added oil droplets. Figure 8.27 shows the droplet size distribution for this emulsion. In summary, we can conclude that acoustic spectroscopy is able to provide information about size of the solid active ingredient as well as the size of the emulsion droplets of sunscreen moisturizers. In addition, it yields information about the rheological properties of these sunscreens.

8.6. MILLING There are several papers published presenting results of controlling the milling process using acoustic particle sizing [28–34]. One of the most convincing comparative studies has been performed by S.Menze from Netzsch GmbH [34]. He performed milling of alumina slurry and monitored results with three different instruments: l l l

MasterSizer 2000 of Malvern Instruments (laser diffraction) Ultrafine Particle Analyser of Leeds & Northrup (dynamic light scattering) DT-1200 from Dispersion Technology, Inc.

He stated that For the analysis with the ultrasonic spectrometer with an increasing specific energy input an increasing comminution progress is obtained. In contrast, the measurement with the MasterSizer 2000 of Malvern Instruments (laser diffraction) leads to a plateau around 150 nm and the analysis with the Ultrafine Particle Analyser of Leeds & Northrup (dynamic light scattering) shows a plateau around 250 nm. This statement is illustrated with graphs in Figure 8.28. Table 8.4 presents the average values of the particle size after milling was completed. As can be seen, the final particle sizes of about 154 nm were measured with the laser diffraction and 115 nm with the dynamic light scattering instrument, whereas sizes of about 32 nm were detected with the ultrasonic spectrometer. In order to verify these results and determine which instrument is more suitable, transmission electron microscopy (TEM) photographs were taken. The TEM images clearly show that there is a large amount of very fine particles (50 nm and smaller) already produced besides some larger partly agglomerated ones. The latter has a strong influence on the volume distribution but less on the number distribution. Therefore the conclusion was derived that the attenuation of ultrasound seems to be more sensitive to the primary particle size, whereas the signal in photon spectroscopy is influenced by a small amount of large particles. Nevertheless, the TEM images prove that it is possible to grind fused corundum down to the nanometer size range.

337

Applications for Dispersions

Median particle size X50,3/µm

5

1

0,1

Median particle size X50,3/µm

1

Analyse UPA UPA UPA MS 2000 MS 2000 MS 2000 DT 1200 DT 1200 DT 1200

vt 8 m/s 10 m/s 12 m/s 8 m/s 10 m/s 12 m/s 8 m/s 10 m/s 12 m/s

Grinding media: Al2O3 dGM = 900 µm without stabilization cm = 0.2

2⫻103

FIGURE 8.28

vt 8 m/s 10 m/s 12 m/s 8 m/s 10 m/s 12 m/s 8 m/s 10 m/s 12 m/s

Grinding media: Al2O3 dGM = 1100 µm without stabilization cm = 0.2

5; 0.05

0,1

Analyse UPA UPA UPA MS 2000 MS 2000 MS 2000 DT 1200 DT 1200 DT 1200

104 Specific energy Em,v / kJ·kg⫺1

105

2⫻105

Comparison of different optical particle size analysis with ultrasonic spectroscopy.

TABLE 8.4 Results of the Final Particle Size Analysis with Different Devices Measurement device

Analyzed x50,3, mm

MasterSizer 2000

0.154

UPA

0.115

DT-1200

0.032

338

CHAPTER

2:02:27 PM

6

8

0.9

0.7 4

0.6 0.5

3 0.4 2

0.3 0.2

4:06:17 PM

1

Large particles content

Attenuation at frequency 65 MHz

0.8 5

0.1 0

0.0 0

5

10 15 20 Measurement number

25

30

FIGURE 8.29 Continuous milling of alumina slurry monitored with DT-1200.

We have our own experience with monitoring milling using acoustic particle sizing. Figure 8.29 illustrates milling of the alumina slurry starting from 12 mm down to 22 nm. It was done also with Netzsch mill, but in USA. Milling took about 2 h. We organized the flow of the slurry from the mill through the DT-1200 acoustic sensor and back to the mill. This allowed us continuous measurement of attenuation and calculation of the particle size. Graphs show progressive evolution of attenuation and also gradual reduction of the large particle content. These two examples and many others given in the above-mentioned papers suggest that acoustic spectroscopy is at least very promising for adequate monitoring of the milling process, especially down to nanosizes.

8.7. MONITORING DISSOLUTION One of the attractive features of acoustic spectroscopy is its ability to perform very fast measurements. It can be achieved by simply monitoring the intensity of sound after propagation through the sample at single frequency and single gap. This setup makes possible to capture at least one measurement per second. In some instances, we were able to make five measurements per second. This high speed of measurement is very useful for monitoring dissolution processes. One of the examples for this application is given in Ref. [35]. Here we present our own experience with Nestle instant drinks. We had two samples of chocolate milk powders. These powders were different as a result of the surface treatment. One of the powders contained the surface layer

339

Applications for Dispersions -55 -60

Attenuation @ 100 MHz

-65 -70 -75 Sample 2A

-80

Sample 2B

-85 -90 -95

1

51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 Measurement #

FIGURE 8.30 Kinetics of dissolution of two chocolate milk powders.

of the other material. This should bring some implications for kinetics of these powders dissolving. Figure 8.30 illustrates this kinetics with measured ultrasound pulse intensity at 100 MHz. For this particular sample, it was possible to perform five measurements per second. These fast measurements are useful for characterizing the kinetics of dissolution, which depends on the structure of the surface layer. Sample 2B has a special coverage layer. Its kinetic curve has two distinctive times that can be associated with dissolving the surface layer and then the bulk of the particles. This effect is well reproducible and can be used for similar purposes with other materials.

8.8. EFFECT OF AIR BUBBLES Air contained in sample may affect both the attenuation and speed of sound. Air may be present either as bubbles or as air trapped at the interstices between particle agglomerates. Here we consider how such air might affect the measurements and how we can gain further information about this air by interpreting such data. An acoustic theory describing sound propagation through a bubbly liquid was formulated by Foldy [36] and confirmed experimentally [37–39].

340

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8

The effect of air bubbles on the attenuation and sound speed depends both on the bubble size and the sound frequency. For instance, a 100-mm bubble has a resonance frequency of about 60 KHz. This frequency is inversely proportional to the bubble diameter. A bubble of 10 mm diameter will have a resonance frequency of about 0.6 MHz. Acoustic spectroscopy typically measures the attenuation spectra over the range of frequencies from 1 to 100 MHz. The size of the bubbles must be well below 10 mm in order to affect the complete frequency range of acoustic spectrometer. Bubbles of sizes below 10 mm are very unstable, as is known from general colloid chemistry and the theory of flotation. Colloid-sized gas bubbles have astonishingly short lifetimes, normally between 1 ms and 1 ms [39]. They simply dissolve in the liquid because of the high curvature. Thus, air bubbles can only affect the low-frequency part of the acoustic spectra, typically below 10 MHz. The frequency range from 10 to 100 MHz is available for particle characterization, even in bubbly liquids. For bubblefree samples, the entire spectral range can be used, providing more detailed particle sizing capabilities and more robust characterization. The low-frequency part of the attenuation spectra, from 1 to 100 MHz, can be used to characterize the amount of air in the system. Air in colloidal systems can have a surprising and dramatic effect on the rheological properties, which is often not understood. Hydrophobic systems, such as carbon black or graphite, may contain air trapped at the interstices of particle agglomerates. For instance, bubbles can be centers of aggregation, which might make them an important factor in controlling stability. The low-frequency portion of the attenuation spectra can be used to quantify the wettability of such dispersions, while at the same time the high-frequency portion can be used to determine the particle size. For the vast majority of samples, the presence of air can be ignored. This conclusion was confirmed with thousands of measurements performed with hundreds of different systems.

REFERENCES [1] A.S. Dukhin, P.J. Goetz, Characterization of aggregation phenomena by means of acoustic and electroacoustic spectroscopy, Colloids Surf. 144 (1998) 49–58. [2] M. Kosmulski, J.B. Rosenholm, Electroacoustic study of adsorption of ions on anatase and zirconia from very concentrated electrolytes, J. Phys. Chem. 100 (1996) 11681–11687. [3] M. Kosmulski, Positive electrokinetic charge on silica in the presence of chlorides, J. Colloids Interface Sci. 208 (1998) 543–545. [4] M. Kosmulski, J. Gustafsson, J.B. Rosenholm, Correlation between the Zeta potential and rheological properties of anatase dispersions, J. Colloids Interface Sci. 209 (1999) 200–206. [5] M. Kosmulski, S. Durand-Vidal, J. Gustafsson, J.B. Rosenholm, Charge interactions in semi-concentrated titania suspensions at very high ionic strength, Colloids Surf. A 157 (1999) 245–259.

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341

[6] J. Gustafsson, P. Mikkola, M. Jokinen, J.B. Rosenholm, The influence of pH and NaCl on the zeta potential and rheology of anatase dispersions, Colloids Surf. A 175 (2000) 349–359. [7] S.-i. Takeda, P.J. Goetz, Dispersion/flocculated size characterization of alumina particles in highly concentrated slurries by ultrasound attenuation spectroscopy, Colloids Surf. 143 (1998) 35–39. [8] S.-i. Takeda, Characterization of ceramic slurries by ultrasonic attenuation spectroscopy, in: V.A. Hackley, J. Texter (Eds.), Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, American Ceramic Society, Westerville, OH, 1998. [9] S.-i. Takeda, T. Chen, P. Somasundaran, Evaluation of particle size distribution for nanosized particles in highly concentrated suspensions by ultrasound attenuation spectroscopy, Colloids Surf. (1999). [10] A.S. Dukhin, P.J. Goetz, Characterization of chemical polishing materials (monomodal and bimodal) by means of acoustic spectroscopy, Colloids Surf. 158 (1999) 343–354. [11] S.-i. Takeda, H. Harano, I. Tari, Particle size characterization of highly concentrated alumina slurries by ultrasonic attenuation spectroscopy, Improved Ceramics Through New Measurement, Processing and Standards 133 (2002) 59–65. [12] S.-i. Takeda, H.H. Suenaga, I. Tari, Particle size characterization of highly concentrated barium titanate slurries by ultrasonic attenuation spectroscopy, Improved Ceramics Through New Measurement, Processing and Standards 133 (2002) 53–59. [13] M. Guerin, J.C. Seaman, C. Lehmann, A. Jurgenson, Acoustic and electroacoustic characterization of variable-charge mineral suspensions, Clays Clay Miner. 52 (2004) 158–170. [14] M. Guerin, J.C. Seaman, Characterizing clay mineral suspensions using acoustic and electroacoustic spectroscopy, Clays Clay Miner. 52 (2004) 145–157. [15] A.S. Dukhin, P.J. Goetz, S.T. Truesdail, Surfactant titration of kaolin slurries using (z-potential probe, Langmuir 17 (2001) 964–968. [16] A.S. Dukhin, P.J. Goetz, Characterization of concentrated dispersions with several dispersed phases by means of acoustic spectroscopy, Langmuir (2000). [17] A. Bleier, Secondary minimum interactions and heterocoagulation encountered in the aqueous processing of alumina-zirconia ceramic composites, Colloids Surf. 66 (1992) 157–179. [18] M. Kikuta, Method and apparatus for analyzing the composition of an electro-deposition coating material and method and apparatus for controlling said composition, US Patent 5,368,716 (1994). [19] A.S. Dukhin, P.J. Goetz, X. Fang, P. Somasundaran, Monitoring nano-particles in the presence of larger particles in liquids using acoustics and electron microscopy, J. Colloids Interface Sci. (2009). [20] M. Behra, H. Krug, Nano-particles at large, Nat. Nanotechnol. 3 (2008) 253–254. [21] Groups encourage nano-material stewardship, Chem. Eng. News (2008) 34, July 21. [22] G. Miller, Nano-materials, sunscreens and cosmetics: Small ingredients-big risks, Friends of the Earth, www.foe.org (2006). [23] The Appropriateness of Existing Methodologies to Assess the Potential Risks Associated with Engineered and Adventitious Products of Nanotechnology, European Commission, Health & Consumer Protection Directorate-General, SCENIHR, 2006. [24] Nanotechnology, A Report of the U.S. Food and Drug Administration Nanotechnology Task Force, July 25, 2007. [25] P. Somasundaran, S.C. Mehta, L. Rhein, S. Chakraborty, Nanotechnology and related safety issues for delivery of active ingredients in cosmetics, MRS Bull. 32 (2007) 779–786.

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[26] K.M. Tyner, A.M. Wokovich, W.H. Doub, L.F. Buhse, L.P. Sung, S.S. Watson, et al., Comparing methods for detecting and characterizing metal oxide nanoparticles in unmodified commercial sunscreens, Nano-medicine 4 (2009) 145–159. [27] D. Fairhurst, A.S. Dukhin, K. Klein, Stability and structure characterization of cosmetic formulations using acoustic attenuation spectroscopy, Particle Size and Characterization, Chapter 16, ACS Symposium Series, vol. 881 (2004) pp. 249–269. [28] F. Stenger, S. Mende, J. Schwedes, W. Peukert, Nano-milling in stirred media mills, Chem. Eng. Sci. 60 (2005) 4557–4565. [29] N. Bell, J. Cesarano, J.A. Voight, S.J. Jockwood, D.B. Dimos, Colloidal processing of chemically prepared zinc oxide varistors. Part 1. Milling and dispersion of powder, J. Mater. Res. 19 (5) (2004) 1333–1340. [30] P. Wilhelm, D. Stephan, On-line tracking of the coating of nano-scaled silica with titania nano-particles via zeta potential measurements, J. Colloids Interface Sci. 293 (2006) 88–92. [31] X.Z. Wang, L. Liu, R.F. Li, R.J. Tweedie, K. Primrose, J. Corbett, et al., Online characterization of nano-particles suspensions using dynamic light scattering, ultrasound spectroscopy and process tomography, Chem. Eng. Res. Des. 87 (2009) 874–884. [32] R. Weser, F. Hinze, B. Wessely, A. Richter, Online Characterization of High Concentrated Particulate Systems by Using Ultrasonic Pulse Method, Particulate System Analysis, United Kingdom, 2008. [33] M. Tourbin, C. Frances, Monitoring of the aggregation process of dense colloidal silica suspensions in a stirred tank by acoustic spectroscopy, Powder Technol. 190 (2009) 25–30. [34] S. Mende, Representative on-line measurement of comminution results for nanogrinding in stirred media mills, NETZSCH-Feinmahltechnik GmbH, S. Mende, “Mechanische Erzeugung von Nanopartikeln in Ru¨hrwerkskugelmu¨hlen”, Thesis. Technische Universita¨t Carolo-Wilhelmina, Braunschweig, 121pp. (2004). [35] R. Saggin, J.N. Coupland, Ultrasound monitoring of powder dissolution, Food Eng. Phys. Properties 67 (2002) 1473–1477. [36] L.L. Foldy, Propagation of sound through a liquid containing bubbles, OSRD Report No.6.1-sr1130-1378, 1944. [37] E.L. Carnstein, L.L. Foldy, Propagation of sound through a liquid containing bubbles, J. Acoust. Soc. Am. 19 (1947) 481–499. [38] F.E. Fox, S.R. Curley, G.S. Larson, Phase velocity and absorption measurement in water containing air bubbles, J. Acoust. Soc. Am. 27 (1957) 534–539. [39] S. Ljunggren, J.C. Eriksson, The lifetime of a colloid sized gas bubble in water and the cause of the hydrophobic attraction, Colloids Surf. 129–130 (1997) 151–155.

Chapter 9

Applications for Nanodispersions 9.1. Reference Nanomaterial for Particle Sizing and z-Potential in Dilute and Concentrated Systems: Colloidal Silica Ludox 344 9.2. Large Particle Content Resolution using Acoustics 349 9.3. Monitoring Presence of Large Particle using Electroacoustics 354

9.4. Monitoring Nanoparticle Content in Systems with a Broad Polydisperse Size Distribution 356 9.5. Limitation of UltrasoundBased Method for Characterizing Nanodispersions 364 References 367

Nanodispersions are dispersed systems that contain nanoparticles. There is almost complete consensus with regard to the definition of the nanoparticles: these are particles with sizes under 100 nm. This definition would include microemulsions and other soft particles. However, we would retain old term “microemulsions” for such equilibrium soft particles systems. We present applications for such systems in Chapter 10. This chapter is devoted to the solid, rigid nanoparticulates. Ultrasound techniques allow easy and very precise characterization of particle size and z-potential in such systems. This is reflected in several publications describing industrial applications of these techniques in nanotechnology. There are three publications regarding online characterization of nanodispersion: by Stenger and others [1], Bell and others [2] on nanomilling, by Wilhelm and Stephan [3] on monitoring coating by nanoparticles, and by Wang and others [4] about online particle size measurement. General particle size and z-potential measurements of nanodispersions are presented in Refs [5–7]. We will discuss here in some detail some other examples of characterization of nanodispersions performed in our laboratory. The first one is on reference material for nanoparticulates. There is one specific nanodispersion that Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23009-0 Copyright # 2010, Elsevier B.V. All rights reserved.

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has been used as a reference material for a long time. It is silica Ludox. We present characteristics of this dispersion measured in collaboration with Horiba, USA, and discuss its differences with the other potential reference/ certified material—gold sols. Silica Ludox is currently under consideration by the European Union Institute of Standards as certified material for particle sizing of nanodispersions. Then we present our studies of polydisperse systems containing nanoparticle fraction. Ultrasound offers the possibility to characterize these systems without any special sample treatment. This allows monitoring the fraction of nanoparticles in such systems with broad size distributions, see Ref. [8]. Alternatively, there is the possibility of measuring the content of large particles when nanoparticles dominate the size distribution. There are two possible ways to achieve this: one using an acoustic probe [9] and the other with the electroacoustic z-probe. We determine the sensitivity of ultrasound-based methods for these two extreme tasks, which present important industrial applications in cosmetics, chemical-mechanical polishing (CMP) slurries, nanotoxicology, etc. In the end, we discuss the possible limitations of ultrasound-based methods for characterizing nanodispersions.

9.1. REFERENCE NANOMATERIAL FOR PARTICLE SIZING AND z-POTENTIAL IN DILUTE AND CONCENTRATED SYSTEMS: COLLOIDAL SILICA LUDOX Instruments based on acoustic attenuation are “first principle” techniques that do not require calibration, similar to light scattering instruments. However, they should be subjected to verification tests on a regular basis. Verification in this sense involves challenging the system with a known sample and comparing the calculated result with the expected value. If the calculated versus expected result varies beyond the determined maximum deviation, then the system typically requires service attention (alignment, source or detector replacement, etc.) to return the system to the proper operating condition. On the other hand, electroacoustic instrument do require calibration, as specified in Chapter 7. It would be very desirable to have a material that could serve both purposes, verification and calibration at once. It would be even better if such material could be tested with other methods as well. It turns out that there is a candidate that could meet these requirements—colloidal silica Ludox. We discuss this material here. Different nomenclature can be used when discussing materials used to calibrate or verify particle characterization instruments. Terms used include standards, reference materials, and certified reference materials. For the purposes of this chapter, the terminology defined by ISO guide 30 [10] will be used.

Applications for Nanodispersions

345

A reference material is “sufficiently homogeneous and stable with respect to one or more specified properties, which has been established to be fit for its intended use in a measurement process.” A certified reference material is “characterized by a metrologically valid procedure for one or more specified properties, accompanied by a certificate that provides the value of the specified property, its associated uncertainty, and a statement of metrological traceability.” Using these definitions, what many people refer to as a standard is a certified reference material since this includes a certificate of analysis and statement concerning traceability. A reference material can be any particulate sample meeting the defined conditions of being homogenous, stable, and established to be fit for a given use. The current supply of certified reference materials available for use in particle characterization can also be further classified as either monodisperse or polydisperse. Monodisperse (or monosized) reference materials are typically polystyrene latex spheres with extremely narrow distributions. An example distribution may be 1.020
0.022 mm. These materials have such narrow distributions that typically only a midpoint (such as the mean) to the distribution is used for calibration or verification. Polydisperse reference materials have a broader distribution so that other points of the distribution (such as the d10 and d90) can also be cited and used during the verification process. Many polydisperse reference materials are glass beads [11, 12] with certified values including the d10, d25, d50, d75, and d90. Although several standard practice documents [13, 14] suggest using polydisperse reference materials “over one decade of size” (1–10, 10–100, etc.) for the verification process, in practice none of the available materials meets this criteria. The National Institute of Standards and Technology (NIST) standard reference material (SRM) 1003c may be the best characterized sample of this type and would be accepted as appropriate by any regulatory or enforcement agency. The NIST certificate for 1003c provides certified diameters from d5 to d95 in 5% increments ranging from 18.9 to 43.3 mm, hardly a 10–100 mm spread. Unfortunately, NIST SRM cannot be used for concentrated systems and is not well suited for calibrating z-potential instruments. Instead, manufacturers of ultrasound-based instruments have been using colloidal Ludox Silica for at least 20 years. This same material is a potential candidate for certification as SRM by the European Union Institute of Standards. Ludox is a trademark first issued to DuPont, and later acquired by Grace in 2000, for a stable colloidal silica suspension. Scientific references to the particle size of Ludox dates back to the 1960s [15]. Ludox is used for a variety of industrial applications including coatings, ink-receptive papers, metal casting, refractory products, catalysts, and as a clarifying agent. The suspension is

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electrostatically stabilized, preventing agglomeration over time and ensuring easy mixing. Ludox-TM silica has been used by particle characterization manufacturers and end users for many years because of the several unique properties of this material including the following: l l l l l

Availability and low cost Extremely long shelf life Known particle size (30–34 nm) [15, 16] Known z-potential (near 38 mV) Known concentration (50 wt%)

Until recently, very few well-characterized materials in the 30-nm size range were available. For all these reasons, instruments based on acoustic spectroscopy have been using Ludox-TM as a particle size and z-potential reference material for over 20 years. In order to have statistically verified parameters of this material, we measured its particle size with two different techniques: acoustic spectroscopy (DT-1200) and dynamic light scattering (HORIBA LB-550). We present the results of the sizing with DT-1200 first. Ludox-TM-50 was first diluted from 50 to 10 wt% using the following procedure: l

l l l

Prepare 1 l 0.01 M KCl by adding 0.745 g KCl to a 1-l flask and fill to 1 l with DI water. Tare a 250-ml beaker on a balance. Add 25 g of 50% Ludox concentrate. Dilute to 125 g total wt using 0.01 M KCl.

The 10 wt% sample was measured in DT-1201 without further dilution. The only input required for the DT-1201 measurements was the weight percent of the sample. Figure 9.1 demonstrates the typical particle size distribution (PSD) and attenuation spectra measured with DT-1200. The attenuation spectrum is from raw data and its presentation serves the important purpose of confirming the validity of the calculated PSD. Dynamic light scattering measurement requires more dilution. In order to achieve a proper signal level, the five drops of the 50 wt% sample were added to a standard DLS cuvette containing 3 ml of the 0.01 M KCl solution. The particle size is calculated on a volume basis using the refractive index for colloidal silica: 1.45 real, 0.0 imaginary. The sample is dilute enough so as to approximate the viscosity to that of water. Figure 9.2 demonstrates a typical PSD of Ludox measured with LB-550. The PSD of Ludox is fairly narrow, although not as narrow as a latex reference material, with a span (d90-d10/d50) near 0.5 or a standard deviation near 0.1. For this reason, only the mean values are reported for this study.

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Applications for Nanodispersions

3.0 0.4 Attenuation (dB/cm/MHz)

PSD, weight basis

2.5 2.0 1.5 1.0 0.5

0.3

0.2

0.1

0.0

0.0 0.01

0.1 Diameter (mm)

100

101 Frequency (MHz)

102

FIGURE 9.1 Typical particle size distribution and attenuation spectrum of 10% silica Ludox at 25  C measured with DT-1200.

23.00

Undersize (%)

Frequency (%)

100.0

0.0 0.001

0.010

0.100 Diameter (mm)

1.000

0.0 6.000

FIGURE 9.2 Typical particle size distribution measured with LB-550.

This study was performed with five different DT-1200 instruments and three different LB-550 instruments. Samples were measured five times with each of these instruments. The averaged median particle size and standard deviation are shown in Table 9.1. For most cases, the results obtained by different particle characterization techniques can generate a wide spread of mean values. Ludox appears to be a very rare sample where different particle sizing techniques give very similar results. The reasons for this could include the nature of the particles themselves (round, narrow size distribution) and the stability of the suspension at

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TABLE 9.1 Averaged Median Size and Std Deviation of the Silica Ludox Sample Measured with the Acoustic Spectrometer DT-1200 and with Dynamic Light Scattering Instrument LB-550 D50 (mm)

Std dev. (mm)

DT-1200

31

0.9

LB-550

30.08

0.46

various concentrations. This unique combination of availability, low cost, and similar results using multiple techniques suggests that Ludox could be used as a nearly universal particle size reference material in many industries. In addition, silica Ludox can serve as reference and calibration material for z-potential measuring instruments. Small particle size simplifies significantly calculation of the z-potential because inertia effects are not important. This explains why this material has been used for this purpose in electroacoustic instruments for 20 years. When diluted as described above, this material has pH ¼ 9.3 and conductivity 0.23 S/m. Under these conditions, the value of Smoluchowski z-potential equals 38 mV. Statistical tests with five different DT-1200 instruments show that this parameter is measured with a standard deviation of 0.36 mV. Value of z-potential for this material was measured independently with Lazer ZeeMeter model 501 by PenKem, Inc. This instrument reports the value of z-potential calculated according to the Smoluchowski theory. This makes results of two instruments consistent. The modern electroacoustic instrument DT-1200 reports also results calculated using advanced electroacoustic theories described in Chapter 5. This includes the following parameters for silica Ludox: l l l l l l l

Debye length ¼ 2.4 nm ka ¼ 6.5 Maxwell-Wagner frequency ¼ 52 MHz Dynamic mobility ¼ 4.53 Surface charge density ¼ 1.7  106 C/cm2 Dukhin number ¼ 0.4 z (surface conductivity included) ¼ 46.3

These parameters indicate that silica Ludox is at the border of the thin double layer (DL) model. Surface conductivity becomes important and contributes substantially to the absolute value of z-potential. This material is also serves well for testing titration capability. Figure 9.3 shows a typical titration curve measured with DT-1200.

349

Applications for Nanodispersions

pH 0

2

3

4

5

6

7

8

9

0.9

Zeta (mV)

zeta

0.7 0.6

-20

0.5 0.4

-30 conductivity -40

Conductivity (S/m)

0.8 -10

0.3 0.2

FIGURE 9.3 Typical titration of 10% silica Ludox at 25  C measured with DT-1200.

In conclusion, we can state that silica Ludox is very useful universal reference material. We hope that eventually it would be certified as nanoparticle standard.

9.2. LARGE PARTICLE CONTENT RESOLUTION USING ACOUSTICS Characterization of abrasive slurries and modern CMP materials present a specific challenge for the measurement techniques—monitoring the content of large particles that could ruin the quality of polishing. One can find extensive investigations of various characterization techniques for CMP slurries in the publications by Anthony et al. [17, 18] as well as by us [9]. Three aspects of this application cause difficulties when using instruments based on traditional techniques. First, the particle sizes of many abrasive slurries are too small to be measured by sedimentation-based instruments or electric zone instruments. Typically, the mean size of such materials is approximately 100 nm, with no particles or, preferably, just a few larger than 500 nm. The desired range over which particle characterization is desired is very large, which rules out most classical techniques. Third, abrasive systems are typically shear sensitive. Shear caused by the delivery system, or the polishing process itself, may induce an unpredictable assembly of the smaller particles into larger aggregates. However, these aggregates may be weakly formed and easily destroyed by subsequent sonication, high shear, or dilution. Therefore, any technique that requires dilution or other sample preparation steps may in fact destroy the very aggregates that one is attempting to quantify by measurement. We suggest that abrasive systems must be characterized as they are, without any dilution or sample preparation.

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Acoustic spectroscopy provides an exciting alternative to the more classical methods; this technique resolves all three issues mentioned above. This is the conclusion of the Bell Labs tests [17, 18], and we completely agree with it after our own extensive work using Dispersion Technology instruments. In addition to particle sizing, ultrasound offers a simple and reliable way to characterize z-potential using electroacoustics. The importance of electrokinetic characterization of CMP slurries has been stressed by Osseo-Asare [19]. In this section, we present results of our work with model slurries using Dispersion Technology instruments for characterizing both particle size using acoustics and z-potential using electroacoustics. We emphasize here one feature of acoustic spectroscopy that thus far has not been described sufficiently in the literature: namely, the ability to characterize a bimodal PSD. Although Takeda et al. [20–22] demonstrated that acoustic spectroscopy is able to characterize bimodal distributions of mixed alumina particles, the ultimate sensitivity in detecting one very small subpopulation in combination with another dominant mode was not studied. Yet, it is just this feature, the ability to recognize a small subpopulation, that is most critical for studies of abrasives. This section addresses this important issue. Unfortunately, there is no agreement in the literature as to the number of larger particles that might be acceptable in an abrasive slurry. A set of experiments was made to test whether the attenuation spectra changed reproducibly when a small amount of larger particles was added to a singlecomponent slurry of smaller particles. We used three silica materials altogether. For smaller size particles, we used silica Ludox-TM (50 wt%). Two large size particles were employed, namely, Geltech 0.5 mm and Geltech 1.5 mm. We assumed a density of 2.1 g/cm3 for all silica particles. Slurries were prepared at 12 wt% for each material as follows. The silica Ludox was diluted to 12 wt% with 0.01 M KCl solution, resulting in a sample pH of 9.3. The Geltech samples were supplied as a dry powder, which was dispersed in 0.01 M KCl solution, and adjusted to pH 9.6 with KOH. The dispersion was repeatedly sonicated, stirred, and allowed to equilibrate for 5 h before measurement. Table 9.2 presents the particle size data provided by the manufacturer for each of these samples and those measured with the acoustic spectrometer DT-1200. The goals of the experiment were met in three steps. Step 1—Reproducibility: The objective was to prove that the DT-1200 acoustic spectrometer is able to measure the attenuation spectra with the required precision of 0.01 dB/cm/MHz. To prove this, we measured five different fillings of the same 12 wt% silica Ludox-TM. Each filling was measured nine times. A statistical analysis of the measured attenuation spectra yielded an average variation of the attenuation measurement over the frequency range from 3 to 100 MHz.

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Applications for Nanodispersions

TABLE 9.2 Particle Size of the Initial Silica Samples, Expected and Measured, Acoustically Manufacturer

Acoustics

Ludox-TM (nm)

22 (area basis)

31 (weight basis)

Geltech 0.5 (mm)

0.5

0.65

Geltech 1.5 (mm)

1.5

1.72

Step 2—Accuracy: The goal here was to prove that acoustic spectroscopy can characterize the particle size with sufficient accuracy. To achieve this goal, all five silica slurries were measured individually. The measured particle size was then compared with independent information provided by the manufacturer. Step 3—Bimodal Sensitivity: This is the key step in the investigation. Model systems with bimodal PSD were prepared by mixing a small-particle slurry (Ludox-TM) with increasing doses of the larger particle slurries (Geltech 0.5 or Geltech 1.5). The objective was to evaluate the accuracy of the PSD calculated by the DT-1200 software for these known systems. We used Ludox-TM as the major component of two test slurries. The Geltech 0.5 or Geltech 1.5 was added as “large” and “larger” particles to the above test slurries to form various mixed slurries. In each case, the minor fraction was added to the Ludox-TM in steps. Each addition increased the relative amount of the larger particles by 2%. The attenuation spectrum was measured twice for each mixed system in order to demonstrate reproducibility. Figures 9.4 and 9.5 demonstrate the results of these mixed system tests. It can be seen that the attenuation increases with increasing amounts of the “large” or “larger” particles. The increase in the attenuation with increasing doses of the Geltech content is in all cases significantly larger than precision of the instrument. This demonstrates that the DT-1200 data contains significant information about the small amount of the large particles. The final question is to determine whether the calculated PSD shows a correct bimodal distribution for these mixed model systems. Table 9.3 answers this question. The DT-1200 always calculates a lognormal and bimodal distribution that best fits the experimental data. These two PSDs are the best in the sense that the fitting error between the theoretical attenuation (calculated for the best PSD) and the experimental attenuation is minimized. These fitting errors are important criteria for deciding whether the lognormal or the bimodal PSD is more appropriate for describing a particular sample. For instance, the PSD is judged to be bimodal only if the bimodal fit yields substantially smaller fitting error than a lognormal PSD. The values

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0.5

Attenuation (dB/cm/MHz)

0.4

9% 9% 7.4% 7.4% 5.7% 5.7% 3.8% 3.8% 2% 2% Ludox only

0.3

0.2

0.1

0.0 100

101 Frequency (MHz)

102

FIGURE 9.4 Attenuation spectra measured for Ludox-TM-50 silica with various additions of Geltech 0.5 silica. Total solid content is 12 wt%. The legend shows the fraction of the total solid content corresponding to the Geltech silica.

