Monodispersed Particles

Monodispersed Particles

Monodispersed Particles

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Monodispersed Particles Tadao Sugimoto Institute for Advanced Materials Processing Tohoku University Katahira 2-1-1, Aobaku Sendai 980-8577, Japan

2001

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First edition 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. ISBN: 0 444 89569 8 Transferred to digital printing 2005 Printed and bound by Antony Rowe Ltd, Eastbourne

Preface The term "monodispersed particles" as the antonym of "polydispersed particles" means a group of particles made highly uniform in size and shape. If they are defined in terms of size distribution, one may expect the coefficient of variation (= standard deviation / mean size) to be within 10 %. The major significance of monodispersed particles is attributed to the uniformity in physico-chemical properties of individual particles in a dispersion system, which allows us to directly correlate the properties of a whole system with those of each particle and facilitates theoretical approaches. Their practical importance seems to be due to the potential capability for precise control of their own properties. Thus, they are of obvious importance not only in the fields of physical chemistry, dealing with the dynamic behavior and stability of particulate systems, but also in the industries including catalysts, ceramics, electromagnetic materials, photographic emulsions, pigments, medicines, etc. Complying with so many scientific and practical demands, great efforts have been devoted to the preparation of well-defined colloidal particles for a long time. One may be able to trace the history back to the beginning of the twentieth century. For example, uniform gold particles were prepared by Zsigmondy as early as in the 1900s using a seeding technique; monodispersed barium sulfate particles of different shapes by Andreasen et al (1943); spherical sulfur particles by LaMer and Barnes (1946); polystyrene latex by Bradford and Vanderhoff (1955); cubic and octahedral silver bromide particles by Berry et aim the early 1960s; spindle-like akaganeite (P-FeOOH) by Watson et al (1962); spherical silica particles by Stober et al (1968); spherical chromium hydroxide particles by Demchak and Matijevic (1969); platinum nanoparticles in microemulsions by Boutonnet et al (1982); synthesis of hematite particles of different shapes in the gel-sol system (1992). However, the substantial progress in this field, including theory and practice, is rather recent and particularly remarkable for the last two decades. While the fundamental and practical importance is generally accepted, it may not seem easy to attain monodisperse systems exactly serving individual purposes, because many factors have decisive influences on the monodispersity and other relevant characteristics. In fact, a large iiuiiiber of monodisperse particles have been prepared only by trial and error or by

vi

PREFACE

optimization of preparation conditions, and it appears rather rare to find studies which genuinely shed light on the fundamental formation mechanisms, leading to general principles, or design synthetic systems on the basis of reasonable guidelines. Nevertheless, it is also true that steady efforts have been continued to investigate the underlying mechanisms of the formation of monodispersed particles, so that the formation processes are gradually being clarified. Such efforts will finally lead us to reasonable and more precise control of monodispersed systems. Moreover, in response to the persistent demand for the industrial use of monodispersed particles of enormous potential benefits, highly condensed monodisperse systems, socalled the "gel-sol systems," was invented in 1992 to resolve the essential problem, the extremely low productivity, in existing dilute monodisperse systems. The new method is based on an idea of using highly condensed precursor gels as a matrix of growing final particles for preventing their coagulation, which is eventually transformed into a condensed sol of the final particles through a dissolution-recrystallization process. This idea was first realized for the synthesis of monodisperse hematite particles, as referred to above. Since a monodispersed sol is formed from a gel, in contrast to the sol-gel process, the method has been named "gel-sol method." This invention is a kind of challenge to the conventional physical chemistry for colloidal systems, on which existing monodispersed systems have been fabricated. Since the backgrounds of the colloid chemistry in condensed systems essentially differ in a sense from those in dilute systems, establishment of the new colloid chemistry is an urgent issue. In view of these situations, it seems timely to publish a comprehensive book on the preparation, characterization, and application of monodispersed particles, which will be useful for systematic understanding of the underlying principles of general monodispersed systems and mechanisms of individual practical cases and for creating novel systems. This book covers diverse aspects of monodispersed particles, consisting of four parts: Le,, Fundamentals, Preparation, Analyses, and Applications. In Part 1 (Fundamentals), I will introduce the fundamental concepts of nucleation, growth, habit formation, recrystallization, and solid-solution formation in its individual chapters, in each of which theories and the corresponding experimental results will be given. In Part 2 (Preparation), the general principles for the formation of monodispersed particles will be summarized in the first place, and then all examples of monodispersed or fairly uniform systems will be classified into homogeneous or heterogeneous systems, being reviewed systematically according to the characteristics of the individual reactions. In particular, considering the convenience for readers.

PREFACE

vii

I tried to compile as many examples as possible from available literature for uniform inorganic and organic particles, with sizes ranging from a few nanometers to several hundreds of micrometers. As a result, even such particles as fairly unifoim but not so complete as to be called "monodispersed particles" are also covered in this part, in accord with the comprehensive nature of this book. Also, the techniques and underlying principles for the control of particle characteristics, such as mean size, shape, internal homogeneous structure, composition, layered structure, hollow structure, porous structure, and heterojunction, will be described in detail. In Part 3 (Analyses), methods for the analysis of the formation processes of monodisperse particles and for the characterization of the products will be explained comprehensively. In Part 4 (Applications), applications of monodispersed particles as models of colloidal systems to fundamental studies and as advanced particulate materials for practical use will be delineated. For the practical use of monodispersed particles, special emphasis is placed on their potentialities and infinite possibilities in future rather than the current applications, since it is only the latest event that we have found some general clue to their manufacture for practical purposes. I hope that this book, probably the first comprehensive book dealing with general aspects of monodispersed particles, will serve as a guide to the fascinating field of uniform colloidal systems for students of senior and graduate-level courses up to advanced specialists in both academic and industrial centers. Perhaps, the readers will find that this book is not an ordinary textbook, but a unique one which involves numerous new ideas of the author in every chapter, and may notice his implicit intention to lead the readers naturally to the essential principles ruling the wonderland of monodispersed particles. Particularly, I have developed a number of new theories for this book, whenever 1 felt it necessary. A book such as this could not have appeared without the sustained help of many people. I am especially indebted to Prof. E. Matijevic of Clarkson University in the United States for his inspiring encouragements and kind offer of his latest publications. Special thanks are also due to Drs. A. Muramatsu and H. Itoh in my laboratory for their help in searching for the literature and in the preparation of many electron micrographs and the name index. Tadao Sugimoto Sendaiy Japan September, 2000

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Contents PART 1. FUNDAMENTALS 1. Nucleation 1.1. Surface Energy 1 1.1.1. Intrinsic Surface Energy of Amorphous or Poly crystalline Spherical Particles 2 1.1.2. Intrinsic Surface Energy of Single Crystal Particles 5 1.1.3. Surface Energy under the Influence of Adsorption 8 1.1.4. Surface Energy of Silver Halide Microcrystals 11 Silver bromide^ Silver chloride^ Silver iodide 1.2. Equilibrium Concentrations of Three-Dimensional Nuclei 26 1.3. Nucleation Rate 30 1.4. The Nucleation Process in Closed Monodisperse Systems 40 1.4.1. Theory 41 1.4.2. AgCl and AgBr Systems 49 a) AgCl Systems 49 Nucleation under the standard conditions, Effects of QQ b) AgBr Systems 54 Nucleation under the standard conditions, Effects of Qo c) Discussion 56 1.5. The Nucleation Process in Open Monodisperse Systems 59 1.5.1. Theory 60 Nucleation model, The distribution function and number of the unstable nuclei. The number and mean radius of the stable nuclei, Nucleation period, Discussion 1.5.2. AgBr Systems 74 Final particle number, Nucleation period. Size of the embryos References 83

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CONTENTS

2. Growth 86 2.1. Equilibrium Concentrations of Two-Dimensional Nuclei 86 2.1.1. On a Bulk-Solid Surface 86 2.1.2. On a Particle Surface 88 2.2. Nucleation Rate of Two-Dimensional Nuclei 90 2.3. Growth Rate by Surface Reaction 93 2.4. Growth of Polyhedral Particles by Surface Reaction 100 2.4.1. Surface Energy of a Two-Dimensional Nucleus 100 2.4.2. Growth Rate 103 2.5. Diffusion-Controlled and Reaction-Controlled Growth Modes 105 2.6. Criteria for the Distinction of Growth Modes 108 2.6.1. Magnitude of the Growth Rate 108 2.6.2. Activation Energy of the Growth Rate 109 2.6.3. Size Dependence of the Growth Rate 109 2.6.4. Evolution of the Size Distribution Width 109 2.6.5. Effect of the Particle Number Concentration 112 2.6.6. Solubility Dependence of the Growth Rate 113 2.6.7. Chronomals 113 References 117 3. Habit Formation 3.1. Surface Chemical Potential of a Crystal Face 118 3.2. Stable Forms 121 3.2.1. Equilibrium Forms 122 3.2.2. Steady Forms 125 a) Steady Forms in the Reaction-Controlled Growth Mode 125 b) Steady Forms in the Diffusion-Controlled Growth Mode 133 References 138

118

4. Recrystallization 4.1. Phase Transformation 139 4.2. Ostwald Ripening 140 4.2.1. Diffusion-Controlled Ostwald Ripening 141 4.2.2. Reaction-Controlled Ostwald Ripening 142 4.3. Self-Recrystallization 144 4.4. Reversed Ostwald Ripening 144 4.5. Contact RecrystaUization 148 References 153

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CONTENTS

5. Solid-Solution Formation 5.1. Equilibrium Compositions 155 5.1.1. Theory 155 5.1.2. Ag(Cl,Br) Systems 159 5.1.3. Ag(Br,I) Systems 161 5.2. Conversion by Intra-Particle Recrystallization 5.2.1. Conversion of AgCl Particles by Br' Ions 5.2.2. Thermodynamics 176 5.2.3. Kinetics 181 References 185

xi

155

167 167

PART 2. PREPARATION 6. General Principles for the Formation of Monodispersed Particles 6.1. Separation of the Nucleation and Growth Stages 6.2. Inhibition of Random Coagulation 192 a) Use of Electric Double Layers 193 b) Use of Gel Networks 193 c) Use of Protective Agents 193 6.3. Reserve of Monomers 198 6.4. Choice of Growth Modes 199 6.5. Introduction of Alternative Mechanisms 202 References 202

187 188

7. Monodispersed Systems 7.1. Classification of Monodispersed Systems 208 7.2. Homogeneous Systems 210 7.2.1. Homogeneous Redox Reaction 210 a) Reduction 210 b) Partial Reduction 218 c) Oxidation 219 7.2.2. Precipitation by Poor Solvents 219 7.2.3. Precipitation by Cooling 220 7.2.4. Direct Reaction of Ions 221 7.2.5. Dissociative Reaction of Inorganic Complexes 223 7.2.6. Dissociative Reaction of Organic Complexes 224 7.2.7. Dissociative Reaction of Organic Compounds 231

208

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CONTENTS

7.2.8. Decompositional Reaction of Compounds 233 7.2.9. Hydrolysis of Alkoxides 243 7.2.10. Forced Hydrolysis of Metal Ions 252 7.2.11. Dispersion Polymerization 258 7.3. Heterogeneous Systems 265 7.3.1. Phase Transformation of Solids 265 a) Dilute Systems 266 b) Condensed Systems 270 7.3.2. Ostwald Ripening 289 7.3.3. Emulsion Polymerization 291 Growth Mechanism in Emulsion Polymerization 7.3.4. Reaction in Microemulsions 309 7.3.5. Precipitation from Liquid Crystals 314 7.3.6. Inhomogeneous Hydrolysis 316 7.3.7. Hydrolysis in Nonaqueous Emulsions 319 7.3.8. Reaction on Solid Surfaces 322 7.3.9. Reaction in Solid Matrices 329 7.3.10. Reaction in Solid Templates 330 7.3.11. Firing of Solid Precursors 334 7.3.12. Conversion of Aerosol Droplets 336 7.3.13. Oscillatory Nozzle-Jet Techniques 338 References 341 8. Control of Particle Characteristics 8.1. Size Control 368 8.1.1. Feed Rate 369 8.1.2. Temperature 370 8.1.3. Reservoirs 371 8.1.4. Solvents for Solids 371 8.1.5. Restrainers of Growth 372 8.1.6. pH 373 8.1.7. Seeding 374 8.1.8. Supersaturation Quenching 375 8.2. Shape Control 376 8.2.1. Shape Control by Adsorption 376 a) Inorganic Controllers 376 b) Organic Controllers 384 8.2.2. Shape Control by Anisotropic Acceleration a) Twinning Control 388 b) Dislocation Control 389

300

368

388

CONTENTS

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c) Catalytic Control 391 d) Magnetic Control 392 8.2.3. Shape Control of Polymer Particles 393 8.3. Internal Structure Control 396 8.4. Composition Control 406 8.4.1. Mixing 407 8.4.2. Doping 411 8.4.3. Conversion 413 8.5. Layered Structure Control 417 8.5.1. Inorganic Coating 417 a) Coating of Inorganic Particles 417 b) Coating of Organic Particles 419 8.5.2. Organic Coating 420 a) Coating of Inorganic Particles 420 (1) Adsorption 420 (2) Initiation of Polymerization from Solid Surfaces 422 Anionic polymerization initiated by surface-bound groups] Radical polymerization initiated by surfacebound groups. Radical polymerization initiated by adsorbed initiators (3) Copolymerization with Surface-bound Vinyl Groups 427 (4) Living Polymerization Terminated by Surface-bound Groups 428 (5) Coating via Intermediate Adsorption Layer 428 (6) Polymerization Induced by Solid Surfaces 429 (7) Direct Reaction of Polymer Coupling Agents 429 b) Coating of Organic Particles 431 8.5.3. Epitaxial Multilayers 432 8.6. Hollow Structure Control 432 8.7. Porous Structure Control 434 8.8. Heterojunction 436 8.8.1. Epitaxial Heterojunction 436 8.8.2. Non-epitaxial Heterojunction 437 References 441

PARTS. ANALYSES

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9. Analyses of Formation Processes 9.1. Electron Microscopy 453 9.2. X~ray Diffractometry 455 9.3. Infrared Spectroscopy 456 9.4. Ultraviolet-Visible Spectroscopy 459 9.5. Potentiometry 461 a) pH Measurement 461 b) Metal-Ion Potentiometry 463 9.6. Inductively Coupled Plasma Spectrometry 9.7. Gas Chromatography 466 9.8. Ion Chromatography 467 9.9. Radiochemical Analysis 470 9.10. Seeding Analysis 477 References 480

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453

465

10. Characterization of Products 10.1. Transmission Electron Microscopy 482 10.2. Scanning Electron Microscopy 485 10.3. Electron Diffractometry 486 10.4. Energy Dispersive X-ray Spectrometry 488 10.5. Powder X-ray Diffractometry 491 10.6. Oriented Particulate Monolayer X-ray Diffractometry 10.7. X-ray Photoelectron Spectroscopy 500 10.8. Infrared Spectroscopy 502 10.9. Ultraviolet-Visible Spectroscopy 504 10.10. Photon Correlation Spectroscopy 507 10.11. Turbidimetry 512 10.12. Coulter Principle 517 References 518

482

496

PART 4. APPLICATIONS 11. Application to Fundamental Studies 11.1. Determination of the Avogadro Number 520 11.2. Measurement of Zeta-Potential 523 11.3. Determination of Hamaker Constants 529 11.4. Measurement of Interparticle Forces 531 11.5. Studies of Particle Adhesion 535

520

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XV

11.6. Studies of Colloidal Ordering 545 11.7. Studies of Light Scattering 552 11.8. Studies of Optoelectronic Properties of Fine Particles 559 11.9. Studies of Ionic Properties of Fine Particles 561 11.10. Studies of Magnetic Properties of Fine Particles 564 References 573 12. Industrial Applications 12.1. Photographic Materials 580 12.2. Ceramic Materials 588 12.3. Magnetic Recording Materials 596 12.3.1. Fundamentals of Magnetism 596 a) Magnetic Force and Its Unit Systems 596 b) Classification of Magnetisms 600 Ferromagnetism, Ferrimagnetism, Parasitic ferromagnetism, Metamagnetism, Paramagnetism, Antiferromagnetism, Diamagnetism c) Size Effects of Fenomagnetic Particles 603 d) Effects of the Magnetic Anisotropics 608 Shape Anisotropy, Magnetocrystalline Anisotropy, Strain Anisotropy 12.3.2. Relations between the Magnetic Anisotropics and Coercive Force 615 a) Shape Anisotropy 615 b) Magnetocrystalline Anisotropy 616 c) Strain Anisotropy 617 d) Some Reductive Factors for Coercive Force 617 12.3.3. Magnetic Particles Used for Recording Media 619 a) Maghemite 619 b) Cobalt-Modified Maghemite 620 c) Iron 620 d) Chromium Dioxide 620 e) Barium Ferrite 621 12.3.4. Monodispersed Particles for Magnetic Recording Media 622 12.4. Catalysts 628 12.4.1. Preparation and Uses of Metal Catalysts 629 a) Dispersed Catalysts 629 Condensation in gas phases 629

580

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CONTENTS

Precipitation in liquid phases 630 b) Supported Catalysts 632 12.4.2. Size Effects 635 a) Size Control of Metal Particles 636 b) Size Effects on the Electronic States of Metal Particles 637 c) Size Effects on Activity and Selectivity 642 (1) Reactions of a Constant TOP with Size Change 643 (2) Reactions of an Increasing TOP with Size Reduction 643 (3) Reactions of a Decreasing TOP with Size Reduction 644 (4) Reactions of a Maximum TOP with Size Change 644 12.4.3. The Roles of the Support 645 a) Controller of the Mean Size and Size Distribution of Metal Particles 645 (1) Effect of the Specific Surface Area 645 (2) Effect of the Surface Roughness 646 (3) Effect of the Pore Size 646 (4) Effect of the Affinity to Metal Particles 648 (5) Effect of the Affinity to Metal Complexes 649 b) Stabilizer 652 c) Adsorption Medium 655 d) Controller of Electron Density of Metal Particles 656 e) Synergistic Catalyst 657 12.4.4. Application of Monodispersed Particles to Catalysts 658 Appendix: Growth of Metal Particles on Supports by Stepwise Coalescence 663 12.5. Pigments 676 12.5.1. Relationship between Particle Size and Color 677 12.5.2. Shape Effect on Optical Properties 685 12.5.3. Hiding Power 686 12.5.4. Composite Pigments and Their Optical Properties 687 12.5.5. Other Properties Required for Pigments 689 12.6. Biological and Medical Uses 691 12.6.1. Applications to Cytology and

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Diagnostic Examinations in Vivo 692 a) Specific Staining of Proteins in Cells 692 b) Magnetopneumography 695 c) Mechanistic Studies of Endocytosis 696 12.6.2. Applications to Cell Separation and Diagnostic Assay 697 a) Use of Microspheres for Cell Separation 703 b) Separation of Cancerous Cells 705 c) Diagnosis of Virus Diseases 705 d) Chemiluminescent Enzyme-hnmunoassay for Tumor Markers 707 12.6.3. Remedial Uses 708 a) Drug Delivery Systems 709 b) Bioreactors 710 c) Biocleaners 711 d) Hyperthermia 711 12.6.4. Roles of Monodispersed Particles 712 References 714 Name Index

733

Subject Index

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LIST OF SYMBOLS A A(t) a

surface area, Hamaker constant, or light absoq)tion relative amplitude to the average level in dynamic light scattering particle radius, or solid surface area divided b y the number o f the adsorbed particles UQ lattice parameter of fl-axis a^ surface area o f a m o n o m e r a„ surface area of a cluster consisting of n monomers ( n - m e r ) , or electric multipole moment B magnetic flux density b„ magnetic multipole m o m e n t C total concentration o f all monomeric species CQ initial C Cfj C in the bulk in the liquid phase o f a particle growth system C^if C at the critical supersaturation C at the solid/liquid interface in a particle growth system Cj C in equilibrium with a particle of radius r Cj. C„ C in equilibrium with a bulk solid C(„) C in equilibrium with a n - m e r Cjv C in equilibrium with the stable nuclei at the end of nucleation C close to C^i, but slightly below C^^ C* A C * C*'C^ C ( T ) autocorrelation function in dynamic light scattering C^ extinction cross-section of a particle Q^j absorption cross-section o f a particle C^c^ scattering cross-section of a particle c concentration of monomers, or weight concentration of particles CQ lattice parameter of c - a x i s C(oo) concentration o f m o n o m e r s in equilibrium with a bulk solid C(„) concentration of monomers in equilibrium with a n - m e r c„ number concentration o f n - m e r s c„^ c„ in equilibrium with supersaturated monomers c^ number concentration of radicals in m o n o m e r - s w o l l e n polymer particles per unswoUen unit volume c* number concentration of precursor radicals in the aqueous phase of an

XX

LIST OF SYMBOLS

emulsion polymerization system diffusion coefficient of solute molecules or particles transverse diffusivity of particles diameter of a monomer or a particle electric field, or energy magnetic energy of crystalline anisotropy magnetic energy of external magnetic field magnetic energy of shape anisotropy magnetic energy of strain anisotropy electric charge of an electron Faraday constant, or force size distribution function, frequency of a laser beam, or activity coefficient G(T) distribution function of T in photon correlation spectroscopy g normalized size distribution function of embryos, or particle concentration in gram per one gram of the dispersion H external magnetic field, or peak height of a pyramid extrapolated from a two-dimensional nucleus in the form of a truncated pyramid HQ separation between the Stem layers of two particles H^ coercive force HJ^Q) horizontal component of a Rayleigh ratio, RJ^Q) h dimensionless size distribution function, dimensionless magnetic field, surface-to-surface interparticle distance, or height in general / magnetization, intensity of transmitted light, or initiator of polymerization /• initiator radical /Q spontaneous magnetization per unit volume of a particle, or intensity of incident light Ij. residual magnetization /j saturation magnetization J nucleation rate K absorption coefficient of a particle KQ stability constant of initiator radicals Kj) rate constant in the diffusion-controlled growth mode K^ constant in Nielsen's chronomal of diffusion-controlled growth K^ constant in Nielsen's chronomal of mononuclear-layer growth Kj^ rate constant in the reaction-controlled growth mode Kj, constant in Nielsen's chronomal of (polynuclear-layer) reactioncontrolled growth K^ magnetic shape-anisotropy constant

D Dj d E E^ Efj E^ E^ e F /

LIST OF SYMBOLS

K^ K^ K^ k k^ k„ kj k^ kf k^ k^ k^

k„ kj^ k^ k^ L LQ / M m N NQ

Np N^ n

n* n^ HQ n^

xxi

solubility product uniaxial magnetocrystalline anisotropy constant magnetic strain-anisotropy constant Boltzmann constant, or growth rate constant (= k^) reaction rate constant of initiator radicals with monomer k^ of a bulk solid (n = oo) kj" of a bulk solid (n = oo) overall deposition rate constant of all monomeric species per unit area deposition rate constant of monomers per unit area number of radicals entering a polymer particle per unit time averaged k^ for all polymer particles in a system number of monomers polymerized by a radical per unit time in monomer-swollen polymer particles, or reaction rate constant of mononuclear-layer particle growth absolute dissolution rate of a n-mer per unit area absolute release rate of monomers from a n-mer per unit area in the absence of complexing agents recombination rate constant of radicals in monomer-swollen polymer particles rate constant of the surface process for the entry of radicals into a monomer-swollen polymer particle per unit area ligand circumference of a two-dimensional nucleus in the form of a truncated pyramid length magnetic dipole moment of a particle or monomer for polymerization refractive index of a particle relative to that of the medium, magnetic pole strength, or mass of a particle Avogadro number, or demagnetization factor number of embryos generated per unit time (= QVJVQ) number concentration of polymer particles surface density of monomers number of monomeric subunits of a n-mer, real part of the relative refractive index, m, of a particle, or number concentration of cations or anions of 1-1 electrolytes number of monomeric subunits of a critical nucleus n* at the maximum supersaturation QRT/SnDyV^C^, or number concentration of particles in a sol at the ground level under the influence of gravity number or number concentration of stable nuclei in a system

xxii

LIST OF SYMBOLS

«^" final number or final number concentration of stable nuclei in a system after the nucleation period n_ number of unstable nuclei in a system n_* final number of unstable nuclei in a system n^ number concentration of particles n^ number concentration of particles at a height h n^ number concentration of ion / n,^ « of a nucleus at the front of a quasi-steady size distribution n^ average number of radicals in a polymer particle n^ number of bond-fi:ee open sites of a free monomer n/ number of bond-free open sites of a surface monomer P molarity of precipitate P ''lu/^'ioo' or r/rp Q

generation rate of monomeric species in a unit volume, feed rate of monomeric species, total electric surface charge of a particle, or capillary charge of a particle. QQ generation rate or feed rate of monomeric species, assumed to be constant during the nucleation stage q magnitude of a scattering vector in dynamic light scattering R gas constant, or center-to-center interparticle distance R^ infinite reflectivity Rj^ recombination rate of radicals in a polymer particle 7?^(e) Rayleigh ratio for unpolarized incident light r particle radius, circle radius, distance from a particle center, or degree of ion exchange r* radius of a particle in equilibrium with the monomeric species in a solution phase ^min* ''* ^^ ^he maximum supersaturation f mean particle radius TQ mean radius of embryos or initial degree of ion exchange r^oo distance from the center of a tetradecahedral particle to a {100} surface r^ii distance from the center of a tetradecahedral particle to a {111} surface r+ mean linear growth rate of a stable nucleus Too ultimate particle radius at the end of the growth Tg geometric mean radius Tp maximum radius of an embryo, or unswoUen radius of a polymer particle swollen with the monomers