0.5

9% 9% 7.4% 7.4% 5.7% 5.7% 3.8% 3.8% 2% 2% Ludox only

Attenuation (dB/cm/MHz)

0.4

0.3

0.2

0.1

0.0 100

101 Frequency (MHz)

102

FIGURE 9.5 Attenuation spectra measured for Ludox-TM-50 silica with various additions of Geltech 1.5 silica. Total solid content is 12 wt%. Legend shows the fraction of the total solid content corresponding to the silica Geltech.

Calculated for Geltech 0.5 Actual Geltech content (%)

Calculated for Geltech 1.5

Content (%)

Larger size (mm)

Lognormal fitting error (%)

Bimodal fitting error (%)

Content (%)

Larger size (mm)

Lognormal fitting error (%)

Bimodal fitting error (%)

9

14

0.7

14.1

7.3

11

1.6

24.5

4.4

9

15

0.9

10.9

8.7

12

1.7

17.7

5.5

7.4

8

0.9

13.6

4.5

10

1.9

17.8

6

7.4

12

1

15.4

7.1

10

1.9

17.3

5.4

5.7

7

0.9

13.8

4.3

7

1.9

17.2

4.1

5.7

10

1.3

12.9

7.2

6

1.6

18.5

4.7

3.8

4

0.9

9.5

3.7

5

1.3

18.1

3.9

3.8

4

0.7

12

4.3

6

1.6

16.9

3.8

2

4

0.6

10.8

3.6

3

1.9

12.2

2.6

2

5

1.2

12.2

3.7

4

1.6

12.3

3.9

Applications for Nanodispersions

TABLE 9.3 Characteristics of the Larger Particles (Geltech Silica) Calculated from the Attenuation Spectra of Figures 9.4 and 9.5

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of the lognormal and bimodal fitting errors are given in Table 9.3. It can be seen that the fitting errors for the bimodal PSD are better than the lognormal fits for all the mixed systems over the whole concentration range and for both the large and larger sized particles. According to the fitting errors, all PSDs in the mixed Ludox-Geltech systems are bimodal, which of course is correct for these known mixed systems. At the same time, our experience tells us that this fitting error criterion alone is not a sufficient test for the bimodality of the PSD. It is always possible to obtain a better fit to a dataset by allowing more degrees of freedom in the solution. A bimodal PSD provides at least four adjustable parameters for fitting the experimental attenuation curve, whereas a simple lognormal PSD provides only two parameters. The value of the larger particle content is another important parameter that must be taken into account. We have demonstrated that the precision of the DT-1200 is sufficient to detect larger particles present at concentrations larger than 2%. Therefore, the software will not claim a bimodal PSD if the content of the large particles is less than 2%, even if this yields a marginally better fit. Referring to Table 9.3, we see that the calculated large particle content in all of the mixed Ludox-TM/Geltech systems is well above this 2% threshold and gives therefore an additional argument to claim a bimodal PSD for these systems. It is important that the calculated content of the large particles increases with the actual content of the added Geltech particles, which confirms consistency of the PSD analysis. In conclusion, we can declare that acoustic spectroscopy is capable of resolving at least 2% of large particles in concentrated nanodispersions with no dilution and practically no sample preparation.

9.3. MONITORING PRESENCE OF LARGE PARTICLE USING ELECTROACOUSTICS Electroacoustic z-potential probe DT-300 is usually used for measuring the z-potential of concentrated particulates. It turns out that it can be used for a completely different purpose—monitoring presence of large particles in nanodispersions. In order to use it for this purpose, the probe must be placed vertically in a suitable stand that orients it such that the face of the probe with the gold electrode is on top, as shown in Figure 9.6. A cylindrical fixture around the top of the probe forms a cup, with the probe face serving as the bottom of the cup. This cup can be filled with the nanodispersion. Ultrasound pulses generated by the electroacoustic probe enter the sample through the gold electrode. These sound pulses generated by the probe propagate directly into the sample and generate an electroacoustic signal. The magnitude of the electroacoustic signal is proportional roughly to the volume fraction of the particles, as was shown in Chapter 5. In the case of a

355

Applications for Nanodispersions

Sample Plastic cup

Ultrasound pressure wave Deposit

Hmax

Zeta probe

FIGURE 9.6 Electroacoustic probe with novel sample handling system that allows measurement of particulate deposits and porous bodies.

stable system with no sedimentation, the volume fraction of the particles near the surface of the probe is constant and the signal should be unchanged in time. When the nanodispersion contains large sedimenting particles, they will settle on the surface of the probe, which will lead to an increase in the volume fraction there. This will lead, in turn, to an increase in the magnitude of the electroacoustic signal. This time dependence can be used as a measure of the presence of large sedimenting particles. In order to verify the sensitivity of this method, we used two different solid material particles: silica Ludox and silica Geltech 0.5. Each was prepared at a solid concentration of 10% by weight dispersed in distilled water. The pH was adjusted in each case to provide good stability. Figure 9.7 shows the time trend of the CVI signal for five 10 wt% Ludox samples, each successive sample containing increased amounts of Geltec from 0% to 1%. The electroacoustic signal of the clean Ludox sample does not change with time, which is expected since these very small 31-nm particles do not sediment appreciably during the period of measurement. However, the small amount of added larger silica Geltech particles causes a change of the signal with time which is proportional to the amount of the added large particles due to an accumulation of these particles near the probe surface. We see that the measurement is very sensitive to even small amounts of the larger particles, as little as 0.1%. This test yields just a qualitative result regarding the large particle content, but with higher sensitivity than the acoustic method from the previous section.

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Ludox Ludox Ludox Ludox Ludox

5.4⫻106

CVI magnitude (mV(s/g)1/2)

5.2⫻106 5.0⫻106

9

only + 0.1% Geltech + 0.3% Geltech + 0.5% Geltech + 1 % Geltech

4.8⫻106 4.6⫻106 4.4⫻106 4.2⫻106 4.0⫻106 3.8⫻106 3.6⫻106 3.4⫻106 0

50

100 150 Time (min)

200

250

FIGURE 9.7 Kinetics of evolution of the magnitude of the electroacoustic signal due to deposition of silica Geltech particles added in small amounts to silica Ludox.

It has also advantage being much simpler compared to the acoustic method, as there are no special calculations required for verifying the presence of large particles. This simplicity makes it very attractive.

9.4. MONITORING NANOPARTICLE CONTENT IN SYSTEMS WITH A BROAD POLYDISPERSE SIZE DISTRIBUTION Nanoecology and nanotoxicology are rapidly growing scientific disciplines for studying the impact of nanotechnology on human health and the environment [23–28]. A growing number of consumer products, cosmetics in particular, that contain engineered nanoparticles has created an immediate need for careful studies. This is reflected in the law recently adopted by the European Parliament, on the March 24, 2009, and supported by the European Cosmetics Association. One problem that must be addressed is the selection of an accurate method for monitoring the nanoparticle content of various products, many being dispersions containing mixtures of not only nanosized particles but also particles of much larger sizes. While there are known methods for measuring systems containing only nanoparticles, there are no well-established techniques for monitoring nanoparticles mixed with larger ones. Development of such methods is imperative for reliable investigation of nanoparticles in natural and engineered systems, such as cosmetic products [29].

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The first logical step in this direction is an investigation with existing commercially available instruments preferably by an independent institution, which has in possession all the instruments that are relevant for this task. In the absence of such institutions for undertaking a project, instrumentation companies can carry out such studies with their instruments in collaboration with academic institutions. To make the study reliable, it is necessary that it be conducted with the same set of nanoparticles. This approach has also the advantage of the tests being done by staff trained for the particular instrument. Towards this purpose, as a first step, researchers at the Langmuir Center for Colloids and Interfaces at Columbia University and Dispersion Technology, Inc. have undertaken this study to compare different samples with acoustic spectroscopy and transmission electron microscopy (TEM), since these techniques have been reported to be well suited for characterizing nanoparticles with rather narrow size distributions [1–7, 10]. The goal of this study is to determine the sensitivity of the acoustic spectroscopy to nanoparticles in the broad PSDs with large amount of coarser particles. Acoustic spectroscopy might have significant advantages over light scattering methods in studying samples with broad, polydisperse PSDs. Weight basis is innate for this technique because attenuation of ultrasound due to viscous dissipation by submicrometer particles is proportional to the weight of the particles. This means that raw data for calculating particle size depends on only the third power of the particle size. On the contrary, scattered light depends on a much higher power of the particles size, as large as the sixth. This makes scattering by large particles dominating the raw data and overshadowing the contribution of the smaller particles in the nanosize range. The weaker dependence of acoustic attenuation on the particle size makes the contribution of the small (nano) and larger particles more even and the method potentially more sensitive to the nanoparticle content even in very broad size distributions. ZnO powder was selected for this work since this substance, widely used in sunscreen products, has played an important role in recent nanoecology discussions. In order to make samples easily available for all future studies, they were purchased from Fisher Scientific, Inc., which offers powders from a variety of sources. The eight different powders produced by various manufacturers are listed below and in Table 9.4. l

l l l l l

Zinc oxide, reagent ACS and zinc oxide, 99.5þ% manufactured by Acros Organics Z50-500 and Z52-500 USP powder packaged by Fisher Scientific S80249 by Fisher Scientific Zinc oxide ACS reagent grade by MO Biomedicals, LLC Zinc oxide Polystormor™ by Mallinckrodt Chemicals Zinc oxide Nanopowder by American Elements

For adjusting pH of ZnO dispersions, we use KOH 1.000 N from VWR International. For enhancing the electric surface charge of ZnO particles, we

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TABLE 9.4 Median Particle Size and Percentage of Nanoparticles (<100 nm) in Various ZnO Samples at 5 wt% Stabilized at pH 10 with Hexametaphosphate Powder name, Manufacturer

Median size (mm)

Cumulative % of nanoparticles according to acoustics, <100 nm

Zinc oxide, 99.5þ% by Acros Organics (5 loads, 25 measurements)

0.273
0.01

11.0
1.4

Zinc oxide, reagent ACS by Acros Organics (1 load, 10 measurements)

0.430
0.02

7.0
0.5

Z50-500 USP powder packaged by Fisher Scientific (1 load, 10 measurements)

0.561
0.017

2.7
0.39

Z52-500 USP powder packaged by Fisher Scientific (4 loads, 28 measurements)

0.660
0.037

1.9
0.4

S80249 by Fisher Scientific (1 load, 5 measurements)

0.398
0.001

6.1
0.2

Zinc oxide ACS reagent grade by MO Biomedicals, LLC (1 load, 5 measurements)

0.349
0.017

8.2
2.1

Zinc oxide Polystormor™ by Mallinckrodt Chemicals (3 loads, 46 measurements)

0.223
0.009

19.6
1.8

Zinc oxide Nanopowder by American Elements (3 loads, 15 measurements)

0.631
0.1

4.7
2.5

use 10 wt% solution of sodium hexametaphosphate from Fluka dissolved in distilled water. We use three different methods: one based on ultrasound (DT-1200), the second one electron microscopy, and the dynamic light scattering. Electron microscopy photographs were taken with a JEOL TEM. The particles were first dispersed in pH 10 sodium hydroxide solutions. After equilibrating and mixing the suspension overnight, we prepared the TEM samples by placing a droplet of the dispersion on a 300-mesh gilder copper TEM grid (Electron Microscopy Sciences, Inc.). After the water in the droplet had evaporated, particles precipitated on the grid. This grid was then placed

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on the TEM sample holder and inserted into the TEM instrument. The accelerating voltage applied was 100 kV. The magnifications for all the samples were kept similar for comparison. We used two different dynamic light scattering instruments. Instructions for both of them suggest using filtration for removing large particles. Strong scattering by even a few large particles could cause artifacts with this technique. Filtration is not acceptable for the broad PSDs of the current study with large quantities of large particles. Nevertheless, we have performed some trial measurements, which failed. The results contradicted those from TEM photographs even qualitatively. We do not discuss them here. All samples were prepared at 5 wt% by adding 5 g of powder to 95 g of distilled water and sonicating the solution for making it homogeneous. The pH became naturally from 6 to 7. This sample then was poured into the DT-1201 measuring chamber with a magnetic stirrer for preventing sample sedimentation. All the measurements were performed with this sample without taking it out of the measuring chamber. The main purpose of sample preparation is to break up flocs and disperse the particles. This is achieved by increasing the surface charge of the particles, and z-potential measurements allow optimization of this procedure. The software option for running multiple measurements continuously was used for z-potential measurements. Initial value of the z-potential was about þ10 mV for all samples. As it was too low for preventing flocculation, as a first step of dispersion the pH was adjusted to 10 by adding small amounts of 1 N KOH. The z-potential then became negative, between 10 and 15 mV, which was not enough to keep the particles from aggregating. Hence as a second step small increments of 10% hexametaphosphate (HMPH) solution were added and the sample sonicated after each addition. Simultaneous continuous measurement of the z-potential displays a gradual increase in its absolute value. Under HMPH saturation conditions, the z-potential eventually reaches a sufficiently significant negative value. This saturation value differs from one powder to another, but it is more than 30 mV for all of them but not exceeding 40 mV. Monitoring of the increase in z-potential is an important part of the sample preparation procedure since HMPH overdosing can cause the particles to flocculate again. This preparation usually takes about 1 h, although some particles exhibit much longer equilibration time. This might show up during the particle size measurement as a drift of the measured attenuation spectra with time. In such cases, the sample was mixed for a longer equilibration time. Multiple measurements of PSD were performed with every sample. Some of the samples were prepared several times. The term “load” is used in Table 9.4 for indicating different preparations of the same powder. Each “load” has been measured at least five times. All these data have been statistically treated. Table 9.1 presents the arithmetic average of the median

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particle size and average absolute deviation of this parameter for each ZnO sample. The tested powders were observed to have very different sizes in the range from 220 to 660 nm. One powder (Nanopowder by American Elements) was expected to contain the smallest particles, as it was the only one that had the term “nano” in its name. This sample was prepared three times, and three different loads and multiple sonication were tried to disperse it without success. In this case, the size was observed to vary from one preparation to another, which is reflected in the relatively large variation coefficient. In any case, the amount of nanoparticles of this powder dispersed in water was still very small. The breadth of all the PSDs is illustrated in Figure 9.8. Interestingly, all distributions are broad, with sizes in the nanoscale as well as submicrometer and micrometer scales. Table 9.4 presents also the averaged content of nanoparticles for each ZnO sample. Precision of this measurement is defined as the ratio of the “absolute average deviations” to the “average content.” It is clear that this precision varies significantly from sample to sample. The worst number is 2.5% for Nanopowder by American Elements. However, this sample is not stable. It is strongly aggregated, which reduces reproducibility of PSD from one sample preparation to another. Precision of practically all other samples is better than 2%. We would claim this number as characteristic of the method. The powder with the largest particle size is Z52-500. This powder has the least amount of nanoparticles, about 2%, according to Table 9.4. The powder with the highest amount of nanoparticles is Mallinckrodt. According to

1.0

Cumulative, weight basis

0.9 0.8 0.7 0.6 0.5

Zinc oxide, ACS Acros Organics Zinc oxide, 99.5+% Acros Organics Z50-500 USP Z52-500 USP S80249 by Fisher Scientific Zinc oxide 99.99% by Alfa Aesar Zinc oxide ACS MO Biomedicals, LLC Polystormor, Mallinckrodt Chemicals Nanopowder America Elements

0.4 0.3 0.2 0.1 0.0 10-2

10-1

100 Diameter (mm)

101

FIGURE 9.8 Cumulative particle size distributions for eight different ZnO samples.

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Table 9.1, it contains 20% of nanoparticles. We used these two powders for making artificial mixtures for the consistency test. Comparison of the PSDs produced by acoustics and electron microscopy is complicated by the difference in their innate basis. Acoustic yields PSD on the weight basis, whereas innate basis of electron microscopy is number distribution. Difference between weight and number distributions can be tremendous for a broad PSD. Figure 9.9 shows as an example size distribution for Z52-500 measured using acoustics and recalculated on both weight and number basis. Weight basis indicates the weight of the particles with a certain size. For instance, Z52-500 contains only 2% of ZnO by weight in particles with sizes under 100 nm. Number basis presents number of particles with a certain size. For Z52-500, the number of nanoparticles with size under 100 nm is about 75%. The same cumulative number is more than 35 times different if expressed either on a number or on a weight basis. TEM photographs provide information for calculating PSD on a number basis. In order to compare this statistical data with acoustic PSD, it should be converted to the weight basis. Unfortunately, this approach is not realistic because of the complexity of generating complete and representative PSD using TEM. There is an International Standard describing this procedure, ISO 13322-1 [30]. In particular, this standard specifies the number of treated particles required to reach 95% of the “correct” PSD. In the case of broad PSDs, it is several thousands. For comparison, the photographs shown on Figures 9.3–9.6 contain only up to 60 particles. Statistical treatment of

1.1

Number basis

Weight basis

1.0

Particle size distribution

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10-2

10-1

100 Diameter (mm)

FIGURE 9.9 Particle size distribution of Z52-500 on weight and number basis.

101

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hundreds photographs for generating a single, reliable PSD is not a realistic approach for solving the problem of monitoring nanoparticle content. The situation will become even more hopeless if we take into account the possibility of various artifacts associated with interpretation of complex shapes of particles and aggregates on the photographs. It is possible that TEM sample preparation results in the aggregation of initially separate nanoparticles. That is why calculation of the number of the nanoparticles on each TEM photograph includes particles within aggregates. This might lead to overestimating the nanoparticle content. This short analysis brings us to the conclusion that statistical histograms similar to the ones presented on Figure 9.10 and calculated using just a single photograph for each sample cannot be directly and quantitatively compared with the weight-based results produced by acoustics. We can use these Mallinckrodt

Probabilities of particles in the bin

1.0

Mallingckrodt

0.8 0.6 0.4 0.2 0.0

0

Z52-500

Probabilities of particles in the bin

1.0

100 200 300 400 500 600 Size (nm)

Z52-500

0.8 0.6 0.4 0.2 0.0

0

100 200 300 400 500 600 Size (nm)

FIGURE 9.10 TEM photographs and size histogram of the Malincrodt and Z52-500 ZnO dispersion. Bar is 500 nm.

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photographs only for some qualitative comparison in these two methods. Photographs of all other ZnO samples can be found in Ref. [8]. These TEM photographs yield some information that correlates with results of acoustic measurement. Here are some conclusions derived from a comparison of the results from TEM photographs that agree with the data given in the Table 9.4. First of all, the Mallincrodt sample contains clearly the smallest particles. There are two samples by Acros Organic. Sample ACS contains particles of much larger sizes than sample 99.5%, which agrees with Table 9.4. Samples MO Biomedicals and S80249 have similar size distributions, which agrees with the data in Table 9.4. Sample of Nanopowder by American Elements is highly aggregated. It looks as if the aggregates consist of nanoparticles. We could not find any contradiction between the TEM photographs and the PSD results presented in the Table 9.4. This comparison of the TEM photos and the results in Table 9.1 allows us to conclude that TEM confirms particle sizing by acoustics, at least qualitatively. There is a possibility of verifying the consistency of acoustic particle sizing by mixing two samples together. Also, this mixing test would yield some information on the sensitivity of the method to the nanoparticle content. The most informative would be additions of small increments of the sample with the largest nanoparticle content to the sample with the least nanoparticle content. There are two ways one can estimate the nanoparticle content in such mixtures. First of all, one can use known nanoparticle content in the individual samples and also the known ratio of the individual samples in a particular mixture. For instance, the Mallinckrodt sample contains 20% nanoparticles, whereas the Z52-500 sample contains only 2% nanoparticles. The content of the Mallinckrodt sample in each mixture is also known. This is sufficient for calculating the expected nanoparticle content in each mixture. On the other hand, measurement of the attenuation spectra for each mixture yields the raw data for calculating the PSD. This is the same procedure as for individual samples. It provides the nanoparticle content in mixtures similar to individual samples. If everything is consistent, both methods should yield the same nanoparticle content in a particular mixture. We have performed this consistency verification test using ZnO sample Z52-500 as the base. This sample contains the least amount nanoparticles according to acoustics measurements, only 2%. The weight of the initial sample was 100 g, which is sufficient for filling the DT-1200 sample chamber. Additions of the Mallinckrodt sample, with the largest nanoparticle content, to Z52-500 were made in increments of 2 g and then 4 g. The final

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mixture contained 26 g of the Mallincrodt sample and 100 g of the Z52-500 sample. We can calculate nanoparticle content in each mixture using following algorithm. The initial 100 g of Z52-500 samples contains 5 g of ZnO, which in turn has only 0.1 g (2%) of nanoparticles. Let us assume that we have added M grams of the Mallincrodt sample. This addition contains 5  M/100 g of ZnO because the sample was prepared at 5 wt% of ZnO. The nanoparticle content would be (5  M/100)/5 because only 20% of Malincrodt ZnO is nano. Total weight of ZnO in such mixture would be (5 þ M/20)g. Total weight of nanoparticles, assuming known PSD of individual samples, would be (0.1 þ M/100)g. The percentage of nanoparticles in the mixture of 100 g Z52-500 samples and M g of Mallincrodt samples, P, equals: P¼

0:1 þ M=100  100 5 þ M=20

Alternatively, we can estimate the percentage content of nanoparticles in each mixture from the acoustic attenuation spectra measured for the mixture. These two numbers should be the same. This is the essence of the performed mixing test. Figure 9.11 presents graphically results of the test. Values on the X-axis correspond to the value of percentage P calculated for each mixture. The actual results of the acoustic sizing measurements are represented by circles in Figure 9.11. The solid line represents the ideal case when the measured content equals exactly the expected nanoparticle content, P. It is seen that deviation of the acoustically measured nanocontent from the expected values is on the order of 1%. It is significantly less than the 2% precision level, which we determined as the precision threshold on the basis of study of the individual samples. We can conclude that acoustic spectroscopy yields nanoparticle content in these mixtures that agrees well with the expected values. Finally, we can conclude that particle size characterization using acoustic spectroscopy allows monitoring the nanoparticle content in the dispersions with a broad PSD with a precision of at least 2%. It can be achieved in concentrated dispersions with no dilution. This conclusion is confirmed by multiple measurements of individual samples of eight different ZnO dispersions and their mixtures. Electron microscopy confirms qualitatively the results of PSD characterization using acoustic spectroscopy.

9.5. LIMITATION OF ULTRASOUND-BASED METHOD FOR CHARACTERIZING NANODISPERSIONS We consider ultrasound-based methods as well suited for characterizing various nanodispersions and having many advantages over other traditional techniques such as dynamic light scattering. However, there are some

365

Applications for Nanodispersions

7

% nano particles, measured

6

5

4

3

2

1 1

2

3 4 % nano particles, estimated

5

6

FIGURE 9.11 Percentage of nanoparticles in the Z52-500 sample after incremental additions of Mallinckrodt Chemicals sample. X-axis shows the percentage calculated from known amounts of the added Mallinckrodt sample, assuming that it contains 20% nanoparticles, according to the Table 9.1. Y-axis shows the percentage calculated from the attenuation spectra, which is measured for the mixture.

limitations of these methods that we would like to point out here for preventing potential confusion. The first limitation is the smallest particle size. Specifications of acoustic instruments usually place this number at 10 nm. We would like to justify this number. In order to do this, we have plotted the attenuation spectra calculated for silica nanodispersions with sizes 31, 20, and 10 nm in Figure 9.12, together with background intrinsic attenuation of water. It is seen that a reduction of the particle size leads to a decay of the attenuation. Consequently, the attenuation spectra for smaller particles approach background attenuation of water because contribution of particles to attenuation diminishes. At some point, the contribution of the particles becomes comparable with precision of attenuation measurement and becomes indistinguishable from the liquid background. Figure 9.12 indicates that for 10% silica dispersion this smallest size limit can be attributed to the top 10 nm, with the attenuation very closely approaching that of water. This limit is valid for certain density and volume fraction of particles. Increasing the density contrast relative to 1.2 for silica would allow measurement of somewhat smaller particles. The same thing would happen with

366

Attenuation (dB/cm/MHz)

CHAPTER

0.4

31 nm Ludox

0.3

20 nm Ludox

0.2

10 nm Ludox water

9

0.1

0.0 1

10 Frequency (MHz)

100

FIGURE 9.12 Theoretical attenuation spectra for 10% silica with different sizes shown in the legends and intrinsic attenuation of water.

increasing volume fraction. However, we would not recommend applying this particle sizing method for particles smaller than 10 nm. The precision of the measurement must be improved, which would require thermostabilization of the sample. This analysis is valid for rigid particles. The situation with soft particles might be different. For instance, we present the data on microemulsions with size down to 5 nm in Chapter 10. This size is confirmed by independent methods. This makes us optimistic of applying this particle size method for microemulsions. The question remains open for other soft particles, like proteins. Conversion of the attenuation into the soft particles size distribution requires knowledge of the thermal properties of the material of the particle, such as thermal expansion, heat capacity, and heat conductance. These quantities are easily measurable for oils in case of microemulsions. In the case of proteins and other macromolecules, these quantities are unknown. This is the main obstacle of applying this method for sizing of proteins. There is one more possible complication in measuring the sizing of very small particles. Intrinsic attenuation of the liquid is somewhat dependent on chemical composition of the liquid and temperature. Contributions of these factors could become comparable to contribution of particles when particle size becomes very small. It is our experience that this possibility must be taken into account when the particle size becomes 10 nm and lower. There are also some warnings regarding using ultrasound-based methods for measuring the z-potential. Measurement of the electroacoustic signal does

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not present a problem even for the smallest particles. Calculation of the z-potential, on other hand, does require some caution. The problem arises because the particle size might become comparable with the Debye length. Traditional Smoluchwski theory is limited only for a thin double layer, when the particle size exceeds the Debye length at least by 10 times. This condition becomes somewhat violated even for silica Ludox in an aqueous solution, as follows from numbers presented in Section 9.1. Decreasing the value of the parameter ka would require eventually application of the advanced theory for calculating the z-potential, if the user needs accurate and absolute value of this parameter. This theory eventually must take into account not only surface conductivity effects but also the potential overplay of the double layers. An appropriate theory is given in Chapter 5. At the end, we would like to repeat that these points of caution should not affect the optimism regarding the use of ultrasound-based methods for characterizing nanodispersions.

REFERENCES [1] F. Stenger, S. Mende, J. Schwedes, W. Peukert, Nano-milling in stirred media mills, Chem. Eng. Sci. 60 (2005) 4557–4565. [2] N. Bell, J. Cesarano, J.A. Voight, S.J. Jockwood, D.B. Dimos, Colloidal processing of chemically prepared zinc oxide varistors. Part 1. Milling and dispersion of powder, J. Mater. Res. 19 (2004) 1333–1340. [3] P. Wilhelm, D. Stephan, On-line tracking of the coating of nano-scaled silica with titania nano-particles via zeta potential measurements, J. Colloid Interface Sci. 293 (2006) 88–92. [4] X.Z. Wang, L. Liu, R.F. Li, R.J. Tweedie, K. Primrose, J. Corbett, et al., Online characterization of nano-particles suspensions using dynamic light scattering, ultrasound spectroscopy and process tomography, Chem. Eng. Res. Des. 87 (2009) 874–884. [5] N. Bell, M. Rodriguez, Dispersion properties of an alumina nano-powder using molecular polyelectrolyte and steric stabilization, J. Nanosci. Nanotechnol. 4 (2004) 283–291. [6] Y.P. Sun, X. Li, J. Cal, W. Zhang, H.P. Wang, Characterization of zero-valent iron nanoparticles, Adv. Colloid Interface Sci. 120 (2006) 47–56. [7] G. Petzold, R. Rojas-Reyna, M. Mende, S. Schwartz, Application relevant characterization of aqueous silica nano-dispersions, J. Dispers. Sci. Technol. 30 (2009) 1216–1222. [8] A.S. Dukhin, P.J. Goetz, X. Fang, P. Somasundaran, Monitoring nano-particles in the presence of larger particles in liquids using acoustics and electron microscopy, J. Colloid Interface Sci. (2009). [9] A.S. Dukhin, P.J. Goetz, Characterization of chemical polishing materials (monomodal and bimodal) by means of acoustic spectroscopy, Colloids Surf. 158 (1999) 343–354. [10] ISO Guide 30:1992(E)/Amd.1:2008 Terms and definitions used in connection with reference materials. [11] NIST 1003c Standard Reference Material, see https://www-s.nist.gov/srmors/tables/ view_table.cfm?table¼301-1.htm for a listing of NIST particle size Standard Reference Materials. [12] Whitehouse PS 202 material, see www.whitehousescientific.com for a listing of Whitehouse particle standards.

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[13] ISO Standard, Particle size analysis—laser diffraction methods—part 1: general principles, ISO 13320. [14] USP <429> Light Diffraction Measurement of Particle Size, USP30, NF25. [15] G. Dezˇelic´, M. Wrischer, Z. Devide´, J. Kratohvil, Electron microscopy of Ludox colloidal silica, Colloid Polym. Sci. 171 (1) (July 1960). [16] H. Bergna, W. Roberts, Colloidal Silica: Fundamentals and Applications, CRC Press, Boca Raton, FL, 2006, ISBN: 0824709675. [17] L. Antony, J. Miner, M. Baker, W. Lai, J. Sowell, A. Maury, et al., The how’s and why’s of characterizing particle size distribution in CMP slurries, Electrochem. Soc. Proc. 7 (1998) 181–195. [18] M. Carasso, J. Valdes, L.A. Psota-Kelty, L. Anthony, Characterization of concentrated CMP slurries by acoustic techniques, Electrochem. Soc. Proc. 98–97 (1998) 145–197. [19] K. Osseo-Asare, A. Khan, Chemical-mechanical polishing of tungsten: an electrophoretic mobility investigation of alumina-tungstate interactions, Electrochem. Soc. Proc. 7 (1998) 139–144. [20] S.-i. Takeda, P.J. Goetz, Dispersion/flocculated size characterization of alumina particles in highly concentrated slurries by ultrasound attenuation spectroscopy, Colloids Surf. 143 (1998) 35–39. [21] S.-i. Takeda, in: V.A. Hackley, J. Texter (Eds.), Characterization of Ceramic Slurries by Ultrasonic Attenuation Spectroscopy, Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, American Ceramic Society, Westerville, OH, 1998. [22] S.-i. Takeda, T. Chen, P. Somasundaran, Evaluation of particle size distribution for nanosized particles in highly concentrated suspensions by ultrasound attenuation spectroscopy, Colloids Surf. (1999). [23] M. Behra, H. Krug, Nano-particles at large, Nat. Nanotechnol. 3 (2008) 253–254. [24] Groups Encourage Nano-material Stewardship, Chem. Eng. News (July 21, 2008) 34. [25] G. Miller, Nano-Materials, Sunscreens and Cosmetics: Small Ingredients-Big Risks, Friends of the Earth, www.foe.org, 2006. [26] The appropriateness of existing methodologies to assess the potential risks associated with engineered and adventitious products of nanotechnology. European Commission, Health & Consumer Protection Directorate-General, SCENIHR, 2006. [27] Nanotechnology. A Report of the U.S. Food and Drug Administration Nanotechnology Task Force, July 25, 2007. [28] P. Somasundaran, S.C. Mehta, L. Rhein, S. Chakraborty, Nanotechnology and related safety issues for delivery of active ingredients in cosmetics, MRS Bull. 32 (2007) 779–786. [29] K.M. Tyner, A.M. Wokovich, W.H. Doub, L.F. Buhse, L.P. Sung, S.S. Watson, et al., Comparing methods for detecting and characterizing metal oxide nanoparticles in unmodified commercial sunscreens, Nano-medicine 4 (2009) 145–159. [30] ISO Standard, Particle size analysis—image analysis methods—part 1: static image analysis methods, ISO 13322-1, 2004.

Chapter 10

Applications for Emulsions and Other Soft Particles 10.1. Particle Sizing of Emulsions and Microemulsions 10.2. Monitoring Emulsion Stability 10.3. Water-in-Oil Emulsion Evolution Controlled by Ion Exchange 10.4. Dairy Products 10.5. Biological Cells, Blood

371 374

377 383 388

10.6. Soft Particles: Latex 391 10.7. Micellar Systems Particle Sizing and Rheology 392 10.8. CMC, Polymers, Gelation 395 10.9. z-Potential Measurements of Soft Particles 396 References 397

In this chapter we discuss the means of characterizing heterogeneous systems that contain soft particles, such as latices, emulsions, and microemulsions. We emphasize applications for which ultrasound provides a “cutting edge” over more traditional colloidal-chemical methods. We start with a review of existing literature describing acoustic and electroacoustic techniques for characterizing such soft particles. Here we briefly present some of them, describing our own results in more detail in later subsections. The application of acoustics to emulsions is associated to a great extent with names of McClements [1–11] and Povey [12, 13]. Each leads a group that makes “Acoustics for Emulsions” a central theme of their research. Some of these papers are listed in the Applications Table at the end of this chapter. We mention here, in particular, experiments by McClements’s group with water-in-hexadecane emulsions made in the early 1990s [2, 5, 6], the patent by Povey’s group for characterizing asphaltines in oil [14], and papers by McClements’s group for testing a theory for thermal losses in concentrates (see Chapter 4) using corn oil-in-water emulsions [15, 16]. There are several reviews by these authors that summarize these results. In general, these studies showed that acoustic methods can provide droplet size distribution, and are suitable for monitoring surfactant effects, flocculation, and coalescence, as well as phase transitions [17]. Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23010-7 Copyright # 2010, Elsevier B.V. All rights reserved.

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There is an interesting feature of these works: the raw data is most often a velocity measurement, whereas we prefer to use the measured attenuation. We suggest that readers interested in sound-speed-related results read the original papers of these authors. We illustrate such features of acoustic methods later in this chapter by describing our own attenuation-based experiments. The first application of electroacoustics to emulsions, as far as we know, was the colloid vibration potential (CVP) experiment conducted by Babchin, Chow, and Samatzky [18] using relatively dilute (<10 vol%) bitumenin-water and water-in-crude oil emulsions. They showed that it was possible to monitor the effectiveness of a surfactant in achieving emulsion stability. Similar experiments by Isaacs, Huang, Babchin, and Chow [19] showed that one could monitor the effectiveness of demulsifiers in water-in-crude oil emulsions, and that the change in CVP correlated well with the results from centrifugation and photomicrography. Later, Goetz and El-Aasser [20] measured the electrosonic amplitude (ESA) and CVP using a Matec 8000 instrument for toluene-in-water microemulsions that were stabilized by cetyl alcohol and sodium laurel sulfate. Although the trends were similar, whether measured electroacoustically or using microelectrophoresis, the amplitude of the z-potential measured electroacoustically was much lower than that measured by microelectrophoresis. This discrepancy was probably a result of using the O’Brien relationship as well as his dilute case theory for dynamic mobility. The most recent work in the electroacoustics of emulsions is associated with O’Brien. For instance, O’Brien et al. [21] tested O’Brien’s ESA theory for emulsions and other particles having low-density contrast [22]. Particle size and z-potential data were reported using the Acoustosizer for dairy cream, intravenous fat emulsions, and bitumen emulsions at volume fractions up to 38%. Their results supported the possibility of characterizing concentrated emulsions using electroacoustics. These investigations were continued by Kong, Beattie, and Hunter [23]. They applied ESA for characterizing the z-potential and particle size of sunflower oil-in-water emulsions. Using O’Brien’s theory for their calculations, they obtained meaningful results, including variations with temperature and surfactant content, and at volume concentrations up to 50%. Our view on the best way to use ultrasound for characterizing soft particles such as emulsion droplets is quite different. For obtaining information on droplet size distribution and other related properties, we believe that acoustic attenuation measurements [54–62] are much more accurate, robust, and reliable than electroacoustics measurements. We will support this statement with several experiments that follow. However, droplet size measurement is not sufficient for an adequate description of the emulsion stability. Characterization of the electric surface properties is essential for understanding factors that control emulsion stability. Electroacoustics is a very useful tool for collecting this information. We will

371

Applications for Emulsions and Other Soft Particles

illustrate this on the example of water-in-kerosene emulsion, whose stability is related to ion exchange between the droplet and the bulk.

10.1. PARTICLE SIZING OF EMULSIONS AND MICROEMULSIONS One of our first studies with soft particles was made using a 5 vol% waterin-car oil emulsion [24]. We prepared this emulsion by simply adding water to the car oil, and then sonicating the mixture for 2 min. The emulsion remains stable for at least several hours. The two topmost curves in Figure 10.1 show the attenuation spectra for this emulsion; the bottommost curve shows the intrinsic attenuation of the pure car oil, which plays the role of background attenuation for the emulsion. The rightmost curve in Figure 10.2 shows the droplet size distribution calculated from the attenuation spectra of the initial emulsion. The median particle size is about 1 mm, which correlates with our visual observation; pure car oil is transparent but becomes white after adding water and sonicating. This whitish appearance results from the light scattered by the relatively large water droplets. In order to prove that the increase in the measured attenuation comes from the water droplets, we added AOT surfactant to reduce this emulsion to a microemulsion. The addition of just 1% AOT dramatically changed the attenuation spectra, as can be seen from the middle curves in Figure 10.1. At the same time, the whitish emulsion became completely transparent again, just like the initial untainted car oil. This visual appearance also proves that the droplet size became very small, indeed much smaller than the wavelength of light. The leftmost PSD curves in Figure 10.2 show that the addition of just 1% AOT reduced

Attenuation (dB/cm/MHz)

2.0 1.6 no AOT 1% AOT 1% AOT pure oil no AOT

1.2 0.8 0.4 0.0 100

101 Frequency (MHz)

102

FIGURE 10.1 Attenuation spectra of the 5% water-in-car oil emulsion and microemulsion.

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1.0

PSD, weight basis

0.8

1% AOT 1% AOT no AOT

0.6

0.4

0.2

0.0

10-4

10-3

10-2 10-1 100 Diameter (mm)

101

102

FIGURE 10.2 Droplet size distributions of 5% water-in-car oil emulsion and microemulsion.

the droplet size by more than two orders of magnitude, from 1 mm down to approximately 30 nm. The initial emulsion had become a microemulsion. This simple test reflects the main peculiarities essential for acoustics of emulsions and microemulsions. Attenuation of emulsions contains substantial contribution at low frequency. This is a fingerprint of the large micrometersized droplets—measurable attenuation at the frequency range below 10 MHz. This part of the attenuation spectra would depend on the thermal expansion coefficient of oil and yields information on the droplet size. On the contrary, the attenuation of microemulsion goes to zero at the lowfrequency range. They contribute to attenuation mostly at high frequencies. This conclusion immediately stresses the importance of intrinsic attenuation for characterizing emulsions and microemulsions. Let us assume that we do not know exactly the intrinsic attenuation of the oil phase. Then, we would have error in estimating the intrinsic attenuation of the system. If the theoretical intrinsic attenuation is lower than the experimentally measured attenuation at high frequency, the software calculating droplet size distribution would add the microemulsion fraction for compensating the deficit in attenuation at high frequency. Error in intrinsic attenuation would show up on the droplet size distribution as a microemulsion fraction. Underestimation of the intrinsic attenuation for microemulsions leads to overestimation of the droplet size. In order to confirm these conclusions, we continued our study of surfactantrelated effects with classical water-in-heptane microemulsions using AOT as

373

Applications for Emulsions and Other Soft Particles

Attenuation (dB/cm/MHz)

4

water to AOT ratio R and water weight fraction % R=80, 16.58 R=70, 14.82 R=70, 14.82 R=60, 12.97 R=50, 11.05 R=40, 9.04 R=30, 6.94 R=20, 4.73 R=40, 9.04 R=10, 2.4 heptane +6% AOT heptane, 0 water, 0

3

2

1

0 100

101 Frequency (MHz)

102

FIGURE 10.3 Attenuation spectra of water-in-heptane microemulsion. The AOT content is constant, but the water content varies.

an emulsifier [25]. The idea of these experiments was to keep the surfactant content fixed while increasing the amount of water in controlled steps. As the water content is increased, the limit of the available surfactant causes the droplet size to grow larger, based on the amount of available surfactant for stabilizing the droplet surface. Figure 10.3 shows the attenuation spectra for these water-in-heptane microemulsions. It is seen that the attenuation increases with increasing amounts of water. This extra attenuation is certainly related to heterogeneity because the intrinsic attenuation of heptane and water is much lower. Their superposition in the mixtures is much lower than the measured attenuation. Figure 10.4 shows the droplet size distributions calculated from the attenuation spectra of Figure 10.3. The parameter R represents the ratio of water to AOT. The droplet sizes do indeed increase with increasing water content relative to the amount of AOT. The droplet size distribution becomes bimodal when the water content ratio R exceeds 50. This test confirms again that increasing droplet size eventually generates attenuation at the low-frequency range. This is thermal losses described in Section 4.3.3 in Chapter 4. The total attenuation curves of Figures 10.1 and 10.3 are the superposition of thermal losses and intrinsic losses. Scattering losses are not important because the droplet size for these systems is too small. Scattering becomes a factor affecting attenuation within frequency range from 1 to 100 MHz when the droplet size exceeds roughly 3 mm. For droplets above 10 mm, scattering dominates the attenuation spectra and the effect of thermal losses becomes negligible.

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R=50

4

4

10

R=50

R=40

R=50, 7/30/98 4:34:49 PM R=40, 7/30/98 4:15:24 PM R=30, 7/30/98 3:54:51 PM R=20, 7/30/98 3:38:08 PM

2

R=30

R=10, 7/30/98 2:44:40 PM

R=20

1

R=60

3

PSD, weight basis

PSD, weight basis

3

R=80, 7/30/98 5:51:45 PM R=70, 7/30/98 5:14:43 PM R=60, 7/30/98 4:53:06 PM R=50, 7/30/98 4:34:49 PM

2

1

R=70

R=10 60

0 10-4

10-3 10-2 10-1 Diameter (mm)

100

0 10-3

R=80

10-2 10-1 Diameter (mm)

100

FIGURE 10.4 Droplet size distributions calculated from the attenuation spectra in Figure 10.3.

The acoustics of large droplets was studied in detail by Richeter and others [26]. They came to the conclusion that the general scattering theory described in the Chapter 4 predicts correctly the dependence of the critical frequency on the droplet size. However, the absolute value of the attenuation deviates from the experimental one. They stated that it looked like the wrong volume fraction. We suggested using the empirical parameter “scattering coefficient” for matching magnitude of the theoretical and experimental attenuations, see Section 4.4.6. This scattering coefficient must be used for large emulsion droplets with size 10 mm and higher similar to solid, rigid, large particles.

10.2. MONITORING EMULSION STABILITY Acoustic spectroscopy is suitable for characterizing not only stable emulsions but unstable ones as well. For instance, it is possible to monitor the changes in the attenuation spectra with time. Figure 10.5 shows an example of such time dependence for a 10 wt% orange oil-in-water emulsion. It is seen that the attenuation changes continuously over the course of the 18-h experiment. There are two effects occurring simultaneously. First, the general attenuation decreases during the experiment; this decrease reflects the creaming of the orange oil, and the resultant decrease in the weight fraction of the oil phase which is still within the path of the sound beam. Secondly, in addition to this obvious creaming effect, the attenuation spectrum changes its shape: a reflection of the evolution of the droplet size due to coalescence.

375

Applications for Emulsions and Other Soft Particles

Attenuation (dB/cm/MHz)

1.3 4/20/2001 4/20/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001 4/19/2001

1.2

1.1

12:24:33 AM 12:04:01 AM 10:48:01 PM 9:38:46 PM 8:49:45 PM 8:09:31 PM 7:20:43 PM 6:59:59 PM 6:46:44 PM 6:39:40 PM 6:11:24 PM 6:03:13 PM 5:51:05 PM

1.0

0.9

0.8 100

101 Frequency (MHz)

102

FIGURE 10.5 Attenuation spectra of the unstable 10 wt% orange oil-in-water emulsion.

0.60 0.59

size

0.58

0.6

0.57 0.56

0.5

0.55 0.54

std.deviation

Median lognormal size, mm

0.7

0.53

0.4

std.deviation

0.52 0.51

0.3

0.50 0

50

100

150

200

250

300

350

Time, min FIGURE 10.6

Time variation of the mean size and standard deviation for orange oil droplet.

Figure 10.6 shows the time dependence of the median droplet size and its standard deviation, as calculated from the attenuation spectra depicted in Figure 10.5. It is seen that the median size grows almost twofold, from 400 to 700 nm, in the first 6 h. It is interesting that during this time the width of the droplet size distribution becomes narrower, which implies that smaller droplets coalesce at a faster rate than the larger ones. In general, the stability of an emulsion depends not only on the surfactant content but also on any applied shear stress. By applying a shear stress to the

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emulsion, we might cause two opposing effects: coalescence of smaller droplets into larger ones and disruption of larger droplets into smaller ones. Quite often, it is not clear which process might predominate. Acoustic spectroscopy offers a useful way to answer this question. To address the question concerning the effect of shear, the attenuation spectra for a 19 wt% anionic oil-in-water emulsion was continuously measured, while the emulsion was pumped continuously for 20 h with a peristaltic pump. The topmost curves in Figure 10.7 show the first and last attenuation spectrum from this long experiment. The gradual time dependence during this experiment is illustrated in the left-hand plot in Figure 10.8, which shows the attenuation at only two selected frequencies (4 and 100 MHz) versus time. It is seen that the shear-induced attenuation changes quite smoothly from the first to the last measurement. Even more significantly, however, we see that there are two quite different characteristic time constants associated with this shear-dependent change. The attenuation at 4 MHz changes rather quickly during the first 200 min, and then remains more or less constant. In contrast, the attenuation at 100 MHz changes exponentially over the 20-h experiment, and apparently still has not reached a steady-state value at the end. We think that the variation of the low-frequency attenuation reflects the breakdown of larger particles into smaller ones, whereas the change in the high-frequency attenuations reflects coalescence of smaller particles into larger droplets. The right-hand graph in Figure 10.8 illustrates the droplet size distribution at the beginning and end of the experiment. 7

after shear, 4/4/2001 2:11:48 PM oil only initial emulsion, 4/3/2001 5:37:53 PM water only

Attenuation (dB/cm/MHz)

6 5 4 3 2 1 0 100

101 Frequency (MHz)

102

FIGURE 10.7 Attenuation of the 19 wt% oil-in-water emulsion under continuous shear.

377

Applications for Emulsions and Other Soft Particles

5 7

4

5

A, 100 MHz

4 3 2

B, 4 MHz

PSD, weight basis

Attenuation (dB/cm/MHz)

6

3

after 30 h shear

2

1

1

initial emulsion

0 0

200

400

600

800 1000 1200

Time, min

FIGURE 10.8 due to shear.

0 10-2

10-1 100 101 Diameter (mm)

102

Variation of the attenuation and droplet size of the 19 wt% oil-in-water emulsion,

A similar study has been performed with another acoustic spectrometer— Ultrasizer built by Malvern, by Kippax, and Higgs [27]. They used calcium chloride for modifying the droplet size of an unknown emulsion. This salt affects the z-potential of the droplets and consequently electrostatic factor of their stability. It turns out that the isoelectric point for this chemical is about 4 mM. At this concentration, the emulsion is supposed to be the most unstable. Acoustic attenuation measured with Ultrasizer confirmed that the changes correlated with amount of additive. Droplet size calculated from these attenuation frequency spectra reaches a maximum value at the isoelectric point, indeed. These authors published also data on the droplet size distribution of diesel fuel inverse emulsions in the same paper.

10.3. WATER-IN-OIL EMULSION EVOLUTION CONTROLLED BY ION EXCHANGE Freshly prepared water-in-oil emulsion has initially the water droplets with an electroneutral interior. Ions are trapped inside the large water droplets in about equal amounts. The surface charge induced by the adsorbed surfactant molecules or other means is screened with an external diffuse layer (DL) in the oil phase. The external DLs are very thick due to the low ionic strength. We could estimate the amount of energy Wext required to charge this external DL to a certain level of surface charge Q as:

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CHAPTER

Wext ¼

Q2 Q2 ¼ 2Cext ep e0 Skext

10

ð10:1Þ

where S is the surface area and kext is the Debye parameter for the external DLs. Obviously, there is another possibility to screen the surface charge. It could be done with screening the DL inside of the droplet. This second opportunity would also take a certain amount of energy, which we could estimate as a free energy required for charging the DL: Win ¼

Q2 Q2 ¼ 2Cin ep e0 Skin

ð10:2Þ

Comparison of these two charging energies yields the following approximate result: sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi Win em kext em Km Km ¼ ¼ ¼  0:001 ð10:3Þ Wext ep kin ep Km 40Kp where we have assumed that the conductivity of water is about 106/40 times larger then conductivity of the oil phase, the effective diffusion coefficients Deff are the same, and the Debye parameter can be given by the following equation: k2 

K e0 eDeff

ð10:4Þ

This is certainly a very approximate estimate. However, it clearly indicates that charging of the interior DL to a certain electric charge is orders of magnitude more energy efficient than charging the exterior one. It is important to mention that charging up to a certain level of the electric potential, instead of the electric charge, would lead to the opposite conclusion. We believe that in this case of surfactant-stabilized emulsions we are dealing with the constant surface charge case because it is related to the adsorption of thermodynamically determined amount of the surfactant. The above-mentioned difference in energy for charging the DLs creates a driving force for the co-ions to leave the interior of the droplet. This simply means that there is a gradient of the co-ions electrochemical potential that drives out the co-ions from the droplet. This statistical process is slow because ion should become solvated by surfactant molecules on the oil-water interface. It is known (see Section 2.3.8) that only sufficiently large ions can exist in nonpolar liquids. This slow leakage of co-ions from the droplet generates an internal electric charge that would screen the adsorbed surface charge. Eventually, this would lead to the complete collapse of the external DL. The surface charge of the adsorbed SPAN (the surfactant sorbitan mono-oleate) would be completely compensated by the interior charge of the water droplet.

379

Applications for Emulsions and Other Soft Particles

This slow ion exchange process affects surface tension and can affect emulsion stability and droplet size. Can we observe this? It turns out that ultrasound-based techniques make possible studying this interesting, but not well known, factor of emulsion stability. This is described in detail in Ref. [28]. We present here some main features of that article. Emulsion of water in kerosene was stabilized with 1% SPAN 80 (Fluka). For emulsion preparation, water was added to the kerosene solution with a certain concentration of SPAN. Then it was sonicated for 2 min. The water content was 5 vol% in all samples. Then emulsion was placed into the sample chamber of the acoustic and electroacoustic spectrometer DT-1200 by Dispersion Technology. These emulsions require continuous mixing to prevent settling, which is provided by the built-in magnetic stir bar in the DT-1200 sample chamber. Measured parameters are attenuation frequency spectra, electroacoustic CVI signal, and conductivity. The measurements were performed continuously over 48 h. The attenuation frequency spectrum is the raw data for calculating the droplet size distribution. Figure 10.9 shows the attenuation curve as it evolves over time. Increase of the high-frequency attenuation is the most striking feature of this process. This evolution of the attenuation spectra reflects the variation of the droplet size distribution.

1.4 1.2 1.0 0.8 0.6 0.4 0.2 11 10 9

8 7 Time,

6 5 4 5 h pe r unit

3

2

70 80 50 60 30 40 MHz) ( 20 y c en 0 10 Frequ

90 100

Attenuation (dB/cm/MHz)

1.6

0.0

FIGURE 10.9 Evolution in time of the attenuation frequency spectra measured for waterin-kerosene emulsion under condition of continuous mixing.

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4 start 0

4

4

0

0

4

4

0

0

4

4

0

0

4

6h 11 h 16 h 21 h 26 h 36 h 41 h

0 10-3

10-2

10-1 100 Diameter (mm)

101

102

FIGURE 10.10 Evolution in time of the droplet size distribution calculated from the attenuation frequency spectra measured for water-in-kerosene emulsion under condition of continuous mixing.

Figure 10.10 presents the droplet size distribution calculated at 5-h intervals. It is seen that during the first 10 h the droplet size monotonically increases, corresponding to simple coalescence of the original emulsion droplets. During this initial period, the median droplet size changes roughly from 0.4 to 2 mm. Then, suddenly after 10 h, a fraction with an average size of 25 nm appears. The content of this fraction increases during the next 30 h. The emulsion fraction continues to coalesce during this time, eventually reaching a size of about 7 mm. The third period in the emulsion evolution begins roughly after 40 h. At this point, the emulsion converts completely to the small-size state. After this, the attenuation spectra become stable and the evolution of the droplet size stops. Comparison of these droplet size distributions with attenuation spectra reveals a very simple correlation. The attenuation at low frequencies below 10 MHz corresponds to the emulsion droplets, whereas the small size droplets contribute to the high-frequency attenuation. It would be very hard to explain such evolution of the droplet size and attenuation if we did not have the opportunity to measure simultaneously the electroacoustic signal and the conductivity. Both these parameters reveal important correlations with droplet size.

381

400

100

300

60 200

emulsion content

40 100

20

12,000.0

CVI, absolute units

80

conductivity

10,000.0

0

200

400

Emulsion content, %

Conductivity, 10⫻10-10 S/m

Applications for Emulsions and Other Soft Particles

0 600 800 1000 1200 1400 1600 1800 2000 2200 Time (min)

8000.0 6000.0 4000.0 2000.0 0.0

FIGURE 10.11 Correlation between emulsion droplet content and measured conductivity and electroacoustic signal.

Figure 10.11 shows the actual measured electroacoustic signal generated by the water-in-kerosene emulsion. With time, the CVI magnitude gradually decreases. From this CVI behavior with time, we can conclude that: l l

the microemulsion droplets do not contribute to the CVI signal and the emulsion droplets do contribute to the CVI signal

The question arises about the mechanism of this effect. If observed decay of the CVI is associated with reduction of the surface charge, what is the reason of its change? Does this mean that the adsorbed SPAN molecules are losing charges they carry to the surface? Is there any other explanation that would not involve restructuring of the SPAN molecules? We would like to stress that a similar effect was observed independently for a completely different emulsion: water-in-diluted-bitumen [29]. The authors also observed a decay of the electroacoustic signal with time. This can be considered as indication that the observed effect reflects some general features of the interaction of the emulsion droplets with the medium. In answering these questions, we should keep in mind that a clear correlation exists between the apparent surface charge decay and the conductivity increase, as shown in Figure 10.11. This increase in conductivity could be

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explained only with the release of ions from water droplets into the kerosene while the droplets are coalescing. It is important to mention that the conductivity of the initial keroseneSPAN 80 solution is about 310  1010 S/m. The conductivity of the initial emulsion is less than this value. It appears that the water droplets adsorb SPAN and reduce the conductivity. However, with time the conductivity increases, eventually exceeding the initial value in the kerosene-SPAN solution (see Figure 10.11). The “ion exchange model” presented at the beginning of this section could explain all the observed features of the emulsion-microemulsion transition. First of all, it explains why the conductivity increases during the coalescence period and eventually significantly exceeds the conductivity level of the original kerosene-SPAN solution. This excess conductivity comes from the counterions released from the water droplets into kerosene. An alternative model that could potentially explain some of the observed facts with restructuring of the adsorbed SPAN molecules would not explain the excess conductivity. This is a serious argument supporting the ion exchange model. The ion exchange model also explains the lack of CVI signal for the microemulsion. It happens because the surface charge of the adsorbed SPAN molecules is completely screened by the internal charge of the microemulsion droplet. Consequently, it looks from the outside as being electroneutral. There is one possible alternative explanation discussed in Section 5.4.5. According to the theoretical analysis presented there, CVI should not exist at all for thin isolated DLs. The fact that it exists in our case for initial emulsions can be explained only by the overlapping of their DLs. Reduction of the droplet size should enforce this overlap. Consequently, CVI for the microemulsion should be larger, not smaller. However, we observe just opposite. This means that the simple theoretical mechanism of Section 5.4.5 is not valid in this transition. In this model, redistributing the screening electric charge from the outside DL to the inside one is a limiting factor of the emulsion droplet breakup into much smaller mini- and possibly microemulsion droplets. Apparently, this breakup occurs when a sufficient critical amount of energy is released because of the collapsing external DL. This breakup is not exactly spontaneous, because it strongly depends on the stirring. However, stirring with magnetic stirrer bar in the DT-1200 chamber is not vigorous. It is clear that reduction of the surface tension is the most important factor. In this sense, the observed phenomenon is similar to the “negative interfacial tension” mechanism of spontaneous emulsification [28]. It is also possible that the combination of stirring and reduction of the surface tension intensifies “interfacial turbulence” described in Ref. [30].

Applications for Emulsions and Other Soft Particles

383

10.4. DAIRY PRODUCTS Many dairy products are oil-in-water (milk) or water-in-oil (butter) emulsions. Acoustics allows characterization of the droplet size and fat content as well as monitoring the spoiling process. Results of this study are published in Ref. [31]. We present here a brief report. All dairy products used for measurement were purchased from the local A&P supermarket. There was no special sample preparation involved. Milk and butter samples were measured as obtained, with no dilution. Milk was simply pored into the DT-100 sample chamber with a volume between 20 and 100 ml depending on the instrument setup. There was no mixing or pumping while measuring attenuation of the stable products. DT-100 has a built-in magnetic stirrer, which could be used for mixing unstable samples. It is used for measuring sound speed during the milk spoiling process. Butter is pushed into the chamber at a temperature around 20  C when it becomes a soft gel-like substance. Measurement must be performed in a mode where the gap between the transducer and the receiver closes. It is opposite to the normal mode of DT-1200 functioning. We also observed that attenuation of butter is much less stable and reproducible than attenuation of milk. It is not clear whether we should attribute this to the instability of the sample or inability to fill measuring chamber properly. Attenuation frequency spectra for various dairy products are shown in Figure 10.12. Legends on this figure give the type of the dairy product, the relative fat content according to the manufacturer’s label, as well as the date and time of the measurement. This figure shows only one curve for each product. Actually, each product was measured several times for ensuring reproducibility. The calculated droplet size distribution of fat for whole milk is given in Figure 10.13. The theoretical fit of the experimental attenuation is shown in Figure 10.14 with individual contributions of intrinsic attenuation (fat-free milk) and thermal attenuation (fat droplets) to the total measured attenuation. Thermal attenuation is calculated using the following thermal properties of the milk fat: thermal conductivity ¼ 4 [mW/cm  C], Cp ¼ 1.3 [I/g C ], thermal expansion ¼ 6.2 [10-41/C ]. The more interesting feature of acoustic attenuation characterization procedure is that it allows characterization of droplet size distribution even in gel-like systems, like butter with no dilution or melting. This is possible because thermal attenuation effect is thermodynamic in nature, and it does not require relative droplet-liquid motion. These attenuation curves contain information for calculating the droplet size distribution and fat content. Milk is an obvious candidate for verifying the particle sizing capability of ultrasound-based techniques. It is an oilin-water emulsion with known fat content. In addition to fat droplets, water

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margarine, 11 g/14 g, 6 g/14 g, light butter, double devon cream, 11 g/14 g, 5 g/15 ml, heavy cream, 3 g/30 ml, half-half, 3 g/15 ml, light cream, 6 g/240 ml, milk whole, milk reduced fat, 5 g/240 ml, 2.5 g/240 ml, milk low fat, milk fat free, water,

8

Attenuation (dB/cm/MHz)

7 6

10

4/22/2003 5:58:50 PM 4/22/2003 4:38:45 PM 4/22/2003 2:06:06 PM 4/7/2003 5:09:43 PM 4/7/2003 4:00:25 PM 4/7/2003 3:24:14 PM 4/7/2003 2:45:39 PM 4/7/2003 2:10:35 PM 4/7/2003 1:33:02 PM 4/7/2003 12:41:04 PM 4/7/2003 10:47:12 AM

5 4 3 2 1 0 100

101 Frequency (MHz)

102

FIGURE 10.12 Attenuation of sound in various dairy products.

2.0

PSD, weight basis

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 10-1 Median size = 0.5804

100 Diameter (mm) Mean size = 0.701 Std.deviation = 0.312

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Cumulative, weight basis

2.2

size 0.2000 0.2328 0.2709 0.3153 0.3670 0.4272 0.4972 0.5787 0.6736 0.7839 0.9124 1.0620 1.2361 1.4387 1.6745 1.9489 2.2684 2.6401 3.0729 3.5765 4.1628

percent 0.0001 0.0007 0.0111 0.0474 0.1198 0.2266 0.3575 0.4976 0.6319 0.7489 0.8420 0.9094 0.9536 0.9794 0.9925 0.9980 0.9997 1.0000 1.0000 1.0000 1.0000

101

D16% = 0.3840 D84% = 0.9303

FIGURE 10.13 Size distribution of the fat droplets in whole milk calculated from the attenuation spectra shown on Figure 10.15 with theoretical fit.

385

Applications for Emulsions and Other Soft Particles

experiment intrinsic attenuation thermal attenuation total theoretical attenuation

Attenuation (dB/cm/MHz)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

thermal

intrinsic

0.0 100

101 Frequency (MHz)

102

FIGURE 10.14 Theoretical fit with contributions from individual mechanism of whole milk attenuation spectra. Droplet size is shown in Figure 10.14.

contains controlled amounts of proteins and sugars. Usually it is considered as complicating factor. However, it is possible to include contribution of proteins and sugars into intrinsic attenuation of the liquid. Figure 10.15 shows that there is a certain reproducible difference in the measured values between water and fat-free milk. This is the contribution from proteins and sugars. We can use the attenuation of 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03 4/7/03

0.6

Attenuation (dB/cm/MHz)

0.5 0.4

1:33:02 PM 1:23:15 PM 1:12:49 PM 12:51:29 PM 12:41:04 PM 12:31:02 PM 12:22:25 PM 12:13:49 PM 12:05:12 PM 11:56:29 AM 10:47:12 AM

low-fat milk, Americas Choice 1% Sugars 12 g, Proteins 8 g, per 240 ml

0.3 0.2 0.1

fat-free milk, Americas Choice Sugars 12 g, Proteins 8 g, per 240 ml

water

0.0 100

101 Frequency (MHz)

FIGURE 10.15 Precision test with low-fat-content milk samples.