LIST OF SYMBOLS

xxiii

r^v r„ r^ r^ S

mean radius of stable nuclei at the end of a nucleation stage number-average radius surface-average radius weight-average radius supersaturation ratio (= c|c^^^ = CIC^, surface area, or scattering coefficient of a particle S^^ maximum supersaturation ratio T absolute temperature t time tj^ nucleation period U^ magnetostatic energy of a particle u^ electric mobility of a particle V volume of a particle FQ molar volume of solvent molecules V^ London - van der Waals attractive energy Vg B o m repulsive energy V^ Electric repulsive energy V^ magnetic energy V^ molar volume of a solid Vj total potential energy Vj[Q) vertical component of a Rayleigh ratio, Rj(Q) v^ volume of a monomer unit of a linear polymer chain Vp unswollen volume of a polymer particle swollen with the monomer X monomer in a liquid phase X^ species / [ZJ molarity of species i X^ cluster consisting of n monomers (n-mer) X° monomer in a surface molecular layer X mole fraction of monomers in a liquid phase, or distance in general jc* fraction of the number of the embryos whose radius is greater than r* Xo*

X* at r, = 0 (=

VQ/V^)

x^ x^ JC(„) JC(«) z^

mole fraction o f monomeric species i in a liquid phase mole fraction o f n - m e r s in a liquid phase m o l e fraction of monomers in equilibrium with n - m e r s mole fraction of monomers in equilibrium with a bulk solid (n = oo) valency of ion /

Greek letters YIU/YIOO r^^io» aspect ratio of a prolate particle (= l/w), optical size a parameter (= 2jrr/X.'), magnetic energy parameter (= MH/kT), direction

xxiv

LIST OF SYMBOLS

cosine of saturation magnetization, or collision efficiency of particles to a solid in a magnetic field P shape factor of a particle defined by 44^271^, the ratio of the swollen volume of a polymer particle to its unswollen volume, or ratio of the equatorial radius to a half length of the revolution axis of an oblate particle P„ stability constant of complex ML^ (= {ML^]l{M\[Lf) P^ ratio of the scattering cross section of a particle to its geometric cross section r half-width at the half-height of an amplitude peak as a function of frequency at a certain scattering angle in dynamic light scattering Y specific surface energy, or specific wall energy of a magnetic domain 6 thickness of a Stem layer, or thickness of a diffusion layer e optical extinction t surface energy of a bond-free site, or molar optical absorption coefficient eg vacuum permittivity, or width parameter of a size distribution of embryos e^ relative permittivity ^ zeta potential ri viscosity 6 scattering angle, diffraction angle, slope angle of the inscribed circular cone of a two-dimensional nuclei, or angle between magnetic field and primary axis of an anisometric particle K Debye-Hiickel parameter, or light absorption index of a particle X Wj7«j, or wavelength in vacuum X' wavelength in a medium (= \l\x^\ m, = refractive index of the medium) ^i chemical potential of a monomer in liquid phase, refractive index of particles in vacuum, or magnetic permeability II magnetic moment vector of a particle [I relative permeability (= \kj\k^, \x^ = vacuum magnetic permeability) ti® standard chemical potential of a monomer in a liquid phase \j^ refractive index of a medium in vacuum, or vacuum magnetic permeability moo surface chemical potential of the {100} face of a polyhedral particle jiiii surface chemical potential of the {111} face of a polyhedral particle tJL° chemical potential of a monomer in the interior of a pure bulk solid \k^^^ chemical potential of a monomer in a n-mer (X(^) chemical potential of a monomer in a liquid phase in equilibrium with a monomer in a n-mer

LIST OF SYMBOLS

pi(„)

xxv

chemical potential of a monomer in a liquid phase in equilibrium with a monomer in a bulk solid (n = oo) \ii chemical potential of species /, or surface chemical potential of the ith face of a polyhedral particle \i^ chemical potential of a n-mer in a liquid phase |x„° chemical potential of a n-mer in the interior of a bulk solid |i^® standard chemical potential of a n-mer in a liquid phase \f chemical potential of a monomer in a surface molecular layer H^" '^ standard chemical potential of a monomer in a surface molecular layer Vj. number of adsorption sites occupied by an adsorbate molecule / ^ degree of reaction in Nielsen's chronomals jt„ angular distribution of afieldradiatedfrom electric multipoles in light scattering p space charge density, or density of a solid p' overall density of a particle dispersion Po density of a medium p„(e) polarization ratio (= Hj[Q)/V^{Q)) o surface charge density, mass specific magnetic moment, standard deviation of a size distribution, or collision parameter OQ mass-specific spontaneous magnetization, or breadth parameter of a particle size distribution Gg geometric standard deviation of a size distribution o^ modal value of the optical size parameter a Qp mass-specific magnetic moment RTVp/SiiDyV^C^ (a time constant), time constant in autocorrelation X function, or turbidity T^ angular distribution of the field radiatedfi^ommagnetic multipoles in light scattering V volume, or velocity VQ molecular volume of a solvent, or mean volume of embryos v^ volume of a monomer v^ mean volume of stable nuclei i;/ initial v^, equivalent to the v^^ at the maximum supersaturation TJ^ mean volumic growth rate of stable nuclei '^max volume of a nucleus at the front of a quasi-steady size distribution v^ volume of a n-mer Vp maximum volume of an embryo (j) work for transferring a monomer from a bulk solid (n = «>) into a liquid phase and dispersing it to a mole-fraction level (= kTinS), or angle between the directions of magnetization and a magnetic field

xxvi

([)„

LIST OF SYMBOLS

supersaturation energy parameter of the solubility of a n-mer against that of the bulk solid (= mn(C(„/C(«))) X magnetic susceptibility X relative magnetic susceptibility (= X/MO) \p \p„ per n^^ for a n-mer (= ^l^^n^^), or angle between the direction of magnetization and the primary axis of an anisometric particle \po potential difference between the bulk solid and the solution phase in an inhomogeneous system \|)° surface energy of a free monomer \|)„° intrinsic surface energy of a n-mer \p„ surface energy of a n-mer \p§ Stem potential \p°''' intrinsic surface energy of a surface monomer \p^ surface energy of a surface monomer

PART 1. FUNDAMENTALS INTRODUCTION The so-called "monodispersed particles" are generally defined as particles which are uniform in size, shape and internal structure. Their size distribution is exceedingly narrow - say, less than 10% in coefficient of variation (= standard deviation / mean size). Because of these characteristics, they are not only used widely as ideal models for fundamental studies in colloid science, but are also expected to possess infinite potentialities as the most ideal particulate materials in industry for ceramics, electromagnetic devices, catalysts, photographic materials, pigments, medicines, etc. For these applications, it is necessary to establish general principles for the synthesis of monodispersed particles perfectly controlled in their mean size, shape, and internal structure. As an introduction to this final goal, the backgrounds of the fundamental unit processes including nucleation, growth, habit modification, recrystallization, and solid-solution formation will be described in Part 1. Although the arguments in Part 1 are focused on the particle formation in liquid systems, they may mostly be applied to gaseous systems as well.

CHAPTER 1 NUCLEATION 1.1. Surface Energy Probably, the surface energy, or more rigorously the interfacial energy, is one of the most fundamental quantities in the elementary processes of particle formation, including the nucleation event, particle growth, Ostwald ripening, coagulation, etc. In this sense, it seems quite reasonable to deal

2

FUNDAMENTALS

with this quantity first of all m this book. However, despite its crucial importance, this essential quantity in coUoid and interface science and all other sciences dealing with phenomena in inhomogeneous systems had never been defined quantitatively or formulated as a function of more elementary quantities, such as chemical potentials, in thermodynamics of inhomogeneous systems since its advent in the late 19th century. This was the very reason why the author started to consider, presumably, the greatest fundamental problem of modem thermodynamics left unsolved, hi the course of the reconsideration on the background of the interfacial energy, the author became aware of serious inconsistencies involved in the two principal theorems of modem thermodynamics for inhomogeneous systems, which seemed to have inhibited the sound development of thermodynamics of their own; ie,, (1) the chemical potential of each component is equal throughout an inhomogeneous system including its interfacial chemical potential when the system is in equilibrium; (2) the interfacial energy is an excess energy of an interface independent of the chemical potentials of the interfacial components. Hence, what the author had to do first was to disprove logically these conventional theorems.^ After that, on the basis of the deductive conclusion that the origin of the interfacial energy is the difference in the chemical potential of each component between the interface and bulk phases, a new theoretical approach to the interfacial energy has been proposed.^"^ The formulation of the surface energy of solid in liquids, or solid-liquid interfacial energy, to be described in this book is based on this general theory. I J . l . Intrinsic Surface Energy of Amorphous or Polycrystaiiine Spherical Particles Let us call the minimum subunit of a particle a "monomer," corresponding to the molecular formula of the solid, Afip such as AgBr, Si02, FCjOj, CU2O, etc., and a cluster consisting of n monomers a "n-mer." In the solution phase, some of the monomers may exist in different forms such as complexes or dissociated ions. All of them including the monomer in the original molecular form of Afij will generally be referred to as "monomeric species" in this book. The term "monomer" will also normally indicate a monomeric species in the molecular form of A^Bj without dissociation or complexation. First, let us direct our attention to the monomers in the molecular form ofAJSj in the solution phase, at equilibrium with the bulk solid (n = 00). If the monomers in the bulks of the solid phase, a, and liquid phase, p, are

1. NUCLEATION

3

represented by IC and Z, respectively, the formation of a free monomer X in the liquid phase from the solid phase a and solvent molecule, L, in the liquid phase may be written as

where XL^ is the monomer X coordinated or solvated by the molecules L in the liquid phase. The standard free energy of formation of XL^ per single molecule, \p°, is given by

where fx^Ln® is the standard chemical potential of JSLL^ per single molecule m the liquid phase, (i^"'" is that of Z" in the solid phase a, and ]x^ is that of L in the liquid phase p. The superscript e of (x^Ln® indicates [x^Ln ^^ "^^le fraction x-^^^ 1 obtained by extrapolation from x^^« 1, while the superscript 0 of [ix°'" or ^ means the chemical potential of each pure substance at %°= 1 or jc^ = 1. The op" is equivalent to the interfacial energy of the monomer X or, conventionally, the surface energy of the monomer X If the system is in equilibrium between phases a and p, it follows from Eq. (1.1.1) that li°'« = ^ ^ ^ - « ^ , .

(1.1.3)

If the p-phase is a dilute solution, the mole fraction of U ^L> ^^^ ^^ approximated as x^ « 1, and thus \x^ « ^^, Hence, it holds that ^«>« = ^ ^ - „ H ° .

(1.1.4)

If we redefine V-xii^-ny^.^ and V^xurn\k^ as the standard chemical potential, [i®, and chemical potential, ji, of the monomer in the solution phase, respectively: le.,

Eq. (1.1.4) is rewritten as

4

FUNDAMENTALS

^0,a^^

(1.1.6)

In this treatment, the solvent is regarded as only a medium, and the formation energy of the coordinated n solvent molecules from the free molecules is incorporated into the standard chemical potential of the monomer X. Such a treatment is allowed only for dilute solution phases, such as we deal with in this book. On the other hand, if the mole fraction of Z in phase P in equilibrium with the sparingly soluble solid phase a is denoted by x^^y \i can be written as \x=\i-^kThix^^y

(1.1.7)

Hence, the surface energy of a free monomer is given by ^^=^^-/=-i{:7Tnjc^^,

(11.8)

where [x® and [x° are abbreviations for jx®*^ and pi°'°, respectively. Strictly speaking, the surface energy of a free monomer, ip^, should be referred to as the "liquid-solid interfacial energy" of a free monomer. However, we will use the term "surface energy" hereafter in this book, according to the convention. If c^^^ and VQ denote the solubility of the bulk solid in number concentration of the monomers and the molecular volume of the solvent, respectively, x^^^ is given by

Hence, \p° in Eq. (1.1.8) can be calculated readily using this relationship. On the other hand, if the monomers composing an amorphous n-mer of a sufficiently large n are randomly oriented in the n-mer, the n-mer may assume a spherical shape, owing to its amorphous structure, and the surface area of such a n-mer, <2„, may be proportional to n^^, so that if a^ and v^ denote the surface area of a monomer in the liquid phase and its volume in the solid, respectively, then a„ = a^n^ = (36izv\)"'ny\

(1.1.10)

Therefore, the intrinsic surface energy of the n-mer, \p„°, is given by

1. NUCLEATION

i|f^t|;V/3

(1.1.11)

regardless of the shape of the monomers. Here, the intrinsic surface energy means the surface energy of a pure surface without any adsorption of ions or foreign molecules. The intrinsic specific surface energy, or intrinsic surface energy per unit area of the spherical n-mer particle, y^, is given by

.oA.H.zH!^. ^n

^1

(1.1.12)

(36711)^^^

Polycrystal particles consisting of randomly oriented subcrystals are normally spherical. The surface energies of such particles may be formulated in the same way as that of the amorphous particles, as represented by Eq. (1.1.11) or (1.1.12). However, in the case of spherical single crystals, which are occasionally obtained in the course of dissolution or intraparticle Ostwald ripening, the theoretical treatment is more or less different, as shown in the section 1.1.4. 1.1.2. Intrinsic Surface Energy of Single-Crystal Particles Single-crystal particles normally assume their own characteristic habits. In this case, the treatment may differ from that of amorphous or polycrystalline spherical particles, since the surface monomers are regularly oriented. For the treatment of polyhedral microcrystals, we first direct our attention toward the coordination numbers of atoms in crystals. Each atom in a crystal has its own coordination number, and each molecule or monomer is bonded together through the definite bonding spots of its constituent atoms. Since the surface energy is due to the excess free energy of the bond-free open sites of the surface monomers, one may assume that the surface energy of a surface monomer is allocated to each bond-free site of the surface monomer. If we interpret the surface energy of a solid in this manner, we will be able to avoid the difficulty in specifying the originally ambiguous concepts of the microscopic surface areas of a n-mer and free monomer, since we can clearly defme the numbers of their bond-free open sites. If we consider that a crystal is constructed through the repeated addition of each neutral monomer unit onto the crystal surface, a neutral bond-free open site, or simply an "open site," of a monomer may be defined as the minimum unit of the "bond-free spots" of its constituent atoms, through

FUNDAMENTALS

Bonding of Atomic Molecules

#

Bonding of General Molecules a negative bond-free spot (S-)

a positive bond-free spot (S+)

An open site = A pair of S+ and SFig. 1.1. The concept of an open site of an atomic molecule or of an ionic molecule. which the monomer lying on a surface is connected with another monomer in the surface layer. For example, if a crystal consists of only one kind of atoms, like metal crystals, the number of the open sites of the monomer is equal to the coordination number of the atom. On the other hand, the coordination number of a Ag atom or a Br atom in the face-centered cubic crystal of AgBr is six. Hence, a Ag atom has six positive bond-free spots, while a Br atom has six negative bond-free spots. In this case, one open site of a AgBr monomer consists of a pair of each one bond-free spot of Ag and Br. Thus a AgBr free monomer in the solution phase of a suspension

1. NUCLEATION

7

of AgBr microcrystals has five open sites, since a pair of a bond-free spot of Ag and Br is already used for the bonding of the AgBr monomer of its own. Also, the surface monomers of the {100}, {110}, and {111} faces of the AgBr microcrystals have one, two, and three open sites, respectively, per a surface monomer, since the bond-free spots of Ag^ or Br" on the {100}, {110}, and {111} faces are one, two, and three, respectively (see Fig. 1.4). Generally, if a monomer is composed of cations and anions, the cations and anions have, in total, an equal number of bond-free spots, which is also equal to the number of the open sites of the monomer, regardless of the states of the monomer - such as a free monomer or surface monomer of a certain surface layer. In this treatment, we may not need to take into account the other monomeric species in the forms of dissociated ion-pairs and complexes coexisting in the solution phase, since the dissociation and complexation of a monomer are events which are independent of the formation of a free monomer from the solid. We will thus be able to deal with any kinds of crystals including ionic, covalent, and metallic ones, through this general treatment. Figure 1.1 illustrates the concept of an open site for an atomic or ionic molecule. Based on the above guidelines, let us consider the intrinsic surface energy of a polyhedral microcrystal bound by surfaces of a specific face index. Generally, the intrinsic surface energy of a surface monomer, \|)°'^, is given by ^O,o^j^0.o_|j^0^

(1.1.13)

where \i^'° is the standard chemical potential of the surface monomer, and \i^ is that of a monomer in the interior of the solid. If n / and n^ denote the numbers of the open sites of a surface monomer in the specific surface layer, and a free monomer in the liquid phase, respectively, the surface energy of an open site, e, common to both monomers, is given by e =i t ! : . l ! .

Using \|)° in Eq. (1.1.8), one obtains

(1.1.14)

FUNDAMENTALS i|;^'^ = Xt'=-AJknnX(„^,

(1.1-15)

X-^.

(1-1-16)

where

The intrinsic specific surface energy of the crystal, y^, is thus obtained as Y^ = iV>^»^ = -N'^kkTlnx^^y

(1.1.17)

where A^ is the surface number density of the surface monomers.^ The A^ and n / depend on the surface structure of a given crystal, while the n^ depends on the internal crystal structure, as will be discussed in section 1.1.4. 1.1.3. Surface Energy under the Influence of Adsorption While the effect of adsorption on interfacial energy, y, is already involved in the formula of y in the fundamental theory,^ it can be derived also from a concept of the surface complex, in a manner similar to the derivation of the absolute value of solid-liquid interfacial energy in the preceding sections 1.1.1 and 1.1.2. In this book the latter approach is adopted to show that the same result as that of the fundamental theory is obtained in a different way of derivation."* If the monomeric species and/or dissociated ions of the solid or foreign molecules are adsorbed to the solid surfaces, the surface energy will be reduced. In this case, the adsorption sites of an adsorbate are assumed to be limited to either of the cationic or anionic bond-free spots, and not both. Thus the number of the adsorption sites of a surface monomer is equal to the number of the open sites of a surface monomer, «/. In addition, we further assume that an arbitrary adsorbate molecule Y) can be adsorbed to v^. cationic adsorption sites (v,. ^1), but a solvent molecule can be adsorbed to only one adsorption site (v^ = 1). In this case, the free energy of formation of one surface complex Y^"^, %"",fromv^. sites of a monomer in the bulk solid and an adsorbate molecule of Y^ in the bulk liquid may be written as

1. NUCLEATION

i|;r=tir-^ti°-^,,

(1.1.18)

where m"", \i^, and ^ are the chemical potentials of a surface complex l^"^, a monomer Z in the bulk solid phase, and a free molecule of Y^ in the liquid phase, respectively; « / is the number of the cationic adsorption sites of a surface monomer. It should be noted that ^ 7 ^ / corresponds to the chemical potential of a site of a bulk solid monomer to be reacted with a solvent molecule or Y^ if the bulk monomer is assumed to be located at the solid surface. Since the ip^"^ corresponds to the interfacial energy of a surface complex Y^^, the specific interfacial energy y is given by

where 2 means the sununation for all surface complexes including the surface complex of the solvent, A^^"" is the maximum surface density of Y^"" and y^^ is the coverage of Y-^ over the cationic adsorption sites. At equilibrium of adsorption, dy equals 0 for the changes of any y^"^. Since Idy;" = 0 from ly,"" = 1, JY = S A f > W

(1120)

where the ip^^ is the interfacial energy of a surface complex of a solvent molecule with an adsorption site (z=l), and I! means the summation for all surface complexes with the adsorption sites except the surface complex of a solvent molecule. Since Eq. (1.1.20) must be satisfied for independent changes of all y."^ for i^2 at equilibrium, it must holds for any kinds of X^" with i^2 that

where Nf is the surface number density of the cationic adsorption sites. Hence, y at equilibrium is given by

FUNDAMENTALS

10

Y=w:*i.

(1.1.22)

The tpi" is given by ,0

V^"

M"

(1.1.23)

where \f and pi^ are the chemical potentials of a solvated surface monomer but free from adsorption of Y^ and a solvent molecule in the liquid phase, respectively. If [x^'"^"^ and [i^^ denote the standard chemical potential of a part of a surface monomer in the solid-side surface layer a^ and the chemical potential of a coordinated solvent molecule to the surface monomer, respectively, (1.1.24) = 1^

-^n^V^L ^n^kTbiaL,

where [ij^'"^ is the standard chemical potential of the coordinated solvent molecule, and a / is the activity of the coordinated solvent molecules at the surface. Since jx^-Mi" for Mz, ^ Eq. (1.1.23) where tx/ is the chemical potential of a solvent molecule of the pure solvent, one obtains from Eqs. (1.1.23) and (1.1.24) that i|r^ = :!!LL+jt7Tna^,

(1.1.25)

o

where xp^'"^ is the standard free energy for the formation of a surface monomer of the clean solid surface without the adsorption of non-matrix components, given by ^o.o.^o.a,_^o,„;(^o.<,_^0) Thus,

(1.1.26)

1. NUCLEATION

11

"^^kTlaal

y=K -,.0

=

(1.1.27)

y''+N°kThial,

where y" is the intrinsic specific interfacial energy without the adsorption of non-matrix components N^n:=N:

(1.1.28)

The relation of Eq. (1.1.27) can be derived directly from the fundamental equation of y as well.^ On the other hand, if the adsorbate is only species z, and if T^ and T^ denote the surface densities of species i and solvent L, respectively,

^L-^PrNl

(1.1.29)

Hence, the mole fraction of the adsorbed solvent molecules, jc/, is given by X, = I

r.-r,

(1.1.30)

iVs-(v,-i)r, If Xj° is close to unity, A / may be approximated as a / - JC/. In this case, Y is given from Eqs. (1.1.27) and (1.1.30) as (1.1.31)

y = y°+N°kTln\ 1 - -

(

Jv;-(v,-i)rj

1.1.4. Surface Energy of Silver Halide Microcrystals Silver halide particles, including silver bromide microcrystals in particular, may be among the most popular colloidal particles for their excellent monodispersity and characteristic variety in crystal habit. Furthermore, a good deal of reliable thermodynamic data on the stability

FUNDAMENTALS

12

constants of their complexes in the solutions and some experimental data on the specific surface energies are also available. They are thus likely to be among the most suitable materials for the comparison of the above theory on the surface energy of colloidal particles with the corresponding experimental results. Silver Bromide The silver bromide crystal belongs to the face-centered cubic system, as illustrated in Fig. 1.2. Also, the coordination numbers of Ag and Br atoms are both 6 and the number of the open sites of a monomer, n^, is 5, as is obvious from the scheme of the monomer shown in Fig. 1.3. On the other hand. Fig. 1.4 depicts the atomic arrays of the {100}, {110}, and {111} faces, along with the shapes of AgBr microcrystals bound by the respective crystal faces. The hypothetical regular {111} surfaces of an octahedral particle consist of four triangular planes covered with Ag atoms only and the corresponding four Br-planes, where the Ag- and Br-planes are located alternately, as shown in Fig. 1.5. Hence, if a surface is a Ag-plane, the

• Ag

O Br

Fig. 1.2. Unit cell of the facecentered cubic crystal.

Ag+

Br" Fig. 1.3. Scheme of a AgBr free monomer with n, = 5.

1. NUCLEATION

13

[001]

[100] Fig. 1.4. Atomic arrays of {100}, {110}, and {111} faces of AgBr, corresponding to the respective shaded planes. The closed and open circles represent cations and anions, respectively. The black and white arrows stand for the directions of the respective cationic and anionic bond-free spots of a surface monomer on each surface. The « / values for {100}, {110}, and {111} faces are 1, 2, and 3, respectively.

Fig. 1.5. Hypothetical regular {111} faces of an octahedral particle. The left-hand model is an octahedron, bound alternatively by positive (black) surfaces of cations and negative (white) surfaces of anions. The right-hand model is one bound only by neutral (gray) surfaces with equal numbers of unifonnly distributed cations and anions. reverse side of the octahedron is a Br-plane, and thus one may consider that one Ag-atom in a surface layer and one Br-atom in the surface layer of the reverse side constitute one AgBr surface monomer. However, the actual {111} surfaces of an octahedral particle of AgBr are supposed to be neutralized by randomly distributed ions of the opposite charge through

FUNDAMENTALS

14

occupying about a half of the adsorption sites of the regular surfaces to minimize the total surface energy. Even so, the surface monomer density, A^, and the number of open sites of a surface monomer, «/, are regarded to be the same, because the total number of open sites is unchanged by this procedure. As a consequence, n / = 1, 2, and 3 for the {100}, {110}, and {111} faces, respectively, and thus the >^ values (= AI//AZ^) of these faces are 1/5, 2/5, and 3/5, respectively. If the lattice parameter of AgBr is denoted by flo» the A^ values of the {100}, {110}, and {111} faces are given by 2la^, V2/flo^ and ima^, respectively, where a^ = 5.7748 A at 25 °C. If Pii» ^'(«)» ^ ^ ^sp denote the stability constant of monomers in aqueous solution, defined by [^]/[v4][5], the solubility of the bulk solid in terms of molarity of the monomers (not the total solubility for all monomeric species), and solubility product of a AS-type bulk crystal, c\^^ is given by <«)=Pii^^

(1.1.32)

Furthermore, the mole fraction of the monomers in equilibrium with the bulk solid is given by Eq. (1.1.19) (JC^^^ = c\^^V^, where V^ is the molar volume of the solvent (KQ = 1.807x10"^ dm^ mol'^ for water at 25 ^C). For AgBr at 25 °C, p^^ = 10^^ and K^ = 10"'^^' (see ref. 5 or 6). If we calculate the values of the intrinsic specific surface energy -f of the {100}, {110}, and {110} faces of AgBr in water using Eq. (1.1.17), we obtain Y^QQ^ = 109, Yuo° = 154, Yni' = 188 mJ m'^ at 25 X , where YIOO' : Yno" : Yin' = 1 : V2 : V3. On the other hand, cubic, regular dodecahedral, and regular octahedral microcrystals of AgBr are bound by only {100}, {110}, and {111} faces, respectively, as shown in Fig. 1.6. If we define the shape factor )3 of a particle as

Fig. 1.6. Cubic, dodecahedral, and octahedral AgBr microcrystals bound by only {100}, {110}, and {111} faces, respectively.