102

386

CHAPTER

experiment intrinsic attenuation total theoretical attenuation for unimodal PSD thermal attenuation total theoretical attenuation for bimodal PSD

7

Attenuation (dB/cm/MHz)

10

6 5

unimodal

4 3 2

thermal

1 0

intrinsic

100

101 Frequency (MHz)

102

FIGURE 10.16 Theoretical fit with contributions from individual mechanisms for butter.

the fat-free milk as intrinsic attenuation when calculating the droplet size of fat in the milk samples with the same protein and sugar content. Figure 10.16 presents the theoretical attenuations with intrinsic and thermal contributions. The intrinsic contribution is much higher in this case because it corresponds to pure fat, not to water. These calculations use the same thermodynamic properties of fat as in the case of the whole milk. It turns out that attenuation measurement offers a simple way to characterize it with no special treatment of the sample. Fat content is a very important quality control parameter of dairy products. Figure 10.12 shows the attenuation spectra of various dairy products. It is clear that attenuation increases with increasing fat content. It is convenient to characterize the fat content by plotting attenuation versus fat weight fraction for all measured products at a certain selected frequency. Figure 10.17 shows this dependence at the frequency of 44 MHz. It is seen that there is practically a linear correlation between attenuation and fat content. This might be used as a simple and direct way of determining this parameter. We might expect a high reproducibility of this procedure because the attenuation spectra measured for different brands of the same dairy product are practically the same. Acoustics also makes possible characterization of the chemical reactions using sound speed when milk is spoiling. Spoiling of milk is clearly related to complex biochemical reactions. This means that sound speed must vary in time when milk is getting spoiled because, as we stated earlier, sound speed is sensitive to the chemical composition of the system. The expected sound speed changes are small. Monitoring of these changes requires stabilization of temperature. In the current experiment, we maintain the temperature within 0.05  C and measured with a precision of 0.001  C. Samples of whole milk with a volume of 100 ml were used. A magnetic stirrer mixed the samples in order to keep them homogeneous. It turns out that homogeneity of the

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Applications for Emulsions and Other Soft Particles

double devon cream

Attenuation (dB/cm/MHz) at 44 MHz

5

4 light butter

3

2

1

0 0

10

20

30 40 50 60 Weight fraction of fat (%)

70

80

FIGURE 10.17 Attenuation of oil-in-water dairy products at one frequency versus the fat content.

sample could be maintained only for roughly 12 h. After this time, oil separates from the water and the measurement becomes meaningless. There is one more opportunity to verify the data. It is the temperature dependence of biochemical reactions. It is clear that the reaction must be faster at higher temperature. That is why we made measurements at two different temperatures: 26 and 34  C. Figure 10.18 shows results of the sound speed measurement. It is a phase method measurement. Data collection is fast, about 1 min for a single point. It is seen that the curves shift to shorter times with temperature, indeed, as expected. 1.4

Sound speed variation (m/s)

1.2

T=26 ⬚C

1.0 0.8 0.6 0.4 0.2

corresponding minima points

0.0

T=34 ⬚C

-0.2 0

100

200

300 Time (min)

400

500

600

FIGURE 10.18 Sound speed versus time for whole milk at two different temperatures.

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It is interesting that there is practically no change in attenuation. This confirms our statement made in the Chapter 3 that sound speed is more suitable for characterizing chemical reactions, whereas attenuation is sensitive to the processes and properties with the large space scale of nanometers and micrometers.

10.5. BIOLOGICAL CELLS, BLOOD Characterization of biological cells is complicated in terms of interpretation of attenuation spectra because of their complicated internal structure. Even density of these objects is not clear due to the interior water content. However, there have been a number of studies indicating that measurement of acoustic properties of these systems is possible [32–39]. Authors of these papers have managed to extract some useful data. These papers are decades old. That is why we have performed tests with the main purpose of verifying the possibility of measuring acoustic attenuation and sound speed in cell suspensions with modern instruments [40]. Blood is a good model system. We wanted to show that the acoustic spectrometer DT-100 is capable of measuring blood with high precision, which should be more than adequate for characterizing properties of blood components. From initial blood samples of a human donor, we collected plasma and erythrocytes. We performed multiple measurements of the plasma, followed by an erythrocyte sample at 95 wt% fraction. After this, we made sets of dilution by plasma, measuring erythrocyte dispersions at different concentrations. The attenuation spectra of the blood plasma are shown in Figure 10.19. It can be seen that the attenuation exceeds substantially the attenuation by water. The precision increases with frequency, reaching about 0.002 dB/cm/ MHz at 100 MHz. The attenuation difference between water and plasma is 150 times larger than the measurement precision. This opens up the possibility to characterize plasma proteins with ultrasound attenuation.

Attenuation (dB/cm/MHz)

1.8 1.6

95%

95%

1.4 1.2

18%

1.0 0.8 47.5%

0.6

9%

0.4 0.2 0.0

Plasma, 0% 100

101 Frequency (MHz)

102

FIGURE 10.19 Attenuation spectra of the samples with various erythrocyte concentrations in blood plasma.

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Applications for Emulsions and Other Soft Particles

Attenuation increases with increasing erythrocyte concentration, which indicates that the observed differences can be attributed to the erythrocytes. These differences exceed the measurement precision 100 times, showing that this measurement can be used for erythrocyte characterization. Differences are most pronounced at high frequency. The results clearly indicate that such measurements must be done at frequencies above 10 MHz, preferably at 100 MHz and higher. Interpretation of these attenuation spectra is simplified by the fact that erythrocytes mostly scatter ultrasound because of their rather large size. Also, it is (conveniently) not too large, which allows application of the simple Rayleight theory, Equation (4.88). In order to use this theory, we need to know the compressibility of erythrocytes. This parameter can be calculated from the sound speed in the erythrocyte. There are two ways to calculate this parameter. The first one is associated with fitting the attenuation spectra and searching for the value of the sound speed in erythrocytes that would yield the best fit. The other one is by the using measured sound speed in erythrocyte suspension and then calculating this parameter with the Wood equation, Equation (4.21). Then we can compare values obtained from these two approaches and derive some conclusions as to how this acoustic theory would work for biological cells. It turns out that the best fit of the 9.5% erythrocyte suspension using the speed of sound in the suspension as an adjustable parameter yields a value of 1900 m/s for the speed. Results of the best fit are shown in Figure 10.20. The speed of sound in the erythrocyte dispersion shows a progressive change with the concentration of erythrocytes, as shown in Figure 10.21. 1.0

Attenuation (dB/cm/MHz)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

10 Frequency (MHz)

100

FIGURE 10.20 Experimental attenuation of the sample with 9.5% erythrocytes and theoretical fit assuming compressibility of erythrocytes that corresponds to a sound speed of 1900 m/s.

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1620 1610

Sound speed (m/s)

1600 1590 1580 1570 1560 1550 1540 1530 1520

plasma

1510 0

10

20

30

40

50

60

70

80

90

100

Weight fraction of erythrocytes, % FIGURE 10.21 Experimental sound speed of erythrocyte solutions versus weight fraction of erythrocytes.

It can be seen that the speed increases linearly with volume fraction for low concentrations. Comparing the initial slope with Equation (4.21), using rp ¼ 1.08 g/cm3 for the density of erythrocyte, yields cp ¼ 1700 m/s. This value is close to the value obtained from attenuation data, the difference likely due to some effects other than the scattering contribution to the attenuation and the approximate nature of Equation (4.88) for large particles. Additional fits for the speed of sound in erythrocytes obtained from Equation (4.21) and Figure 10.21 are shown in Table 10.1. Values for the speed of sound in biological cells can vary from 1600 to 1850 m/s [41]. The sound speed of 1700 m/s for erythrocytes falls within this range. Applying the Wood equation to a single erythrocyte, neglecting scattering from the membrane, and assuming a protein concentration of 32% [42] yield a sound speed of 2500 m/s in the protein fraction, a value typical of proteins [41].

TABLE 10.1 Sound Speed of Erythrocyte Solution at 9.5 wt%, Experiment and Theory Assuming Various Values of the Sound Speed in Erythrocytes, Which Corresponds to Different Compressibility of Erythrocytes Experiment, sound speed (m/s) 1528.8

Theory, Cp ¼ 1700 (m/s)

Theory, Cp ¼ 1900 (m/s)

Theory, Cp ¼ 5000 (m/s)

1529

1539

1576

Applications for Emulsions and Other Soft Particles

391

This analysis leads us to the conclusion that acoustic measurement of blood can reveal the variations in erythrocyte compressibility, which may be used for diagnostic purposes. Also, it opens up the possibility for characterizing suspensions of other biological cells.

10.6. SOFT PARTICLES: LATEX Sizing of submicrometer soft particles, such as latices, vesicles, and micelles, is the most complicated task for acoustic spectroscopy for two reasons. First of all, conversion of the measured attenuation spectra into particles size distribution require information about the thermodynamic properties: thermal conductivity, t; heat capacity, Cp; and thermal expansion, b. Determination of these parameters presents problem for all soft particles, including microemulsions. This was also pointed out in the independent sensitivity analysis performed in Refs [43, 44]. The second problem is unique for the soft particles mentioned above. In addition to the thermal properties, calculation of the particle size requires information on the intrinsic attenuation. Soft particles are made from materials that often attenuate sound at megahertz frequencies. In the case of emulsions and microemulsions, such attenuation can be easily measured because the oil phase is usually available. In the case of latex particles, micelles, etc. it is impossible, in principle, to extract and measure independently the material of the particle. This fact distinguishes these particles from all others and makes them the most challenging systems for acoustic particle sizing. Absence of the particle material leaves us with only one possible solution—assumption that the particular latex or micelle particles are so rigid that the corresponding attenuation in the body of the particle is negligible. This assumption does work in some cases reported in the literature [45, 46]. We have reported another example in the Chapter 4. It was regarding the successful determination of the thermal expansion coefficients for several latex particles. To demonstrate this, we measured a set of five polymer latices that had been specially synthesized to have different rheological properties, as characterized by a “hard-to-soft ratio” parameter. The thermal expansion coefficient for these latices, calculated from the measured attenuation spectra, correlates with the measured particle rheology (Figure 4.21). Another example is presented in the next section for certain micelles. The next section offers also another possibility of characterizing these complicated soft particles. Instead of converting attenuation into particle size distribution, we can use longitudinal rheology presentation. This might be sufficient, as is shown in the next section, for understanding many properties of these systems. It is definitely sufficient for comparing samples with regard to reproducibility. On the other hand, the uncertainty in the intrinsic attenuation could explain, perhaps, some observed deviation from theory. For instance, Stieler

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and others [47] observed peculiar excess attenuation of latex dispersions at frequencies above 30 MHz.

10.7. MICELLAR SYSTEMS PARTICLE SIZING AND RHEOLOGY There has been a detailed study of micellar solutions using acoustic spectroscopy by the group of G. F. Palmieri in Italy [48]. Poloxamers are triblock copolymers of poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) available in different molecular weights and PPO/PEO ratios. The presence of PEO and PPO blocks in a single polymer chain gives rise to essentially amphiphilic molecules whose self-assembling properties display a wide range of phase behavior. This ability to form micelles and liquid-crystalline phases is strongly temperature dependent since increasing the temperature allows self-association which decreases the critical micelle concentration (CMC). Another important property of Poloxamers is their thermogelling behaviour: in fact, water dispersions of some of these polymers are generally in the liquid phase at low temperatures but become a strong gel at increased temperatures. This sol/gel transition have been correlated to the intrinsic changes in the micelle properties, or to the entropic variation in the ordered water molecules close to the PPO segments, or to the possibility of formation of a cross-linked and three-dimensional structure able to entrap water in this network. Neutron scattering studies have demonstrated the formation of a gel structure for a micelle concentration reaching the critical volume fraction of 0.53, which allows locking of the micelles in a hard-sphere, crystalline structure due to their high volume density. Both micellization and gelation depend on different factors: temperature, polymer concentration, and PEO block length. It is for this reason that the Poloxamer 407 phase transitions and the effect of hydroxypropyl b-cyclodextrin (HP b–CD) on them were studied using acoustic spectroscopy with purpose of verifying the relevance of this method in the pharmaceutical field (Figure 10.22). As the first step, the hydrodynamic diameter of the micelles of Poloxamer 407 in the concentration range of 3–25% (w/v) was investigated by measuring the attenuation and propagation velocity of ultrasound at different temperatures. Then the effect of the addition of HP b-CD on the Poloxamer 407 water systems was monitored by adding different amounts [5–20% (w/v)] of HP b–CD, which is widely used in oral and parenteral pharmaceutical dosage forms since it increases the stability and solubility or poorly water-soluble drugs through the formation of inclusion complexes. Previous studies had demonstrated that the addition of different glycols and polyalcohols, as well as the addition of HP b–CD, influenced both the gelation and micellization temperature of Poloxamer 407, outlining a shift of this parameter towards higher values. In this case, acoustic spectroscopy allowed a

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Applications for Emulsions and Other Soft Particles

6 9 12 15 17.5 20

0.7

Mean diameter (µm)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 10

12

14

16

18

20

22

24

26

28

30

Temperature (°C) FIGURE 10.22 Particle size versus temperature plots of Poloxamer 407 samples at different concentrations [6–20% (w/v)].

better characterization of the microstructure and behaviour of these systems at increasing temperatures. Figure 10.21 shows the mean micelle diameter at different copolymer concentrations plotted against temperature. These micelle diameters were calculated from the measured attenuation spectra using certain input parameters listed for soft particles in the Chapter 4. Values of these parameters were selected using the best theoretical fit of the experimental data. The evolution of the micelles size allows the identification of the different phase transitions. At low temperatures, very small values of the mean diameter can be associated with the presence of unimers species in solution, which then aggregate forming the micelle structure at increased temperature. These data have been used for analyzing chemical changes in the micelles and the macromolecules forming them. In addition to converting attenuation spectra into particle size, some authors have also used rheological presentation of the the measured acoustic raw data. They have stated that the same transitions may be seen from the analysis of the rheological parameters, which are strictly related to the sound speed (G0 ) and attenuation data (G00 ) according to the Chapter 3. Both moduli aid the identification of the system’s phase behavior which shows different trends depending on the temperature of analysis. It is necessary to make clear that the dynamic moduli obtained from acoustic spectroscopy measurements are not comparable with those derived from rotational rheometers working in the oscillation mode since the applied stress is not tangential as in an oscillatory experiment, but “longitudinal” and the tested frequencies are much higher.

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6 9 12 15 17.5 20

2.55 2.50 G¢ (Pa ⫻ 109)

10

2.45 2.40 2.35 2.30 2.25 12

14

16

18

20 22 24 Temperature (⬚C)

26

28

30

FIGURE 10.23 G0 values of Poloxamer 407 samples at different concentrations [6–20% (w/v)] at different temperatures calculated at the frequency of 50 MHz.

The value of modulus G0 for Poloxamer 407 (Figure 10.23) decreases during micellization until it reaches a plateau. This trend is more evident in concentrated systems, but is practically not detectable for the dilute ones. For the 17.5% and 20% samples, it is also possible to identify a slight inflexion after the plateau, which may be identified with the sol/gel transition since the corresponding values of the temperature are in agreement with those determined rheologically and by thermal analysis. The decrease in the values of G0 with micellization and thermogelation may be explained by the reduction, during this process, of the micelle/water interaction, which brings about an increased amount of “free” water in the system. Thus, the desolvation gives rise to a reduction of sound speed since the speed in free water is lower than that shown by water engaged in the interaction with micelles. This reduction in speed is connected with an increase in micelle rigidity and a decrease in their compressibility. Anyway, this change in G0 is not considerable since the water involved in the desolvation process is only a small part of the total water present in the system. Besides, the difference in the sound speed between the “free” water and water interacting with the Poloxamer is not relevant. So, at a microrheological level, the interaction with water at the water/ micelle interface is definitely lower compared to the unimer interactions before micellization giving rise to a less structured interface. Therefore, despite G0 being a “bulk” parameter, it is also strictly connected with the phenomena present at the interface and for this reason it tends to decrease with micellization and thermogelation.

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Applications for Emulsions and Other Soft Particles

6 9 12 15 17.5 20

140

G² (Pa ⫻ 104)

120 100 80 60 40 20 12

14

16

18

20 22 24 Temperature (⬚C)

26

28

30

FIGURE 10.24 G00 values of Poloxamer 407 samples at different concentrations [6–20% (w/v)] at different temperatures calculated at the frequency of 50 MHz.

More evident are the results obtained from the analysis of the G00 curves (Figure 10.24). As already explained, this parameter is strictly related to the variation of sound attenuation. In this case, the increase in the modulus values corresponds to the unimer/micelle transition, which is accompanied by an initial increase of G00 followed by a plateau at the end of the transition. This can be explained with the microrheological modification undergone by the system, which is related to a lower interaction with water at micelle interface. It causes a destructuration of the sample and an increase in “free” water. Also, the gelation visible for the 20% (w/v) sample can be identified with an increase in the G00 modulus, in agreement with the reduction of the G0 modulus previously described. Thus, G0 and G00 are related to the microstructural changes of the investigated system. The decrease in G0 in presence of two important phase transition (micellization and gelation) is related to sound speed reduction during these phenomena. At the same time, the corresponding increase in attenuation leads to an obvious increase in G00 .

10.8. CMC, POLYMERS, GELATION Measurement of acoustic properties of a system that contains soft particles can yield important information by using a simple empirical correlation method. Here we present some examples. An important application is in the study of critical phenomena in surfactant solutions. One of the first such works was by Borthakur and Zana [49]

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of several nonionic surfactants. They were looking for correlation between acoustic properties and fluctuation of concentrations near CMC. They were able to derive some important conclusions regarding rate constants of the association and disassociation of the surfactant molecules from the micelles. Similar studies were pursued by Hauptmann and others [49] with polyvinyl alcohol and polyvinyl acetate polymer solutions. The possibility of direct acoustic measurement of the CMC was demonstrated by Durackova et al. [50]. Acoustic measurement can be used for studying polymers and monitoring gelation processes. In general, the usefulness of ultrasound for characterizing polymers was formulated by Mason almost 50 years ago [51]. As one of the existing practical applications, we can cite Ref. [52] on the gelation of tofu. In the conclusion of this paper, the authors stated that the results of their work confirm the possibility of using acoustic spectroscopy as an alternative to rheology in the characterization of tofu gel.

10.9. z-POTENTIAL MEASUREMENTS OF SOFT PARTICLES Electroacoustic measurement of the z-potential of soft particles might seem to present a problem at first sight due to the low-density contrast between the dispersed phase and the dispersion medium. However, it turns out to be not a problem for modern, highly sensitive electroacoustic spectrometers. Density contrast even close to 0.01 may be sufficient if the volume fraction of soft particles is high enough. The most convincing experimental evidence was obtained in Japan by the group of Prof. Ohshima [53]. The present authors studied two polymethacrylate latex dispersions. They measured the z-potential using electrophoretic method and electroacoustics— DT-1200 from Dispersion Technology, Inc. The measured values are given in Table 10.2. It is seen that agreement between the two methods is very good.

TABLE 10.2 z-Potential of Two Latices Measured with Two Different Methods Method

z-Potential (mV)

Latex A

Electroacoustics Electrophoresis

40.5 38.6

Latex B

Electroacoustics Electrophoresis

44.1 43.3

Applications for Emulsions and Other Soft Particles

397

As the next step, they made measurements at different concentrations of NaCl. The agreement remains practically at all measured concentrations. This work can serve as a justification for the use of electroacoustics for characterizing the z-potential in dispersions of soft particles, and especially in emulsions where dilution is critical.

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[41] J. Bereiter-Hahn, C. Blase, Ultrasonic characterization of biological cells, in: T. Kundu (Ed.), Ultrasonic Non-Destructive Evaluation, CRC Press, Boca Raton, FL, 2004, pp. 725–760 (Chapter 12). [42] M. Toubal, M. Asmani, E. Radziszewski, B. Nongaillard, Acoustic measurement of compressibility and thermal expansion coefficient of erythrocytes, Phys. Med. Biol. 44 (1999) 1277–1287. [43] A.K. Hipp, G. Storti, M. Morbidelli, Particle sizing in colloidal dispersions by ultrasound. Model calibration and sensitivity analysis, Langmuir 15 (1999) 2338–2345. [44] F. Babick, F. Hinze, S. Ripperger, Dependence of ultrasonic attenuation on the material properties, Colloids Surf. 172 (2000) 33–46. [45] J.R. Allegra, S.A. Hawley, Attenuation of sound in suspensions and emulsions: theory and experiments, J. Acoust. Soc. Am. 51 (1972) 1545–1564. [46] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth study of ultrasound velocity and attenuation in suspensions: comparison of theory with experimental measurements, J. Colloid Interface Sci. 156 (1993) 261–269. [47] T. Stieler, F.-D. Scholle, A. Weiss, M. Ballauff, U. Kaatze, Ultrasonic spectrometry of polysterene latex suspensions. Scattering and configurational elasticity of polymer chains, Langmuir 17 (2001) 1743–1751. [48] G. Bonacucina, M. Misici-Falzi, M. Cespi, G. Palmieri, Characterization of micellar systems by the use of acoustic spectroscopy, J. Pharm. Sci. (2007) 1–11. [49] A. Borthakur, R. Zana, Ultrasonic absorption studies of aqueous solutions of nonionic surfactants in relation with critical phenomena and micellar dynamics, J. Phys. Chem. 91 (1987) 5957–5960. [50] S. Durackova, M. Apostolo, S. Ganegallo, M. Morbidelli, Estimation of critical micellar concentration through ultrasonic velocity measurements, J. Appl. Polym. Sci. 57 (1995) 639–644. [51] W.P. Mason, Dispersion and absorption of sound in high polymers, in: Handbuch der Physik., vol. 2, Acoustica part1, 1961. [52] C.H. Ting, F.J. Kuo, C.C. Lien, C.T. Sheng, Use of ultrasound for characterizing the gelation process in heat induced CaSO42H2O tofu curd, J. Food Eng. 93 (2009) 101–107. [53] Yu. Ishikava, N. Aoki, H. Ohshima, Characterization of latex particles for aqueous polymeric coatings by electroacoustic method, Colloids Surf. 46 (2005) 147–151. [54] A.S. Dukhin, P.J. Goetz, Evolution of water-in-oil emulsion controlled by droplet-bulk ion exchange: acoustic, electroacoustic, conductivity and image analysis, Colloids Surf. A 253 (2005) 51–64. [55] P. Hauptmann, R. Sauberlich, S. Wartewig, Ultrasonic attenuation and mobility in polymer solutions and dispersions, Polym. Bull. 8 (1982) 269–274. [56] V.J. Stakutis, R.W. Morse, M. Dill, R.T. Beyer, Attenuation of ultrasound in aqueous suspensions, J. Acoust. Soc. Am. 27 (1955) 3. [57] G.K. Hartmann, A.B. Focke, Absorption of supersonic waves in water and in aqueous suspensions, Phys. Rev. 57 (1940) 221–225. [58] P.A. Allison, E.G. Richardson, The propagation of ultrasonics in suspensions of liquid globules in another liquid, Proc. Phys. Soc. 72 (1958) 833–840. [59] A.K. Hipp, G. Storti, M. Morbidelli, On multiple-particle effects in the acoustic characterization of colloidal dispersions, J. Phys. D Appl. Phys. 32 (1999) 568–576. [60] T. Stieler, F.-D. Scholle, A. Weiss, M. Ballauff, U. Kaatze, Ultrasonic spectrometry of polysterene latex suspensions. Scattering and configurational elasticity of polymer chains, Langmuir 17 (2001) 1743–1751.

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[61] M.A. Barrett-Gultepe, M.E. Gultepe, E.B. Yeager, Compressibility of colloids. 1. Compressibility studies of aqueous solutions of amphiphilic polymers and their absorbed state on polysterene latex dispersions by ultrasonic velocity measurements, Langmuir 34 (1983) 313–318. [62] P. Hauptmann, R. Sauberlich, S. Wartewig, Ultrasonic attenuation and mobility in polymer solutions and dispersions, Polym. Bull. 8 (1982) 269–274.

Chapter 11

Titrations 11.1. pH TITRATION 11.2. Surfactant Titration 11.3. Salt Titration. High Ionic Strength 11.3.1. Double Layer at High Ionic Strength 11.3.2. Electroacoustic Background

402 403 409

410 413

11.4. Cement. Surfactant Titration at High Ionic Strength 11.5. Time Titration, Kinetic of the Surface-Bulk Equilibration 11.6. Importance of Mixing, Agitation During Titration References

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420 421

Webster’s dictionary [1] gives the definition of the word “titration” as it relates to general chemistry as “the process of finding out how much of a certain substance is contained in a solution by measuring how much of another substance it is necessary to add to the solution in order to produce a given reaction.” This term has a slightly different meaning in colloid chemistry. It emphasizes the second part of Webster’s definition: observing the reaction caused by adding a certain substance to the solution. Colloid chemistry deals with interfaces, and consequently the reaction of interest must somehow be related to the surface properties. An obvious “reaction” is the change in B-potential due to additions to the chemical composition. We use this modified definition of the word “titration” later in this chapter. Ultrasound-based methods are especially suitable for titrations because they do not require sample dilution and allow mixing. As a result, practically all ultrasound-based instruments are equipped with titrators. We will discuss four different titration types here: pH titration, surfactant titration, salt titration, and time titration; hence, this is essentially a kinetic study. We also discuss three major factors that must be taken into account for performing a proper titration experiment: 1. Agitation 2. Equilibration time 3. Background electroacoustic subtraction One section is dedicated to each one of these. Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23011-9 Copyright # 2010, Elsevier B.V. All rights reserved.

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11.1. pH TITRATION We start with the most widely known version of colloidal titration: pH titration. In this case, acid and/or base are the chemicals used to affect the chemical composition of the colloid, namely the pH value. Measuring the variation of pH with changes in the acid-base content can itself yield useful information. However, this type of simple pH titration is not yet widely accepted in the field of colloid science. The more popular version of a pH titration is the simultaneous measurement of both pH and B-potential, after the addition of either acid or base. In this case, the pH plays the role of a liquid solution chemistry indicator, whereas B-potential characterizes the surface properties. The output B-potential versus pH relationship characterizes the chemical balance between surface and bulk chemistry, and this determines its value for the researcher. Some practically important examples of pH titrations can be found in Refs [2] (magnetic fluids), [3] (alumina slurries), [4, 5] (minerals and clay dispersions), and [6, 7] (nanoparticles). Ultrasound is an important tool for pH titration because it provides a means for measuring the z-potential without dilution. Traditional microelectrophoresis techniques require extreme dilution, which significantly affects the surface-bulk chemical balance. The electroacoustic method eliminates the need for dilution, and allows us to obtain more meaningful and useful titration data. We have already presented several examples of pH titration using the DT-300 electroacoustic probe in Chapter 8. One more example is demonstrated in Figure 11.1, which shows three titration sweeps of a 40 wt% kaolin slurry.

-24 -26

zeta (mV)

-28 -30 -32 -34 -36 -38 4

5

6

7 pH

FIGURE 11.1 pH titration of a 40 wt% kaolin slurry.

8

9

10

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The most important point on these pH titration curves is the pH value at which the z-potential equals zero. This is called the isoelectric point (IEP). If pH is the controlling factor, then we call this the isoelectric pH. The value of the IEP depends on the chemistry of the surface and the chemistry of the solution. The IEP defines the range of instability of the sample being tested. As a general rule, a colloid is unstable if the magnitude of the z-potential is less than approximately 30 mV. Generally, this unstable region may extend over a range of 2 pH units in the vicinity of the IEP, but obviously this depends upon the slope of the B-potential versus pH curve around the IEP. In the examples mentioned above, the isoelectric pH of rutile is about 4, alumina about 9, anatase 6.3, and silica possibly about 2. There is a listing of IEPs for many materials in the books published by Kosmulski [8, 9]. Kaolin shown in Figure 11.1, under these conditions, does not have an isoelectric pH. Stability of the dispersion correlates strongly with its rheological properties. This means that there should be a correlation between the measured z-potential and rheology, which is known from classical books on colloid science [10, 11]. Electroacoustically measured z-potentials also correlate well with those from rheology as was shown by Franks [12] with silica dispersions at different concentrations of electrolytes with the Cl ion. He used the AcoustoSizer from Colloidal Dynamics, Inc. The IEP is important and useful because it is independent of the instrument calibration in the sense that it does not depend on the absolute values of the computed B-potential. Consequently, the IEP can be measured with fairly high accuracy and precision. An international project is now under way by groups of scientists from the National Institute of Standards and Technology (NIST) in the United States, the Federal Institute for Material Research in Germany, and the Japan Fine Ceramics Center for setting IEP standards using electroacoustics [13]. All groups are using the same alumina sample, Sumitomo AKP-30, prepared in the same aqueous solution. Statistical analysis of multiple measurements made with the Matec ESA-8000 and Dispersion Technology DT-1200 instruments has led them to conclude that the IEP can be measured electroacoustically with a precision of 0.1 pH unit. Such pH titrations might be useful not only in water but in other liquids as well. For example, Van Tassel and Randall applied this titration to alumina in ethanol [14]. They used two methods to measure the B-potential: microelectrophoresis and CVI. The results were practically identical, making suitable Henry correction.

11.2. SURFACTANT TITRATION Automatic “surfactant titration” is a simple way for determining the optimum dose of surfactant required to stabilize the system. In this case, a surfactant solution of specific concentration is automatically dispensed by a burette. The software protocol specifies the number of points to be measured and

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the total amount of surfactant to be added. The equilibration time is important here as well as in pH titration. This is the time required for the injected surfactant molecules to reach the particle surface and establish surface-bulk equilibrium. It is important to keep in mind that we use term “surfactant” for describing a broad spectrum of possible surface-active agents. This would include protein molecules and related titrations performed in Refs [15, 16] and even nanoparticles in Ref. [7]. We will present here some results of titrations performed by ourselves; as examples, titration of PCC and for kaolin concentrated slurries are shown in Figures 11.2–11.4. This method allows us to compare the efficiency of different surfactants. For instance, Figures 11.3 and 11.4 illustrate a “surfactant titration” of the same 40 wt% kaolin slurry with two different agents: hexametaphosphate (HMPH) and silicate. It can be seen that HMPH is almost three times more efficient than silicate; it reaches the maximum B-potential at 0.5 wt% relative to the kaolin, whereas silicate requires 1.4 wt%. In addition, Figure 11.3 stresses the fact that the efficiency of a surfactant often depends on pH. In this particular case, for instance, HMPH efficiency improves with increasing initial pH. At the same time, the addition of HMPH changes the pH value, as shown in Figure 11.5. This leads us to the conclusion that “surfactant titration” and “pH titration” are in many cases related. In order to represent the surface modification properly, we should use three dimensions: plotting B-potential as a function of pH and the surfactant content. Figure 11.6 represents such a three-dimensional fingerprint that has been constructed from the data shown in Figures 11.3 and 11.5.

10.3

10 pH

10.2 10.1

-10

10.0

-20

9.9

-30

9.8 zeta

-40

9.7 9.6

-50 0.0

0.1 0.2 0.3 0.4 Percent of the added hexametaphosphate relative to the PCC weight

9.5 0.5

FIGURE 11.2 Titration of a PCC slurry with 0.1 g/g hexametaphosphate solution.

pH

zeta (mV)

0

405

zeta (mV)

Titrations

-12

initial pH

-16

6.5 8.8 9.5 10.5

-20 -24 -28 -32 -36 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Hexametaphosphate to kaolin ratio by weight in % FIGURE 11.3 Titration of a 40 wt% kaolin slurry using hexametaphospate with different starting pH.

-10

11

-12 -14

10

-18

9

pH

zeta (mV)

-16

-20 -22

8

-24 -26

7

-28 -30

6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Silicate-kaolin ratio by weight, in %

FIGURE 11.4 Titration of 40 wt% kaolin slurry using silicate.

Such a fingerprint presentation allows us to determine the optimum chemical composition for stabilizing a particular slurry. Electroacoustic fingerprints might have different axes, not necessarily pH and surfactant content. Conductivity, for example, is another important parameter which characterizes the total ionic strength.

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11 10

pH

9 8 7 6 0.0

0.2

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Hexametaphosphate to kaolin ration by weight in %

2.0

2.2

10 12 14 16 18 20 22 24 26 28 30 32 3 0.0 4

6 7 8

pH

9 10

0

11 2.

zeta (mV)

FIGURE 11.5 Variation of pH during hexametaphospate titration of 40 wt% kaolin slurry at different starting pH.