1. NUCLEATION

15

(1.1.33)

P=

where A and V are the surface area and volume of a polyhedron, p = 32, 16V2, and 16V3 for a cube, dodecahedron, and octahedron, respectively. Since

^27pr^>'^

27Pu,n^

(1.1.34)

where v^ is the volume of a monomer and n is the number of the monomers composing the particle, the ratio of the surface areas per unit volume of a cube, dodecahedron, and octahedron is given hyA^^ :A^^ :A^^f = 1: (1/2)^^^ : (3/4)^^^ = 1 : 0.891 : 0.953. Since the intrinsic surface energy of a n-mer, ip^''^, is given by ,0,0

0^

(1.1.35)

the ratio of the xp^^ values of these polyhedra is tp^cu^^ : '^n,dof • '^n,oct^ = 1:2^^ : (9/2)^^ = 1 : 1.26 : 1.65. The ratio is independent of the number of the constituent monomers, n. Thus, although the difference m %^ is smaller than that in y^, a cubic particle is the most stable of these regular polyhedral microcrystals. If the AgBr particle is a spherical single crystal, A/^n/ is given by (1.1.36) «o

since a hemisphere, viewed from a position perpendicular to a {100} face of a spherical f.c.c. crystal, contains open sites directed normal to the {100} face whose number is the same as that of a {100} circular plane of the same radius, and the total number of such open sites of the sphere is equal to that of six {100} circular planes of the same radius (N^nf = 6itr(llaQ)IAT(r^). (Detailed calculation gives A^ values with n / = 1, 2, and 3 as 0.603/^0^, 0.7005/flo^ and 0.332/ao^ respectively; 1 x 0.603/^0^ + 2 x 0.7005/ao^ + 3 X 0.332/^0^ = 3^0^.) Although the spherical f.c.c. single crystals of AgBr

16

FUNDAMENTALS

and AgCl were treated in the same way as amorphous spheres in ref. 3, the treatment with Eq. (1.1.36) may be, at least, theoretically more reasonable. Hence, •f of a spherical AgBr is calculated as y" = 163 mJ m'^ from Eqs.(1.1.17) and (1.1.36). Since the shape factor of a sphere is P = 16n/3, A^ :A^, = 1: 0.806 and ^^^J : %J' = 1 : 1.21. Thus, the reduction of the total surface area is quite effective in lowering the total surface energy of a particle. Also, it is expected that if the difference in y" among the facets of a polyhedron is smaller than a certain limit, the total surface energy of the polyhedron may be lowered by the reduction of the total surface area even if the overall y" is raised, as wOl be shown later. The above calculations for AgBr microcrystals in water at 25 °C are summarized as foDows: AgBr: K^ = 10"^"^ moP dm'^ p^ = lO'-^ moK^ dm^ c\^) = lO"'*^ mol dm-^ x^^^ = 2.79x10-^"; \jj° = 9.06x10-2° J; a^ = 5.7748 A; Tjj =fl„3/4= 4.8145x10-2 nm^ n, = 5 {100} face: A^ = 2/aQ^ = 5.997 ran-^, « / = 1, X = 1/5 Yioo" = 109 mJ m-2 {110} face: N° = VI/OQ^ = 4.241 nm'^, n / = 2, X = 2/5 Yuo" = 154 mJ m-2 {111} face: N° = 2/V3ao^= 3.463 nm-^, « / = 3, X = 3/5 Yui" = 188 mJ m-2 Spherical surface: N°k = 3/500^ = 1.799 nm'^ Y^" = 163 mJ m-2

iVioo" : ^110" : ^111" = 1 : W 2 : 1/V3

Aa^ • A^:A^:A^

= 1: (l/2)i'« : (3/4)^"* : (jt/6)^^ = 1 : 0.891 : 0.953 : 0.806

1. NUCLEATION

17

Yioo° : Yuo" : Yui" : Y^." = 1 : V2 : V3 : 3/2 = 1 : 1.41 : 1.73 : 1.50

^..J

•• %J

•• %J

•• %J

= 1 : 2^'' : m r

•• (9:t/16r

= 1 : 1.26 : 1.65 : 1.21 The value of v^ was obtained from the lattice parameter. Silver Chloride The AgCl crystal belongs to the same crystal system as the AgBr, facecentered cubic (f.c.c.) system, so the treatment is identical to that for AgBr. The results of the calculations for AgCl microcrystals at 25 °C are summarized below. AgCl: K^ = 10-"^ mol^ dm'*; p^ = lO^-^ mol"^ dm^ c\„^ = lO"*"^ mol dm-^ X(„) = 6.39x10-'; ^\)'' = 7.77x10"^ J; a^ = 5.5502 A; Vi = 4.2743x10-^ nm^; «, = 5 {100} face: AT = 6.493 nm'^ nf =l,k=

1/5

Yioo" = 101 mJ m"^ {110} face: iV" = 4.591 nm-^ « / = 2, X = 2/5 Yiio" = 143 mJ m-2 {111} face: AT = 3.748 nm'^, n° = 3, X = 3/5 Yni" = 175 mJ m-^ Spherical surface: N°X = 2,l5a^ = 1.948 nm"^ Y^; = 151 mJ m-^ The v•^ was obtained from the lattice parameter. Silver Iodide The wurtzite-type (p-type; hexagonal) and zinc-blende-type (Y-type; cubic) crystals are known to be stable up to 137 °C and from 137 to 146 °C,

FUNDAMENTALS

18

respectively, as shown in Fig. 1.7. However, they are usually observed as their mixture in normal precipitation around room temperature. Since each coordination number of Ag and I is 4 for both crystal types, w^ = 3 for both P- and Y-types, as shown in Fig. 1.8. If we assume that the solubility product, solubility in terms of the Agl monomer, and molecular volume are common to both types, we can calculate the Y° values of their individual faces in water. For the calculation of Y° of spherical particles, we used the molecular volume {v^ = 6.88x10"^ nm^) calculated from the specific gravity (d = 5.67) (C/. V calculated from the lattice parameters: 6.86x10"^ nm^ for the P-type; 6.80x10'^ nm^ for the Y-type). The following are the properties of Agl and calculated "f values of each face of the p - and Y-types and of the spherical surface. The calculation of Y° of spherical Agl particles is based on Eq. (1.1.12), as is different from the cases of f.c.c. crystals of AgBr and AgCl. K^ = 10-^^^^ moP dm-^ p,, = 10'' mol"^ dm^ c\^) = 10" 8.59 mol dm-^ x^^^ = 4.63x10"^^; ^ = 9.80x10-2° J; v, = 6.88x10-^ nm^ /(-type Agl: a^ = 4.59 A, CQ = 7.52 A, n, = 3

• Ag

O I

Wurtzite-type (P-type) hexagonal crystal

Ag

O I

Zinc- blende- type (y- type) face- centered cubic crystal

Fig. 1.7. Unit cells of the P-type and y-type of Agl crystals.

1. NUCLEATION

19

{0001} face: N° = l/VSoo^ = 2.80 nm-^ « / = 1*, X = 1/3 Yoooi" = 91.5 mJ m-^ (*«/ = 1 and 3 are possible, but nf = 1 was taken because of its higher stability.) {lOlO} face: A^ = 1/a^o = 2.90 nm-^ nf = 2,X = 2/3 Yioro" = 189 mJ ni-2 Y-type Agl: a^ = 6.48 A, n, = 3 {100} face: N" = lla^^ = 2.38 nm-^ n° = 2, X = 2/3 Yioo" = 156 mJ m"^ {110} face: iST = ^/2|a^ = 3.37 ^m-^ n / = 1, X = 1/3 Yiio° = 110 mJ m-^

A P-AgI

^ y-Agi

Fig. 1.8. Schemes of a monomer in P-AgI or Y-AgI (w^ = 3 for both types).

20

FUNDAMENTALS

{111} face: AT = 2/V3ao' = 2.75 nm-^ nf = 2, X = 2/3 Yni° = 180 mJ m'^ Spherical surface: a^ = {367w^y^ = 8.12x10"^ nm^ Y^,' = 121 mJ m-^ Table 1.1 shows comparison between the theoretical Y° values and the experimental ones obtained by potentiometric measurement on the basis of the Gibbs-Thomson relation.^ Normally, the comers and edges of a microcrystal are not completely sharp on the atomic scale, and this effect becomes conspicuous as the size is reduced. For example, the most stable form of a AgBr or AgCl microcrystal is cubic, but, as the size decreases, the rounded comers and edges become prominent and the final shape looks like a sphere in a size range below several hundred angstroms. This may be because the surface monomers at the comers and edges, having high chemical potentials with greater nf values than those at the flat surfaces {100}, tend to rearrange themselves, e.g., with partial interbonding among the surface monomers, to reduce their overall chemical potentials to some extent within the geometric

Table 1.1. Comparison between theoretical and experimental Y values of AgCl, AgBr, and Agl {Source: Ref. 3) Y (mJ m"^) Temp.

AgBr

AgCl

CC)

Exper.

Theor.

Exper.

10

104

102

112

20

102

101

25

102

30 40 Facets

Agl Exper

Theor.

111

128

125

107

109

120

122

101

104

109

116

121

102

101

105

108

117

119

100

100

102

107

112

116

{100}

Theor.

{100}

sphere

1. NUCLEATION

21

limitations. In this case, the total surface energy of a particle, ip„, may be reduced to some level even below % of a perfect cube having the same n, despite the increase in overaU y of the particle. For example. Fig. 3.1 in chapter 3 illustrates octahedral, intermediate tetradecahedral, and cubic AgBr microcrystals, where a tetradecahedral particle is bound by eight {111} faces and six {100} faces. If we refer to the general form of the tetradecahedra from the tetradecahedron M to the cube as "B-type," the total surface energy % and volume F of a B-type tetradecahedron are given by * . = 12[2r^oo-3(\/3rioo-rni)']Yioo

(1.1.37)

and

where r^oo and r^^^ are the distances from the center of the particle to the {100} and {111} serfaces.^ Eliminating r^oo from Eqs. (1.1.37) and (1.1.38), g^ [2-y/3(yl3-a)(y/3-pff

^

(1 1 39)

where g s ^\^„V(6V^ym)^ « = Ym/Yioo. and/7 s r,J^^, Here, g^^ is the % of a tetradecahedron, relative to that of a perfect cube of the same volume (= 6P^^yioo)- If/? is varied at constant V, Y^QQ, and a, then g or \p„ assumes the minimum at /? = a from dg/dp = 0. Hence, the minimum value of g is given by

Figure 1.9 shows g^.„ as a function of a. If a = V3, g^^ = 1 and thus \p„ = 6F^Yioo- This means that, when a = V3, the most stable form is a perfectly sharp-edged cube. However, if the chemical potentials of the monomers at the comers are reduced to some extent by their self-rearrangement as usual, a may become less than V3. In this case, g^^ in Eq. (1.1.40) is still below unity, and hence \p„ of the tetradecahedron of a less than V3 is lower than

FUNDAMENTALS

22

T

r—ll-2

Fig. 1.9. g^ and Y/Y«J, as functions of a. that of a cube of the same volume. On the other hand, the total surface area of the tetradecahedron, A, is given by A = 12[2rJoo-(3-^/3)(^/3r,oo-r,ll)2].

(l-l-^l)

Thus, the overall y at the minimum \p„ is given by 100

^

(1.1.42)

2-i3-^)0-af

Since YIOO = Yc,*.

_l_-i = _&llMlf)L ycub

(1.1.43)

2-(3-^/3)0-ccf

and thus y/y^j, > 1 for a tetradecahedron of type B (2/V3 < a < V3). Figure

1. NUCLEATION

23

1.9 shows Y/YCU^, as a function of a together with g^„. Therefore, \|)„ of the tetradecahedral particle of type B is lower than that of the cubic particle of the same volume, while the overall y of the former is significantly higher than that of the latter. In other words, if a < V3, the reduction of the total surface area per unit volume outweighs the increase of the total surface energy per unit surface area. This means that the stability of the shape of a particle should be assessed from its total surface energy, \|)„°, and not from the overall specific surface energy, y- Also, when the chemical potentials of the surface monomers at the comers and edges are somewhat lowered by the intermolecular rearrangement, ip„ is efficiently reduced by the degeneration of the sharp comers and edges. This seems to be the reason why exceedingly small particles look like spheres, although not perfect spheres whose \p„ is somewhat higher than that of a cube of the same n, owing to some contribution of the facets of much higher Y° in the spherical surface. In this context, Berry^ estimated Y° of AgBr particles to be ca. 140 mJ m"^ from the data of the size-dependent solubility using the well-known GibbsThomson relationship [y = (kTr/2v2)]n(c/c^„^); r - particle radius]. This value is much higher than the calculated y° of the cubic particles (=109 erg cm"^), but it is not an unreasonable data if one considers the shape effect of a particle on its total surface energy. AgCl particles having sizes of the order of submicrons prepared by the ordinary double-jet technique^ are almost always cubic, reflecting the lowest surface energy of the cubic particle. In the case of AgBr, however, octahedral particles are readily obtained despite the calculated high \p„° of the octahedral particles, when the excess concentration of bromide ions is simply adjusted to a level slightly above that for the preparation of the cubic particles. In addition, dodecahedral AgBr particles cannot be obtained unless some specific adsorptives are added.^ Although these particle shapes are mostly accounted for in terms of the steady growth form, they are closely related to the equilibrium form,^ so that there seems to be some special reason for the easy formation of the octahedral AgBr particles, skipping the dodecahedral particles of an intermediate Tp„°. As mentioned above, the specific surface energies of the {111} and {110} faces, corresponding to the comers and edges, respectively, of cubic microcrystals, are thought to be lowered to some extent by the self-rearrangement of the surface monomers at these sites. However, the exceptionally high probability of generation of the octahedral AgBr particles bound by the {111} faces cannot be explained by the reduction of the specific surface energy of the {111} faces caused by the self-rearrangement of the surface monomers.

24

FUNDAMENTALS

Neither can it be explained by adsorption of Br" ions, since it is consistent neither with the case of AgCl with only a cubic shape nor the impossibility of the formation of the dodecahedral AgBr particles without adsorptives. Thus, this phenomenon is a consequence of the specific nature of the {111} faces of AgBr. Presumably, it may be ascribed to some specifically allowed electronic relaxation of the Ag and Br atoms in the top layer of the {111} faces of AgBr, based on the significant covalent content of the Ag-Br bonds. Therefore, there seems to be a possibility of a considerable overestimation involved in the high theoretical values of y^^^^ (= 188 mJ m' ^) and Ygph° (= 163 mJ m"^), for which the specific nature of the {111} faces of AgBr is not taken into account. To date, the liquid-solid interfacial energies have been evaluated by experiments on the nucleation or growth rates of precipitates or sizedependent solubilities, on the basis of the Gibbs-Thomson relationship. For AgBr particles in aqueous solutions, several data for y are available; Le,, 140 mJ m"^ by Berry^ 120 mJ m"^ by Ljalikow^°, 200 mJ m"^ by Nielsen^\ etc. For AgCl, Nielsen^^ estimated y to be ca. 100 mJ m'^ from the measurement of the nucleation rate. On the other hand, Walton^^ proposed the foUowing relation by correlating the surface energy with the lattice energy of ionic crystals: y^-

'—^,

(1.1.44)

eF^mN^r^^ where y;^ is the intrinsic specific surface energy of a {hM} face, Uj is the lattice energy at temperature T, r^ is the mean ionic radius, F ^ is the structure factor depending on the crystal face {hid} and atomic arrangement (e.g., F^oo = 9 and F^^^ = 3 at 25 **C for alkali hahde crystals), and mNg is the number of ions in one mole of solid. If we pick up the approximate surface energies y for AgBr, AgCl, and Agl calculated from Eq. (1.1.44) with Fioo = 9 and listed in a table of ref. 12, in comparison with the surface tensions extrapolated from the molten salts to 25 °C in the same reference, 156 mJ m-^ (161 mJ m"^) for AgBr, 172 mJ m"^ (170 mJ m'^) for AgCl, 121 mJ m"^ for Agl, where the values in the parentheses are those of the surface tensions extrapolated from the melts. However, the good agreement of the values, calculated for the {100} faces of AgBr and AgCl on the basis of Eq. (1.1.44), with the respective surface tensions extrapolated from the melts may not necessarily mean that the theory is valid, since the surface tensions

1. NUCLEATION

25

obtained above the melting points of these salts may correspond to the specific surface energies of the randomly oriented surface molecules. Benson and his coworkers^^ performed a number of calculations for the surface energy of alkali halides, taking into account the electrostatic, dipoledipole, and dipole-quadrupole interactions and the Bom repulsive forces among the surface ions: e,g., NaCl (YIQO = 188 mJ m"^; YUO = 445 mJ m"^), KCl (Yioo = 163 mJ m"^; YHO = 352 mJ m"^), LiCl (YIOO = 209 mJ m"'; Yuo = 558 mJ m"^), LiF (YIQO = 169 mJ m"^; YUO = 832 mJ m"^), etc. These YIOO values are considerably different from those calculated from Eq. (1.1.44): Le., 169, 108, 193, and 336 mJ m"^ for NaCl, KCl, LiCl, and LiF, respectively.^^ In addition, the differences between YIOO ^ ^ Yno seem too large from the viewpoint of the present theory (YIOO°^ YIIO° = 1 • ^2). The above inconsistencies in the foregoing theoretical treatments may imply the inherent difficulty in theoretical approaches to the surface energy of solids from the the complex interactive factors of the solid surface molecules. Moreover, in the existing theoretical treatments, the interaction of the surface molecules with the solvent, which seems to be one of the decisive factors to be considered, is not taken into account. In contrast, the present theory, formulated on the basis of the equilibrium concentration of the monomers of a bulk solid in the solution phase, may be of advantage in its straightforward approach to the chemical potentials of the surface monomers exposed to the same environment. On the other hand, probably the most reliable experimental approach to the liquid-solid interfacial energy available at present may lie in the precise measurement of the size-dependent solubility of ultrafine particles, with the simple Gibbs-Thomson equation firmly based on thermodynamics. This is because the other methods represented by the analysis of the nucleation or growth processes, with the simplified classical theories of these processes, contain many assumptions and approximations. In this sense, the excellent agreement between the present theory and experimental determination of Y^ of AgBr particles, on the basis of the Gibbs-Thomson equation, may suggest the feasibility of the present theory.^ However, the possible effects of atomic and/or electronic relaxation of the surface monomers on the surface energy are disregarded in the present theory. These effects must also be taken into consideration in some cases as suggested by the specific behavior of the {111} faces of AgBr. In such cases, the calculated values of Y° are regarded as the maximum values of each crystal surface, since all relaxation processes, including atomic displacement, molecular distortion, electronic depolarization, etc., always lead to the reduction of the surface

26

FUNDAMENTALS

energy.

1.2. Equilibrium Concentrations of Three-Dimensional Nuclei For the start of nucleation of fine particles, the concentration of the solute must be sufficiently higher than the solubility of the solid. In other words, the solution must be in a supersaturated state. If we consider the free energy of formation, AG„, of a n-mer at the standard state (jc„ = 1) from n monomers in the supersaturated solution, AG„ may be given by AG„=ti:-n^ = |i:-n(n^^A:nnjc),

(1-2.1)

where |x® and [x^^ are the standard chemical potentials of a monomer and nmer at mole fractions x = 1 and jc„ = 1 in the liquid phase, respectively; and X is the mole fraction of the monomers in the supersaturated state. If we use the mole fraction of the monomers in equilibrium with the bulk solid, x^^y Eq. (1.2.1) may be rewritten as AG„= -nkThi—

+ |x^-n^x^

(1.2.2)

where \i^ is the chemical potential of a monomer in the bulk solid (n = oo). On the other hand, the surface energy of the n-mer, \|)„, is given by .0^

(1.2.3)

Thus, if a supersaturation energy parameter ^ is defined by (1.2.4)

*=

Eq. (1..2.2) can be written as = -n + i|r„.

(1.2.5)

The supersaturation energy parameter, (j), is the difference of the chemical potentials of a monomer at supersaturation and that at equilibrium with the bulk solid. In other words, it is equivalent to the work needed to transfer

1. NUCLEATION

27

a monomer from the interior of the bulk solid to the liquid phase and disperse it to the supersaturated concentration. Since xlx^^-^ is the supersaturation ratio, S, i.e.. S,

(1.2.6)

\-)

(|> may be written, (1.2.7)

^ = kT]nS. Since the surface area of a n-mer, a„, is given by Eq. (1.1.34), 27Pui 1'n = V«n = Y

I 4

\l/3 ,2/3

(1.2.8)

j

If we define xp as f27PUi * =Y

\l/3

(1.2.9)

\p„ is written as (1.2.10)

i|;^=i|;n 2/3

Although \p defined by Eq. (1.2.9) corresponds to \|J„ extrapolated to n = 1, it is generally not equal to the surface energy of a monomer, ij)°, but is a function of the shape and crystal structure of the n-mer (see section 1.1.2). Nevertheless, if the n-mer can be regarded as an amorphous sphere particle, and free from adsorption of its own ions or any other foreign molecules, \p is equal to ip° [see Eq. (1.1.8)]. In this special case. i|r = i|rO = -JkTTnx

H'

(1.2.11)

as can be confirmed if one inserts P = 16jr/3 and y" of Eq. (1.1.12) into Eq. (1.2.9). For fee single-crystal particles, ip/ij)" = 0.952 (cube), 1.20 (dodecahedron), 1.57 (octahedron), 1.15 (sphere), from ip/xp^ = {3/4)aQY^\

FUNDAMENTALS

28

Nf\, As a consequence, AG^ can generally be written as AG = -(t)n + \|rn^^

(1.2.12)

If AG„ consisting of the components, -^n and \|)n^^^, is illustrated as a function of n at a given c|), one may find a peak as shown in Fig. 1.10.^"^ Normally, the n-mer which gives the peak of AG„ is referred to as a "critical nucleus." If the number of monomers composing the critical nucleus is denoted by n*, n* is given from d^GJdn = 0 by

-(f

(1.2.13)

If the maximum AG„ is denoted by AG*, AG* is given by (1.2.14)

On the other hand, when n-mers are dispersed, so as to be equilibrated

J

60r ;

2

/ Jt!lL

50

I kT

4G'

kT 2

30

kT

kT

n

\ 3 « \

'^ 1 20

10

'

0

-10

-20 •

-30

-40

•

.• 10

\ \ .'.

20

kT

•

\

Fig. 1.10. AG„/A:T and its components as functions of « at a given ^IkT (From Ref. 14.)

29

1. NUCLEATION

with the supersaturated monomers, \i'n\i = AG+it7Tnx = 0,

(1.2.15)

where [t^ is the chemical potential of a n-mer and x„ is the mole fraction of the n-mers. If c„" and VQ denote the number-concentration of the n-mers in equilibrium with the supersaturated monomers and the molecular volume of the solvent, respectively, x„ is given by (1.2.16)

^«=^<^»

and thus c„' is given by

(-AG.

c„' = Uo'exp = UQ exp [ kT )

f^n-ijrn^ kT

(1.2.17)

Hence, the equilibrium concentration of the critical nuclei, c„*, is the lowest of all kinds of the n-mers, since AG* is the maximum of AG„ for all kinds of n-mers. The potential barrier, AG*, corresponds to the activation energy for nucleation, so the nucleation is enhanced if AG* is lowered with reduction of \p by adsorption or with increase of (j) due to increasing supersaturation (see Eq. 1.2.14). For example. Fig. 1.11 shows AG„ as a function of n with variation of (|), in which AG* is inversely proportional to

Fig. 1.11. AGJkT as a function of n with variation of ^/kT, (From Ref. 14.)

30

FUNDAMENTALS

1.3. Nucleation Rate as a Function of Supersaturation If jX(^)° denotes the chemical potential of a monomer in a n-mer, it is given by 0 e e fA(n)=l^i,-JA;,-l-

(1.3.1)

Hence, it follows from Eq. (1.2.1) that dAG

n

(1.3.2)

dn On the other hand, from Eqs. (1.2.5) and (1.2.12),

9n

^ = -(j)+—1^ = -|x + |IO+_IJ^. dn dn

(1.3.3)

Comparison between Eqs. (1.2.2) and (1.2.3) yields ^»„,= , o . ^ = , o . | * n - > / 3 .

(1.3.4)

If the dissolution of this n-mer is in equilibrium with the deposition of the monomers in the solution phase, ^ ^

= ^ " - , = 0.

(1-3.5)

TTius, dAGJdn = 0 is equivalent to [i^„°= M- In this case, one obtains from Eq. (1.3.4) <|,= H - n 0 ^ 2 ^ „ - i / 3

(1.3.6)

This is identical to Eq. (1.2.13). If the mole fractions of monomers in equilibrium with a n-mer and bulk solid are represented by JC(„) and x^„y and

1. NUCLEATION

31

their number concentrations by c^^^ and c^^^, respectively, (1.3.7) SO that C(^) is given by

^^">=^->^^Hi]^)

(1.3.8)

from Eq. (1.3.6) and ^ = kT[n{X(^y/Xf^^) in Eq. (1.2.4). This equation and Eq. (1.2.13) correspond to the so-called Gibbs-Thomson equation. If the equilibrium condition between dissolution of a n-mer and deposition of monomers given from Eq. (1.3.5) as

is rewritten from a kinetic standpoint, k^a c

=k^a

(1.3.10)

where kf is the rate constant of the deposition of monomers to a unit surface area of a (n-l)-mer; kj^ is the absolute release rate of monomers from the resulting n-mer per unit surface area; a^_^ and a^ are the respective surface areas of the (n-l)-mer and n-mer. If a denotes the surface area of a monomer and if a n-mer can be assumed to be a sphere, a„ = a,n^

(1.3.11)

SO that

When the deposition of the monomers is in equilibrium with the dissolution of bulk solid (n = oo),

32

FUNDAMENTALS

k^c

^k^

(1.3.13)

where kj^ is the absolute release-rate of monomers from the bulk solid per unit area. Since c^^^ and kj are constants, kf is a constant independent of n as the number of monomers composing a n-mer. On the other hand, kj corresponds to the release rate of monomers from a bulk solid when the monomers are totally absent in the solution phase. Hence kf, given by ^Qo%(oo)» is not the collision frequency of the monomers of a unit concentration to the surface of a solid per unit area, but the rate constant of the incorporation of the monomers into the surface layer of the solid. As a consequence, it follows from Eqs. (1.3.8), (1.3.12) and (1.3.13) that rO

sOln-l .2/3

2i|f

*n = *«| — 1 exp [3kTn'f^)

(1.3.14)

Hence, k^^ is a function of n. Normally, some monomeric species other than monomers in the molecular form coexist in the form of dissociated ions and complexes coordinated by anions or cations in the solution phase of a heterogeneous system. If c^^o/, kj, and kj denote the number concentration of an arbitrary monomeric species i in equilibrium with bulk solid, its deposition rate constant, and its release rate from the bulk solid, respectively, the overall dissolution rate of the bulk solid per unit area, k„, may be written as Tlk'x' k^^EkL = Ek!,cU=-^C„=k,C^,

(1.3.15)

where x^^^' is the mole fraction of the monomeric species / in equilibrium with the bulk solid, C„ is the total number concentration of the monomeric species in equilibrium with the bulk solid, and k^ is the overaD deposition rate constant for all monomeric species defined by

1. NUCLEATION

33

^d

E ^ _

(1.3.16)

Although k^ and k^ are constants in a given system, each of them may be a function of C„ owing to the change of the activation energy for deposition of the monomers and dissolution of the solid by adsorption of anions and/or cations to the solid surface. Thus, k^ should be determined as a function of C^ by the measurement of the dissolution rate of sufficiently large particles, while k^ can be determined by k^ = kJC„. If C^^^ denotes the total concentration of the monomeric species in equilibrium with a n-mer, C^^^C^ = c^^^c^^^ so that the equation for the overaU absolute dissolution rate of a nmer per unit area, k^, corresponding to Eq. (1.3.14) is given by

K-K

( exp

2^^ I \'ikTn'^^)

(1.3.17)

where K=KC.-

(1-318)

Let us consider the following unit process consisting of a forward reaction of a monomeric species X with a (n-l)-mer, Z^_p and reverse reaction of a n-mer: X . X „ . , . X„.

(1.3.19)

If the average surface charge density of n-mers is assumed to be the same as that of the bulk solid regardless of the n value, the forward reaction rate J^_^^ and reverse reaction rate J^^_^ may be given by

and

34

FUNDAMENTALS

(1.3.21)

^«,n-l = *n^n^n'

where C and c^ are the total concentration of the supersaturated monomeric species and the concentration of the n-mers, respectively. If a quasi-steady state is assumed in the following sequential reaction, in which the concentration of each class of the clusters keeps a steady level, but slowly changes with the change of C: ^1

^2

^n-1

^n

(1.3.22)

^n*

the overall forward reaction rate, J, is given by /i-i,n

ii,/i-i

a n-i

n-l

n n n^

(1.3.23)

where J corresponding to the nucleation rate is independent of n, from the assumption of a quasi-steady sequential reaction. On the other hand, if c^ denotes the concentration of the n-mers in equilibrium with the supersaturated monomeric species, it holds that H-i

n n

(1.3.24)

^A-i from Eq. (1.3.23) with J = 0. (1.3.24), J can be written as

Eliminating it„ from Eqs. (1.3.23) and

J=K°n-f*^n-l

c , n-l _ €

\ ^«-l

c II e

(1.3.25)

^n I

Since J is independent of AZ, the value n in Eq. (1.3.25) can be replaced by w+1, so that Eq. (1.3.25) is rewritten from Eqs. (1.2.7), (1.2.17), and (1.3.18) as

1. NUCLEATION

35

ka

J=

n

( ^n \

I (|)

«xp|-g;|exp

dn

kT

(1.3.26)

e

where the vSurface area of a n-mer, a^, is given by (1.3.27) Eq. (1.3.26) can also be obtain by eliminating k^ from Eqs. (1.2.23) and (1.2.24) and using k„ in Eq. (1.3.17) with Eq. (1.2.17). Since c^/c/ must approach zero as n becomes infinite, it follows from Eq. (1.3.26) that JVn

<

(36nvl)'fXoxp(mT)

(1.3.28) 213^

xjn'^^exp

>MJ^U_ kT

J n

If we use n* in Eq. (1.2.13) and AG* in Eq. (1.2.14), AG„ in Eq. (1.2.28) may be transformed as ( \ / \2/3 Ag„ _ -(|)n+i|rw^/3 _ AG* n + 3' n ' -2 kT kT kT Kn') \n )

(1.3.29)

If we take the first three terms of the Taylor expansion of Eq. (1.3.29) for (nln*)'-'^ at {jiln*)"-" = 1, \2

where

AG„

AG-

kT

kT

1-3

-^-1

INn' ) \

AG* 2 ^ . kT

(1.3.30)

36

FUNDAMENTALS

x=.

3AG' N kT

N

-^-1

dx =

i|r

dn

(1.3.31)

the latter of which is obtained by using AG* = Mf{n*)^l3 in Eq. (1.2.14). As a consequence, Eq. (1.3.28) is transformed as ^n

-^"o i

3

\l/3

{kT\l^ kT

(1.3.32)

3AG' N kT

(1.3.33)

\Anv\] xfexpi-y^ydy. At /I = 1,

3AG" N kT

X = ,

i\/^ J

Let us estimate the magnitude of jc at n = 1. For example, tj> - \|>P" =9.06x10-2" J and S = 2 for AgBr at 25 °C, (|) = 2.85x10-^1 J and AG* = 1.36x10-1' J, so that AG*/A:r = 3.30xlO^ Even if 5 = 10, (|) = 9.48x10-^^ J and AG* = 1.23xl0-i* J, so that AG*/itr = 2.98x10". Thus, J: at n = 1 is normally of a sufficiently large negative value, so that j""exp(-/)rfy - J"exp(-y-)rfy = /ir.

(1.3.34)

In addition, c^/c^' = 1 at /i = 1, so that one finally obtains from Eq. (1.3.32)

4u N kT^^^l oV3^/^j

(1.3.35) kT

If we use AG* = 4\pV27(|)2 in Eq. (1.2.14), (t> = kT[T\S in Eq. (1.2.7), and \|) - V = -^71n(C(„)Vo) in Eq. (1.1.8) when the effect of adsorption on \|> can be ignored, J may be written more concretely as

1. NUCLEATION

k

37

(A

2 \ 1/3

j = — ^0

4[ln(C(.)Uo)]' +lnS 27(ln5)2

^-ln(C(.)Uo) exp

(1.3.36)

From Eq. (1.3.35) with Eqs. (1.2.13) and (1.2.14) it follows that dlogJ _ dhiJ _ d dlogC dhiS ~ 7^ [ 274,^

(2±]\i ,

13<|)J

= „'^l.

(1.3.37)

This equation means that (1.3.38)

JocC n%l

in a Hmited range of C. Thus, at least theoretically, it is possible to determine n* from the plot of log J vs log C. Once n* is determined, AG* (= ^n*/2) and \|; (= 3^n*^^^/2) can be calculated. The ratio of steady concentration of n-mer to its equilibrium concentration, cjc^', is given from Eqs.(1.3.32) and (1.3.35) as (1.3.39)

If we use the error function defined by erAx) ^ - ^ f'exp(-)'2)rfy = A

fW")'')^.

(1-3.40)

it follows that

l\xv{-f)dy=^-j\xv{-f)dy=^{l-erm. Hence,

(1-3.41)

38

FUNDAMENTALS

c'n'/2

Fig. 1.12. Distributions of c„ and c/ as functions of fu

(1.3.42)

Specifically, at n = n*, x = 0 and thus cjc^"" = 1/2. Figure 1.12 illustrates c^ and c^"" as functions of n. With respect to the nucleation rate, Becher and Doring^^ derived the following formula: y=

\l/2 ' AG*' 2D In5 exp kT ) d\. \6Tin* ^

(1.3.43)

where D is the diffusion coefficient of solute, d is the diameter of a monomer, and the other notations are the same as those in this book. Afterwards, Zeldovich^* derived the same formula using a mathematically

1. NUCLEATION

39

different method. If we rewrite this equation using Eq. (1.2.13) in order to facilitate the comparison between this equation and Eq. (1.3.35), J ID /=

34)^

exp

'-LG^'

(1.3.44)

kT

One may find a considerable difference between Eq. (1.3.44) and Eq. (1.3.35). As far as the assumption of a quasi-steady state in the sequential reaction of Eq. (1.3.22) is concerned, both treatments are based on the same assumption. They represented J^_^^ and J^^.^ as Jn-,.n = K-l.nClC„.,

(1-3.45)

and

respectively, where c^ is the concentration of the monomers. Since they did not distinguish monomeric species from monomers, c^ in Eq. (1.3.45) may correspond to our C. However, the most serious problem is that the dissolution rate of a n-mer, k^„.i, was postulated to be independent of n. Since A:^^.^ corresponds to k^a^ in our formulation, neither the effect of the increasing surface area a^ nor the size dependence of k^ was taken into consideration. As it is obvious that the dissolution rate of a whole n-mer must increase proportionally with its increasing surface area, this effect cannot be ignored. Also, since ^^^.^ was assumed to be independent of the n value of the n-mer in their theory, k^_^^ was assumed to be size dependent in order to satiwsfy k^_^^c^c^_^^ = k^^_^c^\ However, this treatment basically contradicts the original basis of the Gibbs-Thomson relationship in Eq. (1.3.8), since both the size dependence of k^ Qxk^^_Ja^ in Eq. (1.3.17) and the size independence of k^ or k^,i^/a^_^ have been derived from the GibbsThomson relationship which is based on the greater dissolution rate of a nmer per unit area than that of the bulk solid, owing to the chemical potential of a monomer in the n-mer originally being higher than that in the bulk solid. In addition, k^^_^ was related with the diffusion coefficient of solute, D, and the diameter of a solute molecule, d, as

40

FUNDAMENTALS

This relationship was derived by equating ^^^.^ with the inverse of time, r\ and the root-mean square displacement, )^ with d in the Einstein diffusion equation, k^/^Zt) = D. However, the dissolution process may not be represented by the diffusion model of free monomers, since some activation process of the monomers bound in the surface layer is generally needed for their detachment from the original lattice positions. Incidentally, although the content of jc in Eq.(1.3.42) for cjc^"" is different from that in the corresponding equation of the Becher-Doring theory (x = (lnS/6n*f\n-n*) = {AG/3kT)'^(n/n*-l)), the apparent form of their equation is the same and, similarly, cjc^^ = 1/2 at JC = 0 or n = «*. Experimental measurement of the nucleation rate is often performed by turbidimetry after instantaneous mixing of the reagents,^^'^^ multichannel spectrophotometry in combination with the stopped-flow technique,^^'^ spectrophotometry of products with a streak-camera and photomultiplier synchronized with pulse-radiolysis of a medium,^^'^ etc. However, for the analysis of the nucleation rate or evaluation of the size of the critical nuclei, n*, on the basis of the nucleation theories described in this section, the condition of the quasi-steady state of the sequential reaction of the stationary nuclei must be established, since these nucleation theories are all based on this presumption. In order to realize this premise, the monomers must be furnished steadily from some reservoir involved in the system. Otherwise, what we follow is the extremely fast reaction of forming transient complexes, non-steady nucleation at the expense of the precursor complexes, and subsequent Ostwald ripening of the nuclei preformed by the non-steady nucleation, any of which the quasi-steady nucleation theory cannot be applied to. Thus it may be necessary to reassess past experimental data from this viewpoint. In addition, it must be noted that the quasisteady nucleation process is not independent of the concurrent growth of the nuclei. Rather, the nucleation rate is determined by the growth of the preformed nuclei, and is closely related to the size control of general monodisperse particles, as will also be shown in the following section.

1.4. The Nucleation Process in Closed Monodisperse Systems

1. NUCLEATION

41

In this section 1.4, the theory of the spontaneous nucleation process in closed monodisperse systems and its application to closed systems for the formation of monodisperse AgCl and AgBr particles are described.^^'^'' 1.4.1. Theory In a typical monodisperse particle system, the monomeric species are initiaUy accumulated in the system through some reaction from a reservoir of the monomeric species involved in the system. In this stage, no appreciable nucleation occurs, even if the concentration of the monomeric species exceeds the solubility level of the bulk solid (prenucleation stage). When the concentration reaches a critical level for nucleation (critical supersaturation level), the nucleation virtually starts to be observed (the start of the nucleation stage). In this stage, the nucleation rate is rather low, so the concentration of the monomeric species continues to increase, though the increasing rate is a little lowered due to their consumption for nucleation. However, as the concentration still increases, the nucleation is drastically accelerated and, finally, the concentration of the monomeric species reaches the maximum level (maximum supersaturation). At this stage, the generation of the monomeric species and their consumption by nucleation and the growth of the preformed nuclei is balanced. After that, the concentration of the monomeric species starts to be reduced, due to the increasing consumption of the monomeric species by growth of a number of the preformed nuclei. When the concentration of the monomeric species goes down to the critical supersaturation level again, the nucleation is terminated (the end of the nucleation stage). Even after the end of the nucleation, the remaining stable nuclei can continue to grow with a supersaturation below the critical level for nucleation (growth stage), because the two-dimensional nucleation on the growing particles does not need as high a supersaturation as the three-dimensional nucleation (see chapter 2). When the concentration of the monomeric species becomes close to the solubility level of the bulk solid, the particle growth finishes. The change of concentration of the monomeric species may be illustrated by Fig. 1.13, where C^^ corresponds to the critical level for nucleation. This is the so-called LaMer diagram proposed by LaMer and Dinegar^ for presenting the formation process of monodisperse particles in a simple diagram. For the preparation of monodisperse particles, the nucleation stage should be as short as possible and the renucleation in the growth stage should strictly be inhibited. The distinct separation between the nucleation and growth stages is normally the first requirement for the formation of monodisperse particles of an excellent uniformity. It

42

FUNDAMENTALS

Time Fig. 1.13. LaMer diagram. The regions I, II, III represent prenucleation stage, nucleation stage, and growth stage, respectively. (From: Ref. 23.) must be noted, however, that a clear-cut critical supersaturation level cannot be defined, because the nucleation event is basically a probability process. Thus, when the concentration of the monomeric species exceeds the solubility level of the solid, there is always some probability of nucleation. Nevertheless, the simple LaMer diagram is helpful for understanding the general rule of monodisperse particle formation. From the above insight into the nucleation stage of monodisperse particles, the duration of the nucleation period is obviously governed by the rate of consumption of the monomeric species by growth of the preformed nuclei. Hence, we have to formulate a nucleation theory in conjunction with the growth of the nuclei. The study of ref. 26 is a trial to formulate the nucleation theory from this viewpoint. The aim of this study was to establish the background for the mechanism of automatic size control of the monodisperse particles in open systems, in which the reactants, as sources of the monomers, are introduced from outside and the particles are grown by a kind of the Ostwald ripening through the dissolution of a part of the nascent nuclei instantly generated in a limited zone into which the reactants are introduced. This theory was verified in the formation process of monodisperse AgBr particles in the controlled-double-jet precipitation

1. NUCLEATION

43

system. For the extension of the theory of nucleation in open systems to general closed systems in which the particles are generated through the spontaneous nucleation, it seems necessary to modify the theory using the concept of the quasi-steady spontaneous nucleation theory described in the foregoing section. Figure 1.14 illustrates an evolution of the size distribution of the stationary nuclei with the increasing concentration of the solute continuously generated or supplied from an internal reservoir. Figures 1.14(a) and (b) correspond to the prenucleation stage up to the onset of the nucleation of stable nuclei and the nucleation stage of the stable nuclei, respectively, where / is the frequency or distribution function of the size distribution, r is the particle radius of a nucleus, r* is the particle radius of a nucleus whose solubility is equal to the current concentration of the solute, as related by the Gibbs-Thomson equation, and r^* is the minimum r* at the maximum supersaturation. If the solute concentration is denoted by C, r* decreases with increase of C and increases with the reduction of C. The stable nuclei are defined as relatively large nuclei, mostly greater than r* in

Fig. 1.14. The evolution of the size distribution of the stationary nuclei and the change of r* with time in the prenucleation and nucleation stages of the stable nuclei. The dashed curve in (a) represents the extension of the predetermined steady distribution of the stationary nuclei at the maximum supersaturation. The D.F. is the distribution front of the propagating stationary nuclei: (a) prenucleation stage; (b) nucleation stage. (From Ref. 23.)

44

FUNDAMENTALS

radius, to be grown to the final product, in contrast to the relatively smaU unstable nuclei with r around r* or mostly less than r*, to be eventually dissolved. However, this distinction between the stable and unstable nuclei is only a matter of probability. Namely, some nuclei in a size class are stable nuclei to be grown to the final product, while the other nuclei in the same size class are unstable nuclei to be finally dissolved. Nevertheless, the probability to become stable nuclei increases with the increase in size. According to the classical nucleation theory in section 1.3, the size distribution of the stationary nuclei in a quasi-steady state is in the form of a monotonously decaying curve from the monomer level to the infinite size. In a real nucleation system, however, there must be an upper size limit in the distribution, and the upper size limit or distribution front (D.F.) must continue to advance to complete the predetermined steady distribution curve proper to the supersaturation. The nucleation rate, J, is equivalent to the generation rate of the largest nuclei at the distribution front. Thus, if the particle volume of the largest nuclei is denoted by v^^^y the consumption rate of the solute for the quasi-steady propagation of the stationary nuclei is equal to Jv^JV^ in molarity. In other words, the consumption rate of solute increases with increasing v^^^, even if C and thus J are unchanged, and hence the increasing consumption rate with the increase of C is accelerated by this effect. However, since the supply rate of monomer, Q, is greater than the consumption rate, Jv^^JV^, in the prenucleation stage, the concentration of the solute, C, must increase as

dt

V^

As C rises up to the critical supersaturation level close to the maximum supersaturation, r* is lowered to be close to r^* and the nucleation may start to be observed. Due to the critical dependence of the nucleation rate on supersaturation, the nucleation is dramatically pronounced with a slight increase of C in this stage and C soon reaches its upper limit or the maximum supersaturation, corresponding to f^ in Fig. 1.1.4. At this moment, r'^'^r^* and dC/dt=0, so that ^

m

max

For maintaining the J constant at the maximum supersaturation, Q is required to increase with the increase of v^^^^. However, if Q is constant or

1. NUCLEATION

45

its increasing rate is small, J must start to be declined, since v^^^ unconditionally increases with the growth of the largest nuclei, owing to the much lower solubility of the largest nuclei with radius r^^^ than the supersaturation level (/.e., r^^^ > r*). With the declining steady nucleation rate, J, the distribution function in the relatively small size range is lowered, but the distribution front continues to advance. Hence, a considerable amount of the stationary nuclei, preformed and grown into the range greater than r*, may return to be dissolved even though r* remains close to r^*. It seems noteworthy that a considerable number of once generated nuclei of r > r* are dissolved through a kind of Ostwald ripening even in the closed system. In contrast to the ordinary Ostwald ripening, even a nucleus of r > r* has a high probability of dissolution, since the absolute value of r is sufficiently small. Finally, J becomes virtually zero at the end of the nucleation stage and the size distribution is split into two and the distance between the two distribution groups is enlarged with the progress of the growth process. The supersaturation kept near the maximum supersaturation level during the nucleation of the stable nuclei gradually declines in the growth stage with the growth of the stable nuclei. As is obvious from the description of the nucleation process, the starting point of the nucleation is not definite. However, if one considers that the stationary nuclei always exist even at a solute concentration level below the critical supersaturation and that some nuclei once generated are dissolved again, it seems rather reasonable to define the "nucleation" in general as the "generation of the stable nuclei." One may assume that the nucleation in this definition starts when C reaches the maximum supersaturation. In any case, soon after the maximum supersaturation, the relation of Q = Jv^JV^ loses its quantitative significance, since the system is not in the quasi-steady state of the stepwise propagation of the stationary nuclei anymore, on which the relation is based. Hence, we need to find a new approach to the nucleation process after the maximum supersaturation. For this purpose, one may divide the consumption of the supplied solute into two components; Le., the generation of the stable nuclei and the growth of the generated stable nuclei:

The first term in the right-hand side is the component for the generation of the stable nuclei and the second term is the component for the growth of

46

FUNDAMENTALS

the generated stable nuclei, where VQ" is the initial particle volume of the stable nuclei, equivalent to v^^^ at the moment of the maximum supersaturation, v^ is the mean volumic growth rate of the stable nuclei, n^ is their number density, and QQ is the supply rate of the solute which is assumed to be kept constant during the short nucleation period. As we set the initial conditions as n = 0 and dnjdt = J = QQVJV^^ at r = 0, the stable nuclei in this model are defined as those generated after r = 0 and thus greater than v^^ in particle volume. The increment of the particle size of the stable nuclei, relative to the initial size (= v^^ in particle volume), is thought to be sufficiently small in the nucleation stage, since the supplied solute is mainly consumed for the increase in the number of the stable nuclei. Thus, i)^ may be regarded as nearly a constant during nucleation stage, since v^ is a function of the maximum supersaturation and the mean particle volume of the stable nuclei. Hence, one may describe the behavior of n^ as a function of time by using the time-averaged i)^ in Eq. (1.4.3) as

„..«^ 1-exd

-vj

(1.4.4)

0

/J

As a consequence, the final particle density at r = <», AI^°°, is given from Eq. (1.4.4) as O V n:=-^^.

(1.4.5)

This is the fundamental equation for size control of monodisperse systems.«'^^'2^'^« As has been mentioned earlier, the nucleation rate, J(= dnjdt\ is given by J^ = Go'^m/'^max ^^ * ^ maximum supersaturation. If <2o is varied, J may adjust itself to the change of j2o by only a slight variation of the maximum supersaturation level by virtue of the critical dependence of J o n supersaturation. In other words, the difference in the maximum supersaturation level with the variation of j2o is very small. However, even if the difference is extremely small, the difference between the maximum supersaturation level and the critical or minimum supersaturation level for nucleation may be proportional to j2o- ^^ addition, if dCldt is proportional to go, the time required for reaching the maximum supersaturation from the critical supersaturation must be the same, irrespective of the Q^ value. In this case.

1. NUCLEATION

47

'^max ^^ ^he maximum supersaturation, or v^^, is a constant, irrespective of QQ, and thus the increase of the mean particle volume of the stable nuclei thereafter is independent of QQ. When both the maximum supersaturation and the mean particle volume of the stable nuclei are independent of QQ, V^ is also independent of QQ, The variation of QQ without the change of v^ may be possible by controlling the release rate of the reactants from an internal reservoir for the generation of the monomer, without changing the initial composition and temperature. On the other hand, i)^ may be varied independently of QQ by addition of some accelerators of growth such as solvents of the solid, or restrainers such as strong adsorptives to the solid surfaces. In these cases, Ai^* is expected to be proportional to QQ and inversely proportional to v^, leading to the precise control of the final particle size by changing a single factor of either go or TJ^.. On the other hand, if we vary QQ by changing the initial composition or the temperature, v^ is normally varied at the same time and hence, in this case, the hnear relationship between «^" and QQ or 1/v^ may not be observed. Nevertheless, there is no direct interdependence between QQ and v^ during the nucleation period, and Eq. (1.4.5) may always hold even if v^ and QQ are varied simultaneously by changing the initial composition or the temperature, as is obvious from the derivation procedure of Eq. (1.4.5). Eq. (1.4.5) suggests an important conclusion that the final particle density, w^°°, is determined only by QQ and i)^, irrespective of the momentary nucleation rate, dnjdt, and the size of the generated stable nuclei. This is in contrast to the conventional elucidation of the final size of colloidal particles in closed systems, in which the nucleation rate is treated as the major factor for determination of the final particle density or, at least, as an essential competitive factor on equal terms with the growth rate, and the supply rate of solute is usually not taken into account. Probably, this is because the concept of the mass balance at a constant feed rate of solute is not involved in conventional nucleation theories. The growth-limited nucleation is the key concept in the fundamental approach to the principle of size control of coUoidal particles. Figure 1.15 illustrates the change of n^ as a function of time. Here tj^ in Fig. 1.15 is a time constant given by tj^ = n^'^/(dnjdt\^ = n^"v^7Qo^m = v^^/v^. As long as QQ can be regarded to be constant throughout the nucleation and early growth periods, it holds for the time t from the start of the nucleation that

48

FUNDAMENTALS n. r1 : 1 \

'1 1 :

•^ CO

c O

'^ CO

/

1 / 1 /:

'/ •

//

if

Q.

^

Time

Fig. 1.15. Change of «+ as a function of time, where r^ is a time constant corresponding to the nucleation period. (From Ref. 23.)

QoVJ = ".^.-

(1.4.6)

where v^ is the average volume of all nuclei at time t. Thus, Eqs. (1.4.4) and (1.4.6) yield

(i) J] 0 +

(1.4.7)

0 +

1-exp

n

\

1

'* }

The change of v^ during nucleation is small, as shown in Fig. 1.16. However, the end of the nucleation is more indefinite than its start. If t until the intersection between the straight lines, vjv^^ = (vjv^^t and vjv^^ = 1, is defined as the nucleation period /^, r^ is given by

^N=—•

(1.4.8)

Since tj^ is determined only by u^° and i)^, the nucleation period as a

1. NUCLEATION

49

determinant of monodispersity is basically independent of QQ,

vjv:

D.