0.5 1.0 o, % rati 1.5 n i l ao H-k SHP

FIGURE 11.6 Fingerprint of the z-pH-hexametaphosphate titration of the 40 wt% kaolin slurry.

It is known that B-potential depends on the ionic strength; it generally becomes smaller with increasing ionic strength. This factor affects B-potential values, because ionic strength changes during a titration. For instance, this factor is responsible for the difference in kaolin B-potential at pH 9.25 between the three sweeps of the titration shown in Figure 11.1.

407

Titrations

60 50

zeta (mV)

40 30 20

0.01 M KCl 0.1 M KCl

10

1 M KCl 2 M KCl

0 0

5

10 15 20 wt% TWEEN 80 to alumina AKP30

25

30

FIGURE 11.7 Dependence of alumina AKP30 (3 vol%) z-potential on concentration of Tween 80 at different KCl concentration.

Another example illustrating the importance of the ionic strength for the surfactant titration is shown in Figure 11.7. The role of the ionic strength is discussed in the next section in more detail. Titration experiments can be based on the measurement of other parameters, not necessarily z-potential only. For instance, particle size distribution would be affected by surfactant content as well. It is possible to use this parameter—particle size distribution, as the basic measure for the titration procedure. This can be done using acoustic particle sizing. Figures 11.8 and 11.9 present the results of such particle size surfactant titration. The sample was a 3 vol% dispersion of alumina AKP-30 in an aqueous solution of 0.01 M KCl. The surfactant was Tween 80. It is seen that the attenuation spectrum changes dramatically with increasing concentration of the surfactant. This evolution reflects the aggregation of the initially stable alumina particles into larger particles. Figure 11.9 shows the particle size distribution calculated from these attenuation spectra. It clearly reflects the appearance of larger aggregates. The ability to measure particle size distribution together with electroacoustic signal is important for adequate calculation of the z-potential for aggregated systems. It is known (Chapter 5) that CVI depends on the particle size. This dependence becomes significant if the particle size exceeds roughly 300 nm when CVI is measured at 3 MHz, as with the DT-300 instrument. Particle size independently measured using the acoustic sensor of DT-1200 enables performing this correction for large aggregates. This becomes especially important for electroacoustic measurements at high ionic strength, when system is unstable due to low z-potential values.

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Attenuation (dB/cm/MHz)

3

11

wt% of TWEEN 80 to alumina AKP30 28 26.6 25.1 23.6 22.1 20.7 19.2 17.7 16.3 14.8 13.3 11.8 10.3 8.9 7.4 6 4.5 3 1.5 0

2

1

0 100

101 Frequency (MHz)

102

FIGURE 11.8 Attenuation curves for alumina dispersion with different concentrations of Tween 80 at 0.01 mol/dm3 KCl solution.

wt% of TWEEN 80 to alumina AKP30 26.6 22.1 17.7 13.3 8.9 4.5

1.0

PSD, weight basis

0.8

0.6

0.4

0.2

0.0 10-2

10-1

100 Diameter (mm)

101

102

FIGURE 11.9 Particle size distribution of the alumina with different concentrations of Tween 80 at 0.01 mol/dm3 KCl solution.

409

Titrations

11.3. SALT TITRATION. HIGH IONIC STRENGTH Ionic strength is important not just as a factor affecting B-potential, as was shown in the previous section. Figure 11.10 provides one more illustration, which stresses one important advantage of the electroacoustic method. It is capable of conducting titrations and z-potential measurements over an extremely wide range of conductivities, from nonpolar liquids like toluene up to several moles per liter aqueous solutions. The titration experiment illustrated in Figure 11.10 was performed at the ionic strength range from 0.01 up to 2 M. Variation of the ionic strength due to changes in the salt content might affect not only the absolute value of z-potential but even its sign. For instance, Kippax and Higgs [17] use calcium chloride for modifying the droplet size of unknown emulsions. This salt affects z-potential of the droplets and consequently the electrostatic factor of their stability. They report that the IEP for this chemical is about 4 mM. We would like to emphasize here that from the two modes of electroacoustic measurements, CVI and ESA, the first one might have some advantage from the viewpoint of the measurements at extreme ionic strengths. It is our experience that DT-300, based on CVI measurement when ultrasound is the driving force, does not require any recalibration when the ionic strength changes from that of nonpolar liquid to highly conducting aqueous solutions. This is apparently not true for the ESA method, according to some early publications on studies of high ionic strength by Rosenholm at Abo University and Kosmulski at Lublin University. They and their coauthors published a series of papers [18–23] presenting their results obtained with the ESA technique using a Matec instrument. 70 0.01 M 60

zeta (mV)

50 40 30 20 10 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

KCl concentration in M/l FIGURE 11.10 z-potential of alumina AKP30 at 3 vol% versus KCl concentration.

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They observed measurable electroacoustic effects at high ionic strength, which indicated the presence of a double layer beyond the limits of the classical DL theory. They also observed an interesting shift of the IEP to higher pH. They also discovered two problems that are inherent to the ESA technique at high ionic strength. Later developers of the Acoustosizer [24] confirmed their conclusions. Briefly, these problems are as follows: l

l

The electric field, the driving force in the ESA mode, causes a very large current at high ionic strength, which overloads the driving amplifier. The high electric current sets up magnetic effects.

Neither of these problems exists in the CVI mode of electroacoustics that is used in Dispersion Technology instruments. The total acoustic power introduced into the sample is extremely small, and the resultant electric currents are correspondingly weak. Nevertheless, the sophisticated pulse techniques that are employed in these instruments allow such weak signals to be accurately measured.

11.3.1. Double Layer at High Ionic Strength Electroacoustic measurement at high ionic strength sheds new light on the old problem of the double layer structure with increasing salt concentration. For many decades, indeed since the creation of the classical Goyu-Chapman-Stern (GCS) model, the electric surface charge has been considered the main factor in generating a double layer. Lyklema [11] describes in detail many of the sometimes chemically complex ways that charge might appear at the interface. However, if we exclude the more complex situations, the double layer, according to the GCS model, is a simple equilibrium structure resulting from the balance between electrostatic interactions (i.e., between the surface charge and bulk ions) on one hand and the thermal motion of the ions on the other. In this sense, the electric surface charge is the cause or the driving force of the double layer formation. However, it became clear long ago that the GCS theory has a restricted range of validity. Recent papers by Mancui and Ruckenstein [25, 26] present a very comprehensive review of various attempts to generalize this theory. One of the most interesting conclusions of these new DL models is that a separation of electric charge within the interfacial water layer might occur even for uncharged surfaces, that is, a DL might exist even without any surface charge. Dukhin, Derjaguin, and Yaroschuk mentioned this possibility for the first time, as far as we know, about 20 years ago [27–29]. These papers are available in English but apparently were unknown to Mancui and Ruckenstein, who reached the same conclusion in their recent papers [25, 26] with much better justification. All these authors pointed out that the peculiar properties of any surface water layer can affect the solvation energy of ions. Early authors mentioned

Titrations

411

two potential mechanisms: variation of the dielectric permittivity and image forces. Later authors suggested additional mechanisms such as van der Waals interaction between the ions and media, ion size effects, and changes in the dielectric constant with the magnitude of the electric field. From a phenomenological viewpoint, all these mechanisms, and perhaps others as well, lead to the same result, namely that cations and anions may have quite different abilities to penetrate the surface water layer. This in turn might lead to differences in their concentration within this layer, and, consequently, to a finite electric charge. This finite charge would in turn generate an electric field, which then builds a screening diffuse layer in the bulk. Figure 2.11 in the Chapter 2 presents a simple cartoon that illustrates the difference between GCS model and this new “zero surface charge” (ZSC) DL. Interfacial charge in the ZSC model is associated with the water phase. Structural peculiarities of the surface water layer diminish towards the bulk. Deviations in ion concentrations induced by this structure would diminish correspondingly. For instance, if the water interfacial structure allows higher concentration of cations and lower concentration of anions, there would be excessive positive charge within the interfacial layer as shown in Figure 2.11. Density of this interfacial charge would decline with the structure. This interfacial charge would electrostatically attract counterions (negative in Figure 2.11). Concentrations of both ion species become equal at some distance from the surface. Charge density becomes zero at this distance as shown in Figure 2.11. After this point and further away from the surface, screening charge of counterions builds up. Residual peculiarities of the water interfacial structure would still affect a spatial distribution of the screening charge at shorter distances. This ZSC DL can be useful for interpreting several experimental facts. In particular, it explains the existence of the electrokinetic effects at high ionic strength. According to the classic GCS theory, increase in the ionic strength must suppress electrokinetic phenomena because of the collapse of the DL. However, there are several experimental papers suggesting that electrokinetic phenomena do exist at high ionic strength. The first work known to us was published in 1985 by Deinega et al. in Russian Colloid Zh., and later translated into English [30]. The more recent works are by the Kosmulski and Rosenholm group and summarized in their recent review [23]. There are also papers published by Johnson, Scales, and Healy reporting z-potential value on the range of 20–30 mV even at 1 M ionic strength [31, 32]. This large value of z-potential at high ionic strength is very surprising because it cannot be explained within the scope of the classical GCS theory. Modern modifications to this theory, such as correction for ionic sizes, image forces, and other factors listed by Lyklema [11], make the situation even worse. Figure 3.18 in Lyklema’s book shows that all these factors lead to a reduction of the surface potential for a given surface charge compared to the GCS classic theory.

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The experimental work by Johnson, Scales, and Healy might be an important milestone in studying interfaces at high ionic strength. It certainly deserves independent experimental confirmation, preferably with a different instrument. This is one of the purposes of this paper. These scientists do not offer any theoretical description of the DL structure that leads to the high z-potential value at high ionic strength. We think that the ZSC DL model offers a possible explanation of these experimental facts. Interfacial charge associated with the water structure could extend away from the surface beyond the slipping plane, as shown in Figure 2.11. Screening charge has collapsed due to high ionic strength. However, it is located in the moving water phase and could generate electrokinetic effect. There are several other experimental facts that could be explained using the ZSC DL model. For instance, Derjaguin, Dukhin, and Yaroschuk applied this model to explain the phenomenon of “coagulation zones” in the stability of colloids with adsorption layers of nonionic polymers [28]. Later, Dukhin et al. applied the same ideas for explaining reverse osmosis on uncharged membranes [29]. Mancui and Ruckenstein applied these ideas for describing surface tension [25], and later to the stability problem [26]. More recently, Yaroschuk [33] suggested that the shift in the IEP observed by Kosmulski and Rosenholm at high ionic strength [23] is also related to this new DL model. Alekseyev and others [34, 35] applied similar ideas for explaining their observation of electro-osmosis at high ionic strength. It is clearly very important to verify the existence of this “zero-charge DL” and develop methods for studying its structure. There was an attempt to achieve this goal using CVI measurement in the dispersion with nonionic surfactants as a probe to modify the structure of the interfacial layer and thereby shed light on the distribution of charge within this structure [36]. As described in many publications on this subject, nonionic surfactants have the capacity to shift the position of the slipping plane. We mention here the early work by Glazman in 1966 [37] and the most recent paper by the Somasundaram’s group in 2005 [38]. This idea of a nonionicsurfactant-induced shift of the slipping plane dominates the corresponding chapter in Lyklema’s book [11]. In this work, we suggest using the shift in the slipping plane simply as a tool to localize the position of the charge within the interfacial layer. This attempt was successful in terms of the measuring technique. Figures 11.7–11.10 are taken from that article. Also, these tests have confirmed the existence of the electrokinetic effect at high ionic strength. However, attempts to modify structured surface layer have failed because the nonionic surfactant apparently does not adsorb at high ionic strength. Further studies would be required.

413

Titrations

11.3.2. Electroacoustic Background The measured electroacoustic signal (total vibration current, TVI) contains a signal contribution from ions (ion vibration current, IVI) and a signal contribution from the colloidal particles (colloid vibration current, CVI). In many real-world systems, IVI is negligible compared to CVI, and can be neglected. However, this assumption may not be valid when: l l l l

there is a low particle-liquid density contrast there is a low volume fraction of particles the particle size is large the ionic strength is high, meaning a high concentration of ions

The first three factors decrease CVI, whereas the last one increases IVI. When the IVI is of comparable magnitude to CVI, it should be subtracted from the measured electroacoustic signal to allow proper calculation of the B-potential. This subtraction is not trivial since the measured alternating current is a vector, not a scalar. Figure 11.11 demonstrates one possible relationship between CVI and IVI. They are presented as vectors on the complex plane of real and imaginary components. We intentionally chose the case when the magnitudes of IVI and CVI are approximately the same but their phases are opposite. This allows us to illustrate several potential complications caused by the IVI. First of all, one can see that the measured TVI is much smaller in magnitude than the actual CVI. This means that we would get a much smaller absolute value for z-potential if we used TVI for calculating it. The error in the Im I negative

TVI

positive

TVI error

IVI

Re I

CVI

negative

positive

FIGURE 11.11 Scheme illustrating the relationship between CVI and IVI on the complex current plane.

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absolute value might be tolerable in many cases. Unfortunately, the IVI affects not only the magnitude but also the phase. This creates problems with the apparent sign of the z-potential. There is a convention in determining sign of the B-potential from the TVI phase. This convention is introduced when we calibrate the instrument using 10 wt% silica Ludox TM-50 at pH 9.3. We know from independent microelectrophoretic measurement that these silica particles have a B-potential of 38 mV. The absolute value of this known B-potential allows us to calibrate the magnitude of the measured electroacoustic signal. The fact that the B-potential is negative allows us to calibrate the phase. We assign a phase of 180 to these negative silica particles. Positive particles would therefore generate a signal with a phase close to 360 (or 0 ). According to this convention, the B-potential changes sign when the phase of the measured electroacoustic signal crosses either 90 or 270 , that is, the vector crosses the boundary between the right and left half planes of Figure 11.11. This analysis indicates that if the measured TVI phase is in the vicinity of the 270 or 90 , we might have problem in determining the sign of the B-potential. These problems become especially pronounced near the IEP where the CVI becomes smaller because of the decrease in the B-potential. Failure to subtract any IVI near the IEP might lead to errors in the apparent position of the IEP. These errors would increase with increasing IVI. This is one of the reasons why the case of colloids with the high ionic strength is of general interest for electroacoustic characterization. Measurement of IVI yields information on the solvation number of ions. It is discussed in some details in the Chapter 12. Usually, the IVI is a very small fraction of the Ludox CVI used for calibration purposes. However, when the salt concentration approaches 0.1 M, it becomes comparable with the contribution from the particles. This roughly defines the range where the IVI should be taken into account—ionic strength above 0.1 M.

11.4. CEMENT. SURFACTANT TITRATION AT HIGH IONIC STRENGTH Cement is very peculiar and challenging system for characterization purposes. The properties of the cement particles in water change over several hours because of long-lasting hydration processes. This time evolution depends strongly on particle concentration. The ability to study this process at concentrations that are similar to the industrial situation is very valuable. Several authors well known in the field of cement science have published papers describing studies of various cement dispersions using ultrasound

Titrations

415

[39–44]. The main advantage of using an ultrasound-based technique is the ability to characterize a concentrated cement dispersion without diluting it. We present here results of our own study in some detail, which are important for this complicated system. We used both acoustics and electroacoustics. The first method, acoustics, is important because a study by NIST [40] presents clear evidence that the attenuation frequency spectra change during cement hydration and in fact can be used as a fingerprint of this process. The second method is electroacoustics, which is usually applied for characterizing the z-potential in concentrated dispersions. Authors of Refs [41–44] applied this method successfully for monitoring the interaction between a super-plasticizer and cement. All these groups used instruments designed and manufactured by Dispersion Technology, Inc. We used the same instruments on several occasions working with various cement samples submitted by our customers. In order to achieve the goals formulated in these studies, we have developed a methodology of handling cement samples at the extremely high concentration of 72% by weight. We have learned how to prepare stable and reproducible samples. We have also learned how to run super-plasticizer titration. This section presents our experience. It is methodological. It is not about the evolution of a particular cement slurry chemistry or a particular super-plasticizer chemistry. It is about the ways of monitoring cement evolution with sufficient precision and reproducibility. We used two different cement samples that we received from one of our customers. We were informed that there were some differences in the chemical composition of the powders. The nature of these differences is not relevant to this paper, which is dedicated to the description of the method but not the peculiar features of the particular cement. We refer to these cement samples as “cement A” and “cement B.” We also received three different plasticizers. The chemical nature of these substances is also not relevant here. We use them just to show that this method is suitable for determining the better super-plasticizer and its optimum dose. We refer to these plasticizers as “plasticizer D,” “plasticizer H,” and “plasticizer G.” All cement samples were prepared at concentrations of 72% by weight. A relatively large sample volume of 200 ml was prepared and an external peristaltic pump was used for sample circulation and mixing. In order to achieve these numbers, we mixed 283 g of cement with 110.09 g of distilled water, assuming a density for the cement particles of 3.16 g/cm3. Comparison of various plasticizers would require having reproducible samples with the same properties. A combination of mixing in a high-speed blender for 1 min followed by high-power sonication for 15 min yielded samples with a reproducible z-potential value. After mixing, the sample was

416

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11

poured directly into the DT-1201 measuring chamber after mixing and then sonication was applied to the sample in the chamber while being circulated with the external peristaltic pump. We monitored the evolution of z-potential and conductivity during the sonication period for samples of both cements A and B. Figure 11.12 shows how the z-potential of the cement A changes with time. It is seen that the initial z-potential is negative. It becomes positive due to the surface hydration after roughly 5 min and then reaches a reproducible value after 15 min. Figure 11.12 shows same results for the cement B. Absolute values, both negative and positive, are smaller for this cement compared to cement A. The main conclusion is that the surface of the cement particles reaches an almost stable condition after 15 min of sonication. These samples could be used for comparative studies with different chemical additives. In order to verify the effect of water dilution, we added a small amount of water to one of the samples of “cement A” 30 min after it was prepared. The corresponding z-potential curve is shown in Figure 11.13. Dilution with water reduced the weight fraction of the cement from 72% down to 66%. It is seen that the z-potential has changed not only in its value but also in its sign. Then, with time it slowly recovers back to positive values but with a much smaller magnitude.

four additions of plastisizer G measured with Setup 4

sonication treatment

2

Zeta potential (mV)

1

0 0

10

20

30 time (min)

40

50

60

-1

-2

Setup 2 Setup 1

-3 FIGURE 11.12 Evolution of z-potential in time (after 1 min preliminary mixing in a blender) for three different samples and two different experimental setups for cement A.

417

Titrations

adding water, cement wfr 66%

8 6

Zeta potential (mV)

4 2 0

0

5

10

15

20 25 time (min)

-2

30

35

40

45

-4 -6

start sonication Setup 1 Setup 1 Setup 2

-8

FIGURE 11.13 Evolution of z-potential in time (after 1 min preliminary mixing in a blender) for two different samples.

This test indicates that dilution of the concentrated cement sample changes the z-potential dramatically. It raises questions on the relevance of the data collected with traditional light-based z-potential instruments that work at extreme dilution. It is interesting that addition of water does not affect conductivity much, as shown in Figure 11.14. 2.3

water injection

6

2.2

4

2.1 2.0

2

1.9

0

1.8

-2

1.7

-4

1.6

-6

1.5

-8

Conductivity (S/m)

Zeta potential (mV)

8

1.4 0

5

10

15

20 25 Time (min)

30

35

40

45

FIGURE 11.14 Evolution of z-potential and conductivity in time (after 1 min preliminary mixing in a blender) for one sample of cement A.

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11

Figure 11.14 shows the variation of conductivity for the cement A. The conductivity increases substantially during the initial period as the z-potential changes sign. It remains almost constant after that. Addition of water does not affect conductivity much. We tested the role of super-plasticizer using cement A. Small amounts of particular super-plasticizers were added to the mixing cement. Some time was required for this added chemical to be spread homogeneously through the mixing sample and adjust the surface properties. We allowed 3 min of waiting period. Then we took a small portion of the mixing sample from the chamber and placed it on the top of the electroacoustic probe for z-potential measurement. After the z-potential measurement was completed, this portion of the sample was returned back to the chamber. Then a new incremental injection of the super-plasticizer was made and the cycle repeated. The measured values of z-potential at different concentrations of all three tested super-plasticizers are shown in Figure 11.15. It is seen that in all three cases the z-potential reverses sign at a specific concentration. This so-called “IEP” depends on the nature of the superplasticizer and varies by a factor of three from the most efficient plasticizer H to the least efficient plasticizer G. The concentration of the super-plasticizer at the IEP is essentially the optimum dose for that particular additive. We also measured the particle size distributions of both cements after they reached the steady state. Figure 11.16 shows the attenuation spectra for both cement slurries as well as the corresponding particle size distribution. It is interesting that these cements have very different z-potentials but practically identical particle size distributions. Ultrasound-based techniques—electroacoustics and acoustics—allow us to characterize concentrated cement dispersion at 72 wt% with no dilution. 4

plastisizer D plastisizer H plastisizer G

Zeta potential (mV)

3 2 1 0 −1

0.005

0.010

0.015

wt% plastisizer to cement

−2 −3 FIGURE 11.15 Values of z-potential for three different samples of cement A at different concentrations of super-plasticizers. Measurements begin after 1 min preliminary mixing in a blender and 15 min of sonication. Equilibration delay after each addition of super-plasticizer is 3 min.

419

Titrations

PSD, weight basis

2.0

cement A cement B

1.6 1.2 0.8

cement A cement B

20 16 12 8 4

0.4 0.0 100

Attenuation (dB/cm/MHz)

24

2.4

0 101 Diameter (mm)

102

100

101 Frequency (MHz)

102

FIGURE 11.16 Particle size distribution and attenuation frequency spectra (raw data) for both cements after reaching steady state due to preliminary mixing with follow-up 15 min sonication.

11.5. TIME TITRATION, KINETIC OF THE SURFACE-BULK EQUILIBRATION There is one important requirement for pH titration: it must go through points where the surface chemistry is in equilibrium with the liquid bulk chemistry. This is the only case where pH titration is meaningful and reproducible. It is in this situation that the IEP demarcates the instability range of the system. This implies that a true pH titration is actually an “equilibrium pH titration.” This means that, in order to run pH titration properly, we should have some idea about chemical kinetics and about the time required for reaching surface-bulk equilibrium after the addition of chemicals. We have come across situations in which investigators neglected this kinetic aspect of the pH titration and therefore measured completely meaningless, nonreproducible titration curves. One of the most striking examples is zirconia. Figure 11.17 illustrates variations in the B-potential and pH of a zirconia slurry with time following a step change of pH. We call this a “time titration” in the Dispersion Technology software. It is seen that equilibration takes more than half an hour. In contrast, the equilibration of a 5 vol% alumina slurry in water takes less than 10 s. This difference between alumina and zirconia requires a large difference in the respective titration protocols used to obtain the data. Automated equilibrium titration of zirconia will take hundreds of times longer than that of alumina. It might require days or even weeks, which makes it, at the very least, inconvenient or impractical. Fortunately, zirconia represents a rather extreme case. Usually the surfacebulk chemistry balance reaches equilibrium in seconds, as with alumina. However, we strongly recommend first performing a “time titration” when

420

CHAPTER

48

4.7

pH

4.6

47

4.5

46

4.4 45

pH

zeta potential (mV)

11

4.3 44

4.2

zeta 43

4.1 4.0 30

42 0

5

10

15 Time (min)

20

25

FIGURE 11.17 Equilibration of 3 vol% zirconia slurry prepared in KCl
10 initially to 4.

2

with pH adjusted

presented with a new material with unknown properties. This would allow one to determine the equilibration time and make some allowance in the titration protocol, if necessary. However, the equilibration time often is itself a function of pH. At some pH values, the surface may equilibrate quickly, whereas at other pH values, it may equilibrate very slowly. For this reason, the Dispersion Technology software automatically adjusts the time allowed for equilibration depending on the dynamic response of the sample to each increment of added reagent. Some important time titrations are discussed in Chapter 13. We can call them “time sedimentation” titrations. Basically, we can monitor sedimentation stability of the dispersion by letting the particles sediment on the surface of the z-potential probe turned upside down, Figure 13.2. This type of titration can be used for studying properties of deposits, at is done in the Chapter 13, or, alternatively, for testing the presence of large particles in a nanodispersion (Chapter 9).

11.6. IMPORTANCE OF MIXING, AGITATION DURING TITRATION There is one factor that might affect the quality of titration data: the degree of agitation or mixing of the sample. According to our experience, the vast majority of problems with titration are related to how well the added reagent is mixed with the sample. This is also the conclusion of the international consortium mentioned earlier.

421

Titrations

-22

9.5

-24 -28 -30

sonication 1

9.4 pH zeta

-32

9.3

pH

zeta (mV)

-26

-34 -36

9.2

-38 9.1 -40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 Hexametaphosphate to kaolin ration in % FIGURE 11.18 Effects of sonication on a hexametaphosphate titration of a 40 wt% kaolin slurry.

There are several reasons for the importance of agitation. l

l l l

It provides a homogeneous spreading of the reagent injected by the burettes into the sample. It prevents sedimentation, especially in the vicinity of the IEP. It prevents the buildup of a particle deposit on the sensor surface. It helps to speed up the surface-bulk equilibration.

It breaks up aggregates and flocs when the chemical composition allows for colloid stability. Figure 11.18 illustrates this last statement regarding the importance of agitation. The incremental addition of HMPH induces a negative electric charge on the kaolin particle surfaces. It can be seen that curves break at a HMPH to kaolin ratio of 0.06 wt%. This happens because of sonication of the sample at this point. Sonication breaks up the aggregates of kaolin particles. This increases the total area of the kaolin-water interface, which in turn amplifies the CVI signal. This increase of CVI signal will be interpreted as increasing the B-potential if one does not make a correction for the increase in surface area caused by the smaller particle size. It is interesting that after several more injections the curve looks like it is recovering back to the initial slope. Still, this experiment confirms the importance of agitation.

REFERENCES [1] Webster’s Unabridged Dictionary, second ed., Simon & Schuster, New York, 1983. [2] Th. Rheinlanderr, T. Priester, M. Thommes, Novel physicochemical characterization of magnetic fluids, J. Magn. Magn. Mater. 256 (2003) 252–261.

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[3] S. Gaydardzhiev, P. Ay, Evaluation of dispersant efficiency for aqueous alumina slurries by concurrent techniques, J. Dispers. Sci. Technol. 27 (2006) 413–417. [4] M. Guerin, J.C. Seaman, C. Lehmann, A. Jurgenson, Acoustic and electroacoustic characterization of variable-charge mineral suspensions, Clays Clay Miner. 52 (2004) 158–170. [5] A.S. Dukhin, P.J. Goetz, S.T. Truesdail, Surfactant titration of kaolin slurries using B-potential probe, Langmuir 17 (2001) 964–968. [6] Y.P. Sun, X. Li, J. Cal, W. Zhang, H.P. Wang, Characterization of zero-valent iron nano-particles, Adv. Colloid Interface Sci. 120 (2006) 47–56. [7] P. Wilhelm, D. Stephan, On-line tracking of the coating of nano-scaled silica with titania nano-particles via zeta potential measurements, J. Colloid Interface Sci. 293 (2006) 88–92. [8] M. Kosmulski, Chemical Properties of Material Surfaces, Marcel Dekker, New York, 2001. [9] M. Kosmulski, Surface Charging and Points of Zero Charge, Taylor and Francis, New York, 2009. [10] R.J. Hunter, Zeta potential in Colloid Science, Academic Press, London/New York, 1981. [11] J. Lyklema, Fundamentals in Colloid and Interface Science, vol. 2, Academic Press, New York, 1995. [12] G.V. Franks, Zeta potential and yield stresses of silica suspensions in concentrated monovalent electrolytes: isoelectric point shift and additional attraction, J. Colloid Interface Sci. 249 (2002) 44–51. [13] V.A. Hackley, J. Patton, L.S.H. Lum, R.J. Wasche, M. Naito, H. Abe, et al., Analysis of the isoelectric point in moderately concentrated alumina suspensions using electroacoustic and streaming potential methods, NIST, Fed.Inst. for Material Research-Germany, JFCC-Japan, University of Main., Report (2001). [14] J. Van Tassel, C.A. Randall, Surface chemistry and surface charge formation for an alumina powder in ethanol with the addition HCl and KOH, J. Colloid Interface Sci. 241 (2001) 302–316. [15] K. Rezwan, L. Meier, M. Rezwan, J. Varos, M. Textor, L. Gauckler, Bovine serum albumin adsorption onto colloidal alumina particles: a new model based on zeta potential and UV-Vis measurements, Langmuir 20 (2004) 10055–10061. [16] K. Rezwan, L.P. Meier, L.J. Gauckler, Lysozyme and bovine serum albumin adsorption on uncoated silica and AlOOh-coated silica particles: influence of positively and negatively charged oxide surface coatings, Biomaterials 26 (2005) 4351–4357. [17] P. Kippax, D. Higgs, Application of ultrasonic spectroscopy for particle size analysis of concentrated systems without dilution, Am. Lab. 33 (2001) 8–10. [18] M. Kosmulski, J.B. Rosenholm, Electroacoustic study of adsorption of ions on anatase and zirconia from very concentrated electrolytes, J. Phys. Chem. 100 (1996) 11681–11687. [19] M. Kosmulski, Positive electrokinetic charge on silica in the presence of chlorides, J. Colloid Interface Sci. 208 (1998) 543–545. [20] M. Kosmulski, J. Gustafsson, J.B. Rosenholm, Correlation between the zeta potential and rheological properties of anatase dispersions, J. Colloid Interface Sci. 209 (1999) 200–206. [21] M. Kosmulski, S. Durand-Vidal, J. Gustafsson, J.B. Rosenholm, Charge interactions in semiconcentrated titania suspensions at very high ionic strength, Colloids Surf. A 157 (1999) 245–259. [22] J. Gustafsson, P. Mikkola, M. Jokinen, J.B. Rosenholm, The influence of pH and NaCl on the zeta potential and rheology of anatase dispersions, Colloids Surf. A 175 (2000) 349–359. [23] M. Kosmulski, J.B. Rosenholm, High ionic strength electrokinetics, Adv. Colloid Interface Sci. 112 (2004) 93–107.

Titrations

423

[24] W.N. Rowlands, R.W. O’Brien, R.J. Hunter, V. Patrick, Surface properties of aluminum hydroxide at high salt concentration, J. Colloid Interface Sci. 188 (1997) 325–335. [25] M. Mancui, E. Ruckenstein, Specific ion effects via ion hydration: 1. Surface tension, Adv. Colloid Interface Sci. 105 (2003) 63–101. [26] M. Mancui, E. Ruckenstein, The polarization model for hydration/double layer interactions: the role of the electrolyte ions, Adv. Colloid Interface Sci. 112 (2004) 109–128. [27] S.S. Dukhin, A.E. Yaroschuk, Problem of the boundary layer and the double layer, Colloid. Zh. 44 (5) (1982) 884–895 (English translation is available). [28] B.V. Derjaguin, S.S. Dukhin, A.E. Yaroschuk, J. Colloid Interface Sci. 115 (1987) 234. [29] S.S. Dukhin, N.V. Churaev, V.N. Shilov, V.M. Starov, Problems of modeling reverse osmosis, Adv. Chem. USSR 55 (1988) 1010–1029 (English translation: 1988, 43, 6). [30] Yu.F. Deinega, V.M. Polyakova, L.N. Alexandrova, Electrophoretic mobility of particles in concentrated electrolyte solutions, Colloid. Zh. 48 (1986) 546–548. [31] S.B. Johnson, P.J. Scales, T.W. Healy, The binding of monovalent electrolyte ions on a-alumina. 1. Electroacoustic studies at high electrolyte concentrations, Langmuir 15 (1999) 2836–2843. [32] S.B. Johnson, G.V. Franks, P.J. Scales, T.W. Healy, The binding of monovalent electrolyte ions on a-alumina. 1. The shear yield stress of concentrated suspensions, Langmuir 15 (1999) 2844–2853. [33] A.E. Yaroschuk, A model for the shift of isoelectric point of oxides in concentrated and mixed-solvent electrolyte solutions, J. Colloid Interface Sci. 238 (2001) 381–384. [34] O.L. Alekseev, Yu.P. Boiko, F.D. Ovcharenko, V.N. Shilov, L.A. Chubirka, Electroosmosis and some properties of the boundary layers of bounded water, Kolloid. Zh. 50 (1988) 211–216. [35] O.L. Alekseyev, Yu.P. Bojko, L.A. Pavlova, Electroosmosis in concentrated colloids and the structure of the double electric layer, Colloids Surf. A 222 (2003) 27–34. [36] A.S. Dukhin, S.S. Dukhin, P.J. Goetz, Electrokinetics at high ionic strength and hypothesis of the double layer with zero surface charge, Langmuir 21 (22) (2005) 9990–9997. [37] J.M. Glazman, Discuss. Faraday Soc. 42 (1966) 255. [38] P.K. Misra, B.K. Mishra, P. Somasundaran, Organization of amphiphiles. Part 4. Characterization of microstructure of the adsorbed layer of decylethoxylene nonyl phenol, Colloids Surf. A 252 (2005) 169–174. [39] D.C. Darwin, R.Y. Leung, T. Taylor, Surface charge characterization of Portland cement in the presence of superplasticizers, in: S.G. Malghan (Ed.), Electroacoustics for Characterization of Particulates and Suspensions, NIST, 1993, pp. 238–263. [40] V.A. Hackley, L.-S. Lum, C.F. Ferraris, Acoustic sensing of hydrating cement suspensions, National Institute of Standards and Technology, US Department of Commerce, NIST Technical Note 1492, 2007. [41] J. Plank, C. Hirsch, Impact of zeta potential of early cement hydraton phases on superplasticizer adsorption, Cement Concrete Res. (2007). [42] J. Plank, B. Sachsenhauser, Impact of molecular structure on zeta potential and adsorbed conformation of a-allyl-o-methoxypolyethylene glycol-maleic anhydride super-plasticizer, J. Adv. Concrete Technol. 4 (2006) 233–239. [43] J. Plank, M. Gretz, Study on the interaction between anionic and cationic latex particles and Portland cement, Colloids Surf. A 330 (2008) 227–233. [44] J. Plank, Ch. Winter, Competitive adsorption between superplasticizer and retarder molecules on mineral binder surface, Cement Concrete Res. 38 (2008) 599–605.

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Chapter 12

Applications for Ions and Molecules 12.1. Ionic Solvation Numbers in Aqueous Solutions 425 12.2. Characterization of Ions in Nonaqueous Media 429

12.3. Protein Electric Charges 437 References 438

This chapter is dedicated to characterization of ion solvation in aqueous and nonaqueous solutions. Characterization of electric charges of complex macromolecules also belongs here because such objects can be considered as large ions. Ultrasound provides a unique means for studying ions in liquids and, in particular, their interaction with liquid molecules. We will discuss these methods initially for aqueous solution, then for nonaqueous solutions, and at the end for complex charged biomolecules.

12.1. IONIC SOLVATION NUMBERS IN AQUEOUS SOLUTIONS Interaction of ions’ electric charges with water molecules’ dipole moment leads to formation of structured water layers around ions. There was intense discussion regarding the nature of this structure of water around ions in the middle of the twentieth century. Readers can find a short overview and description of the models involved in the paper by Bockris and Saluja [1]. It seems that final consensus was achieved with the introduction of two characteristic numbers. The “coordination number” is the total number of water molecules that are in contact with an ion. Solvation number represents the number of molecules that remain associated with the ion during its movement through the solution. The solvation number is more important because all ions undergo constant thermal motion. Usually, the solvation number determines the effective size of the ion.

Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23012-0 Copyright # 2010, Elsevier B.V. All rights reserved.

425

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12

There have been several methods developed for studying the solvation numbers of ions. Bockris and Saluja [1] provide a table comparing results of these methods. Two of these methods are based on ultrasound. We will discuss them here. The first method is based on acoustics, in particular, on sound speed measurement. We have mentioned already in Chapter 3 that compressibility of electrolyte solutions depends on the concentration of electrolyte ions (Section 3.7). This compressibility dependence reflects itself in the sound speed, Figure 3.9, which can be easily measured. Compressibility decay with increasing electrolyte concentration reflects buildup of the solvating layers around ions. Viscoelastic properties of water in these layers are quite different compared to the bulk. This theoretical model is apparently associated with the name of Passynski who derived an expression for the solvation number assuming that compressibility of water in the solvating layer is negligible. This (Passynski) equation [2] links the solvation number nsol with compressibilities of water bpwat and the electrolyte solution bpel as follows:   n1 bpel Vel 1 p ð12:1Þ nsol ¼ n2 bwat n1 Vwat where n1 and n2 are the number of moles of solvent and solute in the solution, Vel and Vwat are the volume of the solution and partial molar volume of the pure solvent, respectively. Passynski [2] and later Bockris and Saluja [1] measured sound speed in various electrolyte solutions, then calculated compressibilities and solvation numbers of these electrolytes. There is one problem associated with this method. The solvation number presented in Equation (12.1) is the total number for an electrolyte. In other words, it is the sum of the solvation numbers for ions in the given electrolyte solution. If one wants to know the solvation number of a particular ion, either some assumptions have to be made or, alternatively, another independent method has to be found. Bockris and Saluja [1] suggested combining the acoustic method with an electroacoustic method employed by Yeager and Zana [3–11]. This second method yields the difference in solvation numbers of ions, instead of their sum. Clearly, it is a well-justified combination. Yeager and Zana developed their method using the ion vibration potential (IVP), which is described in some detail in Section 5.1. Combining together the acoustic compressibility method with the electroacoustic IVP method, Bockris and Saluja arrived at a way of calculating the solvation numbers of individual ions. Their values turned out to be about 2 units less on average compared to other methods. It is a very large difference taking into account that a typical value of the solvation number is around 4–5 units only. This generated a very interesting discussion between Zana and Yeager on one side and Bockris and Saluja on the other, see Ref. [12].

427

Applications for Ions and Molecules

TABLE 12.1 Compressibility of Various Electrolytes at 1 M Concentration from Bockris and Saluja [1] and Dukhin and Goetz [13] Compressibility 1010 m2/N1 Dukhin and Goetz

Bockris and Saluja

KCl

4.02

3.98

LiCl

4.07

4.06

NaCl

3.96

4.04

CaCl2

3.82

3.69

Instead of going into details of this discussion, we can compare the results obtained using more reliable modern techniques with the older experimental data of the Bockris-Saluja [1] and Yeager-Zana [3–11] methods. Table 12.1 presents results of compressibilities calculated from the sound speeds reported by Bockris and Saluja [1] and Dukhin and Goetz [13]. It is seen that these numbers agree within 1–2%. This can be considered as a very convincing confirmation that the acoustic compressibility method is based on reproducible and trustworthy experimental data. In order to perform similar test for the electroacoustic IVP method, we measured electroacoustic signals of several electrolyte solutions using the Dispersion Technology DT-300 z-potential probe. These results are reported in Table 12.2. These results are not comparable directly to the data by Yeager and Zana. They measured electroacoustic ion potential, whereas we measured the electroacoustic ion current. However, we can derive several important conclusions: first of all regarding the concentration effect. Zana and Yeager stated that “. . ..there is usually a large range of concentration, starting from 10 M and extending up to 10 M, where the measured IVP is independent of the electrolyte concentration. . ..” On the contrary, our data presented in the Table 12.2 indicates that IVI varies quite substantially within the same concentration range. This contradiction is a result of conductivity playing different roles in the IVP and IVI measurement modes. However, IVI increases with concentration in a clearly nonlinear fashion, as follows from the data of Table 12.2. It reaches saturation at higher ionic strength. This indicates that there is another factor missing. This factor puts in doubt experimental data of the electroacoustic method applied to the estimation of the solvation number. We can verify at least the reproducibility of the data presented in the Table 12.2. This data was collected in the year 2001. About 6 years later, there was an independent study of the IVI by Muller and Faude [14]. Table 12.3 shows IVI values for two electrolytes with somewhat different concentrations measured by both groups with DT-300. These values are quite

428

CHAPTER

12

TABLE 12.2 Ion Vibration Current for Four Different Electrolytes at Different Concentrations

Silica Ludox, calibration

Magnitude in absolute units

Magnitude relative to Ludox in %

Magnitude measurement error %

351.7

100.0

0.7

180

Phase in degrees

Distilled water

3.28

0.9

1.8

325

0.01 M KCl

8.98

2.6

0.8

343

0.1 M KCl

38.75

11.0

1.2

352

0.5 M KCl

77.63

22.1

0.8

358

1 M KCl

85.04

24.2

0.8

1

0.01 M NaCl

7.72

2.2

3.4

343

0.1 M NaCl

25.96

7.4

1.7

352

0.5 M NaCl

48.04

13.7

1.5

1.5

1 M NaCl

52.3

14.9

0.4

2.5

0.01 M NaI

2.35

0.7

3.0

250.2

0.1 M NaI

21.32

6.1

0.8

204

0.5 M NaI

54.13

15.4

0.2

209

1 M NaI

61.8

17.6

0.2

212

0.01 M RuCl

16.28

4.6

0.6

352

0.1 M RuCl

78.45

22.3

0.1

4.2

0.5 M RuCl

128.09

36.4

0.8

13.5

1 M RuCl

146.17

41.6

0.9

15.5

TABLE 12.3 Ion Vibration Current for Two Electrolytes Measured Independently IVI at 0.75 M Muller and Faude [14]

IVI at 0.5 M Dukhin and Goetz [13]

KCl

68.6

77.63

NaCl

31.1

48.04

Applications for Ions and Molecules

429

close. We can conclude that the most recent studies using IVI instead of IVP for measuring electroacoustic effect in electrolyte solutions yield reproducible experimental data. This new experimental technique can be used for verifying the old data collected by Zana and Yeager. There are two more arguments for revisiting old electroacoustic studies of solvation numbers of ions. One is the new theory for IVI-IVP published by Ohshima [15]. Treatment of the experimental data with this new theory might yield different results for the solvation numbers. The other is the discovery by Muller and Faude [14]. They measured IVI at different pH. It turns out that pH might affect the measured data significantly. This might be one of the reasons for the differences in Table 12.3 because Dukhin and Goetz [13] did not control pH in their measurements. The fact that pH affects IVI and, consequently, solvation numbers of ions is interesting in itself. It is possible that using new experimental data with the new theory would resolve the question regarding the missing 2 units in solvation numbers of typical ions.

12.2. CHARACTERIZATION OF IONS IN NONAQUEOUS MEDIA Electrochemistry of low-conducting nonpolar liquids still presents many challenges, which are discussed in some detail in Section 2.3.8. The main difference is that liquid molecules cannot properly solvate ions because of their either small or nonexistent dipole moment. A deficit of solvating agent controls the electrochemistry of these liquids, in contrast to the deficit of dissociating agent for aqueous solutions. There is consensus among scientists working in this field that some chemical must be added to the solution for providing proper solubility of ions. Usually these are surfactants, which can create solvating micelles around ions. The size of such complex ions is still not well understood. Traditional methods of studying ions sizes and solvation numbers presented in the previous sections are not suitable for nonaqueous solutions. Here we suggest one new method, which is described in Ref. [16]. The first step in studying the electrochemistry of nonaqueous solutions is finding ways of controlling and reproducing it. This means that we should find a simple, reproducible chemical that would provide controllable solvation of ions that can spontaneously appear in the liquid. It turns out that nonionic surfactants present one class of such chemicals. It is clear that solvation should depend on the polarity of the initial molecules that build an outer shell around ions. More polar molecules would provide better shielding for ions, creating stronger and larger micelles. This

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suggests the idea that ionization in low-conducting media could be related to the hydrophobic-lyophilic balance (HLB) number. The larger this number, the more polar are the molecules of the surfactant. This means that we might expect a direct correlation between ionization and HLB number for various substances that are soluble in low-conducting liquids. To verify this hypothesis, we used several nonionic surfactants from the SPAN family, with HLB numbers varying from 1.8 to 8.6. There are two extensive reviews describing the properties of such nonionic surfactants [17, 18], but neither mentions that they could induce electric conductivity in low-dielectric liquids. Yet, we discovered that they do induce electrical conductivity, which means that the so-called nonionic surfactants in the SPAN family are actually ionic. They produce a substantial number of ions and allow one to control the ionic strength and conductivity with high precision. There are two hypotheses regarding the origin of these ions. First of all, it is possible that SPAN molecules dissociate. Their ions might be larger that Bjerrum radius (see Section 3.2.8), which would allow them to exist in a nonpolar liquid. The other hypothesis relates this phenomenon to the “ion pairs” that always exist in nonpolar liquids due to the small contaminations. They consist of simple ions with small sizes. Electrostatic attraction pulls these cations and anions together creating the so-called ion pairs. These entities exist in dynamic balance between electrostatic attraction and destructive thermal motion. Surfactant molecules can interfere with this balance. They can build protective shells around ions, which effectively enlarge size of the ions and prevent their aggregation in pairs. It is not clear yet what mechanism is responsible for the observed ionization. We would use term dissociation in effective meaning, keeping in mind the second possibility of the ions’ origin. The main review on this field written by Kitahara [17] and Morrison [18] noted the use of conductivity measurements as a direct measure of the ionization of ionic surfactants such as AOL. We followed the same logic to characterize the ionic properties of our so-called nonionic SPAN surfactants. Figure 12.1 shows the conductivity of the kerosene samples with various concentrations of different nonionic surfactants. The reproducibility of these measurements is about 1%, probably limited by the precision with which we added the surfactant. There is clearly large difference between the different surfactants. In order to rule out the possibility of impurities in the surfactant as a factor responsible for this conductivity, we tested two different suppliers of SPAN 20. The conductivity curves are practically identical, which proves that the observed effect is related to the chemistry of the surfactant molecule. The differences in the conductivity curves correlate with the HLB numbers of the corresponding surfactants: the higher the HLB number, the larger the

431

Applications for Ions and Molecules

7000 SPAN 20, Spectrum Chem.Co. SPAN 20, Aldrich SPAN 85, Fluka SPAN 80, Fluka Arlacel 83, Aldrich

Conductivity (10⫻10-10 S/m)

6000 5000

HLB=8.6

4000 3000 2000 HLB=3.7

1000

HLB=4.3 HLB=1.8

0 0

1

2

3 4 5 Surfactant content (wt%)

6

7

FIGURE 12.1 Conductivity of kerosene solutions with various surfactants.

conductivity. This agrees well with the micelle model of ionic conductivity induced by the surfactants. The higher HLB number reflects more polarity of the corresponding surfactant. Consequently, the protective micelle shell around the ion must be stronger and, perhaps, even more extended. Statistically, this would lead to a larger number of ions and a higher conductivity. However, the HLB number is not the only factor. We tested a surfactant from a different family for verifying the importance of other details in the surfactant chemistry. For example, the surfactant Arlacel, with a slightly smaller HLB number than SPAN 80, induces a higher conductivity. This might happen because the molecules can split at different binding points. Measurements in different liquids reveal that this conductivity effect depends also on the properties of the liquid. It exists in quite pure liquids, such as 99% pure hexadecane, butyl acetate, toluene, heptane, and hexane (see Figure 12.2). It is interesting that there is no direct correlation with dielectric permittivity. But these conductivity measurements of homogeneous liquids provide only limited information. That is why in the next step we will add insoluble, dispersed particles for creating a heterogeneous system. Although conductivity measurement can confirm ionization, by itself it does not yield sufficient information for characterizing the size of the resultant ions or the apparent dissociation constant. To get more detailed information, we again follow the Kitahara and Morrison reviews, both of which combine data for homogeneous and heterogeneous systems. Addition of

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5000

kerosene, d.p. 2 butyl acetate, d.p. 5.01 hexadecane, d.p. 2.04 hexane, d.p. 1.89 heptane, d.p. 1.91 toluene, d.p. 2.38

4000

Conductivity, 10⫻10-10 S/m

12

3000

2000

1000

0 0

1

2

3

4 5 6 SPAN 80 content, wt%

7

8

9

FIGURE 12.2 Conductivity of the various nonpolar liquids with SPAN 80.

insoluble dispersed particles to the low-conducting liquid allows one to separate ions between the bulk fluid and the particle surface. This can then be used as a source of additional information for further analysis. Following this logic, we prepared dispersions of alumina in the same SPAN solutions. We characterized these dispersions using two techniques: conductivity and electroacoustics. Electroacoustics was only briefly mentioned by Morrison [18], as it became commercially available at just about that time. Actually, it only recently became a useful tool for characterization of the low-conducting dispersions with the formulation of a new electroacoustic theory that takes into account overlap of double layers, see Section 5.6. The electroacoustic method in nonaqueous systems yields directly information about electric surface charge. Combining this data with conductivity results allows us to estimate size of the ions, their concentration, apparent dissociation constant, diffusion constant, and the size of the protective micelle. These parameters were not available before. Figure 12.3 shows the conductivity for the solution with and without alumina versus SPAN 80 content. It is clearly seen that the addition of the alumina particles causes a very large change in the conductivity. This change can be explained only by adsorption. This means that alumina particles adsorb substantial amount SPAN 80. There is clearly break point around 1%. Below this point, alumina particles adsorb practically all SPAN 80 molecules.

433

Applications for Ions and Molecules

800

Conductivity (10⫻10-10 S/m)

700

liquid only

600 500 400 5 vol% alumina dispersion

300 200 100 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

SPAN 80 content, (wt%) FIGURE 12.3 Conductivity of the SPAN 80 in kerosene solutions with and with out alumina AKP-30 particles at 5 vol%.

We could also conclude that particles either adsorb both cations and anions or dramatically change apparent dissociation constant of the SPAN molecules. If dissociation remains the same and particles adsorb only one type of ions, conductivity in this region (<1%) would drop only 2 times. Remaining ions (counterions) would maintain at least the half conductivity level compared to the pure liquid. Here we assume, following Morrison, that sizes and mobilities of the ions are about equal. However, there is still a small conductivity in the dispersion at 1 wt%, about 10 pS/cm. Is it related to just one type of the ions? Is there any difference in adsorption between cations and anions? In order to get answers to these questions, we use an electroacoustic technique for characterizing the surface charge of the particles. Figure 12.4 shows the results of such measurements. We use the following values for the properties kerosene and alumina: rm ¼ 0.8 g/cm3; rp ¼ 3.8 g/cm3;  ¼ 1.5 cp; em ¼ 2; a ¼ 0.5 mm. The CVI was measured at an ultrasound frequency of 3 MHz. It is seen that the surface charge increases with the SPAN 80 content and reaches a saturation value between 1% and 2% of SPAN content. This is in good agreement with the conductivity data presented in Figure 12.3. The sign of the surface charge turns out to be positive. We have two confirmations for this statement. The first is a simple microscopic observation with a microelectrophoretic instrument, PenKem model 501. The particles move in the electric field as they would if they had a positive charge. The

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Surface charge (10⫻10-6 C/cm sq.)

0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

SPAN 80 content (wt%) FIGURE 12.4 Surface charge of alumina AKP-30 particles at 5 vol% in kerosene with various concentrations of SPAN 80.

second is the phase of electroacoustic signal. The values of CVI phase are between 0 and 90 , which correspond to a positive surface charge. These measurements now allow us to answer the set of questions formulated in the previous section. It is clear that the cations are in excess on the surface, whereas anions form the counterion cloud in the bulk between particles. For SPAN 80 content we could assume that all neutral molecules are on the surface, as well as all cations. The anions are in the solution. They are responsible for the small conductivity in the dispersion at 1 wt%, with a value about 10 pS/cm. The electroacoustic data allows us to calculate the amount of these anions. Combining the conductivity and electroacoustic data, we can estimate the diffusivity and size of these anions. In order to combine the conductivity and electroacoustic data, we can use the following equation for conductivity of the medium in the dispersion with 1% SPAN 80 content.
F2 þ þ D C0 þ D C ð12:2Þ 0 RT In the previous section, we established that the conductivity in this case is related to the anions: Km ¼

Km ðat 1%Þ ¼

F2   D C0 RT

ð12:3Þ

435

Applications for Ions and Molecules

The concentration of these anions can be determined from the value of the particle’s surface charge, s, keeping in mind the electroneutrality condition between the particle surface and bulk. This leads to the following expression: 3’s ð12:4Þ aF where ’ is volume fraction, a is particle radius of an assumed monodisperse system, and F is the Faraday constant. Substituting Equation (12.4) in Equation (12.3), we get the following expression for the diffusivity of anions: C 0 ¼

D ¼

KRTa 3F’s

ð12:5Þ

For the dispersion at 1 wt% SPAN, we have the following values for the relevant parameters: K ¼ 10 pS/cm; a ¼ 0.5 mm; RT/F ¼ 0.025 V; s ¼ 5  106 C/m2; ’ ¼ 0.05. The value of the diffusion coefficient for the SPAN 80 anions equals: cm2 ð12:6Þ s For comparison, this means that diffusion coefficient of SPAN 80 anions is 100 times smaller than the diffusion coefficient of Liþ ion in water [19]. This relatively small diffusion coefficient implies that the SPAN 80 ions are much larger than the Liþ ion, perhaps because they are encapsulated in a protective micelle. Having obtained now the diffusion coefficient, we can make an approximate estimate of the ion radius ai using the Einstein expression for the diffusion coefficient: D  107

D ¼

kT 6pai

ð12:7Þ

Using an approximate viscosity of 1.5 cp, this gives a radius for SPAN 80 anions in kerosene of 14.6 nm, or a micelle diameter of about 29 nm, which is surprisingly close to the Bjerrum critical diameter of 28 nm estimated by Morrison [18]. That is why these ionic species exist in the nonpolar kerosene. We can also make an estimate of the SPAN 80 apparent dissociation constant in kerosene using the SPAN 80 data at 1 wt% without particles. The molar concentration, C0, of SPAN 80 at this point is 2.3  102 mol/l because its molar weight is 428 g/mol. The conductivity of the SPAN 80 solution at this point is about 300  1010 S/m. In order to calculate the concentration of ions using these conductivity data, we must assume that the cation diffusivity is approximately equal to the anion diffusivity. This leads to the following values of the ions concentrations:

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KRT ð12:8Þ  7:5  107 mol=L F2 D which in turn leads to the following value of the SPAN 80 apparent dissociation constant Ddiss:  Cþ 0 ¼ C0 

Ddiss ¼

 Cþ 0 C0 ¼ 2:4  1011 C0

ð12:9Þ

This value of dissociation constant is equal to the value of weak acid dissociation in water. This means that SPAN surfactants behave as a weak electrolyte in nonaqueous media. To calculate the size of the cations, we can use the difference in conductivity between plain kerosene and the alumina dispersion at the same 1 wt% of the surfactant. At this concentration, all cations are deemed adsorbed on the alumina surface, so we can attribute this difference in conductivity solely to the cations. This difference in conductivity (300 pS vs.10 pS) is about 30 times larger than the residual conductivity of the dispersion that is associated with anions. This means that cations are about 30 times smaller that anions, and thus we can estimate that the size of the cation is only about 1 nm. Finally, Figure 12.5 illustrates the ions and the dispersed alumina particles with their approximate sizes as follows from the described measurements and calculations.

-

-

-

+

+

+ +

+

alumina particles

+

+

+

cations 1 nm

-

anions 30 nm

+

-

+ + -

-

+

surfactants

FIGURE 12.5 Cartoon illustrating ion structure and position in the dispersed system described in the text.

437

Applications for Ions and Molecules

12.3. PROTEIN ELECTRIC CHARGES

Electroacoustic signal (10⫻104 A/cm/Pa)

Ultrasound offers a simple way of measuring electric charges of proteins and other biomolecules. This can be done either directly or by adsorbing biomolecules on mineral particles with higher density contrast. The last method would amplify the electroacoustic signal. Such an approach was used by Rezwan and others [20, 21]. They adsorbed bovine serum albumin (BSA) on alumina nanoparticles. Electroacoustic measurements performed with the DT-300 z-potential probe allowed them to monitor compensation of the positive surface charge of alumina by negatively charged BSA molecules. They also observed shift of the alumina isoelectric point making pH titrations at different additions of BSA. These experiments allowed them to derive important conclusions regarding BSA properties and its interaction with the alumina surface. For instance, it turns out that only about 30% of the protein negative charges build bonds with positive groups on the alumina surface. This indicates that BSA retains its three-dimensional shape when being adsorbed. However, further studies indicated that the electroacoustic probe DT-300 is sufficiently sensitive for direct measurement of BSA when it is concentrated enough [22]. Figure 12.6 shows the electroacoustic signal (CVI) measured for several different volume fractions of protein. The electroacoustic signal increases with volume fraction in accordance with electroacoustic theory presented in the Chapter 5. This theory allows us to calculate the electric charge of the particles from the CVI. This surface charge should be independent of the volume fraction of particles, because it is related to the electrochemical equilibrium between protein and the bulk liquid. Figure 12.7 shows that this is indeed true. 24 20 16 12 8 4 0 1

2

3

4 5 6 Weight fraction of protein, %

7

8

FIGURE 12.6 Colloid vibration current of bovine serum albumin at various concentrations.

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Elementary charges per protein

8 7 6 5 4 3 2 1 0 1

2

3 4 5 6 Weight fraction of proteins, %

7

8

FIGURE 12.7 Electric charge of bovine serum albumin in elementary charges per particle at various concentrations.

This result agrees with independent measurements of BSA electric surface properties [23, 24]. For instance, at neutral pH the z-potential of BSA has been reported as 23 mV [24], which agrees with surface charges shown in Figure 12.7. Calculating the charge, using Equation (2.34) yields a net charge of 6, in close agreement with the value found here from the CVI, Figure 12.7. These experiments prove that electroacoustics is a suitable tool for the measurement of electric charges of proteins and other biomacromolecules. Electroacoustic measurement of protein charges has a major advantage from the sample handling viewpoint. This method does not apply an electric field to the protein solution, which is known as one of the drawbacks of microelectrophoresis. An electric field can affect the protein conformation either directly or due to heat production.

REFERENCES [1] J.O.M. Bockris, P.P.S. Saluja, Ionic solvation numbers from compressibilities and ionic vibration potentials measurements, J. Phys. Chem. 76 (1972) 2140–2151. [2] A. Passynski, Acta Physicochim. 8 (1938) 385. [3] E. Yeager, J. Bugosh, F. Hovorka, J. McCarthy, The application of ultrasonics to the study of electrolyte solutions. II. The detection of Debye effect, J. Chem. Phys. 17 (1949) 411–415. [4] E. Yeager, F. Hovorka, The application of ultrasonics to the study of electrolyte solutions. III. The effect of acoustical waves on the hydrogen electrode, J. Chem. Phys. 17 (1949) 416–417. [5] E. Yeager, F. Hovorka, Ultrasonic waves and electrochemistry. I. A survey of the electrochemical applications of ultrasonic waves, J. Acoust. Soc. Am. 25 (1953) 443–455.

Applications for Ions and Molecules

439

[6] E. Yeager, H. Dietrick, F.J. Hovorka, Ultrasonic waves and electrochemistry. II. Colloidal and ionic vibration potentials, J. Acoust. Soc. Am. 25 (1953) 456. [7] R. Zana, E. Yeager, Determination of ionic partial molal volumes from ionic vibration potentials, J. Phys. Chem. 70 (1966) 954–956. [8] R. Zana, E. Yeager, Quantitative studies of ultrasonic vibration potentials in polyelectrolyte solutions, J. Phys. Chem. 71 (1967) 3502–3520. [9] R. Zana, E. Yeager, Ultrasonic vibration potentials in tetraalkylammonium halide solutions, J. Phys. Chem. 71 (1967) 4241–4244. [10] R. Zana, E. Yeager, Ultrasonic vibration potentials and their use in the determination of ionic partial molal volumes, J. Phys. Chem. 71 (1967) 521–535. [11] R. Zana, E. Yeager, Ultrasonic vibration potentials, Mod. Aspects Electrochem. 14 (1982) 3–60. [12] J.O.M. Bockris, P.P.S. Saluja, Reply to the comments of Yager and Zana on the paper “Ionic solvation numbers from compressibility and ionic vibration potential measurements”, J. Phys. Chem. 79 (1975) 230–233. [13] A.S. Dukhin, P.J. Goetz, Ultrasound for Characterizing Colloids, Elsevier, The Netherlands, 2002. [14] E. Muller, A. Faude, Investigation of salt properties with electro-acoustic measurements and their effect on dynamic binding capacity in hydrophobic interaction chromatography, J. Chromatogr. A 1177 (2007) 215–225. [15] H. Ohshima, Colloid vibration potential and ion vibration potential in dilute suspensions of spherical colloidal particles, Langmuir 21 (2005) 12100–12108. [16] A.S. Dukhin, P.J. Goetz, How non-ionic “electrically neutral” surfactants enhance electrical conductivity and ion stability in non-polar liquids, J. Electroanal. Chem. 588 (2006) 44–50. [17] A. Kitahara, Nonaqueous systems, in: A. Kitahara, A. Watanabe (Eds.), Electrical Phenomena at Interfaces, Marcel Decker, New York, 1984. [18] I.D. Morrison, Electrical charges in non-aqueous media, Colloids Surf. A. 71 (1993) 1–37. [19] J. Bockris, A.K. Reddy, Modern Electrochemistry, vol. 1, A Plenum/Rosetta ed., 1977. [20] K. Rezwan, L. Meier, M. Rezwan, J. Varos, M. Textor, L. Gauckler, Bovine serum albumin adsorption onto colloidal alumina particles: a new model based on zeta potential and UV-Vis measurements, Langmuir 20 (2004) 10055–10061. [21] K. Rezwan, L.P. Meier, L.J. Gauckler, Lysozyme and bovine serum albumin adsorption on uncoated silica and AlOOh-coated silica particles: influence of positively and negatively charged oxide surface coatings, Biomaterials 26 (2005) 4351–4357. [22] A.S. Dukhin, P.J. Goetz, T.G.M. van de Ven, Ultrasonic characterization of proteins and blood cells, Colloids Surf. B 52 (2006) 121–126. [23] M.K. Manon, A.L. Zydney, Measurement of protein charge and ion binding using capillary electrophoresis, Anal. Chem. 70 (1998) 1581–1584. [24] J.S. Kang, J.K. Shim, H. Huh, Y.M. Lee, Colloidal adsorption of bovine serum albumen on porous polypropylene-g-poly(2hydroxyethyl methacrylate) membrane, Langmuir 17 (2001) 4352–4359.

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Chapter 13

Applications for Porous Bodies 13.1. Streaming Current and Streaming Potential 13.2. Instrument Setup 13.3. Deposits of Solid Particles

445 448 449

13.4. Deposits of “Controlled Pore Glass (CPG)” Samples 458 13.5. Geological Cores 460 References 463

This chapter has been composed from the materials of the patent “Method for determining Porosity, Pore size and Zeta Potential of Porous Bodies” submitted by Dukhin, Goetz, and Thommes to the US Patent Office in October 2009. This chapter deals with a particular kind of heterogeneous system, which can be described as a porous body consisting of a continuous solid matrix with embedded pores that can be filled with either gas or liquid. According to Lowell et al. [1], the spatial distribution between the solid matrix and pores can be characterized in terms of porosity and pore size. According to Frenkel [2] and Biot [3, 4], the mechanical properties of such systems with respect to any applied oscillating stress depend primarily on the viscoelastic properties of the matrix. Lyklema notes [5] that when such porous bodies are saturated with liquid, additional properties are then related to any surface charge on these pores, which in turn is commonly characterized by a z-potential. Although methods exist for characterizing these mechanical and electrical properties, they all have limitations and call out for improvement. For example, Lowell et al. [1] describe in detail several methods for characterizing porosity and pore size. According to International Union of Pure and Applied Chemistry (IUPAC), pores are classified into three groups: micropores, pore size <2 nm; mesopores, pore size is between 2 and 50 nm; macropores, pore size >50 nm. Gas adsorption techniques are applied for micro- and mesopore analysis. Mercury porosimetry has been the standard technique for macropore analysis. Environmental concerns justify a search for alternative methods that might eliminate, or at least minimize, the use of this dangerous material. Lyklema [5] and a JUPAC Report [6] describe a method for measuring the z-potential of porous bodies using the method of streaming current/potential. However, this method is not applicable for characterizing z-potential of small Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23013-2 Copyright # 2010, Elsevier B.V. All rights reserved.