D.

Fig. 1.16. Change of vjv^^ as a function of time. (From Ref. 23.)

1.4.2. AgCI and AgBr Systems a) AgCl Systems The experimental system for the spontaneous nucleation of silver chloride particles is based on an idea that the spontaneous nucleation may be triggered if silver ions are gradually released from their gelatin complex by lowering the pH with hydrolysis of diethyl sulfate (DES), in the presence of chloride ions. In addition, gelatin as a powerful protective colloid is expected to prevent the coagulation of the generated AgCl particles. Moreover, it may be possible to directly follow the change of supersaturation as a function of time with a Ag/Ag2S electrode (Ag2S-coated Ag electrode).^ The reaction is started by adding DES to 400 cm^ of a transparent gelatin solution containing AgN03, KCl, and KOH at pH 10.8 in a 500-cm^ Teflon

50

FUNDAMENTALS

beaker in a water bath at a constant temperature. The solution is continuously agitated by a magnetic stirrer. The change of the silver potential is followed with a Ag/Ag2S electrode connected with a Ag/AgCl reference electrode ([KCl] = 0.010 mol dm^^) through a salt bridge of 1.5 % agar containing 2 mol dm'^ KNO3 solution. The change of pH is followed, at the same time, with a glass electrode with a double junction reference. All experiments are conducted under a red light to avoid the possible photolysis of AgCl particles. Produced particles are all uniform cubes. An inert deionized gelatin is used in all cases as a 2.0 w/v % solution. In the standard system, the concentrations of AgN03, KCl, and KOH in the gelatin solution are 2.0 x 10"^, 2.0 x 10-^ and 7.0 x 10"^ mol dm-^ respectively; the reaction temperature is 25°C; the added volume of DES is 1.0 cm^, which corresponds to 1.8 x 10"^ mol dm"^ in the final solution. In addition, potentiometry was also carried out in a system without KCl, as a control. Out of the results for the effects of the added amount of DES in the range of 0.5 to 2.0 cm"^, of the initial concentration of KCl ranging from 2.5 X 10""* to 4.0 X 10"^ mol dm"^ and of the reaction temperature varying from 25 to 40 °C,^ only the effects of the DES content, or the Q^ value, on the maximum supersaturation ratio and the final particle density, in addition to the results obtained under the standard conditions, will be described in this book. Nucleation under the Standard Conditions Figure 1.17 shows the [Ag*] and pH changing with time in the presence or absence of KCl under the standard conditions. There is almost no effect of KCl on either the pH curves throughout the experiments or [Ag^]-curves until the start of nucleation. We can clearly observe the typical changing pattern of supersaturation in the [Ag*]-curve, like the LaMer diagram illustrated in Fig. 1.13, when KCl is present. Figure 1.18 shows the changes of the supersaturation ratio, 5, and the amount of the precipitated solid, P, obtained from the analysis of the [Ag^]curve of the standard system in Fig. 1.17. The supersaturation ratio, S, is defined as S^Wmi

(1.4.9)

where [Ag^] and [CI"] are the concentrations of free Ag" ions and CI" ions.

1. NUCLEATION

51

Fig. 1.17. Time evolutions of Ag"^ concentration and pH in the standard AgCl system (solid line) and in KCl-free system (dotted line). (From Ref. 23.)

4

2.5 2.0

3

«? E 1.5

CO

2

T3

E 1.0 o

1h

H 0.5

30

60 Time (min)

90

0.0 120

Fig. 1.18. Supersaturation ratio, S, and molarity of precipitated AgCl, P, as a function of time in the standard system. (From Ref. 23.)

respectively; K^ is the solubility product of AgCl. Particularly, the maximum supersaturation, S^, corresponds to S at [Ag*]„,ax- Th^ amount of the precipitated solid, P, can be calculated from the total silver ion

52

FUNDAMENTALS

concentration and the remaining portion linked to gelatin, the latter of which is a function of pH and [Ag*], and thus can be evaluated from the changing pH and [Ag"]. On the other hand, the supply rate of the monomer in the nucleation period, QQ, is obtained from the product of the tangential slope of the curve of [Ag^] vs t at [Ag^]^3^ in Fig. 1.17 and the ratio of the total concentration of free solute to [Ag^], as a function of [CI"]. When Q^ is obtained, i)^ can be calculated from Eq. (1.4.5) with <2o ^uid AZ^" obtained from electron microscopy on the final product. Hence r^ and i;+° (= 4jt(r^^V3) can be calculated numerically from the following relationship with the data of 1)+, S^, y (specific surface energy of AgCl = 100.9 mJ m"^),^ C(«) (solubility of the bulk solid), and D (diffusivity of the solute obtained from the analysis of the growth process): ^»-«p

i'2-yV\ rlRT

(1.4.10)

n

Finally, r^* and the minimum free energy for nucleation, AG„*, are obtained from the Gibbs-Thomson relation; 2Yn, KHaS.

(1.4.11)

and

171

(1.4.12) 3(/?nn5j2

respectively, where AG„* corresponds to the energy barrier for nucleation at the maximum supersaturation. The nucleation time, /^, is obtained from Eq. (1.4.8) with v° and ti,. As a result, one obtains: j2o = 5.95 x 10'* mol dm'^ s"\ 5„ = 3.56, r„* = 1.66 nm, ^GJ = 1.16 x 10''* J, n' = 5.88 x 10^" dm"^ (d' = 552 ran), i)^ = 2.62 X 10^ nm^ s'S r° = 5.28 nm, v+° = 618 nm^ and t^ = 0.236 s in the standard system, in which the total silver ion content [Ag*]^ = 2.0 x 10"^ mol dm-^ [Cl-]o = 2.0 x 10"^ mol dm'^ [DESJo = 1.81 x lO'^ mol dm-\ [gelatin] = 2.0 w/v %, temperature 25 °C, initial pH 10.8, K^ = 1.77 x 10"^°

1. NUCLEATION

53

moP dm"^ initial C(«) = 5.04 x 10"^ mol dm•^ and i) = 1.44 x 10"^ cm^ s'K Effects of QQ Figure 1.19 shows the effect of the DES content on the time evolution

o 4 CO

2.0

1.6

1.0

0.75

0.50 cm3

3 2 1 n

30

60

1 90

1 120

Time (min) Fig. 1.19. Effect of the DES content in 400 cm^ of the gelatin solution on the time evolution of the supersaturation ratio in the AgCl system. (From Ref. 23.)

5

10

15

20

Qo(10-® mol dm-3 s-1) Fig. 1.20. Linear relationship between Qg and n^" by variation of [DES](, vmder otherwise the standard conditions in the AgCl system. (From Ref. 23.)

54

FUNDAMENTALS

of the supersaturation ratio. Obviously, the maximum supersaturation ratio is little affected by the variation of the DES content due to the critical dependence of the nucleation rate on the supersaturation. Figure 1.20 shows an excellent proportionality of n^" to QQ changed by varying the initial content of DES, [DES]o, which supports the assumption that v^ is a constant independent of QQ, b) AgBr Systems The experimental and the analytical procedures of the present AgBr systems are basically the same as the above AgCl systems, based on the gradual release of Ag^ ions from the Ag*-gelatin complex with the reduction of pH by the hydrolysis of diethyl sulfate (DES) in the presence of Br" ions. The change of the supersaturation ratio is monitored by potentiometry with a Ag/Ag2S electrode coupled with a Ag/AgBr electrode ([KBr]=0.010 mol dm~^) as a reference. Produced particles are all uniform cubes.^"* The standard conditions of the AgBr system (400 cm^) are: [AgN03]o = 2.4 X 10-^ mol dm-^ [KBr]o = 2.4 x lO"^ mol dm-^ [KOH]o = 7.0 x 10"^ mol dm-^ (initial pH 10.8), and [gelatin] = 2.0 w/v %, [DES]o = 1.0 cmV400 cm^ (1.8 X 10"^ mol dm"^), and temperature 25°C. Here, K^ = 4.88 x 10'^^ moP dm-^ initial C(„) = 2.04 x 10"^ mol dm"^ and Z) = 1.04 x 10"^ cm^ s'^ Out of the effects of the initial content of DES, initial concentration of KBr, and reaction temperature, examined in the ranges of 0.5 - 3.0 cm^, 3.0 X 10"^ - 3.6 X 10"^ mol dm-^ and 25 - 40°C, respectively,^^ only the effects of the initial content of DES, as well as the results under the standard conditions, will be described. Nucleation under the Standard Conditions The Jesuits in the AgBr standard system are as follows: QQ = 2.95 x 10"^" mol dm-^ s-\ S^ = 3.61, rj = 1.97 nm, ^GJ = 1.79 x 10"^' J, /i," = 1.66 X 10'^ dm"^ {d,^ = 92.9 nm), i), = 5.15 nm^ s\ r," = 2.12 nm, -u^" = 39.9 nm^, and ^^ = 7.74 s. The very low Q^, compared to that in the AgCl system, is due to the much lower solubility product of AgBr, which limits the total silver ion content to a much lower level. The small values of i)^ and r^ are also due to the low solubility of AgBr. However, it is noteworthy that S^ and r^* are rather close to those in the AgCl system. Ejfects ofQo Figure 1.21 shows the time evolutions of the supersaturation ratio with

1. NUCLEATION

55

CO

120

Fig. 1.21. Effect of the DES content in 400 cm^ of the gelatin solution on the time evolution of the supersaturation ratio in the AgBr system. (From Ref. 24.) different contents of DES in the range of 0.50 - 3.0 cm^ in 400 cm^ of the gelatin solution. In a similar manner to the AgCl system, the peak of S appears earlier with the increasing content of DES, due to the earlier arrival at the pH level necessary for the maximum supersaturation. However, in contrast to the AgCl systems, S^ is significantly lowered with the decreasing content of DES or with the decreasing QQ. Since the nucleation rate in general strongly depends on the supersaturation, the maximum supersaturation is almost constant for QQ above a certain minimum value. However, if the supersaturation approaches the maximum supersaturation level with a QQ smaller than the minimum value, QQ must be balanced with the nucleation rate at a supersaturation below the maximum supersaturation. Such a trend is expected in a system in which QQ is very small. In the AgBr systems, the original content of Ag^, [Ag'^Jg, is lower than in the AgCl systems by a factor of 0.12 due to the limitation of the solubility of AgBr, so that QQ values in the AgBr systems are much smaUer than in the AgCl systems. The Qo-dependent S in the AgBr system is observed for this reason. The v^^ and v^ of the stable nuclei generated under a relatively low supersaturation are somewhat larger than those obtained with a sufficiently high supersaturation. Figure 1.22 shows the linear relationship between the final particle density, n^"", and QQ, as in the AgCl systems. The slightly upward deviation of «^" with the increasing QQ is due to the relatively large i)^ in the low QQ

FUNDAMENTALS

56

5

10

15

Qo(10-''0moldm-3s-'») Fig. 1.22. Relationship between Qo and «^" by variation of [DES]o under otherwise the standard conditions in the AgBr system. (From Ref. 24.)

range, which is caused by the enlarged v^^ in the low QQ range. c) Discussion Klein et al}^ determined the critical supersaturation ratio, 5^„.^, by conductometry in a system of homogeneous precipitation of AgCl in the excess of Ag"^ ions, using the hydrolysis of alkylchloride as a source of CI" ions. They found 5,„,= 1.51, 1.73, and 1.85 at 15, 25, and 35°C, respectively, where S = ([Ag'][Cr]/iC,^)^^ by their definition. If this definition of supersaturation ratio is converted into ours {S s [Ag*][Cr]/K^^), the results correspond to 2.28, 2.99, and 3.42, respectively, in fairly good agreement with the results of the present study. However, the temperature dependence of the 5^„.^ is opposite to that of the present study. Though the cause of this contrariety is not clear, the reduction of 5,„.^ with the increase in temperature may be rather reasonable, since the nucleation rate is drastically changed with a slight change of AGJ/kT, and thus AG^*/kT, which is in proportion to r^(ln5^)-^ from Eq. (1.4.12), is deemed to be kept nearly constant with the change of temperature in a given system. Incidentally, the definition of S must be [Ag'][Cl']/K^p, since the supersaturation ratio in terms of the monomer concentration, [AgCl], is originally given by S = [AgCl]/[AgCl](„) = [Ag1[Cr]/X,^.

1. NUCLEATION

57

Tanaka and Iwasaki^^'^^ studied the nucleation process of AgCl and AgBr particles using the stopped-flow technique coupled with a multichannel spectrophotometer, which is useful for studying a rapid process of the order of msec. They found that some specific silver halide complex of AgX^^'""^^" having a rather narrow UV absorption band was formed instantly on mixing a AgC104 solution of the order of IQ-^mol dm"^ with a KX (X = CI or Br) solution of the order of 10"^ to lO'^mol dm"^ followed by its rapid decay of the order of 10 ms while forming an intermediate product having a broad UV absorption and much slower change in the spectrum of the intermediate. They interpreted the intermediate product as monodisperse nuclei, (AgX)„, consisting of n molecules, and determined n from the apparent relationship of -dCldt oc C" at r = 0, for varying initial concentrations, C, of the precursor AgXj""'^- {n = 7-10 for AgCl; n = 3-4 for AgBr). However, since the precursor, AgX^^'""^^", decays in a non-steady way, the quasisteady nucleation theory does not apply in this nucleation process, and thus the so-obtained n value does not correspond to the n* value of the critical nuclei. If the intermediate product comprises monodisperse nuclei of n formed at the expense of the monomeric complex, the relation of -dC/dt oc C may not hold unless we assume that clusters smaller than the n-mers are in the equilibrium, ^f.i/f.iC^ = Z^^^.^c., with each other in the sequential reaction of c^-*C2-*C3-^ **—*c^_^-*c„, and only the last step of c^^^-^c^ is irreversible; Le,, c^^C2^c^^'"^c^_^'^c^. In addition, the concentrations of the clusters from 2 to n-1 must be assumed to have negligibly low equilibrium concentrations. However, there seems to be no reason for justifying these assumptions. On the other hand, we tried to calculate the radii of the critical nuclei and the supersaturation ratios S at the end of the rapid process, using y = 101 mJ m'- for cubic AgCP^ and y = 109 mJ m"^ for cubic AgBr^° on the assumption that the n values thus obtained, 7-10 for AgCl^^ and 3 - 4 for AgBr^°, are equivalent to the size of the monodisperse nuclei, «*. As a result, we found I r* = 0.451 nm and S = 105 for AgCl with n* = 9; r* = 0.358 nm and 5 = 1230 for AgBr with n* = 4. Since their systems are comparable to those of Davies and Jones^° who obtained the critical supersaturation of AgCl around 1.7-2.0 at 25 °C by conductometry on direct mixing of equal volumes of dilute AgNOg and KCl aqueous solutions, the extremely high S values may not represent the actual supersaturation ratios in equilibrium with the intermediate nuclei. It seems more reasonable to attribute the intermediate product with the broad UV absorption to nuclei of a much larger mean size with a broad sizedistribution generated by a non-steady sequential reaction. The subsequent

58

FUNDAMENTALS

slow change of the intermediate nuclei seems to be due to their Ostwald ripening. In the above context, Nielsen^^ also obtained similar n* around 12, and corresponding S greater than 10^ for the nucleation of barium sulfate particles, from measurements of the turbidity as a function of supersaturation, using «* = dlogl/dlogC in the Becher-Doring theory, similar to Eq. (1.3.37) but giving (n* + 1) instead of n*. Since the solutions of barium ions and sulfate ions were directly mixed in this experiment, the fast nonsteady nucleation must have finished immediately after the mixing, and what was measured might be the rate of the slow succeeding Ostwald ripening and/or coalescence of the generated nuclei. This seems to be the reason for the unreasonably small n* value and the extremely high supersaturation. Hence we must note again that some reservoir of monomeric species which releases them in response to their consumption is always needed for the achievement of the quasi-steady nucleation in a closed system. All classical nucleation theories, including the Becher-Doring theory, implicitly presume such a condition. Hayes et al?^ studied the non-steady nucleation process of silver halides by pulse-radiolysis of water containing a methylene halide and silver ions, in which methylene halide was decomposed to release halide ions by the hydrated electrons generated by pulse radiation ranging from 4 ns to 3 [xs. The spectra of the products were recorded with a photomultiplier and streak camera, and it was found that the monomeric precursors of the silver halides were formed within a time of the order of [is, followed by a nucleation process similar to that observed by Tanaka and Iwasaki^^*^^. Their results clearly show that the nucleation process of silver halides by direct mixing of the reactants consists of two elementary steps; le,, the formation of a monomeric precursor in the order of [JLS, and a subsequent transition to nuclei in the order of 10 ms. It should be noted that the mean size of the nuclei is around 10 nm. As to the supersaturation ratios for the nucleation of AgCl particles in open systems, in which high concentrations of reactants such as AgN03 and KCl are continuously introduced into a gelatin solution throughout the precipitation. Strong and Wey'^ reported 1.029 (SOX) - 1.260 (40°C) and Leubner^^ reported 1.024 at 60°C, as estimated from the measurement of the growth rate of AgCl seed particles at the critical supersaturation. On the other hand, for open systems of AgBr particles, some estimated values of the critical supersaturation ratio, S,„.^, have been reported: S,„.^ « 1.5 at 70 °C by Wey and Strong^^ from the size-dependent maximum growth rate

1. NUCLEATION

59

determined by finding the threshold of renucleation in varied addition rates of a AgNOg solution with a KBr solution in the presence of seed particles by the double-jet technique; S^„.^ = 1.2-1.5 at 25°C by Jagannathan and Wey^"^ as the maximum supersaturation ratio determined on the basis of the Gibbs-Thomson equation with their data of the minimum average size of nuclei observed by electron microscopy during the nucleation stage of double-jet precipitation of AgBr. In the calculation of these 5^„., values, they assumed y = 140 mJ m"^. Since the nucleation in open systems corresponds to a survival process of nascent embryos generated in a local zone of an exceedingly high supersaturation near the outlets of the reactants, the critical supersaturation corresponds to the concentration of the solute in equilibrium with the embryos of the maximum size. If we employ the data of Sugimoto^^ on the maximum radius of AgBr embryos in open systems (« 8.3 nm) with the theoretical y for cubic AgBr (= 109 mJ m'^f, the critical supersaturation ratio, 5^„.p is calculated to be 1.36 at 25°C (if y is assumed to be 140 mJ m"^, then S^^.^ = 1.48). Hence, in any case, the critical supersaturations in open systems of silver halide particles are generally much lower than the maximum supersaturations (probably close to the critical supersaturations) in closed systems. This is because the mean radius of the embryos generated in a local zone of an open system is much larger than r^* in a closed system, probably due to instantaneous coalescence of the highly concentrated primary embryos in the local zone of an open system with a high local concentration of the electrolyte.

1.5. The Nucleation Process in Open Monodisperse Systems In this section we deal with the nucleation process in open systems in which reactants are continuously supplied from outside.^ In this sense, open systems can be regarded as particle-formation systems having at least one of their own reservoirs of monomeric species outside them in contrast to closed systems. The theory of nucleation in open systems and its application to the real systems of AgBr particle formation will be described in sections 1.5.1 and 1.5.2, respectively. For simplicity, we assume that the reservoirs of the reactants are both located outside them and that the reactants are introduced simultaneously into the reaction vessel containing a certain volume of well stirred liquid medium. In this case, the outlets of the two reactant solutions are set close to each other in order to confine the initial reaction to a limited local zone. We usually call this kind of

60

FUNDAMENTALS

precipitation mode "double-jet precipitation" from the characteristic way of introduction of the reactants (see section 8.1.1).^'^^'^^ 1.5J. Theory Nucleation Model The most salient characteristic of open systems is that non-steady nucleation occurs instantaneously in a domain into which the reactants are introduced due to the extremely high supersaturation, as is obvious from the discussion in the preceding section. If we call the nascent nuclei "embryos," the introduction of the reactants is equivalent to that of the embryos. The embryos are first dissolved, but as the monomeric species are accumulated by the dissolution of the embryos, some of them may start to grow, owing to the increasing supersaturation. Hence, the growth and dissolution of the embryos must proceed basically through the Ostwald ripening. Since the embryos are constantly furnished and built up as stationary nuclei, the stationary nuclei will reach a quasi-steady state of balance between generation and either dissolution or growth and a part of them will grow to become stable nuclei as the origin of the final product particles. The overall insights may lead to the following general scheme for the nucleation process in an open system. As the addition of the reactants starts, the generated embryos may simply be dissolved to saturate the bulk solution, but only a slight dissolution may be sufficient for a sparingly soluble solid to achieve the saturation. Following a certain time of accumulation of the stationary nuclei in a non-steady state of change in their size distribution function, most of them existing in the size range of the embryos must reach a quasi-steady state of the balance between generation and either dissolution or growth. The nuclei of radius less than the critical radius r* dissolve, while those of radius greater than r* grow. In this stage, the other nuclei that have grown to exceed the size range of the embryos may continue to grow and increase in number at the expense of the stationary nuclei remaining in the size range of the embryos: the former are defined as "stable nuclei," and the latter as "unstable nuclei." Accordingly, the critical radius, r*, initially infinite, must plummet to approach the mean radius of the embryos due to the unilateral dissolution of the generated nuclei, then turn to be lifted by degrees with the growth of the increasing stable nuclei, and eventually reach the maximum radius of the embryos, r^. Since the nucleation may be defined as the formation of the stable nuclei, this is the end of the nucleation stage. After this moment, r* exceeds Tp,

61

1. NUCLEATION

and the growth stage ensues at the same time, during which the growing stable nuclei {r>r^) cease to increase in number and the unstable nuclei (r^p) remain in a quasi-steady state of the balance between their generation and unilateral dissolution. The model of the nucleation process is illustrated in Fig. 1.23, where the bold arrows represent the flows of monomeric species through the solution phase and the growth of unstable nuclei into stable ones. MONOMERS

^-r To

Fig. 1.23. The model of the nucleation process in an open system. (From Ref. 26.)

The Distribution Function and Number of the Unstable Nuclei Generally, the change of the distribution function f(r,t) of the existing nuclei is given by

df

d(ff) .,

,.

(1.5.1)

where NQ is the number of the embryos generated per unit time, and g{r) is the normalized distribution function of the embryos. Since the generation of the embryos occurs in a limited local zone, A^^, is independent of the total volume of the system, and thus it is more convenient to express NQ by the

62

FUNDAMENTALS

total number of embryos rather than by the number concentration in open systems. Therefore, / has a dimension of reciprocal length. In the quasisteady state of the unstable nuclei {dfldt « 0), Eq. (1.5.1) for the unstable nuclei is reduced to diff) ^N.g(r) = 0. dr

(1.5.2)

If we assume that the growth and dissolution rates of the stationary nuclei are limited by the diffusion of the monomeric species, the diffusion-limited growth rate or dissolution rate, r, is given by DV_ r=

IDyVlC^

•(C-C^)-

r-r'

RT

(1.5.3)

r'r'

where C is the concentration of the monomeric species, and Q is the solubility of a particle of radius r, since (

(^yv.\ [rRT)

C = C.exp

2YK„^ 1+——^ r*RT)

(1.5.4)

and C.-C

1+

2Y^] rRTJ

from Eq. (1.3.8) (Gibbs-Thomson equation). (1.5.2) and (1.5.3) yields Jir,t)-

RTN,0

(1.5.5)

The combination of Eqs.

r

r-r ""^

(1.5.6)

-j;,8ir)dr. Vo

r-r

where v^ is the mean volume of the embryos (as (4jt/3)/oVg(r)rfr), given by

1. NUCLEATION

63

«o =

OK

(1.5.7)

and n^ is a dimensionless constant defined by ^ .

QRT

(1.5.8)

Q in Eqs. (1.57) and (1.58) is the feed rate of the reactants, equivalent to that of the monomeric species with dimensions of mol s"\ different from the dimensions of the supply rate of the monomeric species in closed systems, mol dm"^s'\ Here we simulate the nomialized distribution function of the embryos, g{r), by the following cosine function for 0 < r ^ ITQ. sir)

(1.5.9)

=-—cos28A

2

28A

where 2eo is the half width of g{r), and TQ is the meanradiusof the embryos (ro = 28o). For r > ITQ, g{r) = 0. Figure 1.24 illustrates the cosine distribution function. Eq. (1.5.9) can be rewritten with the maximum radius of the embryos r^ (= ITQ) as ^(r)=—sin^ nr

(1.5.10)

\'p;

If we define a dimensionless distribution function of the unstable nuclei, h{p,p*), as h(p,p*) = {rjn^{r,t) with p = rlr^ and/>* = r*/rp, one obtains from Eqs. (1.5.6) and (1.5.10)

Kp,p) = —^'^^-^ «o P-P\ for r* <. Tp and

, sin27rj?-sin2itp* 2lT

(1.5.11)

FUNDAMENTALS

64 g(r) A g(r)= A^cos^

IT mi

2eo

0

r„

Tp

Fig. 1.24. The cosine distribution function. (From Ref. 26.)

Fig. 1.25. The dimensioniess distribution function, h(p, p*), at different p* values. (From Ref. 26.)

h(j>,P ) = — ^ - ^ ^ 1-/7

sin27t/>

(1.5.12)

2TC

for r* £ r„, where u . is the maximum volume of the embryos (= (4jt/3)r ^)

65

1. NUCLEATION

and V/VQ = 5.747. The dimensionless distribution function, h(p,p*), is illustrated in Fig. 1.25 for various p* values, where p* = 0.6814 is the minimum value of/?*, as will be shown later, p * = 1 corresponds to the end of the nucleation, and /?* = oo corresponds to the ultimate stage of the growth at r* = 00. The h(pp*) at/? = 1 becomes zero when/?* = 1, which means that the distribution functions of the unstable and stable nuclei completely split from each other at the end of nucleation. The number of unstable nuclei, «., is given by n.= r7{r,t)dr-nJWp,pldp, Jo

(1-5.13)

JO

Figure 1.26 shows nJriQ as a function of/?*, which has a peak, 4.479, at/?* = 0.6814. The unstable nuclei are retained even in the growth stage (r* > r ), and their number converges to n^ as r* approaches infinity, irrespective of the form of g(r), as derived from Eqs. (1.5.6) and (1.5.13). Thus, if the ultimate n_ is denoted by «_",

u„ •'0 L JO

dr=n^.