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pores because of the low hydrodynamic permeability of these pores, and furthermore it is not suitable for simultaneously determining porosity and pore size. There have been several attempts at developing ultrasound methods for characterizing porous bodies. A sound wave undergoes change as it propagates through a saturated porous body and, in the process, generates a host of secondary effects that can then be used for characterizing the properties of these bodies. To date, most attempts are associated with the measurement of sound speed and attenuation, the two main characteristics of ultrasound waves propagating through a viscoelastic medium. These two parameters are easily measurable and, in principle, can serve as a source of information for calculating porosity and pore size. US Patent No. 6,684,701 issued February 3, 2004, to Dubois et al. [7] describes a method for extracting porosity by comparing the measured attenuation spectra with that of predetermined standards. US Patent No. 6,745,628, issued June 8, 2004, to Wunderer [8] claims to measure porosity based on transmission measurements of ultrasonic waves in air, which might be possible only for very large pores comparable to the sound wavelength, which for the proposed low frequency is perhaps several millimeters. Yet another US Patent No. 7,353,709, issued April 8, 2008, to Kruger et al. [9] suggests some improvements in this method, but still relies on comparison with attenuation standards to extract the porosity information from the raw data. There are also several patents describing the use of ultrasound for characterizing the porous structure of bone. One example is US Patent No. 6,899,680, issued May 31, 2005, to Hoff et al. [10] for estimating the shear wave velocity, but not attenuation. There are also two patents that utilize differences in sound speed between different propagation modes. The first is US Patent No. 5,804,727, issued Sept 8, 1998, to Lu et al. [11], which simply states that a person skilled in the art would recognize that velocities of different modes could be used for determining the physical properties of materials. The second, US Patent No. 6,959,602, issued Nov 1, 2005, to Peterson et al. [12] suggests that, based on a prediction by Biot, one might use the velocity of fast compression waves for calculating porosity and slow compression waves for detecting body defects. However, analysis of the Biot theory raises many concerns about the efficacy of using ultrasound attenuation and sound speed for characterizing porous bodies. Biot in 1956, crediting the earlier work by Frenkel, developed a well-known general theory of sound propagation through wet, porous bodies by including the following set of 11 physical properties to describe the solid matrix and liquid: 1. 2. 3. 4. 5. 6. 7.

Density of sediment grains Bulk modulus of grains Density of pore fluid Bulk modulus of pore fluid Viscosity of pore fluid Porosity Pore size parameter

Applications for Porous Bodies

8. 9. 10. 11.

Dynamic Structure Complex Complex

443

permeability factor shear modulus of frame bulk modulus of frame

Ogushwitz [13] recognized that the last four of these properties present a big problem in applying Biot’s theory and proposed several empirical and semiempirical methods for estimating their value, but none of his suggestions are sufficiently general, and in some cases simply amount to a substitution of one property with another unknown constant. Barret-Gultepe et al. [14] also discuss this problem in their study of the compressibility of colloids, in which they speak of the importance of a “skeleton effect” and the difficulty of measuring the required input parameters independently. This problem of unknown input parameters makes us skeptical of determining porosity and pore sizes from attenuation and sound speed. Furthermore, relying on attenuation and sound speed alone does not yield any information on the electric properties of porous bodies. However, the electric properties can be determined using ultrasound since sound generates electric signals as the sound wave propagates through the porous body by disturbing the electric double layer (DL) surrounding the pore surfaces. This effect is usually called the “seismoelectric phenomenon” because it was first employed in the field of geological exploration. Ivanov first discovered the effect in 1940 [15] and Frenkel developed the first relevant theory in 1944 [2], see Section 5.7. Independently, Williams in 1948 discovered the same effect [16] and later, in 2007, Dukhin described this effect in a dispersion of structured carbon nanotubes [17]. Each used different names for essentially the same effect. Apparently, Muller et al. also observed this effect in chromatographic resins [18], but mistakenly interpreted it as a colloid vibration current (CVI), see Chapter 5. There is a host of papers on the observation of this seismoelectric effect in geology including those of Markov in 2004 [19], Zhu et al. in 2007 [20] and 1999 [21], Pride et al. in 1994 [22] and 1996 [23], Haarsten et al. in 1997 [24], Mikhailov et al. in 2000 [25], and Thomson et al. in 1993 [26]. This high-frequency effect is very similar in nature to the streaming current/potential, which is also the result of coupling between mechanical and electric fields. However, there is one large difference between seismoelectric current and streaming current. Streaming current is typically measured under steady state or very low-frequency excitation. Consequently, it is an isochoric effect, which means that the liquid and the solid matrices are considered incompressible. In contrast, the seismoelectric effect is nonisochoric by its very nature. A relative motion of phases in pores occurs as a result of a difference in compressibility between the liquid and solid. This difference justifies using a new, separate name for this effect. Seismoelectric and reverse electroseismic effects belong to a family of electroacoustic phenomena, which was described in some detail in Chapter 5. Each of these phenomena is associated

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with coupling between electric and ultrasound fields in heterogeneous systems. Nevertheless, there are many similarities between these effects, which is why an overview of streaming current/potential features is an important part of this discussion. This short overview is given in the next section. Frenkel predicted that the electroseismic electric field strength is proportional to porosity and independent of pore size because of the Darcy constant dependence on the square of the pore size which cancels out size dependence in Equation (5.96). There is no conclusive experimental confirmation of this prediction so far. This conclusion is valid only for isolated and thin DLs, which corresponds to the electric potential distribution in pores as illustrated by curve 1 in Figure 13.1. It is quite possible that the electroseismic effect for the “homogeneous” completely overlapped DL, case (2) in Figure 13.1, would become dependent on pore size. This would open the possibility to characterize both pore size and porosity using two different liquids for saturating the porous body. Unfortunately, there is no theory valid for the homogeneous case yet. Frenkel’s equation does not present electroacoustic field strength as a function of the pressure gradient, in contrast with modern electroacoustic theories. It also ignores the phase shift in the electroseismic signal that occurs at ultrasonic frequencies. Existing theoretical developments in this field, instead of following the lines drawn by Frenkel 60 years ago, have shifted

2

3

1/κ

1

1 Φ=0 R

FIGURE 13.1 Possible distributions of electric potential inside of the pores, where case (1) corresponds to thin isolated double layers, case (2) to homogeneous model of completely overlapped double layers, and case (3) to partially overlapped double layers.

Applications for Porous Bodies

445

emphasis to the structure of the acoustic field in the soil. However, they ignore a basic principle underlying all electrokinetic theories at the Smoluchowski limit regarding the similarity of the space distribution of hydrodynamic and electrodynamic fields in electrokinetics. Despite the lack of adequate theories, the electroseismic effect is very promising for characterizing porous bodies since it offers four important advantages: 1. The ability to characterize very narrow pores in bodies with very low permeability, which complicates or even prevents pumping liquid in continuous flow mode. 2. The elimination of concentration polarization. 3. The possibility to characterize porosity and pore size using two different wetting liquids. 4. The simultaneous characterization of electric properties of the pores surfaces. US Patent No. 7,340,348, issued March 2008, to Strack et al. [27] discusses the acquisition and interpretation of seismoelectric and electroseismic data, but is dedicated to empirical characterization of geological structures and makes no claims about the properties of pores. Rather than using the seismoelectric effect on a geologic scale, we propose using it on a smaller laboratory scale, which we can achieve using a commercially available electroacoustic device described in US Patent No. 6,449,563, issued September 2002 to Dukhin and Goetz [28] and described in Chapter 7. This instrument launches ultrasound pulses into a heterogeneous system and measures the electric response. According to said patent, the claimed use of the device was for characterizing particle size and z-potential of dispersions and emulsions. In contrast, we propose using essentially the same device for characterizing the porosity, pore size, and z-potential of porous bodies, employing some modifications in sample handling and calibration. This porosity and pore size characterization is possible because the measured electroacoustic signal generated by ultrasound in porous bodies is essentially seismoelectric current generated on the scale of the laboratory device. We test these ideas here with a number of model porous bodies. This includes deposit of solid particles, deposit of porous particles, and geological cores.

13.1. STREAMING CURRENT AND STREAMING POTENTIAL It is well known that the flow of liquid through a porous body generates an electric current or potential, depending on the method of measurement. This electric response occurs because of the motion of the ionic diffuse layers that screen the electric surface charge covering the pores. This motion would appear with an applied mechanical driving force at any frequency. However,

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13

the mechanism is quite different for a constant or low-frequency driving force, as compared with high-frequency excitation at several megahertz or higher. This difference in applied frequency provides some justification for using two different terms for essentially the same electromechanical effect. Historically, the first experimental observation of this coupling in porous bodies was made for a constant applied driving force. In the field of colloid and interface science, this effect is known as streaming current or streaming potential. It is usually assumed that the gradient of the applied pressure is constant and time independent, and that the liquid is incompressible, which makes this an isochoric phenomenon. Kruyt in 1952 [29], Dukhin et al. in 1974 [30], and Lyklema in 2000 [5] discuss many experimental and theoretical studies of this effect. Streaming current/potential depends strongly on the distribution of the electric potential inside the pores, F. Figure 13.1 illustrates possible space distributions of this potential including two extreme cases: (1) an isolated thin DL and (2) homogeneous, completely overlapped DLs. There are analytical theories that describe the main features of this electrokinetic effect for these two extreme cases. Smoluchowski in 1903 [31] was the first to develop the well-known theory for streaming current/potential for the case of isolated thin DLs. This theory yields the following expression for the electric potential difference DV generated by the pressure difference DP: DV ¼

ee0 z DP Km

ð13:1Þ

where e and e0 are the dielectric permittivity of the liquid and vacuum, z is the electrokinetic potential of the pore surface,  is the dynamic viscosity of the liquid, and Km is the conductivity of the liquid. This theory is valid when the capillary radius, R, is much larger than the DL thickness, k1, that is, kR  1

ð13:2Þ

as illustrated by curve 1 in Figure 13.1. It is also assumed that surface conductivity associated with excess ions in the DL is negligible. According to the Smoluchowski theory, this effect offers little hope for studying porosity and pore size, since these parameters are simply absent in Equation (13.1). This is an unfortunate result of the geometric similarity between the hydrodynamic and electric fields in the pores under conditions where Smoluchowski theory is valid. The introduction of surface conductivity would introduce some dependence on pore size, but even this is uncertain because of the unknown values for the surface conductivity. Decreasing the pore size leads eventually to an overlap of the DLs, which is reflected as a transition from distribution 1 to distributions 3 in Figure 13.1.

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Applications for Porous Bodies

Further decreasing of the pore size would cause complete overlap and the electric potential in the pore becomes constant. This is the case of the “thick” and “homogeneous” DL, when: kR  1

ð13:3Þ

It is curious that the theory corresponding to this case of complete overlap was developed not in field of general colloid science but by scientists dealing with membrane phenomena, such as reverse osmosis and hyperfiltration. The reason for this is that condition (Equation 13.3) is valid in water only for very small pore size, which of course is the case for reverse osmosis membranes. This theory is made quite complex due to “concentration polarization,” a phenomenon that occurs in sufficiently charged membranes when the number of counterions substantially exceeds the number of co-ions in the pores. This concentration polarization leads to a separation of ions at the pore entrance, which in turn generates a concentration gradient in front of the pore. Tanny and Hoffer in 1973 [32] developed a theory of streaming current/potential for the homogeneous case that takes into account this concentration polarization. We present here just the final expression to illustrate the complexity of this theory: 00

00

00

00

00

0 c~ þ X 1 c~s 2 þ c~s ðX  c~s fw Þ  c~s t1 Xfw FE Jv f1w Dx þ ln s0 ¼  ln 0 0 00 RT yfw c~s þ X 2 c~s2 þ c~s ðX  c00s fw Þ  c~s t1 Xfw n 00 on 0 o ð13:4Þ 00 00 00 ~ ~ 2~ c þ X  c f  C 2~ c þ X  c f þ C w w s s s s X þ c~s fw ð1  4t2 Þ  0  ln  00 þ 00 00 2C cs þ X  c~s fw  C 2~ cs þ X  c~s fw þ C 2~

where E is the electromotive force of streaming effect, R is the gas constant, T is the absolute temperature, F is the Faraday constant, X is the effective charge density, Jv is the volume flow, fij is the friction factor between species i and j, y is the tortuosity coefficient, ’w is the water fraction of water in the membrane, Dx is the thickness of the membrane, and cs is the local salt concentration. The Tanny-Hoffer theory clearly indicates the complexity of the concentration polarization phenomenon. In order to use this theory for characterizing pores, it will be necessary to find experimental conditions where this concentration polarization do not develop. Practically speaking, the only available solution is to use an alternating driving force instead of a constant excitation, as is typical in traditional streaming current/potential measurements. To this end, there are several theoretical and experimental studies of the streaming current under oscillating pressure conditions. Packard in 1953 [33], Cooke in 1955 [34], Groves et al. in 1975 [35], Dukhin et al. in 1983 [36] and 1984 [37], and Renaud et al. in 2004 [38], as well as US Patent No. 3,917,451, issued in 1975 to Groves and Kaplan [39] all speak of measurements under oscillatory conditions. Their theoretical analysis indicates the appearance of an additional multiplier, which

448

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13

depends on Bessel functions of kR, but no additional dependence on porosity. Essentially, all these studies assume isochoric conditions and consequently are not directly applicable for describing the electrokinetic effect when isochoric conditions are not valid. This assumption was justified in the above references because rather low frequencies of excitation were applied. However, the use of low-frequency excitation in the kilohertz range does not prevent the formation of concentration polarization. Instead, the frequency must be in the megahertz range for complete elimination of this complex effect. However, isochoric condition does not hold at such high frequency and, instead of streaming current, we should deal with seismoelectric phenomena.

13.2. INSTRUMENT SETUP The proposed instrument for measuring seismoelectric current employs an electroacoustic spectrometer for generating and sensing this effect. There is one embodiment of such an electroacoustic device presented in US Patent No 6,449,563, issued September 2002 to Dukhin and Goetz [28] and commercialized by Dispersion Technology as model DT-300. We use this device for experimental verification of the suggested method. We add a novel sample handling system and novel calibration procedure which allow using this device for characterizing porous bodies rather than dispersions and emulsions, as was intended in the said patent. The novel sample handling system is shown in Figure 13.2. The electroacoustic probe is placed Sample

Plastic cup Ultrasound pressure wave

Deposit Hmax

Zeta probe

FIGURE 13.2 Electroacoustic probe with the novel sample handling system that allows measurement of particulate deposits and porous bodies.

Applications for Porous Bodies

449

vertically in a suitable stand that orients it such that the face of the probe with the gold electrode is on top. A cylindrical fixture around the top of the probe creates a cup, with the probe face serving as a bottom of the cup. This cup can be filled with liquid, and a porous body can be placed in this liquid in contact with the gold electrode. Ultrasound pulses generated by the electroacoustic probe enter the liquid phase through the gold electrode, whereupon they enter the porous body which is placed on top of the gold electrode. The sound wave generates a seismoelectric current as it propagates through the porous body and is sensed as an alternating current between the gold electrode and its surrounding stainless steel shell. The electronics measures and processes these current pulses in a manner similar to the electroacoustic pulses generated by the dispersed particles as described in the said patent.

13.3. DEPOSITS OF SOLID PARTICLES We want to provide proof that this electroacoustic signal is indeed a seismoelectric current. A deposit of solid particles can serve as a model porous body. This approach has the advantage due to the ability of characterizing interfaces when the particles are still well dispersed in the liquid. Properties of the solid-liquid interface must remain the same in the deposit. This provides an additional important constraint on treating experimental data. To achieve this goal, we orient the electroacoustic probe vertically, as shown in Figure 13.2, and allow the particles to build a deposit on the face of the gold sensing electrode. The sound pulses generated by the probe propagate directly into the deposit and should generate a seismoelectric current which would be measured as an electroacoustic signal. We should expect some deviation in the electric field line pattern compared to that obtained during z-potential calibration using a stable dispersion. It turns out that the measurement of a deposit offers a novel calibration procedure which automatically corrects for this deviation. We used three different solid material particles: silica Ludox, silica Geltech 0.5, and alumina AKP-30. Each was prepared at a solid concentration of 10% by weight dispersed in distilled water. The pH, as given in Table 13.1, was adjusted in each case to provide good stability. The conductivity and z-potential of each sample was measured with the DT-300 instrument before any sedimentation occurred. The particle size distribution of the same dispersions was measured with a DT-1200 acoustic sensor manufactured by Dispersion Technology, the results of which are also given in Table 13.1 and shown as PSD curves in Figure 13.3. Figure 13.4 shows the time trend of the CVI signal for five 10 wt% Ludox samples, each successive sample containing increased amounts of Geltec from 0% to 1%. The electroacoustic signal of the unadulterated Ludox sample does not change with time, which is expected since these very small 30-nm particles do not sediment appreciably during the period of measurement.

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TABLE 13.1 Properties of the Dispersions that Form Deposits Material pH Conductivity Debye Zeta D50 (S/m) length potential (mm) (nm) (mV)

D10 (mm)

D90 (mm)

Density (g/cm3)

Ludox TM

9.3

0.22

2.4

38

0.030 0.020 0.045

2.2

Geltech 0.5

8.4

0.078

4.1

59

0.572 0.555 0.590

2.2

Alumina AKP-30

4.6

0.032

6.5

þ56

0.286 0.155 0.530

3.9

Particle size distribution, volume basis

40

Geltech 30

20

10 Ludox 0 0.01

AKP-30 0.10 Diameter (mm)

1.00

FIGURE 13.3 Particle size distributions of the silica and alumina dispersions measured with DT-1200 by Dispersion Technology Inc.

However, the added small amount of the larger silica Geltech particles causes a time rate of change proportional to the amount of the added large particles due to the accumulation of these particles near the probe surface. We see that the measurement is very sensitive to even small amounts of the larger particles, as little as 0.1%. This behavior confirms that the measured signal comes from the area near the probe surface. Figure 13.5 includes the data from Figure 13.4, but on a larger scale, and shows data for even higher concentrations of Geltec at 3% and 5%. Also shown is a 10 wt% Geltec dispersed simply in water.

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Applications for Porous Bodies

Ludox Ludox Ludox Ludox Ludox

5.4⫻106

CVI magnitude (mV (s/g)1/2)

5.2⫻106 5.0⫻106

only + 0.1% Geltech + 0.3% Geltech + 0.5% Geltech + 1 % Geltech

4.8⫻106 4.6⫻106 4.4⫻106 4.2⫻106 4.0⫻106 3.8⫻106 3.6⫻106 3.4⫻106 0

50

100

150

200

250

Time (min) FIGURE 13.4 Kinetics of the electroacoustic magnitude evolution due to deposit of silica Geltech particles added in small amounts to silica Ludox.

9⫻107 Geltech only at 6.6 MHz

CVI magnitude (mV (s/g)1/2)

8⫻107

5 wt%

Geltech only 10 wt%

3 wt%

7⫻107 6⫻107 5⫻107 4⫻107 3⫻107 2⫻107

1 wt% 0.3 wt%

1⫻107

0.1 wt%

0 0

50

100

150

200

250

300

350

400

450

500

Time (min) FIGURE 13.5 Kinetics of the electroacoustic magnitude evolution due to deposit of silica Geltech particles added in substantial amounts to silica Ludox.

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13

These experiments demonstrate that the electroacoustic signal in each case reaches some saturation level at a certain point in time. The question arises as to the reason for such time dependence. Why does the measured signal first reach a maximum and then decrease somewhat? The gradient of pressure determines the direction of the particle motion within the sediment and in turn determines the magnitude and phase of the electroacoustic signal. At the start of the sedimentation experiment, the particles have settled only slightly and all particles experience the same gradient of pressure and move in the same direction thereby contributing signal in the same phase. When the sediment reaches a depth equal to one-half the wavelength, the magnitude of the signal reaches a maximum value since the peak signal occurs at an instant for which all the particles within this depth move in the same direction. However, when the depth of the sediment becomes slightly larger than half a wavelength, the peak signal occurs at an instant of time when the particles in the top layer are moving in the opposite direction to that of particles below this depth and thereby subtract from the measured signal. Therefore, the signal reaches maximum when particles have finished filling this half-wavelength layer and then begin decay when the second half wavelength layer starts filling up. However, particles in the second layer are further away from the surface than in the first layer and therefore the sound is weaker there due to attenuation and particles of the second half-wavelength layer can never completely compensate the contribution of the first layer. Eventually, after filling several half wavelengths, the signal comes to some steady-state level. Let us denote this critical sediment thickness at which the sediment fills to a depth of half wavelength by: 1 H¼ l 2

ð13:5Þ

There several ways to verify this hypothesis of observed correlation between the measured time trend in the electroacoustic signal and sedimentation. The DT-300 typically performs electroacoustic measurements at a frequency of 3.3 MHz, corresponding to a wavelength in water of about 500 mm. Accordingly, the thickness of the sediment layer in which all particles move in the same direction due to same sign of the pressure gradient is about 250 mm. Increasing the frequency twofold would reduce thickness of the first half-wavelength layer from 250 to 125 mm. This thinner layer should be filled twice as fast at 6.6 MHz as at 3.3 MHz. Figure 11.6 compares the time evolution for an identical 10 wt% Geltech sample measured at 3.3 and 6.6 MHz. The time to reach the maximum decreases from 178 to 58 min, somewhat more than the expected two times decrease, but more or less confirming our hypothesis. (We think that the somewhat faster than expected time to reach the maximum at the higher frequency occurs because the fewer

453

Applications for Porous Bodies

particles in the this thinner sediment layer produce a reduced pressure in the deposit leading to a correspondingly smaller packing density and hence require less time to fill the layer.) By the same logic, if we reduce the concentration of Geltec particles by two times, from 10% to 5%, we would expect the critical time required to reach a maximum CVI signal to increase by two times. Figure 13.5 shows that this is indeed the case, as the time to reach a maximum increases from 178 to 395 min. Figure 13.6 shows that a 10 wt% alumina dispersion reaches a maximum even more slowly than the silica Geltech sample due to smaller particle size. The critical time for the Geltec sample is about 6 h. We can test our hypothesis that the signal arises primarily from particles in the deposit on the probe surface by changing the z-potential of the particles. We can achieve this by varying the pH in the solution above this alumina deposit. Alumina is a good choice for this test because it has a well-known isoelectric point at pH ¼ 9. The supernate solution above the deposit initially had a pH of 4.6. At this point, we injected a small amount of 1 M KOH into the supernate which increased the pH to 11. The sample was not mixed in order to keep deposit intact. We relied only on ion diffusion to change the pH in the deposit. Figure 13.7 shows the continuing values of the magnitude and phase of the electroacoustic signal. It is seen that the

Silica Geltech 05 at 10 wt%, 6.6 MHz

8⫻107

Silica Geltech 05 at 10 wt%, 3.3 MHz

CVI magnitude (mV (s/g)1/2)

7⫻107 6⫻107

Alumina AKP-30 at 10 wt%

5⫻107 4⫻107 3⫻107 2⫻107 1⫻107 0 0

50

100

150

200

250

300

350

400

450

500

Time (min) FIGURE 13.6 Kinetics of the electroacoustic magnitude evolution due to deposit of silica Geltech particles and alumina AKP-30 at 10 wt%.

454

Magnitude (mV (s/g)1/2)

CHAPTER

5⫻107

13

isoelectric point

4⫻107 3⫻107 2⫻107 1⫻107 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 360 Time (min) 320 280 240

Phase (degree)

0

200 160 FIGURE 13.7 Kinetics of the electroacoustic magnitude and phase evolution due to re-charging of alumina particles in the deposit due to reduction of pH.

magnitude exhibits an immediate and rapid decay, which reflects a decrease of the z-potential in the deposit due to increasing pH. However, the phase begins to change only after about 4 h. This is an indication that some particles, apparently on the top of the deposit, begin to recharge. It is known from general electroacoustic theory that the most pronounced characteristics of the isoelectric point is a 180 change in the phase of the measured electroacoustic signal. We observe this phase shift here, but spread out over 3 h. This gradual shift in phase agrees with our assumption of slow ion diffusion through the deposit and a slow recharge from the top to the bottom of the sediment. Having confirmed our hypothesis concerning the interpretation of these sedimentation trends, we can now apply these data for calculating the properties of the deposit. Knowing the critical thickness, H, the time, tcr, to reach the maximum signal, and the original volume fraction of the dispersion, ’, allows us to estimate the volume fraction, ’sed. In other words, we can calculate porosity of the deposit using the measured kinetic curve. Let us assume that we have dispersion of spherical particles with an initial homogeneous volume fraction, ’, and density, rp, in a Newtonian liquid with density, rm, and dynamic viscosity . The particle size distribution is assumed lognormal and denoted as P(ai), where ai is radius of particles in the fraction i with a fractional volume fraction ’I ¼ ’P(ai).

455

Applications for Porous Bodies

These particles are settling, under the influence of a specific acceleration g, at a fractional sedimentation velocity Vi given by: Vi ¼

2ðrp  rm Þga2i 9

ð13:6Þ

These settling particles build up a deposit on the face of the probe. It takes time tcr to fill the first half wavelength layer H. Having this time from experiment, we can calculate how many particles have reached surface of the probe due to sedimentation. We should take into account that each fraction of particles contributes individually and presumably independently to this deposit-building process based on the sedimentation velocity of its respective volume fraction. This gives us the following equation for the volume fraction of particles in the deposit: X Vi fi 2gðrp  rm Þf X 2 fX i Vi Pðai Þ ¼ tcr ai Pðai Þ ð13:7Þ ¼ tcr fsed ¼ tcr H H i 9H i We can apply this equation to calculate the volume fraction of the silica and alumina deposits. In the case of silica Geltech, the critical time tcr equals 178 min and the average square of the particle diameter in micrometers is 0.325. This leads to the following value for the volume fraction of the silica in the deposit: fsed ¼ tcr ½ sec

2  9:8  1:2  0:048 X 2 ai Pðai Þ½mm2  ¼ 0:37 9  250  4 i

ð13:8Þ

In the case of alumina AKP-30, the critical time tcr equals 240 min and the average square of particle diameter in micrometers equals 0.152, which yields following value for the volume fraction: fsed ¼ tcr ½ sec

2  9:8  3  0:048 X 2 ai Pðai Þ½mm2  ¼ 0:66 9  250  4 i

ð13:9Þ

These results correlate well with the measured particle size distributions of the silica and alumina materials. Alumina is much more polydisperse, which in principle allows denser packing of particles in the deposit. It is important to stress here that we have not yet used any electroacoustic theory, just the obvious mass balance and the assumption of Stokes sedimentation of individual particles. We can test the applicability of the existing electroacoustic theory that was developed for dispersions using the maximum value of the electroacoustic signal. This theory assumes that particles move relative to the liquid as a result of the density contrast. This electroacoustic effect is called CVI and described in detail in Chapter 5. There is a well-verified theory of CVI for concentrated

456

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13

dispersion, which yields a simple equation for CVI at the Smoluchowski limit for sufficiently small particles: CVIo!0 em e0 zfKs ðrp  rs Þ ¼ AðoÞ rs rP Km

ð13:10Þ

where ’ is volume fraction of the dispersed phase, ’ ¼ 1  Oe, Oe is porosity, and rp and rs are density of the particles and of the dispersion: rs ¼ rp f þ rm ð1  fÞ The parameters Ks and Km are the conductivity of the dispersion and medium, respectively. According to Maxwell-Wagner theory, their ratio at high frequency and with negligible surface conductivity equals: Ks 1f ¼ Km 1 þ 0:5f

ð13:11Þ

The parameter A is a calibration constant determined with calibration procedure using silica Ludox. If we assume that the measured signal generated by the deposit is CVI, then we can use the volume fraction of particles in the deposit as calculated above to estimate the z-potential. If our assumptions are valid, then this estimate of the z-potential would be similar to that measured for the dispersion of the same material as was presented in Table 13.1. However, it turns out that the z-potential value calculated using these data with traditional theory is an unbelievable 244 mV for silica Geltech and an almost equally unbelievable value of 151 mV for alumina AKP-30. These unreasonable z-potential values prove that the traditional theory does not describe generation of the electroacoustic signal in the deposit and that the applicable phenomenon is not CVI. This is hardly surprising at this point. We have already stated that the mechanism of electroacoustic coupling in deposits and porous bodies is completely different from that of dispersions. The propagating ultrasound wave expands and contracts the deposit. Particles in the deposit are pushed together by gravity, but because of the high z-potential they do not aggregate. Rather, they build a flexible network that moves relative to the liquid, in the process displacing ions in the DL and generating a streaming current. This occurs in a nonisochoric mode, and as discussed above, the alternative term for this phenomenon is a seismoelectric effect. There is one more important fact confirming this hypothesis. The phase of the CVI signal is shifted relative to the ultrasound phase because of inertial effects. This phase shift must exist even for submicrometer particles used in this work if the generated, measured signal is CVI indeed. However, we do not observe such phase shift. For positive alumina, the phase is 360 within 1 . For negative silica and recharged alumina, it is

457

Applications for Porous Bodies

exactly 180 within 1 . This indicates that particles size does not play any role. Relative motion of solid and liquid phases occurs in phase with ultrasound. This correlates with the suggested notion of the weak, very flexible network in the deposit. We can now apply the theory derived in Section 5.7 for calculating the z-potential. In addition, the seismoelectric current must be independent of the density of the particles, which distinguishes it from the CVI. This means that instead of Equation (13.10) we can attempt using the following expression for calculating the z-potential from the measured magnitude of seismoelectric current. It was derived in Section 5.7. We will use the expression for high compressibility modulus because the sediment of these highly charged particles clearly is not very rigid. Thus, we would use Equation (5.112):   e0 em z rm Ks Isee ¼ 1 rP rs K m  Comparing this equation with expression for CVI, we can conclude that the z-potential calculated from seismoelectric current (zsee) relates to the z-potential calculated from CVI (zcvi) as follows: zsee ¼ zcvi ð1  fÞ

ð13:12Þ

assuming that ’ corresponds to the deposit and zcvi is calculated for the deposit with this same volume fraction and inertial particle size effect is negligible by setting a small size. The calculated values of z-potentials for silica and alumina deposits are shown in Table 13.2. It is seen that for both materials these new values are much closer to the originally determined z-potentials for stable dispersions using the CVI measurement. The observed deviation could be accounted for by the neglected surface conductivity. Here we can conclude that this experiment with deposit provides a definite confirmation of the simple existing theory of the seismoelectric current.