Fig. 1.26. n./«o as a function of;;*. (From Ref. 26.)

(1.5.14)

66

FUNDAMENTALS

The Number and Mean Radius of the Stable Nuclei In the quasi-steady state of the unstable nuclei, the rate of increase in the number of the stable nuclei, /i^, is given by

K=ff\r..r=NS:sir)dr. p

(1.5.15)

J r

On the other hand, the increment in the total volume of the stable nuclei per unit time can be written as

" " 3 Jr, af

3 Jr, dr

(1.5.16)

Combination of Eqs. (1.5.3), (1.5.15), and (1.5.16) yields (1.5.17)

' : . - i ^ = xA^,

v". where r+ is the mean radius of the stable nuclei defined by

(1.5.18)

-];>• T is a time constant defined as RTv. X =

(1.5.19)

SIIDYKX and X* is a fraction of the number of the embryos whose radius is greater than r*, and is given by

x*^f:gir)dr.

(1.5.20)

1. NUCLEATION

67

Also, the following mass balance generally holds for the total volume of the precipitate consisting of both unstable and stable nuclei,

No%t- ^^fy^r^ I'^/>''''''

(^-^-^^^

where t is the time from the start of the introduction of the monomelic species, including the non-steady state of the unstable nuclei. Here, t is divided into two components, t_ and r^, as (1.5.22)

t=t.^t^.

where t =

471

(1.5.23)

and (1.5.24)

Hence, t_ and t^ are the total volumes of the unstable and stable nuclei, respectively, in a time unit. If (r-r^lr^ is much less than unity, Eq.(1.5.24) can be approximated as

\fdr 0"0 3

3N^

^["frdr-lfydr

1 . 1 < ^ fdr = ^0^0

Thus, one obtains

K

r=nj — - 2

(1.5.25)

The simultaneous equations (1.5.15), (1.5.17), and (1.5.25) forn^, r^, and t^

68

FUNDAMENTALS

lead to the following differential equation for t^ as a function of/?*:

^4K (1.5.26) 3T1P'(3/»*-2)^(P')-2(XO-X*)3

(3p'-2)[x;-(3p'-2)x'] where q(p*) = r^(r*) and (1.5.27) If t^ is set r, = 0, then n^ = 0 from Eq. (1.5.25), jc* = x^* (= v^v^) from Eq. (1.5.17), r^ = Tpfromthe definition of r^. By use of the initial condition, one obtains the solution of Eq. (1.5.26) as — =exp IKFipldp'

(1.5.28)

T

xp;G(p*)exp|-p.>(pVp* dp% 'Po

where pg* is p* at t^ = 0, 3^0

F(p*)

(1.5.29)

(3p*-2)[jCo-(3p'-2)xT

and G(j>*)^

3[p'(3p'-2)g(p')-2«-x')]

(1.5.30)

(3p*-2)[A:o*-(3/''-2)xT On the other hand, njng and rjr^ can be written as follows from Eqs. (1.5.17) and (1.5.25):

1. NUCLEATION

69

« - _ x^t^-3zp*(x^-x*) «o

(1.5.31)

Tx;i3p'-2)

and T. _ />*[V.-2T(JCO'-JC*)] '•p

(1.5.32)

XOX-3TJ7*(^O'-^')

Thus, /jy«o and r^z-p are now given as functions of p* by use of tjx from Eq. (1.5.28). Furthermore, if t^, n^, and r^ at the end of the nucleation (p* = 1) are designated by t°, n", and r°, then n" and r^" are given from Eqs. (1.5.31) and (1.5.32) withp* = 1 by

Z!l = ! l - 3

(1.5.33)

and r"

f:-2T

^P

^"-3T

(1.5.32)

If we use the cosine function of Eq. (1.5.9) for g{r), both nJriQ and rjr^ change with p* as illustrated in Fig. 1.27, where x^* = VQ/V^ = 0.1740 (i.e., PQ* = 0.6814). The njrig rises sharply with p* and then asymptotically approaches 1.567. The values of t^, n^, and r^ at the end of the nucleation (p* = 1) are given by t" = 4.567T =

4.567/?Tu„ ^,

(1.5.35)

70

FUNDAMENTALS

nr=1.567nn

1.561QRT SnOyVC

(1.5.36)

and (1.5.37)

r:=1.638r

While t^^ and r^"" are independent of Q, but dependent on the size parameters of the embryos, v^ and r^, respectively, n^" is proportional to Q, but independent of these size parameters. Here note that the n^" is the total number of the stable nuclei in a given system, while n^* in closed systems described in the foregoing section is the number concentration of the stable nuclei. Since jc* = 0 at/7* = 1, it follows that AI^" = nQ/(rJr*-l) at/?* = 1 from Eq. (1.5.17). Hence, one obtains — - 1 = 0.638.

(1.5.38)

2.0

1.638 1.567

0.6814

Fig. 1.27. n>o and rjr^ as functions of p*. (From Ref. 26.)

71

1. NUCLEATION

As long as Q remains constant, the ratio of rjr* (=1.638) is maintained throughout the growth stage, due to the constant n^", despite the change in the individual variables of r^ and r*. Nucleation Period In general, tjx is given from Eqs. (1.5.6) and (1.5.23) by

t. ^ H r^f p y C qfp^dpldp = php^dp,

(1-5.39)

where q(p) = r^gir). If the cosine function of Eq. (1.5.9) is assumed for g(r), tJx is determined as a function oip*, as shown in Fig. 1.28 together with tJx from Eq. (1.5.28) and tlx (= tJx + tJx). The t. at p* = 1, C, is given by f." = 0.530T,

(1.5.40)

0.530

0.6814

Fig. 1.28. tJx, uh, and t/x- (= tJx + tJx) as functions of p*. (From Ref. 26.)

72

FUNDAMENTALS

Therefore, the nucleation period tf^ is obtained from Eqs. (1.5.35) and (1.5.40) as ?A,= f - + r :

= 5.097T =

5.097/Jru„ ^

(1.5.41)

Hence, t^ is independent of Q, but proportional to v^. Therefore, the knowledge of v^ is required for determining r^. Conversely, if r^ can be determined by some means, the values of i^p, r^, and r^ are obtained by using Eq. (1.5.41). The sum of nJriQ and nJtiQ changes with/?* as demonstrated in Fig. 1.29 and takes 3.858 atp* = 1. If n_ atp* = 1 is represented by nj" (= 2.291no), the ratio of (n_" + AZ^*) to «^* is given by

n.^n, _ 3.858 -2.5 1.567

(1.5.42)

5 U

4 \-

3.858

y

no

1.567

V"

0

T 1 0.6814

2

3 P*

5

Fig. 1.29. nJriQ, njn^^, and HJHQ + HJUQ as functions of p*. (From Ref. 26.)

1. NUCLEATION

73

Thus, it is possible to determine t^ from experimental data on log(«_+n^) vs t and on n^", since tj^ can be specified by the time when \og{nj-n^ = logw^*" + log 2.5. Discussion Since the unstable nuclei must be in a quasi-steady state of generation and dissolution in the growth stage of the stable nuclei, the quasi-steady state must be established at the latest by the end of nucleation. On the other hand, if one sets r^ = 0 or n^ = 0, r* and jc* take the minimum, TQ*, and the maximum, XQ*, respectively, so that n^ assumes its maximum, as is obvious from Eq. (1.5.15). However, since r* has already been below r^ before n^ reaches its maximum, some stable nuclei must have been produced, and thus n^ ^Q prior to the establishment of the quasi-steady state of the unstable nuclei. In other words, there must be some non-steady state of the unstable nuclei beforehand for the production of the stable nuclei, so that t^ is already greater than zero when the quasi-steady state starts. As a result, r* does not actually reach TQ* = 0.6814rp and, when r* reaches its actual minimum, more or less higher than TQ*, the unstable nuclei may enter the quasi-steady state. However, once the unstable nuclei attain the quasi-steady state, the individual values of t^, n^, and r^ are specified by r* which satisfies the initial condition r* = TQ* corresponding to t^ = 0. Consequently, n^", r / , r/, and t^ take the same individual values, regardless of the starting point of the quasi-steady state. Although n^" is independent of r^, it depends on the form of g(r). For example, the effect of the relative distribution width EQ/KQ of the cosine ftinction of Eq. (1.5.9) on n^7wo is shown in Table 1.2, where Tp = TQ + 2EQ. Also shown is w+7«o in the case of g{r) of the normal (Gauss) distribution with the coefficient of variation of EQ/K^ = 1/3 and r^ = 2ro, where EQ is the standard deviation. In general, though AZ^" increases with reduction of EQ/KQ, Table 1.2. Effects of Co/rg on W+^/WQ for cosine and normal distributions (Source: Ref. 26) Normal"

Cosine Co/ro

0.1

0.2

0.3

0.4

0.5

1/3

n"/no

2.40

1.90

1.71

1.63

1.57

1.67

' For the normal distribution, BQ is the standard deviation.

74

FUNDAMENTALS

the effect is rather insignificant unless zjr^ is extremely small. On the other hand, the size distribution of the embryos is expected to be broad, ranging from the dimensions of a monomeric species (r « 0) to r^, due to the instantaneous nonsteady formation process of the embryos, which may involve some coalescence. Thus it seems reasonable to postulate the maximum ^Jr^ for the cosine function, viz., Ejr^ = 0.5. In the meantime, if we simulate g(r) with the nomial distribution of a large standard deviation, g(0) cannot be zero and, besides, r^ is indefinite. These are the main reasons for the choice of the cosine function with ZJTQ = 0.5 to represent the normalized size distribution of the embryos. However, fortunately, the effect of the form of g{r) on «^" is also rather minor as long as the distribution is broad enough. If {r^-r^lr^ « 1, the approximation of Eq. (1.5.24) to Eq.(1.5.25) is unconditionally allowed. In this sense, the {r^-r^lr^ equal to 0.638 appears a little higher than the criterion. However, as is obvious from Fig. 1.27, the increment of n^ is negligibly small after p* exceeds about 0.85, at which nJriQ « 1.50 and (r^-r^lr^ is as low as 0.35, where the approximation is still good enough. Therefore, the final value of n^, /i^" (= 1.567/io), is nearly free from the error due to the approximation, whereas there may be some allowances for r^" and r^" due to it. 1.5.2. AgBr Systems To confirm the theoretical predictions by experiment, open systems of monodisperse AgBr particles were chosen, because of the availability of the well-established controlled double-jet technique,^*^^'^^ in which two solutions of silver nitrate and potassium bromide are introduced simultaneously into a well-stirred aqueous solution of gelatin, while controlling automatically the flow rate of the potassium bromide solution to keep the activity of silver ions or bromide ions precisely at a certain level through detecting the silverion potential with a silver electrode. Figure 1.30 vShows a scheme of the controlled double-jet apparatus.^ The gelatin works as a protective colloid to protect the existing particles against coagulation and secondary nucleation.^^ In this experimental study, 1 mol dm"^ AgNOg coupled with 1.002 mol dm"^ KBr was employed as a source of the monomeric species, and the feed rate of AgN03 was kept constant in each run. The two reactant solutions were introduced into 1000 cm^ of 2 wt % inert gelatin solution of pH 5.0, containing 10"^ mol of KBr, through separate tubes directed toward and ending at the most efficient mixing zone of the impeller. The pBr (= log[Br']) was always set at 3.0 in the middle of the diffusion-controlled

1. NUCLEATION

75

Fig. 1.30. Controlled double-jet apparatus for the preparation of monodispersed silver halide particles. (From Ref. 8.) range for growth,^^ while the diffusion-controlled dissolution is secured irrespective of pBr.9^'^^ Final Particle Number According to the theory, if the addition rate of monomelic species Q is kept constant, the mean sizes of the unstable and stable nuclei as functions of time must be independent of the Q value, since n_ and n^ are both proportional to HQ, and thus to Q, from Eqs. (1.5.13) and (1.5.31). Figure 1.31 for the mean radius of the observed particles plotted against time, for a variety of addition rates of the monomeric species at 70 °C, substantiates the theoretical prediction. The result proves as well that n^^ is proportional to Q as predicted from Eq. (1.5.36), and shown in Fig. 1.32, in which each data-point at ^ = 10 min was used. Here it must be noted that the unstable nuclei are normally not observed together with the particles grown from the stable nuclei by transmission electron microscopy on an aliquot withdrawn from the mother dispersion in the later stage of the growth period due to their prompt dissolution, despite the use of an efficient restrainer which can halt instantaneously the growth and dissolution of AgBr particles. This seems to be due to the fast Ostwald ripening caused by the great difference in solubility between the exceedingly

76

FUNDAMENTALS

Fig. 1.31. The mean particle radius changing with time at different flow rates of 1 A^ AgN03 at 70 °C and pBr 3.0: (o) 10 cm^ min'\ (A) 20 cm^ min'^ (^) 30 cm^ min"\ (n) 40 cm^ min'\ and (•) 60 cm^ min"\ (From Ref. 26.)

fine unstable nuclei and the fully grown particles from the stable nuclei. Thus, the observed particle number apparently goes down to n^ in the growth period, though the unstable nuclei are sufficiently stable to be observed with the stable ones until the end of nucleation and in the very early stage of the growth period, owing to their small difference in solubility. Hence, the measurement of «^* should be done after the apparent particle number becomes constant in the growth stage. If the right-hand side of Eq. (1.5.36) is divided into two terms as n, =

1.567

QRT

(1.5.43)

the Y may be obtained from the slope of the plot of nj" against {QRTI 8jtDK^C„) changing with temperature, since the temperature dependence of Y over a relatively small temperature range, e.g,, from 40 to 70 °C, may be regarded as almost constant compared to the drastic change of w^" and

1. NUCLEATION

77

0.5

1.0

1.5

2.0

Q [mmol-s'^]

Fig. 1.32. Relationship between Q and /z^" at 70 ""C and pBr 3.0. The closed circles stand for the results of repeated experiments. (From Ref. 26.)

{QRmixDVJO^), in view of Eq. (1.1.12) for f. For the temperature dependence of Z), one may use the data of Strong and Wey^^ from 0.88 x 10"^ m^ s-^ at 35 °C to 2.34 x 10"^ m^s'^ at 80 X (activation energy = 4.70 kcal mol'^). V^ is 29.0 cm^ mol'^ for AgBr. For calculation of the total solubility, C^, as a function of pBr and temperature in the presence of 2 wt % gelatin, the data in refs. 37-41 were used. The result is illustrated in Fig. 1.33. On the other hand, Fig. 1.34 shows the changes of the observed particle number with time at 40, 50, 60, and 70 °C, where Q and pBr were fixed at g = 10"^ mol s"^ and pBr 3.0. The values of D, C«„ and n^" at 40, 50, 60, and 70 °C are summarized in Table 1.3. Figure 1.35 shows the plot of n / against QRT/8TCDV^C^, whose slope gives y = 177 mJ m ' l The theoretical n^* values in Table 1.3 correspond to those calculated from Eq.(1.5.36) with y = 177 mJ m"^. For the y value of AgBr there are many proposals ranging from 100 to 200 mJ m'l The rather wide range of the scattered data may be due to the characteristic shape effect of AgBr (see section 1.1.2), the uncertainty of the diffusivity of solute, and the difference

78

FUNDAMENTALS

in growth mode. In any case, there are some limitations in the determination of Y from nucleation processes.^

10"

10-

10"

S

10-

\^

10-

„.

7(n:

•

>

^

A^^/J

.—-":1.

'"^y^

Fig. 1.33. C„ as a function of pBr and temperature in the absence (solid lines) and m the presence (dashed lines) of 2 wt% gelatin at pH 5.0, (From Ref. 26.)

,.^--'

10"

10-

pBr

10 1 8 40''C

50**C

10 1 7 60*'C -no < ^

A *sF

10 1 6

A. -

0

—o— 70°C

0

1

1

1

2

-o • •

t [min]

"^^

1

-Ky

J

Fig. 134. The observed particle numbers changing with time at different temperatures {Q = 10"^ mol s-\ pBr 3.0). (From Ref. 26.)

79

1. NUCLEATION

Table 13. Values of D, C„, and n," at 40, 50, 60, and 70 °C {Source: Ref. 26) ^ "

«+

Temperature CC)

D

40 50 60 70

9.94x10-'° 1.26x10-' 1.56x10-' 1.92x10-'

(mol dm-^)

Theory

Experiment

1.01x10-' 2.14x10"' 4.34x10-' 8.42x10-'

3.15x10" 1.22x10'' 4.97x10'^ 2.14x10'^

3.20x10" 1.25x10" 4.60x10'^ 2.20x10'^

Q = 10"' mol s-\ pBr 3.0, m.s. IN AgNO3/1.002Ar KBr

1

2

3

4

ORT/STcDVrr^Coo [mJ m - 2 ]

Fig. 1.35. Relationship between QRm7d)V„C„ and n," from the data at 40 °C (A), 50 ^C (V), 60 °C (o), and 70 °C (o), yielding the y value. (From Ref. 26.)

Nucleation Period As anticipated from the theoretical curve of nJnQ-^nJriQ in Fig. 1.29, a peak in the particle number is observed for every curve in Fig. 1.34. This is direct evidence of the nucleation process proceeding through Ostwald ripening since, otherwise, the particle number must simply increase up to a certain constant level. According to the method in the theory, the nucleation period r^ is given

80

FUNDAMENTALS

by t at the intersection of the curve of the logarithm of the observed particle number against t with the straight line parallel to the ^-axis above log n^"" by log 2.5, as shown in Fig. 1.36. The nucleation periods at 40, 50, 60, and 70 °C were found to be 82, 38, 12, and 8 s, respectively, by applying the above procedure to the curves in Fig. 1.34. It is noteworthy that the nucleation periods at relatively high temperatures are much shorter than generally believed before, say, 1-2 min^"*^. This is because the nucleation period had been regarded as the time for the observed particle number to reach the limiting value of w^","*^*"*^ which is theoretically inappropriate since the number of the stable nuclei is fixed much earlier. From Eq. (1.5.41), tj^ is independent of Q i£v^ is independent of Q, and vice versa. Since the changing pattern of r with time is independent of Q from Fig. 1.32, it is obvious that t^ is actually independent of Q, Hence v^ is also independent of Q,

log nT

Fig. 1.36. The procedure for the determination of nucleation time tj^ value with log w+" changing with time. (From Ref. 26.)

Size of the Embryos Combmation of Eqs. (1.5.36) and (1.5.41) yields

1. NUCLEATION

81

5.097i)„ t = E ^ 1.567

(1.5.44)

If the plot of tf^, obtained with change in temperature against the corresponding values of n^'^/QV^, results in a straight line passing through the origin, v^ must be kept constant for the change of temperature, and the concrete value will be given from the slope of the straight line. The respective values of r^ and n^^lQV^ at 40, 50, 60, and 70 °C are listed in Table 1.4, and /^ is plotted as a function of n^^lQV^ in Fig. 1.37, which clearly displays ^^ in proportion to n^^lQV^, and v^ independent of temperature. From the slope of the straight line in Fig. 1.37, v^ = 2.43 x 10"^^ cm^ so that r^ = 83.4 A and TQ = 41.7 A. On the other hand, the mean radius of the nuclei, including both unstable and stable nuclei obtained by extrapolation to r = 0 from those observed in the early stage of precipitation, was found to be about 60 A, regardless of temperature from 40 to 70 °C. Figure 1.38 shows the case at 50 °C, as an example. The definite initial radius reveals the instantaneous formation of the embryos. The observed mean radius of the embryos somewhat larger than determined with Eq. (1.5.44) seems to be a result of the fast dissolution of the extremely small embryos by Ostwald ripening, within a few seconds immediately after sampling until quenching, since the particles less than about 20 A in radius were undetectable even by electron microscopy on samples immediately quenched on microgrids in liquid nitrogen, followed

Table 1.4. Dependence of r^ and n^lQV^ on temperature {Source: Ref. 26) Temperature CC)

r^ (s)

n^^lQV^ (s cm-^)

40

82

1.10 X 10'^

50

38

4.31 X 10''

60

12

1.58 X 10''

70

8

7.59 X 10''

Note: Q = 10"^ mol s'\ pBr 3.0, m.s. liV AgNO3/1.002N KBr

FUNDAMENTALS

82

Fig- 1.37. Nucleation period, tf^ as a function of n,'IQV„: (A) 40 °C, (V) 50 °C, (o) 60 °C, and (o) 70 °C. (From Ref. 26.)

tf?/QV« [lO^'s-cm^]

3

Fig. 1.38. Mean radius of nuclei as a function of time at 50 °C. (From Ref. 26.)

1. NUCLEATION

83

by freeze drying. In addition, the apparent mean radius of the nuclei at the end of the nucleation period was about 100 A, irrespective of temperature. It corresponds to r^, because r^ agrees with r* at the end of nucleation, which is always equal to the mean radius of all nuclei in a system of diffusion control."*^ Hence, the apparent mean radius of the embryos is about 50 A. Since the contribution of the particles less than about 20 A in radius is likewise excluded, the actual TQ must be somewhat smaller than 50 A, and thus the deduced value of about 40 A seems reasonable. From the extrapolation to / = 0 in the open systems for the preparation of AgBr particles, Berry"*^, Claes and Berendsen'*^ and Wey"^^ estimated the mean diameter of the embryos to be 275, 200-600, and 140 A, respectively. From these mean diameters, Berriman"^, Hirata and Hohnishi"^^, and Wey and Strong^^ estimated the supersaturation ratio in equilibrium with the embryos as 1.15-1.3, 1.3-1.4, and 1.44-1.54, respectively. Here, it should be noted that these supersaturation ratios are not the critical supersaturation ratios for the formation of the stable nuclei, since the latter correspond to the supersaturation ratio when r* is at r^ and not at TQ. If we use r^ = 41.7 A in our systems, the mean diameter of the embryos is 83.4 A. If y is assumed to be 106 mJ m"^ at 50 °C,^ the corresponding supersaturation ratio [= &xp(2yVJrQRT)] at 50 °C is 1.73 and the critical supersaturation ratio at r* = r^ is 1.32. The r* value which gives the maximum supersaturation ratio is located between 56.8 (= 83.4x0.6814) and 83.4 A. In any case, the mean radius of the embryos is much greater than the r^* of AgBr particles, ca. 20 A, in a closed system as described in section 1.4.2, considerable coalescence among the generated embryos must be involved in the localized reaction zone in the practical open systems as a result of the use of the highly concentrated reactant salts. Hence, the size of the embryos must be affected to a great extent by the agitation strength and positions of the outlets of the reactants. This may be the cause of the rather scattered data reported on the size of the embryos in the literature.

References 1. T. Sugimoto, J. Colloid Interface ScL 181, 259 (1996); ibid. 183, 299 (1996). 2. T. Sugimoto, J. Phys. Chem. B 103, 3593 (1999). 3. T. Sugimoto and F. Shiba, J. Phys. Chem. B 103, 3607 (1999). 4. T. Sugimoto and F. Shiba, J. Phys. Chem. B 103, 3616 (1999).

84

FUNDAMENTALS

5. T. H. James (Ed.), "The Theory of the Photographic Process," 4th Edn., p. 9. Macmillan, New York, 1977. 6. L. G. Sillen, "Stability Constants of Metal-Ion Complexes, Section I," Special Publication No. 17. The Chemical Society, London, 1964. 7. C. R. Berry, Photogr. Sci. Eng. 20, 1 (1976). 8. T. Sugimoto, MRS Bull. 14 (12), 23 (1989). 9 T. Sugimoto, J. Colloid Interface Sci. 91, 51 (1983). 10. K. S. Lijalikow, Wiss. Phot. Photophys. Photochem. 50, 151 (1955). 11. A. E. Nielsen, " Kinetics of Precipitation, " p. 64. Pergamon, New York, 1964. 12. A. G. Walton, "The Formation and Properties of Precipitates," p. 122. Interscience, New York, 1967; A. G. Walton, J. Am. Ceram. Soc. 48, 151 (1965). 13. F. van Zeggeren and G. C. Benson, J. Chem. Phys. 26, 1077 (1957); G. C. Benson and T. A. Claxton, Can. J. Phys. 41, 1287 (1963). 14. T. Sugimoto, in "Koroido Kagaku (Colloid Science)," (Chem. Soc. Jpn., Ed.), Vol. 1, pp. 135-152. Tokyo Kagaku Dojin, Tokyo, 1995. 15. R. Becher and W. Doring, Ann. Physik, Ser. 5, 24, 719 (1935). 16. Ya. B. Zeldovich, J. Exp. Theor. Phys. 12, 525 (1942). 17. A. E. Nielsen, Acta Chem. Scand. 11, 1512 (1957). 18. A. E. Nielsen, Acta Chem. Scand. 15, 441 (1961). 19. T. Tanaka and M. Iwasaki, J. Photogr. Sci. 31, 13 (1983). 20. T. Tanaka and M. Iwasaki, J. Imaging Sci. 29, 86 (1985). 21. K. H. Schmidt and S. Gordon, Rev. Sci. Instrum. 50, 1656 (1979); K. H. Schmidt, S. Gordon, M. Thompson, J. C. Sullivan, and W. A. Mulac, Radiat Phys. Chem. 21, 321 (1983). 22. D. Hayes, K. H. Schmidt, and D. Meisel, J. Phys. Chem. 93, 6100 (1989). 23. T. Sugimoto, F. Shiba, T. Sekiguchi, and H. Itoh, Colloids Surfaces A: Physicochem. Eng. Aspects 164, 183 (2000). 24. T. Sugimoto and F. Shiba, Colloids Surfaces A: Physicochem. Eng. Aspects 164, 205 (2000). 25. V. K. LaMer and R. H. Dinegar, J. Am. Chem. Soc. 72, 4847 (1950). 26. T. Sugimoto, J. Colloid Interface Sci. 150, 208 (1992). 27. T. Sugimoto, Hyomen 29, 978 (1991). 28. T. Sugimoto, Gendai Kagaku 263, 42 (1993). 29. D. H. Klein, L. Gordon, and T. H. Walnut, Talanta 3, 177 (1959). 30. C. W. Davies and A. L. Jones, Dicuss. Faraday Soc. 5, 103 (1949). 31. R. W. Strong and J. S. Wey, Photogr. Sci. Eng. 23, 344 (1979).