TABLE 13.2 Properties of the Dispersions that Form Deposits Real zeta potential (mV)

Density (g/cm3)

Volume fraction

zcvi

zsee

Geltech 0.5

59

2.2

0.37

106

67

Alumina AKP-30

þ56

3.9

0.66

137.3

47

Material

458

CHAPTER

13

13.4. DEPOSITS OF “CONTROLLED PORE GLASS (CPG)” SAMPLES As a next step, we can use large porous particles instead of solid particles for building the deposit. We selected some well-characterized silica-based porous particles manufactured for chromatography use. We used a set of five CPG powders provided by Quantachrome Corporation, each having the same porosity but a different pore size. The particle size of such material usually exceeds 50 mm, so they build up a deposit very rapidly. These materials allow us to verify the prediction of Frenkel’s theory that the magnitude of the seismoelectric signal is independent of the pore size for thin and isolated DLs. In order to perform such a test, we should saturate the pores with a liquid having an appropriate ionic strength. The double layer would be completely overlapped at low ionic strength and for this case we should expect a dependence on the pore size. In contrast, for a high ionic strength liquid, the double layer would be thin compared with the pore size and the dependence of the signal with pore size should diminish. We would like to stress here that these particles are too large for using the theory of O’Brien [40] for the dynamic mobility of a porous particle. They do not move under the influence of an ultrasound wave. Instead, ultrasound penetrates into the particles. We use three different liquids: ethanol, distilled water, and a 0.11 M KCl solution. Each sample was prepared by adding 0.5 g of particles to 10 ml of the liquid. The sample was placed into the sample cup after 10 min of equilibration. The particles build up a complete deposit on the surface of the electroacoustic probe within 1 min. The thickness of this deposit exceeded 1 mm, which eliminates any electroacoustic signal dependence on the deposit height. The experiment protocol consisted of five consecutive measurements of the electroacoustic signal. Each measurement takes about 30 s for a water-based samples and about 1 min for the others. In order to test that the deposit structure is uniform, we use a pipette to resuspend the particles and then allow them to build up yet another deposit. Such redeposition of the sample causes some variation in the measured signal. Usually, this variation is very small, much smaller than the difference between samples. However, it was still larger than precision of the each five measurements set, which is on average 0.015 of 1 million units of the electroacoustic signal. Table 13.3 presents data on the pore size, pore volume, and porosity of these CPG samples as measured by mercury intrusion and extrusion experiments. The experiments were performed over a wide range for pressures starting in vacuum and continuing up to 60,000 psi (1 psi ¼ 6.895  103 MPa) using a Quantachrome Poremaster 60 instrument. The same table also presents the electroacoustic signal magnitude measured for these particles when saturated with 0.11 M KCl solution. According to Frenkel’s theory, this signal should correlate mostly with porosity and become independent of the pore size, because the high ionic

459

Applications for Porous Bodies

TABLE 13.3 Magnitude of the Electroacoustic Signal Measured for Five Different CPG Samples Saturated with 0.11 M KCl Solution Pore size (nm)

12

Pore volume (cc/g) Porosity (%)

40.6

0.478 51.3

Electroacoustic signal magnitude (106 mV  (s/g)1/2)

63.7

1.07

1.18

70.2

72.2

1.5

1.3

0.83

92.4

136

0.79 63.5

1.109 70.9

1.29

1.23

TABLE 13.4 Magnitude of the Electroacoustic Signal Measured for Four Different CPG Samples Saturated with Different Liquids Conductivity (S/m)

Debye length (nm)

Pore size (nm) 40.6

63.7

92.4

136

Electroacoustic signal magnitude (106 mV  (s/g)1/2) Ethanol

2.7  105

226

Distilled water

2.4  104

0.11 M KCl

1.3

0.064

0.22

0.37

0.48

75

2.7

3.3

4.7

5.8

1

1.5

1.3

1.29

1.23

strength makes for a thin, isolated DL. We can see that this prediction of the Frenkel’s theory is valid. Table 13.4 presents the data for the electroacoustic signal for the same particles, but instead saturated with different liquids having widely different conductivities and corresponding Debye lengths. We omit the data for the sample with the smallest 12 nm pores because the size of these pores becomes comparable with size of ions in low conducting ethanol and this might create artifacts. It is seen that decreasing the ionic strength “turns on” the electroacoustic signal dependence on pore size. Figure 13.8 presents this same electroacoustic data, but the signal is normalized such that the value for the smallest pore size of 35 nm is taken as unity. This observation does not contradict Frenkel’s theory because the DL becomes completely overlapped in ethanol. There is no existing theory that would describe seismoelectric current under these conditions. This experiment confirms that we can use a liquid with a high ionic strength for saturating porous bodies in order to eliminate pore size influence

460

CHAPTER

13

Relative value of seismoelectric current

8 7 ethanol

6 5 4 3

distilled water 2 0.1 M KCl water solution

1 0 30

40

50

60 70 Pore diameter (nm)

80

90

100

FIGURE 13.8 Values of the seisomoelectric current magnitude generated by four porous silica samples with the same porosity and different pore sizes in liquids with different ionic strengths.

and determine only the porosity. Then we can saturate the same porous body with a low-conducting liquid and get information on pore size.

13.5. GEOLOGICAL CORES We have measured several cylindrical geological cores of sandstone from different mines. They are marked according to the place of origin as Ohio SS, Berea SS, and Orchard SS. These objects are examples of a truly porous body versus sediment plugs considered above. Initially, these cores are wetted in distilled water. The composition of the covering solution changes over time due to ion exchange and the conductivity eventually increases to 0.0179 S/m, which roughly corresponds to 0.001 M and a Debye length of roughly 8.7 nm, as noted in Table 13.5. This would correspond to an isolated, thin DL condition for these relatively large pores. As the next step, we dried these cores and then wetted them in hexane. The conductivity of hexane, measured with a Dispersion Technology DT-700 instrument, is less than 1011 S/m, which corresponds to a Debye length greater than 6.5 mm. This estimate of Debye length includes correction for the increased ion size in nonpolar liquids. This would correspond to an overlap DL condition for these now relatively small particles. The cores are placed on their sides in order to expose both circular faces of these cylinders to the solution and allow their simultaneous equilibration. After the equilibration process has finished, the same equilibrium solution is used to fill the cup on the top face of the electroacoustic probe, as shown in Figure 13.2.

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Applications for Porous Bodies

TABLE 13.5 Magnitude of Electroacoustic (Seismoelectric) Current Measured for Several Sandstone Cores with Certain Pore Size and Porosity Electroacoustic signal magnitude (106 mV  (s/g)1/2) Water Km ¼ 0.018 S/m; k1 ¼ 8.7 nm

Porosity

Pore size (mm)

Hexane Km < 1011 S/m; k1 > 6.5 mm

Ohio

2.23

0.014

0.086

0.6

Berea

0.83

0.049

0.095

12.8

Orchard

0.57

0.15

0.025

0.34

In order to verify that the measured signal is generated in the core, we measured the pure liquid used for equilibration. The signal was at least 100 times lower than that subsequently measured for the core itself. We also placed a solid Teflon rod on the probe for confirmation that porosity is essential for the measurement. The signal was again more than 100 times smaller than for the core. This was confirmation that the measured electroacoustic signal comes from the core pores and is essentially seismoelectric current. After measuring just the equilibrium liquid, the core is submerged in this liquid on the face of the probe centered on the gold sensing electrode. Each signal from each core was measured continuously many times in order to confirm that the equilibration process has indeed finished. After one side was measured, the core was turned upside down and the opposite side was measured to verify the homogeneity of each core. The value for the measured electroacoustic (seismoelectric) signal magnitude are shown in Table 13.5 together with the porosity and pore sizes for these cores as measured independently by mercury intrusion and extrusion instruments as described previously. There is a good correlation between the porosity and electroacoustic magnitude for the water-saturated Ohio and Orchard cores. The ratio of the core porosity for these two materials is 3.44, which is close to the ratio of the electroacoustic signals of 3.91. This agrees with the theoretical prediction because it is the thin and isolated DL case and pore size difference should play little or no role. However, data for the Berea core does not agree with this trend as the magnitude of the electroacoustic signal is much less than for the Ohio SS core, despite very similar porosity. We think that the anomaly is explained by the nature of the hydrodynamic flow inside the pores. At high frequency in megahertz range, the hydrodynamic flow cannot completely develop if the pore size is larger than “hydrodynamic viscous depth,” which corresponds to the distance from an oscillating surface over which a shear wave decays by

462

CHAPTER

13

a factor of 1/e as it passes into the bulk of the liquid. This depth is approximately equal to 1 mm for the 3-MHz ultrasound frequency used by us. This means that the hydrodynamic field is completely developed inside of Ohio and Orchard cores, but the interior of the Berea core pores remains undisturbed. This shows up in the experiment as a lower porosity. This is why the electroacoustic signal of the Berea SS is lower than for the Ohio SS. Measurement in hexane confirms the theoretical prediction that pore size becomes important with increasing double layer thickness. The magnitude of the electroacoustic signal generated by the Berea core is several times higher than for the Ohio core, which can be explained by the much larger pore size of the Berea core, Figure 13.9. Measurement of the Orchard core is most surprising. It is several times higher than for the two other cores. There is only one explanation: The Orchard core chemistry leads to a much higher surface charge in hexane. In order to confirm this hypothesis that the surface chemistry indeed affects the measured electroacoustic (seismoelectric) signal, we ran a pH titration of an Ohio core. The core was equilibrated in a beaker containing 0.01 M KCl adjusted to different pH values. The signal was then measured by placing the core and its pH-adjusted medium in the sample cup as already described. The pore surface of the sandstone cores is largely silica, which normally exhibits a z-potential, which gradually diminishes toward zero from an initial negative value with decreasing pH. Electroacoustic signal produced by the core should similarly decay with lower pH if it is generated by interior DLs, which in fact agrees quite well with the experimental results shown in Figure 13.10. The final reversal of polarity at the lowest pH might be promoted by Fe ions coming into the solution from the slightly dissolving steel at this very acidic pH.

Lens:X 200

0.25 mm

FIGURE 13.9 Photograph of the Berea sandstone interior structure.

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Applications for Porous Bodies

Electroacoustic magnitude (mV (s/g)1/2)

2⫻105 0 0 -2⫻105

1

2

3

4

5

6

7

8

9

10

11

pH

-4⫻105 -6⫻105 -8⫻105 -1⫻106 -1⫻106

FIGURE 13.10 Dependence of the seismoelectric current magnitude on pH for sandstone Berea core.

These studies of various porous bodies are very encouraging for further developments of this new characterization technique. We have summarized our findings in the patent submitted to the USA Patent office with a title given in the reference [41].

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Subject Index

A Accuracy, 81, 228, 242–243, 265, 271, 275, 278–282, 285, 288–291, 322, 351, 403 Acoustic impedance, 28, 95–97, 128, 172, 178, 189, 262, 293–300 Acoustics, 1–2, 24, 92, 127, 188, 240, 262, 303, 344, 369, 407, 426, 445 Acoustic sensor, 180, 267–268, 270–272, 295, 338, 407, 449 Acoustic spectrometer, 9, 93, 106, 108–109, 132, 160, 176, 240, 251, 267–277, 279, 340, 348, 350, 377, 388 Acoustic spectroscopy, 65, 106–108, 116, 151, 243, 291, 312–313, 326–327, 336, 338, 340, 346, 350–351, 354, 357, 364, 374, 376, 391–393, 396 Acoustosizer, 108, 196, 370, 403, 410 Air bubbles, 171–172, 276, 339–340 Alba, 177, 226 Aldrich, 308 Alekseev, 412 Allegra, 8, 128, 140–142, 243 Alumina, 159, 243–244, 247–251, 253, 255–256, 304–310, 314–326, 336, 338, 350, 402–403, 407–409, 419, 432–434, 436–437, 449–450, 453–457 Alumina, Sumitiomo, 244 Alumina-zirconia, 320, 325 Aluminum hydroxide (ATH), 327–332 Amount of surfactant (AOT), 246, 371–373, 404 Analysis of attenuation, 283–291 Anatase, 304, 308–311, 403 Anderson, 161 Andrea, 265–267, 278 Andreae, 6 Anson, 28–29, 135, 154, 175, 246 Anthony, 349 Apfel, 262 Archimedes, 212 Area-based, 240 Area-weighted, 244 Array, 81–82, 149, 212–213, 268, 277

Asphaltines, 369 Atkinson, 136, 161 Attenuation, 5, 27, 95, 128, 198, 239, 262, 305, 344, 370, 407, 442

B Babchin, 215, 254, 370 Babick, 177 Barodiffusion, 190 BCR silica, 133–134, 243, 251 Beattie, 370 Bell Labs, 350 Berry, 161 Bessel, 165, 448 Beyer, 141 Bhardwaj, 262 Bimodal, 44, 287–290, 306, 313, 320–322, 350–351, 354, 373 Bimodality, 77, 284, 354 Biot, 39–40, 172, 174, 441–443 Bitumen-in-water, 254, 370 Block, 268–269, 392 Bohren, 78, 284 Boltzman, 32, 191 Booth, 6, 188–190, 197 Brinkman, 65, 72 Brownian, 8, 34–35, 55 Bruggeman, 65, 72 Bugosh, 6, 188, 191–192 Busby, 136, 250

C Cabot, 243–244, 288 Calculating the PSD, 286, 319, 363 Cannon, 7, 188, 254 Cant, 295–296 CAPA-700, 247 Carasso, 286 Carhart, 6, 8, 128, 140 Carnie, 190, 197 Case, 5, 22, 92, 129, 189, 242, 265, 306, 347, 370, 402, 434, 443

497

498 Ceramics, 4, 22, 129, 314–327, 403 Challis, 244, 266, 278–279 Chemical-mechanical polishing (CMP), 4, 289, 344, 349–350 Chivers, 7, 28–29, 135, 154, 175, 246 Chow, 254, 370 Chromatographic, 25, 443 Chromatographic hydrodynamic fractionation (CHDF), 244 Coagulation, 38, 55–56, 412 Colloidal dynamics, 3, 108, 188, 196, 267, 403 Colloid chemistry, 200, 305, 340 Colloid loss, 273–274 Colloid vibration current (CVI), 7, 189–205, 207–208, 210–213, 219–227, 255–258, 263, 291–298, 355, 379, 381–382, 403, 407, 409–410, 412–414, 421, 433–434, 437–438, 443, 449, 453, 455–457 Colloid vibration potential (CVP), 188, 190, 193–194, 200, 205–206, 213, 254–255, 263, 292, 370 Compressibility, 6, 10, 27, 40, 92–94, 103–104, 108, 110, 119–120, 155, 162, 168–169, 173, 228, 231–232, 262, 389–391, 394, 426–427, 443, 457 Core-shell model, 60–66, 128, 139–140, 144, 154–156, 247 Coulter, 77, 256, 283 CVI probe, 291–292, 298 CVP for monodisperse, 255

D Dahlberg, 255 Dann, 262 Darby, 115 Dashen, 174 Debye, 3, 6, 8, 10, 32, 46–49, 55, 67, 188, 214, 216–217, 348, 367, 378, 459–460 Debye-Hu¨ckel, 48 Deff, 46, 53, 71, 192, 378 Deinega, 411 Demulsifiers, 370 Denizot, 6, 188 Denko, 247 Derouet, 6, 188 Dielectric, 28–33, 50, 53, 58, 65–72, 116, 174, 193, 203, 208, 216, 219, 221, 263, 411, 430–431, 446 Dielectric spectroscopy, 65–72, 116, 174 Dielectrophoresis, 2, 58 Digital signal processor (DSP), 270

Subject Index

Dipole-dipole interaction, 74–75, 77 Dispergating, 315 Dispersion science, 23 Dispersion technology, 3, 9, 106, 108, 154, 175, 180, 188, 240, 251, 256–257, 267–279, 291, 336, 350, 357, 379, 396, 403, 410, 415, 419–420, 427, 448–450, 460 Dispersion Technology Material Database, 175 DLVO, 54–56 Dohner, 266 Droplet, 10, 38, 44, 80, 129, 161, 176, 208, 246, 262, 332, 335–336, 358, 370–386, 409 Dukhin, 7, 9, 49–50, 52, 67, 69, 72, 100, 102, 135–136, 144–145, 179, 188, 190, 199, 201, 203, 206–208, 222, 228, 232, 240, 244, 266–267, 348, 410, 412, 427–429, 443, 445–448 Dukhin-Semenikhin, 69 Dukhin-Shilov, 50, 228 DuPont, 240, 243–244, 305, 345

E ECAH, 6, 8, 128–129, 131–132, 135, 139–144, 152, 170, 244, 266 Eigen, 6, 8, 116–117, 174 El-Aasser, 370 Electric double layer, 44–45, 48–53, 443 Electric sonic amplitude (ESA), 7, 30, 188–191, 195, 197–198, 210, 213, 217, 254–255, 257, 263, 291, 370, 403, 409–410 Electroacoustics, 2, 26, 99, 130, 188, 254, 262, 305, 344, 370, 402, 426, 443 Electrochemical, 30–34, 36, 44, 54, 378, 437 Electrocoagulation, 72–77 Electrodeposition, 262 Electrodiffusion, 191 Electrodynamic, 26, 28–30, 35–36, 69–72, 74, 76, 132, 193, 198–199, 206–208, 216, 304, 445 Electrohydrodynamic, 75–76 Electrokinetics, 7, 14–15, 39, 50, 52, 57, 66–67, 70–72, 128–130, 188, 195, 198, 203, 214, 217–218, 220–221, 232–233, 296, 308, 350, 411–412, 445–446, 448 Electromagnetic, 78–79 Electromechanical, 2–3, 446

499

Subject Index

Electroneutral, 52, 200, 209, 218, 224, 226, 377, 382, 435 Electroosmosis, 50, 193 Electrophoresis, 2, 8, 11, 14, 22, 49, 53, 60, 66–70, 72–77, 189–192, 194–198, 214, 217, 220–221, 227, 254, 396 Electrorheology, 72–77 Electro-rotation, 2 Electrostriction, 193 Electro-viscosity, 2 Electro-zone, 77, 283 Emulsions, 4, 22, 129, 197, 244, 312, 369, 409, 445 Enderby, 6, 188–190, 197, 254 Enderby-Booth, 189–190, 197, 254 Ennis, 189–190, 197, 255 Epstein, 6, 8, 128, 139–140, 142 Erwin, 266 Euler, 8 Extinction, 13, 78–82, 128, 130–140, 166, 168, 172

F Fairhurst, 7, 189, 254 Faraday, 30, 435, 447 Faran, 140, 161, 250 Far-field, 52, 163–164 Federal Institute for Material Research, 403 Fine, 131, 285, 336, 403 Flocculation, 57, 303–312, 359, 369 Focke, 141 Foldy, 6, 339 Fourier, 267 Fractionation, 42, 77, 283–284 Fresnel, 5, 80, 265

G Galt, 6, 265 Gamma-radiation, 262 George, 116 Gibbs, 44 Glycerin-water, 246 Goetz, 7, 9, 100, 102, 136, 144–145, 179, 190, 222, 240, 244, 266–267, 370, 427–429, 445, 448 Gravimetric, 26, 35–36, 240 Gunnison, 255 Gushman, 266

H Halse, 262 Hamaker, 55

Hampton, 135, 144 Hamstead, 262 Happel, 6, 62–64, 71, 103, 149, 203 Harker, 7 Hartmann, 141 Hawley, 6, 8, 128, 140–142, 243 Helos, 247 Hemar, 136, 144, 154, 247 Henry, 3, 5, 67–68, 214, 256, 403 Henry-Ohshima, 214, 256 Hermans, 6, 8, 188, 254 Hertz, 95, 100 Hexadecane, 32, 244, 247, 369, 431 Hexametaphosphate, 57, 315, 358–359, 404–406, 421 Holmes, 244, 266, 278–279 Hookean, 66, 157, 159, 179, 311 Horiba, 247, 344, 346 Hovorka, 6 Huang, 370 Huffman, 78, 284 Hunter, 69, 115, 191, 254–255, 370 Huygens, 8 Hydrodynamically, 50, 152

I Ill-defined problem, 77, 133, 283–288 Incompressibility, 61 Ingard, 36, 103–104, 173 Interferometry, 264 Interparticle, 130, 135, 152, 179, 216–217, 219, 311–312 Ion vibration current, 30, 115, 188, 413, 428 Ion vibration potential, 188, 426 Irani, 40 Isaacs, 370 Isakovich, 5–6, 153–154 Isoelectric point, 10, 287, 305, 310–311, 377, 403, 437, 453–454

J Jacobin, 117 James, 255 Japan Fine Ceramics Center, 403 Johnson, 174, 411–412 Jones-Dole, 31

K Kaolin-water, 421 KCl, 31, 117, 119, 296, 308–309, 314, 346, 350, 407–409, 420, 427–428, 458–459, 462

500 Kikuta, 262, 327 Kirchhoff, 5 Kong, 370 Koplik, 174 Kosmulski, 308, 403, 409, 411–412 Kostial, 262 Kruyt, 69, 446 Kuvabara, 63, 71 Kuwabara, 6, 62–64, 203, 222 Kytomaa, 136, 161

L Lame, 141 Langmuir, 304, 357 Laplace, 5, 201, 208, 219, 221 Latex/water, 26 Latices, 22, 35–36, 136, 153, 156, 175–176, 255, 391, 396 Legendre, 141–142 Leschonski, 40, 283 Levine, 7, 71, 189, 201–202, 221, 254–255 Levine-Neale, 71 Lindner, 170–171, 251 Lloyd, 161 Loeb-Dukhin-Overbeek, 49 Loewenberg, 196 Lognormal PSD, 43, 285, 287–288, 324, 351, 354 Long-wavelength, 12–13, 61, 135, 144, 161–162, 164, 166–167, 203 Lyklema, 22–24, 32, 44–45, 52, 72, 410–412, 441, 446 Lyophilic, 54–55, 57, 430 Lyophobic, 54–55, 57

M Ma, 161 Macroscopic, 24, 66, 68, 71–72, 77, 127–129, 139–140, 142, 144, 149, 154, 156–157, 193, 201–202, 219, 221, 224, 263–264, 283–286 Macroviscosity, 31 Malvern, 3, 108, 336, 377 Marlow, 7, 9, 57, 135, 189, 211, 254–255 Martin, 262 Matec, 3, 9, 108, 188, 254, 370, 403, 409 Maxwell, 3, 5, 53, 70–72, 190, 192–193, 196, 199, 206–208, 216, 220–222, 231–233, 348, 456 Maxwell-Wagner, 53, 70–72, 190, 192–193, 196, 199, 206–208, 216, 220–222, 231–233, 348, 456

Subject Index

McCann, 157 McClements, 7, 9, 136, 144, 153, 161, 244, 247, 262, 264, 369 McLeroy, 135, 144 Mercer, 161 Microelectrophoresis, 11, 14, 195, 254–255, 370, 402–403, 438 Microemulsions, 78, 175, 246, 366, 370–374, 381–382, 391 Microviscosity, 10, 31 Midmore, 69 Mie, 80–81 Mikhailov, 139, 443 Monodisperse PSD, 44, 286 Morse, 6, 36, 92, 103–104, 132–133, 136, 141, 161–162, 165, 168, 170–171, 173, 253 Multiphase systems, 313 Multiple-scattering, 12, 79, 82, 129–130, 132, 136, 138–139, 161–162, 169–170, 251 Murtsovkin, 73–75

N National Institute of Standards and Technology (NIST), 345, 403, 415 Navier-Stokes, 61, 103–104, 109–111 Neale, 7, 71 Nepers, 95–96, 109, 170, 297 Neumann, 165 Newton, 4–5 Newtonian, 4, 10, 15, 27–28, 66, 92, 94, 102–114, 454 Next Mask, 277–278 Nobel Prize, 6, 8, 116, 174 Nonaqueous, 4, 7, 10, 14, 31–33, 46, 49, 213–228, 232–233, 429–436 Nonconducting, 51–53, 72, 75–76, 204, 208–209 Nondestructive characterization (NDC), 262 Nonequilibrium, 2, 44, 54, 57 Nonlinearity, 72, 240, 254–255 Nonspherical, 136 Nonstationary, 129, 144, 204 Nonsymmetrical PSD, 287 Norrish, 116

O Occam, 286, 288 Ohshima, 9, 67–68, 189–190, 192, 214–216, 256, 396, 429 Oil-in-water, 44, 247, 332, 334, 369–370, 374–377, 383, 387 Oja, 7

501

Subject Index

Onsager, 64, 71, 193, 221 Osseo-Asare, 350 Overbeek, 49, 69–70

P Particle-particle interactions, 55, 62, 66, 73, 128, 130, 135–136, 138–140, 142, 144, 151–154, 189, 195, 197, 224, 240, 242, 244, 255–256, 326 Particle size distribution (PSD), 39–40, 42–44, 57, 66, 77, 128, 133, 149–150, 160, 177, 283–289, 296, 309, 311, 313–314, 319–322, 324–327, 346, 350–351, 354, 357, 359–364, 371, 449 Pascal, 27 PCC-alumina, 321, 324–325 PCC-silica, 321, 325–326 Pellam, 6, 265 Pendse, 7, 9, 189, 254 Pen Kem, 188 Percus-Yevick, 197 Permittivity, 28–30, 32–33, 50, 53, 70, 174, 208–209, 214, 221, 263, 411, 431, 446 Photocentrifugation, 247 Photomicrography, 370 pH titrations, 287, 305, 308, 310, 401–404, 419, 437, 462 Pinkerton, 265 Pogorzelski, 82 Polarization, 51–53, 69, 72, 75–76, 196, 200, 210, 284, 445, 447–448 Polydispersity, 34, 40, 149–150, 205, 228, 253, 255–256 Porter, 116 Portland cement, 268 Povey, 7, 9, 244, 369 Precipitated calcium carbonate (PCC), 314–316, 319, 321–326, 404 Precision (reproducibility), 14, 81, 106, 108, 110, 112, 115, 119, 243, 254, 262, 267–268, 270–272, 275, 278–279, 281, 288–291, 331–332, 350–351, 354, 360, 364–366, 385–386, 388–389, 403, 415, 430, 458 Prediction, 23, 222–228, 253, 263, 283, 288–289, 313, 326, 442, 444, 458–459, 461–462 Pressure, 27, 29–30, 61, 71, 74, 92–93, 95–96, 98, 100, 129, 144–148, 157, 162–163, 169, 171, 173, 175, 180, 189, 191, 193–194, 211–212, 219, 221, 228–230, 262, 265, 295–297, 355, 444, 446–448, 452–453, 458

Pulse-echo, 265 Pycnometer, 39, 119, 177

Q Quasistationary, 193–194

R Radio frequency (RF), 268–270, 275, 292, 294–295 Randall, 256, 403 Rasmussen, 255, 257 Rayleigh, 3, 5–6, 8, 12, 80–81, 92, 131–133, 135, 144, 161–164, 167, 174, 246, 265–266, 389 Rayleigh-Gans, 81 Rexolite, 293 Reynolds, 3, 5 Rheology, 2, 7, 11, 14–15, 21, 36, 60–66, 91–120, 130, 140, 144, 156–157, 176, 250, 308, 312, 391–396, 403 Ricco, 262 Richardson, 136, 246 Richrdson, 26 Rider, 189, 197 Riebel, 7, 9, 266 Rigole, 254 Rosenholm, 308, 409, 411–412 Rouse-Bueche-Zimm, 157 Rowell, 57, 254 Rowlands, 49, 191, 197, 255, 370, 410 Rutgers, 8, 188, 254 Rutile, 136–137, 240–244, 254–258, 279, 287, 304–310, 327–332, 403 Rutile-ATH, 328–329, 331–332

S Samatzky, 254, 370 Saville, 68 Scales, 255, 411–412 Scattering, 2–3, 5–6, 8, 11–14, 22, 35, 39, 78–82, 128–140, 144, 154, 161–172, 175–177, 246, 250–258, 262, 284, 290, 336, 344, 346, 348, 357–359, 364, 373–374, 390, 392 Sedimentation current, 200–204 Semiconducting, 36 Sensitivity, 13, 105, 178, 289, 344, 350–351, 355, 363, 391 Sensor loss, 272–273 Sette, 262, 264 Shilov, 7, 9, 50, 71–72, 190, 198, 201–202, 213, 220, 228 Shilov-Zharkich, 202

502 Showa, 247 Shugai, 190, 197 Sigmund, 251 Sigmund Lindner GmbH, 251 Signal level, 270–272, 275, 346 Signal processor, 270–271, 275 Signal-to-noise, 271, 275–276, 278 Sihna, 264 Silica, 25, 133–134, 178, 240–245, 251, 254–257, 272–274, 276–282, 288–291, 296–297, 314–316, 319, 321–326, 344–353, 355–356, 365–367, 403–405, 414, 428, 449–451, 453, 455–458, 460, 462 Silica, Cabot, 243–244, 288 Silica, Geltech, 243–244, 291, 319, 324–325, 352, 355–356, 449–451, 453, 455–456 Silica, Ludox, 178, 240, 242–244, 254–256, 279, 281, 291, 297, 344–350, 355–356, 367, 414, 428, 449, 451, 456 Silica, Minusil, 272, 279, 288–289 Silicon, 243–244, 255 Smoluchowski, 67–70, 72, 190, 193–199, 214, 228, 256, 348, 445–446, 456 Smoluchowski dynamic electroacoustic limit (SDEL), 190, 193–210 Sols, 40, 254, 344 Sonication, 349, 360, 415–416, 418–419, 421 Sound, 2, 27, 92, 127, 188, 242, 262, 314, 354, 370, 426, 442 Sound speed, 5, 9, 11, 27–28, 31, 36, 93–97, 102, 109–110, 112, 115–116, 118–120, 127, 140, 142–143, 145, 149, 158, 175–178, 180, 192, 228–230, 242–243, 262–291, 296–297, 314, 340, 370, 383, 386–390, 393–395, 426, 442–443 Sowerby, 262 Space-time, 141 Stakutis, 141 Standards, 7, 23, 30, 39, 43–44, 50, 114, 133–134, 242, 249, 251, 255, 278–279, 287–288, 292, 296, 314, 320–322, 324, 326, 330, 344–349, 361, 375, 403, 441–442 Stern, 45–46, 48, 55–56, 69, 216, 255, 410 Stokes, 3, 5, 61, 102–104, 108–111, 157, 333, 455 Streaming current/potential, 2, 15, 441, 444–448 Stress-strain, 102, 157 Structure, 8, 22–23, 27, 32–34, 37–39, 45, 48–51, 73, 75–76, 106, 116, 119, 128,

Subject Index

139, 156, 158–160, 173–174, 198–199, 209, 216, 228, 250, 264, 311, 339, 388, 392–393, 410–412, 425, 436, 442–443, 445, 463 Subpopulation, 289, 350 Sunflower oil-in-water, 370 Sunscreen, 312, 314, 332–336, 357 Supernate, 453 Surface-bulk, 402, 404, 419–421 Surfactants, 10, 33–34, 246–247, 254, 369–373, 375, 377–378, 396, 403–408, 412, 414–419, 429–431, 436 Sympatec, 3, 9, 247

T Takeda, 350 Taylor, 202 Temkin, 92, 104, 106, 129, 142–143 Temperature, 5, 26–28, 30, 32, 46, 92–94, 106, 110–111, 129, 142–143, 171, 178, 263, 296, 366, 370, 383, 386–387, 392–395, 447 Texter, 255 Thermodiffusion, 190 Thermodynamic, 190 Toluene-in-water, 370 Tone-burst, 266–268, 275 Total vibration current (TVI), 413–414 Transducer loss, 272–275 Transmembrane potential, 44, 59–60 Truel, 161 Tsai, 82 Tsang, 161 Tsukuba, 247 Twersky, 161 Tyndall, 3, 5, 8

U Ultracentrifuge, 77, 284 Urick, 144, 246 Uusitalo, 7, 9, 266

V Van der Waals, 54–56, 411 Van Tassel, 256, 403 Viscoelastic, 10, 66, 101, 426, 441–442

W Wagner, 53, 70–72, 190, 192–193, 196, 199, 206–208, 216, 220–222, 231–233, 348, 456

503

Subject Index

Water-ethanol, 115–116 Water-in-car oil, 371–372 Water-in-crude oil, 254, 370 Water-in-heptane microemulsion, 246, 372–373 Water-in-hexadecane, 369 Water-in-oil, 160–161, 208–210, 377–382 Waterman, 161 Wavenumber, 128, 145, 147, 158, 242 Wedlock, 244 Weight-basis, 240 Wettability, 340 Wines, 246

X X-ray, 2, 246, 285

Y Yeager-Zana, 254, 427

Z Zharkikh, 7, 71–72, 201, 220 Zirconia, 243–244, 314–318, 320–321, 323–326, 340, 419–420 Zirconia-alumina, 315, 320–321, 323–326 Zukoski, 68