1. NUCLEATION

32. 33. 34. 35. 36. 37.

38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

85

I. H. Leubner, J. Imaging Sci. 29, 219 (1985). J. S. Wey and R. W. Strong, Photogr. Sci. Eng. 21, 14 (1977). R. Jagannathan and J. S. Wey, J. Crystal Growth 73, 226 (1985). T. Sugimoto, Adv. Colloid Interface Sci. 28, 65 (1987). T. Sugimoto, J. Colloid Interface Sci. 93, 461 (1983). J. Pouradier, A. Pailliotet, and C. R. Berry, in "The Theory of the Photographic Process," 4th Edn.,(T. H. James, Ed.), p. 1. Macmillan, New York, 1977. M. J. Harding, J. Photogr. Sci. 27, 1 (1979). J. E. Maskasky, J. Imaging Sci. 33, 10 (1989). G. Russel, J. Photogr. Sci. 15, 151 (1967). P. Lanza and I. Mazzei, J. Electroanal. Chem. 12, 320 (1966). I. H. Leubner, R. Jagannathan, and J. S. Wey, Photogr. Sci. Eng. 24, 268 (1980). C. R. Berry and D. C. Skillman, J. Phys. Chem. 68, 1138 (1964). R. W. Berriman, J. Photogr. Sci. 12, 121 (1964). C. Wagner, Z Elektrochem. 65, 581 (1961). C. R. Berry, PSA Tech. Quarterly, Nov., 149 (1955). F. Claes and R. Berendsen, Photogr. Korresp. 101, 37 (1965). J. S. Wey in "Preparation and Properties of Solid State Materials," (W. R. Wilcox Ed.), Vol. 6. M. Dekker, New York, 1981. A. Hirata and S. Hohnishi, Nippon Shashin Gakkaishi 16, 1 (1966).

CHAPTER 2 GROWTH 2.1. Equilibrium Concentrations of Two-Dimensional Nuclei 2.1.1. On a Bulk-Solid Surface Based on analogy with three-dimensional nuclei, disk-like twodimensional nuclei may be appropriate for the formulation of the theory of nucleation on a solid surface. The free energy of formation of a disk-like two-dimensional nucleus (n'-mer) on a bulk solid surface, shown in Fig. 2.1, from n' monomers in a supersaturated solution, AG'„,, is given by AG;,= -(j)n^-Ht|r;„

(2.1.1)

where (j) (= kJlriS) is common to the supersaturation parameter for the three-dimensional nucleation, and \|)'„. is the surface energy of a twodimensional n*-mer. The new surface for the two-dimensional nucleus on the bulk solid is only its side surface, so that i|f^^, = Inr^-i = 2y^nhv^n\

1 r 1

,

-*

1

Fig. 2.1. Model of a two-dimensional nucleus.

(2.1.2)

2. GROWTH

87

where r' and h are the radius and height of the two-dimensional nucleus, respectively, and v^ is the volume of a monomer. Since the surface energy of a monomer on the solid surface, \j)', is given by V=2Yy^7rADp

(2.1.3)

i|r'„/ = i|;'n'*'^.

(2.1.4)

\|)'„. can be written by

and hence AG'/=-
(2.1.5)

The number of monomers composing a two-dimensional critical nucleus, n'*, is given from dAG'Jdn' = 0 as n'* =

(2.1.6) \2^}

and AG'*, the maximum of AG'„, is given by AG'* = ^ = ^ ^ 4* 2

= 4>n'*.

(2.1.7)

If we rewrite n'* in terms of the critical radius, r'*, ./•

(2.1.8)

r'* = .

This relationship holds, irrespective of h. Also, AG'* can be rewritten as ,2

AG

(2.1.9)

4)

88

FUNDAMENTALS

On the other hand, r* and AG* for a three-dimensional spherical nucleus are given from Eqs. (1.2.13) and (1.2.14) as

r* =

3«*Uj

2YUI

4%

*

\

(2.1.10)

and AG' =

167CViY^

(2.1.11)

3e

Hence, both r'* and r* are inversely proportional to ^, and there is the following relationship between them: J*.

r_ 2

(2.1.12)

Also, note that AG'* and AG* are inversely proportional to ^ and ^^, respectively. This means that the three-dimensional nucleation rate drops sharply as ^ decreases, while the (l)-dependence of the two-dimensional nucleation rate is much smaUer. The equilibrium number-concentration of the two-dimensional n'-mers per unit area, c'J, is given in a similar manner to Eq. (1.2.17) for threedimensional n-mers as /e

(-AGO

-2/3

(2.1.13)

i kT =

WQ

exp

kT

where Vg is the volume of a solvent molecule and tij,^ corresponds to its partial molecular area on the solid surface, on the assumption that the solvent molecules are randomly oriented on the solid surface. 2.12. On a Particle Surface Let us define a supersaturation energy parameter, (|)„, of the concentration

2. GROWTH

89

of monomers in equilibrium with a three-dimensional n-mer, C(„), against that with the bulk solid, C(„), as <|)„ = i t n n - ^ ,

(2.1.14)

which corresponds to the ^ value of C(„) - see Eq. (1.2.4). If
(2.1.15)

The free energy of formation of a two-dimensional nucleus consisting of n' monomers from a supersaturated solution on a small particle composed of n monomers, as shown in Fig. 2.2, is given by

where |X(^ ° is the chemical potential of a monomer in the small particle consisting of n monomers and \i is the chemical potential of a monomer in the supersaturated solution phase - see Eq. (1.3.5). Since f^'!nrf^ = ^--<^^hn-^^--<^^^. on 5

(2.1.17)

from Eqs. (1.3.5), (1.2.12), and (2.1.15), one obtains

Fig. 2.2. Two-dimensional nucleus composed of n' monomers on a small particle of n monomers.

90

FUNDAMENTALS

AG^=-((l)-(j)>/.V«^^^.

(2.1.18)

Hence, the effective supersaturation energy parameter is reduced by (^^ on a small particle as compared to that on the bulk surface. The «'* value of the critical nuclei and AG'*, the maximum AG'„,, are given by *

n^* =

/

P

(2.1.19)

2(<1>-<|)J

and AG' =

*^

(2.1.20)

4(ct)-,)

As a consequence, the number-concentration of the two-dimensional n'mers, c'^f, is given by /c

-2/3

c'^/=Uo exp

i^'^y-w^

(2.1.21)

kT

This c'^f is of importance when we consider the growth of very small particles, especially the growth of the exceedingly small stable nuclei during, and immediately after, the nucleation period. As n approaches infinity, ^^ approaches zero, and thus Eq. (2.1.21) agrees with Eq. (2.1.13) for c'J on a bulk solid surface.

2.2. Nucleation Rate of Two-Dimensional Nuclei The nucleation rate, J', of two-dimensional nuclei of /i' on a small particle of n may be written in a similar manner to J for three-dimensional nuclei in Eq. (1.3.25) as

91

2. GROWTH

J' = k'a'>,Cc", d

n -I

,

<^n'-l

<^n'

(2.2.1)

n-l n-l

n'}

where a'^,_^ is the surface area of the side surface of the two-dimensional nucleus consisting of AI'-1 monomers, and C is the number concentration of all monomeric species in the solution phase. The k'^ is the deposition rate constant of the monomeric species to a unit area of the vertical side surface of the two-dimensional nucleus, so it may be much greater than k^ in Eq. (1.3.16) for the deposition to a unit area of the overall solid surface, because the steps and kinks of the side surfaces preferentially stabilize the deposited monomeric species. The deposition of the monomeric species and the dissolution of the solid may take place exclusively via such specific sites. However, since the difference between k'^ and k^ is only due to the difference in density of such sites, the equilibrium between the deposition and dissolution may be unaffected. Hence, k'^ and the corresponding k'^ may satisfy the following fundamental relationship: (2.2.2)

If we assume a quasi-steady state for the nucleation of the surface nuclei as weU, the nucleation rate J' on a n-mer is given by J'=-

"

(± exd ii'-KW-W'^

" exp

kT

kT

dn

C'n'

(2.2.3)

v<%

where a'^t = Inr'h = l^Tzh\y^n',

Since djc'j

= 0 at «' = oo,

(2.2.4)

92

FUNDAMENTALS L2li J'y>:

C'n'

c'l,

2^fnhv[k',expii^lkT)

'!>'

1/2.

'exp

(2.2.5)

-i^-^y^^n'"' dn'. kT

In this equation, AGV

/j.,i,/«/l/2 -(4>-<|>>'+iJr'n

kT

kT

(2.2.6)

AG"'

kT

/2 AG^* 71

\n y

= -jc' +

.

kT

where 1

;c'=,

\

(2.2.7)

AG"

kT

and AG'* dn' dx'=^ 2 ^ n'*kT ^'

1 A

4>-„ an dn

kT

(2.2.8)

^/

Thus, Eq. (2.2.5) becomes /,.2/3

<'

J'<

c",

k'^ N "''«i(*-*»)

kT

^AG"-
kT

At «' = 1, c'„/c'„' = 1 and x' - - « , so that

(2.2.9)

2. GROWTH

93

/Jexp(-;c'^)d;«'- [\xp(-x'^)dx'=

sf^.

(2.2.10)

As a consequence,

f.^U^

-„

2/3

M

kT

exa

-AG' +<|> ifej

(2.2.11)

Also, it follows from Eqs. (2.2.9) and (2.2.11) that (2.2.12) c ,

^n

By a method similar to the derivation of Eq. (1.3.42), (2.2.13) n

At«' = n'*, jc' = 0 and thus c'Jc'^f = 1/2, which is the same as the result for three-dimensional nuclei.

23. Growth Rate by Surface Reaction In this section, a new theory on the rate of growth by surface reaction will be formulated on the basis of the above two-dimensional nucleation theory. Figure 2.3 is a scheme for the propagation of the two-dimensional surface nuclei. Let us direct our attention to a growing two-dimensional particle A on the crystal surface. A new two-dimensional particle B starts to grow on the surface particle A after an average time-lag of A^. Each statistical growth pattem of these particles starting from one molecule is assumed to be the same, as illustrated in Fig. 2.4. The growth rate of the two-dimensional particles may be small in the early stage, due to the high probability of dissolution of a small particle, but it may finally become constant when they become sufficiently large under a constant supersatura-

94

FUNDAMENTALS

Particle B

Fig. 2.3. Scheme of the propagation of two-dimensional nuclei.

tion. Also, the average time-lag, A/, for them to reach the same radius level is kept constant throughout the growth process because of the same growth pattern. If a two-dimensional nucleus of a certain size, large enough in the constant growth rate range, is regarded as a stable nucleus generated by the quasi-steady surface nucleation, and if the particle A has grown in radius by /•/ from the stage of the stable nucleus by the time the stable nucleus of the particle B is formed on the particle A, the difference r/ is equivalent to r' on the growth curve of particle A at a time ty This t^ is obtained as the intersection of the straight line extrapolated from the linear part of the growth curve of particle B with the r-axis, as shown in Fig. 2.4. If the surface area of particle A at radius /•/ is A^, one may consider that it takes A/, on average, for the origin of a new stable nucleus to be generated in a surface area A^ In this argument, the choice of the size of a stable nucleus is arbitrary, so long as it is in the constant growth-rate range, since J' is equivalent to the flux of the stationary n-mers, which is independent of the n value in the quasi-steady state for the surface nucleation. Also, r^' or A^ are independent of the choice of the size of a stable nucleus, as is obvious from the above procedure. As a consequence, the nucleation rate of the two-dimensional particles, J', is written as

2. GROWTH

95 Particle A

Particle B

>- t

Fig. 2.4. Growth patterns of two-dimensional nuclei A and B starting with a time lag, Ar, between them.

J'-

A^Lt

(2.3.1)

Since we assume a disk-like two-dimensional particle, we have A, = i r r / ^

(2.3.2)

If Tj' is sufficiently larger than the monomer radius, r„', r / = ;k,Af.

(2.3.3)

where k•^ is the constant growth rate of a sufficiently large two-dimensional particle at a given supersaturation. Hence, A/ is given by Af =

1

(J'nklfl^

(2.3.4)

96

FUNDAMENTALS

The nucleation of the two-dimensional particles may occur everywhere on the original crystal surface and on the growing two-dimensional particles, as shown in Fig. 2.3. Nevertheless, the growing front of a twodimensional particle 5 on a preformed two-dimensional particle A cannot overtake the growing front of particle A until the fringe of particle A attaches to those of other particles in the same layer, so long as the growth pattern of the particles A and B is assumed to be the same. Also, when the growing front of particle A stops, by attachment with other particles in the same layer, the spaces between the attached points will be filled by growth of the attached particles before particle B runs over the halted front of particle A, Therefore, the two-dimensional growth of the particles in the second layer is generally not limited by that in the first layer beneath the second layer. In addition, the state of the second layer, in particle number density and total surface area, is statistically the same as that achieved in the first layer at a time A^ before, since each nucleation event in the second layer happens, on average. A/ after the corresponding nucleation in the first layer. In other words, each layer is completed ist after the layer just underneath is completed. As a consequence, the growth rate perpendicular to the crystal plane, dridt, is given by ^ = A = /,(j/jk^^)i/3, dt At

(2.3.5)

where h is the thickness of the two-dimensional particles equivalent to that of each layer. If the surface area of a three-dimensional particle of radius, r, as the substrate of the two-dimensional particles, AQ, is so small as to be comparable to or less than A^ defined above. At is determined by AQ as Ar=—.

(2.3.6)

This suggests that the first layer is completed before the next nucleation on the same first layer. If At' denotes the At in Eq. (2.3.6) in distinction from the At in Eq. (2.3.1) without the restriction of AQ, then At' ^ At, and AQ ^ A^ happens when

2. GROWTH

97 / ^0^

.2^l/3

(2.3.7) //2

= ^1-

In this case, the growth rate of this three-dimensional particle is given by — = A = j'AJi = 4izr^hf. dt At' "

(2.3.8)

This growth mode may correspond to the so-called "mononuclear-layer growth",^ in which dr/dt is proportional to r. Also, one may find that dr/dt is proportional to T and independent of k^, in contrast to Eq. (2.3.5) in the case of v4o »Ay If the supersaturation is kept constant, J' is constant so that r=

(2.3.9) l-^Tihfr^t

where TQ is r at / = 0. This relationship holds for values of r limited by

r^

N

47C

f^il 2 U^^j

1/3

(2.3.10)

On the other hand, when r is sufficiently greater than (kJJ'itf^^H^ dr/dt is given by Eq. (2.3.5) in which dr/dt is independent of the size of the threedimensional particle. This growth mode may correspond to the "polynuclear-layer growth".^ It is now obvious that the mononuclear-layer and polynuclear-layer growth modes correspond to the earlier and later growth modes, respectively, of any kinds of solid. In fact, if ^^ is sufficiently small, the mononuclear-layer growth mode is finished in the early stage of the growth. However, Hk^ is close to, or smaller than, r^VA/, we must take the size of the monomeric species into consideration and Eq. (2.3.3) must be written as r ^=k Af+r ^

(2.3.11)

in which it is possible for k^ to be zero. In the extreme case of k^ = 0, the

98

FUNDAMENTALS

three-dimensional particle growth proceeds only in the polynuclear-layer growth mode from the beginning, and the nucleation theory on solid surfaces in section 2.2 becomes meaningless, because it is based on the quasi-steady growth of the two-dimensional nuclei. In this case, J' may simply be proportional to the difference between the concentration of the solute and the solubihty of the three-dimensional particle, and dridt is given as drIdt = h/At = J'AJi = J'Up where A^ is the partial molecular area of a monomer. Such a situation may typically be observed in the growth of amorphous particles. Even in the nucleation theory of the three-dimensional crystalline particles, this growth mode is presumed to be a reciprocal process of the dissolution in the quasi-steady growth of the stationary nuclei. This presumption may indeed be reasonable for the stationary nuclei close to, or smaller than, the critical nuclei, since they may not need the two-dimensional nucleation for their growth, owing to their rough surfaces. It may not apply, however, to sufficiently large crystalline particles beyond the size level of the critical nuclei, which need the surface nucleation for their growth. The quantity k^ is a function of the supersaturation, and may be given as k, = k,\iC-C^„,),

(2.3.12)

where /:/ is a constant independent of the supersaturation, v^ is the volume of a monomer, C is the number-concentration of the monomeric species, and C(^) is the number concentration of the monomeric species in equilibrium with the three-dimensional particles consisting of n monomers. Since the supersaturation for the growth of a three-dimensional particle is normally low. — ^

(^-^.] l^t^

= exi^

kT

(2.3.13) kT '

so that k^ may be written as fc/uiC(„)(<|)-4>„) kT

(2.3.14)

Inserting this relationship and Eq. (2.2.11) into Eq. (2.3.5), one finally obtains

2. GROWTH

99

-AG '%*^ 3kT

(2.3.15)

where

K=

kTk'rh-'i\'i\'i'c^,

(2.3.16)

2/9

If the three-dimensional particle is sufficiently large, C^^^ in this equation can be replaced by C^^y In Eq. (2.3.15), even if 5 = 10, (t)/AG'* « 1/10, and thus (j) in the exponent of this equation can normally be neglected. For example, even the critical supersaturation ratio for the nucleation of AgBr in open systems is less than 1.5 (see section 1.5) and, normally, S is around 1.2 or less in the growth stage. Also, if we consider the growth of AgBr particles larger than 0.1 (xm in diameter, (])„ can be regarded to be zero. If 5 = 1.2 and ([)„ = 0 with y = 142 mJ m"^ and v^ = 4.8145x10-2^ cm^ for AgBr at 25 °C, (j) = itJln 5 = 7.50x10-^2 J, ip' = 2Y(jrhi; J^^ « 2yn^'\)^^^ = 6.66x10-2° J, r'* = yu,/(l) = 9.12 nm, «'* = {^^12(S^f = 1.97xl0^ and AG'* = \p'2/4(|) = 1.48x10'^^ J. Since r/ must at least be greater than r'*, the maximum radius of a three-dimensional particle in the mononuclear-layer growth mode, TQ, is at least 4.56 nm, since Anf^ = Jc(r/)2 > jt(r'*)2. In this case, (1)/AG'* = 5.07x10"^, so that ^ in the exponent of Eq. (2.3.15) can be neglected. Thus, in most cases, Eq. (2.3.15) may be approximated by

dt

1 kT

5/6

exp

- A G /*

(2.3.17)

3itT

or, using cj) = kT\x)S and AG'* « \|)'2/4(t). ,/2

— = /i:(lnS)^^Sxp| dt

(2.3.18)

\2{kTf\nS

Hence, it is true that the supersaturation dependence of dridt is much smaller than that of the three-dimensional nucleation rate, J, because AG'*

100

FUNDAMENTALS

is proportional only to (InS)'^ with the smaUer ip'in the former - see Eq. (1.2.14) for the three-dimensional nucleation. The quantity dr/dt is often treated as being proportional to 5-1 or (S-lf, where m is a constant greater than unity. However, this approximation is allowed only in a very small range of variation in 5, except for the case of ^^ - 0 including the growth of amorphous particles, as is obvious from the form of Eq. (2.3.18).

2.4. Growth of Polyhedral Particles by Surface Reaction 2.4.1. Surface Energy of a Two-Dimensional Nucleus Although the simple disk model of two-dimensional nuclei is convenient for general discussion of the growth process by surface reaction, we need a rather more sophisticated model for understanding the formation of the specific shape of crystalline particles. Let us consider a two-dimensional nucleus on a flat plane in the form of a truncated regular pyramid, as shown in Fig. 2.5. If LQ, H, h, r, and 6 denote the circumference of the regular polygonal bottom, the height of the original pyramid, the height of the truncated pyramid, the radius of the inscribed circle of the polygonal bottom, and slope angle of inscribed circular cone, respectively, the surface area of the polygonal bottom, AQ, is given by

^

V ^ j ^ 2 2tane

(2.4.1)

and thus the total surface area of the upper and side planes of the polyhedral two-dimensional nucleus. A, is given by A=

2sine

i-|i-A]%cose(i-A

(2.4.2)

Hence, the increment in the surface area by formation of the two-dimensional nucleus, A/4 (= A-A^), is given by ^^^J^c^ sine

^ ° [

M^ll^^L^A. 2HJ sine °

If we define the shape factor of the polygonal bottom plane, P', as

(2.4.3)

2. GROWTH

101

Fig. 2.5. Two-dimensional nucleus on a flat plane in the form of a truncated regular pyramid.

(2.4.4) 4^0 Eq. (2.4.1) is rewritten as AA =

2^p'u,A(l-cose)

(2.4.5)

,/l/2

sin 6

where v^ and n' are the molecular volume of the solid (monomer volume) and number of the monomer composing the two-dimensional nucleus. If Y denotes the specific surface energy of the nucleus, the surface energy of the nucleus, \|)'^,, is given by wj= AA-v = —^

n'' .

(2.4.6)

sine Hence, the surface energy for a monomer of the nucleus is given by

102

FUNDAMENTALS

2/pV(l-cose)Y sin6

(2.4.7)

This is the general formula for \p' for a two-dimensional nucleus. For example, if the surface nucleus is a disk, P' = jr and 6 = 90°, so that \p' = 2Y(JtUih)^^, in agreement with Eq. (2.1.3). Let us evaluate the ip' values of two-dimensional nuclei bound by {100}, {110}, and {111} faces on a face-centered cubic crystal. Figure 2.6 illustrates the respective surface nuclei on cubic, dodecahedral, and octahedral microcrystals.

tS^

Fig. 2.6. Surface nuclei on cubic, dodecahedral, and octahedral microcrystals.

{100}-nucleus Since h = ao/l (UQ = lattice parameter), YIOO = (2/5ao^)vp° {^^ = surface energy of a free monomer) from Eq.(1.1.17), v^ = UQ^/A, fi' - A for a square, and e = 90°, \p\oo is given by
(2.4.8)

{110}-nucleus In this case, h = aJ2y/2, YUO = (2V2/5ao^)\p°, P' = 3V2 for a rhombus with interior angles sin-\2V2/3) and 180° - sin-^(2V2/3), and 6 = 60°, so that

2. GROWTH

103 (2.4.9)

^no=^r

{lll}-nucleus In this case, h = Ug/lVS, Ym = (2V3/5ao^y\\?°, P' = 3V3 for a regular triangle, and 6 = sin-^(2V2/3) = 70.529°, so that

^'nr-W-

(2.4.10)

Thus, i|''ioo:'»i''iio:'4''iii = 2V2 : 2 : 3. For AgBr at 25 °C, ^ = 9.06x10"^ J (see section 1.1.4), so i|)'ioo = 5.13x10-2" J, ip'no = 3.62x10-2" J, and '^\,, = 5.44x10-2° J. For AgCl at 25 "C, V = 7.77x10-2" J, so Tj^'ioo = 4.40x10-2" J, TJJ'UO = 3.11x10*2" J, and \\>\^i = 4.66x10-2" J. 2A2. Growth Rate For a general two-dimensional nucleus, the vertical component of the side surface area is given by (2.4.11)

a'„,= 2p'hM^n'

in place of Eq. (2.2.4) for a disk-like nucleus, and tp' is given by Eq. (2.4.7). Hence, J' is written analogously to Eq. (2.2.11) as

J' =

hi Vo^N

exp Tz \ kT ) \

'

(2.4.12)

kT

where AG'* is given by Eq. (2.1.20) including ip' of Eq. (2.4.7). On the other hand, A^ in Eq.(2.3.2) is generally given by

A^=^'rl

(2.4.13)

where r, is the inscribed radius of the polygonal bottom of the two-

104

FUNDAMENTALS

dimensional nuclei. Also, Af in Eq.(2.3.4) is rewritten as

As a consequence, dridt is given in the same form as Eq. (2.3.15) or the approximate Equation (2.3.17), but K^

- ^

^

^(n)

(2.4.15)

in which C(„) can be replaced by C„ when the three-dimensional particle is sufficiently large. It is of interest that the order in the magnitude of \|)' is ip^^^ > TJ;\OO » ip'iio for f.c.c. microcrystals since tj;^^^ : xp^^o • "^'loo = 3 : 2 : 2V2 as derived in the preceding section, while ^]f^J > \\)^^f > ip^^.^ since \\^„J : ^\>^^f : 'ypri,cub^ = 1.65 : 1.26 : 1.00 (see section 1.1.4). Thus, from a thermodynamic viewpoint, the most stable shape is a cube, the intermediate one is a dodecahedron, and the least stable one is an octahedron. However, from a kinetic viewpoint, the growth rate of the {110} face must be the highest because of the lowest AG'* [= ip'V4((|)-(t)J from Eq. (2.1.20)], and thus the {110} face must soon disappear, leaving the {111} or {100} faces much lower in growth rate. This deductive conclusion may explain the fact that the {110} face has never been observed in the growth of the typical f.c.c. particles of AgBr and AgCl, unless some specific adsorptives are present (see section 1.1.4). Although \\)\^^ is a little greater than \p\oo» ^ ^ ^^us the probability of the formation of the octahedral particles bound by the {111} faces is more or less higher than that of the cubic particles due to the lower growth rate of the former, the difference in \p' is rather small ('vp'm/ip'ioo = 1.06). In addition, it is quite likely that the highest specific surface energy, Yiip is apt to be reduced to some extent by adsorptives and/or electronic relaxation. In this case, a slight reduction of Ym may be enough for reversing the relationship between \\f\^^ and \p\oQ, and hence we can obtain either octahedral or cubic AgBr particles by relatively minor variation of the preparation conditions such as pBr, concentration of ammonia, etc. In the case of AgCl, we usually obtain only cubic particles. Since AgCl particles normally grow by diffusion-controlled growth mode,-'^ they are thought to

105

2. GROWTH

assume an equilibrium form determined by the ratio of Yioo/Ym and not by the surface nucleation rate/'^ as wUl be described in detail in chapter 3. Thus, as long as YVYIU = 1/^^, we obtain only cubic AgCl particles, inespective of the ip' value.

2.5. Diffusion-Controiied and Reaction-Controlled Growth Modes Particles suspended in fluid actively move around as a result of Brownian motion. In this case, one may assume some stagnant fluid layer of thickness 8 around each particle, and a bulk liquid phase outside the stagnant layer. Figure 2.7 shows the relations between the particle radius r of density d and rib for a particle moving in a fluid of density 1 g cm"^ under the influence of convection, where the left part of each curve in which rib goes down with increase in r is due to the Brownian motion, and the final sharp increase of rib is due to convection.^ One may find from this figure that the thickness of the stagnant layer of colloidal particles of radius less than a few microns is determined by the Brownian motion with almost no influence of convection. If a coUoidal particle grows in conditions of supersaturation, the solute may diffuse from the bulk liquid phase containing a uniform concentration of solute, Q , to the particle surface through a diffusion layer corresponding

/

/ / / /

/ /

i l l

i !^

1 ^

I I !

[

/ / / f

F^^~^?***'**» ~^^--^^S2;2^flB,N.„,^

|_

/ /

!

/

i 1 /

."~ 10

10'

10'

Fig. 2.7. Relationships between the particle radius, r, and rlh in aqueous dispersion at different solid densities, d, (From Ref. 6.)

106

FUNDAMENTALS

to the stagnant fluid layer with some concentration gradient, as shown in Fig. 2.8(a), where C. is the concentration of the solute at the solid/liquid interface, and C^ is the solubility of the particle. Figure 2.8(b) is a macroscopic scheme of the diffusion layer around a spherical particle of radius r, where x is the distance from the center of the particle. The total flux of solute, J, passing through a spherical plane with radius x within the diffusion layer is written by Fick's First Law as (2.5.1)

dx where D is the diffusion coefficient of the solute and C is the concentration of the solute at x. Since J is constant irrespective of x due to the steady state of the solute diffusion, integration of C(x) from (r+8) to r with respect to X gives (2.5.2) 0

As has been shown in Eq. (2.3.17), the linear growth rate of a particle by the surface reaction is not simply proportional to {S - 1). However, it may be allowed to be treated so within a limited range of {S - 1). Thus, if k denotes the rate constant of the surface reaction of solute at the particle surface, the consumption rate of the solute at the particle surface by the

a)

Diffusion Uyer

Bulk Solution

b)

a-

Fig. 2.8. Microscopic (a) and macroscopic (b) views of the diffusion model. (From Ref. 8.)

2. GROWTH

107

surface reaction, which is equal to J, is given by J=47cr2jt(C.-C,).

(2.5.3)

By eliminating Q from Eqs. (2.5.2) and (2.5.3), one obtains the general linear growth rate, dridt (= JVJATir\ as

(2.5.4)

Also, from Eqs. (2.5.2) and (2.5.3), it follows that 1 + -^ . C^,-C^ kr\ 5

(2.5.5)

If there is a great difference between D and kr, either the diffusion process or reaction process becomes the rate-determining step of the overall growth process. If D « hr, C,. - C, from Eq.(2.5.5), and thus the rate-determining step is the diffusion process of the solute, in which drIdt is given by ^^^{uL\v(r-C\.

(2.5.6)

If rlh « 1, this relation reduces to

f = :?^(C,-C,). at

(2.5.7)

r

If D » kr, then C. « Q, and thus the particle grows by the reactioncontrolled growth mode, in which drIdt is given by - = ^ySCb-Cr)dt " * '

(2.5.8)

Figure 2.9 shows the solute concentration profiles in the general growth mode, the diffusion-controlled growth mode, and the reaction-controlled

108

FUNDAMENTALS (Diffusion Layer)

(Bulk Solution)

a

Fig. 2.9. The solute-concentration profiles of diffusion-controlled and reactioncontrolled growth modes. growth mode. Since r/b is mostly kept around 0.25 for a colloidal particle of a few microns or submicrons in radius from Fig. 2.7, we may use Eq. (2.5.7) in place of Eq. (2.5.6) in the diffusion-controlled growth mode, by replacing D with an apparent diffusion coefficient equivalent to D multiplied by a factor of ca. 1.25. As a consequence, the characteristic of the diffusion-controlled growth is that the linear growth rate is inversely proportional to the particle radius, while the linear growth rate in the reaction-controlled growth mode is independent of the particle radius. Here it is noteworthy that the diffusion-controlled growth mode generally switches to the reaction-controlled growth mode with decreasing mean size, when kr becomes much smaller than D.

2.6. Criteria for the Distinction of Growth Modes 2.6.1. Magnitude of the Growth Since the diffusion-controlled reaction rate constant is so high diffusion rate of the solute to the

Rate growth is observed when the surface that the growth rate is limited by the particle, it is the growth mode with the

2. GROWTH

109

maximum conceivable growth rate in a given system. In Eq. (2.5.7), the diffusion coefficient D is nomially of the order of 10"^ cm^ s"^ at room temperature in dilute aqueous solutions. Thus, if (C-Q) or (C-C^) is known, dr/dt can be estimated roughly beforehand. If the linear growth rate is much smaller than estimated from Eq. (2.5.7), the growth mode must be the reaction-controlled one. Also, if the growth rate of a particle is lower than that of one different in crystal habit or structure under the same conditions, the growth of the former must be in a reaction-controlled growth mode, since the diffusioncontrolled growth must be the fastest of all possible growth processes under the given conditions. 2.6.2. Activation Energy of the Growth Rate The activation energy of the diffusion coefficient D is normally around 5 kcal mol"^ (= 21 kJ mol"^) in aqueous solutions,^'^ while that of the rate constant of surface reaction depends strongly on the species of the solid and is usually much higher than that of the diffusion coefficient. Hence, this general trend can be used as a subsidiary criterion for distinction between the two growth modes. 2.6.3. Size Dependence of the Growth Rate As has been stated above, the linear growth rate is inversely proportional to the particle radius in the diffusion-controlled growth mode, while it is independent of the particle radius in the reaction-controlled growth mode. These characteristics of the two growth modes can be used as the criterion. 2.6.4. Evolution of the Size Distribution Width Strictly speaki>:g, the term (C - C^) in Eqs. (2.5.7) and (2.5.8) is not independent of the particle size, inasmuch as Q is a function of the particle radius based on the Gibbs-Thomson relation: C,= Q e x p

f^vi'J

(2.6.1)

rRT )

This Gibbs-Thomson effect is significant when the particle size is very small. If 2yVJrRT « 1, Q is approximated as

no

FUNDAMENTALS

Cr=C, 1 . ^ rRT ;

(2.6.2)

c=c\

(2.6.3)

Likewise, C is written as 1+

r'RTl

where r* is the particle radius in equilibrium with the bulk solution. In the case of the diffusion-controlled growth mode, the linear growth rate is written from Eq. (2.5.7) as (2.6.4)

where

2yDiic, Ko^

(2.6.5)

RT

If the size distribution is sufficiently narrow, the change of the standard deviation, Ar, at a certain r* may be given by d(Ar) dt

Kj,Lr(

2 J_ f

(2.6.6)

r*

where r is the mean particle radius. Accordingly, d(Ar)/dt > 0 for r/r"^ < 2, while d(Ar)/dt ^ 0 for r/r* ^ 2. Thus, if the supersaturation is sufficiently high as r /r* ^ 2, the size distribution becomes narrower with the progress of growth (self-sharpening of size distribution). However, if r/r* is less than 2, Ar tends to increase; Le., the size distribution becomes rather broader under a low supersaturation even in the diffusion-controlled growth mode. Figure 2.10 illustrates the behavior of dln(Ar)/dt and dr/dt with the progress

2. GROWTH

111

of the particle growth.^ Actually, Wey and Strong^ observed the selfsharpening at a high supersaturation and the distribution broadening at a low supersaturation in a AgBr particle system in the diffusion-limited growth mode. Hence, if one wants to promote monodispersity in the diffusioncontrolled growth mode, the supersaturation should be set as high as possible below the critical supersaturation level. In the case of the reaction-controlled growth mode, dridt may be written from Eqs. (2.5.8), (2.6.2), and (2.6.3) as dr

„

(2.6.7) r*

r

where ^«=

(2.6.8) RT

Similarly, d(Ar)/dt is given by
(2.6.9)

dt Thus, d(Ar)/dt is positive for any r so that some broadening of size distribution takes place in the reaction-controlled growth mode due to the Gibbs-Thomson effect, though this effect diminishes with the increase of the mean particle radius, r. In addition, since d{Ar)/dt is independent of r*, the broadening effect cannot be avoided by lifting the supersaturation. As a consequence, in the diffusion-controlled growth mode, the GibbsThomson effect becomes negligible with the progress of the particle growth, and the self-sharpening in size distribution becomes most pronounced when r is around 3r*, as is obvious from Fig. 2.10. However, as the particles grow still more, d(Ar)/dt approaches zero, and thus the self-sharpening also becomes ineffective. In the reaction-controlled growth mode, d(^r)/dt soon becomes virtually zero due to the small Kp, and thus Ar becomes constant with the particle growth. Hence, if we observe some self-sharpening in size distribution with the particle growth, the growth mode is the diffusioncontrolled one or, at least, is very close to it. However, it should be noted

112

FUNDAMENTALS

Fig. 2.10. [dfldt\IKj) and [Jln(Ar)/rfr]/Xp as functions of r for diffusion-controlled growth with the infinite diffusion layer. (From Ref. 8.)

that, even in the diffusion-controlled growth mode, the size distribution can be broadened when the mean size is very small {r"^
2. GROWTH

113 X < r +S

Fig. 2.11. Overlapping of diffusion layers. mode. Hence, it can be used as a criterion for discriminating between the two modes. 2.6.6. Solubility Dependence of the Growth Rate If the total solubility of particles is varied by changing the pH, concentration of the anions or cations, concentration of solvents of the solid, etc. under a constant supersaturation ratio, the growth rate of the particles is virtually proportional to the change of the solubility, as long as the growth of the particles is in the diffusion-controlled growth mode, since the effect of change in the composition of the solution phase on the diffusion coefficient is rather small. However, if the change of the growth rate is far from being proportional to the solubility change, this may suggest that the particle growth is in the reaction-controlled mode, in which the growth rate constant itself is directly affected by the change of composition of the solution phase. But it should be noted that a proportional change of the growth rate with the solubility does not necessarily mean the diffusioncontrolled growth. 2.6.7. Chronomals Nielsen^ proposed the so-called "chronomal analysis" for distinction between the diffusion-controlled growth mode, polynuclear-layer reactioncontrolled growth mode, and mononuclear-layer growth mode. These growth modes may be depicted as in Fig. 2.12.^^ However, as mentioned in section 2.3, the mononuclear-layer growth mode is a special growth

114

FUNDAMENTALS DIFFUSION CONTROLLED GROWTH

REACTION CONTROLLED GROWTH Polynuclear Layer Growth

Mononuclear Layer Growth

O H^ or _ DV„(C.-C m^ b e) dt r r: particle radius D: diffusion coefficient V^: molar volume

— = k V (C.-C^)

f = ^V'(VCe)

k : reaction ^ rate constant

k : reaction rate constant

Fig. 2.12. Schemes of the three growth modes. (From Ref. 11.) mode in the very early stage of particle growth, even if it is actuaUy present. Hence, the chronomal analysis is normally limited to the distinction between the diffusion-controlled growth mode and the polynuclear-layer reactioncontrolled growth mode. First, the degree of reaction, § (the ratio between the amount of a substance, that has reacted, and the total amount able to react), is defined as

i^

(2.6.10)

where Q is the initial concentration of solute, C is the concentration of solute at time t, and C„ is the solubility of the bulk solid of the particles. Here, the particle size is assumed to be large enough for the GibbsThomson effect to be neglected. If r„ denotes the final particle radius at ^ = 1 or C = C„, the particle radius, r, at an arbitrary ^ is given by

r = rj so that

(2.6.11)

2. GROWTH

115

d^^}V_dr dt r^ dt'

(2.6.12)

In the diffusion-controlled growth mode, it follows from Eqs. (2.5.7) and (2.6.10) that "^^ =K,dt, ^^''(1-0

(2.6.13)

where Kj =

"• \ ° — : ; : .

(2.6.14)

f — ^ L _ =^.. •'oxV'(l-x) "

(2.6.15)

2

Hence,

In case of reaction-controlled (polynuclear-layer) growth, if we assume that the linear growth rate of a spherical particle can be described by ^^kVJC-CJ",

(2.6.16)

r

(2.6.17)

one obtains

where

—

= Kt,

116

FUNDAMENTALS

j^^^ 3fc^i«(<^o CJ"

^2.6.18)

If one finds a linear relationship in the plots of the left-hand-side integrals in Eq. (2.6.15), or Eq. (2.6.17) for various values of n against t, one may regard that it represents the growth mode. It should be noted that this method for the discrimination of the growth modes is applicable only to extremely simple monodisperse systems, in which the particles are precipitated from a homogeneous solution without renucleation and coagulation after the initial nucleation, and the ratedetermining step of the total reaction is the final deposition process of the solute onto the growing particles. Thus, it cannot be applied to general monodisperse systems, including the gel-sol systems containing a solid reservoir of monomelic species; or homogeneous closed systems in which the rate-determining step of the total reaction is a preceding step for the formation of the monomeric precursor, such as hydrolysis, reduction, oxidation, or thermal decomposition of the starting materials, including alkoxides in sol-gel systems, metal ions, organic or inorganic precursor compounds, etc., and slow release of a metal ion from a soluble complex; or open systems into which monomeric species is introduced continuously from outside. Therefore, the applicable monodisperse systems are quite limited. When we use this method, previous analysis on the rate-determining step is indispensable. Incidentally, if we formulate the chronomal in the mononuclear-layer growth mode, dr/dt may be given by --kVy(C-CJ.

(2.6.19)

Thus, ff»5

where

dx

^

-KJ,

(2.6.20)

2. GROWTH

117

K-^KV.rSCo-CJ".

(2-6.21)

In addition to the fact that the mononuclear-layer growth is limited only to the exceedingly early stage of particle growth, the assumption of a constant solubility of the particles during their growth may not be allowed, because of the drastic change in their mean size of the extremely small particles. As a consequence, Eq. (2.6.20) may have almost no chance to be used in practical systems.

References 1. A. E. Nielsen, "Kinetics of Precipitation." Pergamon, Oxford, 1964. 2. C. Wagner, Z. Elekrochem, 65, 581 (1961). 3. R. W. Strong and J. S. Wey, Photogr, ScL Eng. 23, 344 (1979). 4. T. Sugimoto, J. Colloid Interface ScL 91, 51 (1983). 5. T. Sugimoto, J. Colloid Interface ScL 93, 461 (1983). 6. T. Sugimoto, AIChE J. 24, 1125 (1978). 7. T. Sugimoto, Photogr. ScL Eng, 28, 137 (1984). 8. T. Sugimoto, Adv, Colloid Interface ScL 28, 65 (1987). 9. J. S. Wey and R. W. Strong, Photogr. ScL Eng. 21, 248 (1977). 10. R. Jagannathan and J. S. Wey, J. Crystal Growth 51, 601 (1981). 11. T. Sugimoto, Hyomen 29, 978 (1991).

CHAPTERS HABIT FORMATION 3.1. Surface Chemical Potential of a Crystal Face Gibbs^ suggested that a polyhedral crystal in equilibrium with the surroundings would assume a shape with the minimum surface free energy. This idea was formulated by Wulff as follows:

l i = l? = ...= Ii = ...= i?:inZ,

(3.1.1)

where Y, is the specific surface energy of the zth face of the polyhedral crystal, r^ is the central distance to the ith face, v^ is the molecular volume of the solid, P is the vapor pressure of the polyhedral crystal, and P„ is the vapor pressure of a crystal of infinite dimensions. The vapor pressures are replaced by the solubilities for a crystal in a solution. This is the concept of the "equilibrium form." Meanwhile, Curie"^ considered that a crystal growing under supersaturation might maintain the equilibrium form with its minimum surface energy. However, his hypothesis has never been verified for the growth of real polyhedral crystals of large dimensions. This is partly because there is virtually no difference in vapor pressure or solubility between a large crystal and the infinite-sized one, Le., P « Poo, so that Eq. (3.1.1) loses its significance, and partly because the shape of a crystal is determined by the difference in growth rate of the individual facets for the reaction-controlled growth ("growth form"). As a consequence, we are liable to believe that the equilibrium form is only a conceptual shape, and that it is independent of the growth form. We know, on the other hand, that as the crystal size is reduced to that of the colloidal particle level of microns or less, the small crystals are subject to Ostwald ripening due to only a slight difference in solubility among them with a small size difference. This means that the equilibrium form of

3. HABIT FORMATION

119

colloidal microcrystals may have at least some significance, even when they grow under some supersaturation. Thus it seems necessary to consider the growth form of colloidal particles in conjunction with the equilibrium form, since the growth form is essentially a kinetic shape created by the deviation from the equilibrium between the surroundings and solid phase. For this purpose, a thermodynamic quantity, the "surface chemical potential" of each crystal facet, may be useful/

[octahedron]

tetradecahedron A

[tetradecahedron M]

tetradecahedron B

[cube]

Fig. 3.1. An octahedron, tetradecahedra A, M, and B, and a cube. (From Ref. 4.) Let us consider a simple tetradecahedral microcrystal bound by six {100} faces and eight {111} faces illustrated in Fig. 3.1. The extreme shapes are the octahedron bound by eight {111} faces only and the cube bound by six {100} faces only. Tetradecahedron A is an intermediate form between the octahedron and tetradecahedron M, while tetradecahedron B corresponds to a shape between tetradecahedron M and the cube. The surface chemical potential (SCP) of a {100} face of a tetradecahedral microcrystal, ^oo' is now defined as a change of surface free energy of the microcrystal by the growth of the {100} face with one mole of the component molecules in the direction normal to the {100} face, where the central distance to the {111} faces is fixed. The SCP of a {111} face is also defined in an analogous manner. In the meantime, the surface free energy of a tetradecahedron, \|)^, is written as *n='SiooYioo^^iiiY 111'

(3.1.2)

where S^oo and S^^^ are the total surface areas of the {100} and {111} faces, respectively, of the particle, and YIOO and y^^^ are the specific surface energies of the respective faces. Thus, (x^oo and \i^^^ are given by

120

FUNDAMENTALS

1^100

(3.1.3)

and

"111

(3.1.4)

where n is the mole number of the component molecules of the particle, TJOO and r^•^^ the central distances to a {100} face and {111} face, respectively, V^ the molar volume of the solid, and v the volume of the particle. For tetradecahedron A (with 1/V3 s r^u^r^gg £ 2/V3), i|»„=12(;/3ri„-rioo)2Yj^

(3.1.5)

*12v^[ri„-(\/3ri„-''ioo)'lYin.

V = 4v/3riii - 4(V^rm - r^^^^,

(3.1.6)

and for tetradecahedron B (with 2/V3 £ '•in/''ioo ^ ^3), l|f„=12[2rfoo-3(^/3r,oo-r,l,)^]Y,oo + 12y3(v/3r,oo-riii)2Yiii,

(3.1.7)

D = 8rfoo-4V3(V3r,oo-riii)3

(3.1.8)

Inserting Eqs. (3.1.5)-(3.1.8) into Eqs. (3.1.3) and (3.1.4), moo and n ^ are obtained as summarized in Table 3.1. The chemical potential of solute in equilibrium with a {100} face, m is given by

121

3. HABIT FORMATION

Table 3.1. The surface chemical potentials of the {100} and {111} faces for tetradecahedra A and B (Source: Ref. 4) Tetradecahedra

M-ioo

M-iii

Type A (l/^/3^^„^/^,^^2/^/3) Type B

| 1 = 1*100* l^(-)^

(3.1.9)

where \i^^^ is the chemical potential of one mole of the component molecules (monomers) in the solution phase in equilibrium with a bulk solid, which s equal to the chemical potential, (x°, of monomers in a bulk solid. On the other hand, ^i and \i^^^ are related to the activities of a monomelic species or the solubility products of the dissociated ions as ^ = lx^+/^7lnx;sp100

(3.1.10)

iX(„^=ii^+/?rhijg,

(3.1.11)

and

where \i^ is the standard chemical potential of solute, iC^^°" is the solubility product of the {100} faces of the particle, and KJ^ is the solubility product of the bulk solid. Similar relationships are also obtained for a {111} face. Thus, K^^^^ and^^^^^ are expressed as functions of/^ioo ^^f^nv respectively: •111 = JCexp[^] sp ^j^TJ ; C = 0^p|RT

i.ioo_

3.2. Stable Forms

(3.1.12)

122

FUNDAMENTALS

3.2.1. Equilibrium Forms If both of the {100} and {111} faces coexist within a tetradecahedron, and are in equilibrium with common solution, (3.2.1)

1^100= ^A 111-

Applying this relationship to ^QQ and \i^^^ in Table 3.1, it follows for both types A and B that (3.2.2)

p=a.

where p = ^m/^'ioo ^nd a s Ym/Yioo- Thi^ ^^ in accord with the Wulff theorem in Eq. (3.1.1). In this case, a must be in the range of 1/V3 < a < V3 due to the geometric restriction of a tetradecahedron, Le., 1/V3 [Xjoo for any value ofp within the range 1/V3 < p < V3, as is obvious from Table 3.1. Hence, {100} faces continue to grow with the dissolution of {111} faces until the {100} faces disappear to form a perfect octahedron (p = 1/V3), since the solubility of the {100} faces is always lower than that of {111} faces - see Eq. (3.1.12). If a ^ V3, on the other hand, {111} faces continue to grow at the expense of {100} faces until

a)

0^^1//T

(a-1/2)

b)

l//T
cj

rf>yT

(o(=2)

P I

'^

I

'^

I

"^

I

"*

H

-5

Fig. 3.2. Reduced surface chemical potentials as functions of p. (From Ref. 4.)

123

3. HABIT FORMATION

Table 3.2. Reduced surface chemical potentials, /^QQ and/^u, as functions of p for tetradecahedra A and B with a constant volume (Source: Ref. 4) Tetradecahedra TypeA(l/V3^p^2/V3)

TypeB(2/V3^p^V3)

Ao ^p'-(}f3p-\?

fu ' Via- \

V3;7'-(V3p-l)'

'Vlp-l V3p - I + (V3 - 2/7)a

i-jiVz-p? V3-a

3V3p - 7 + 3(V3 - p)a

l-yCVI-p)'

2-3(V3-/>)2

'W^p

Table 3.3. Equilibrium forms and their surface chemical potentials with the ranges of Ym/Yioo values (Source: Ref. 4) Yni'Yioo value

^iir^ioo .

value

Equilibrium Form

Surface Chemical Potentials of Equilibrium Fonn M-ioo

l/\/3
l/v/3

Oct.

-00

Yiii'Vioo

Tetradec.

^YlOO'^^m/''lOO

73

Cub.

M-ioo

—00

the {111} faces disappear to form a perfect cube (p = V3), since (JL^^^ < [x^oo for 1/V3

124

FUNDAMENTALS

Cubic Range

(O'lii/O'ioo ^73)

/T

1.5 Tetradecahedral Range

(l//3"<«'iii/ffioo^ ^^)

F n i / n 00 (•
1//T 0.5

Octahedral Range

{
Fig. 3.3. p values of the equilibrium forms of AgBr microcrystals plotted as a function of pBr, which were obtained by aging monodispersed cubic particles of r^oo = 0.242 \im (n) or octahedral particles of r^^^ = 0.286 \xm (A) at 50 ^C for 30 min in the presence of 0.5 mol dm"^ NH3 plus 1.0 mol dm^^ NH4NO3 (pH 8.48). (From Ref. 4.) Since the equilibrium form of a microcrystal is determined by the ratio of the specific surface energies of the facets of the microcrystal, specific adsorption depending on the concentration of adsorptives is the predominant factor. In the case of silver bromide particle systems, the concentration of bromide ions in the solution phase [Br'], or pBr (= -log[Br"]), and coexisting solvent of silver halides, such as anmionia, are the determinants. Figure 3.3 shows the/? value (s r^^^/r^Qo) of the equilibrium form of AgBr microcrystals as a function of pBr, attained by aging cubic or octahedral AgBr particles of ca, 0.5 \xm for 30 min at a constant concentration of

3. HABIT FORMATION

125

ammonia (0.50 mol dm"^ NH3 + 1.0 mol dm"^ NH4NO3 at pH 8.48) and temperature 50°C.'^ The p values were determined by a dye adsorption method.^"^ In this system, p values of the equilibrium form of AgBr particles decreased with increase in [Br"], or decrease in pBr, within the range of 1/V3 to V3, and thus/? = a. The considerably low a values around 1.3 below V3 in the relatively high pBr range around 2.5 seem to be mainly due to the adsorption of ammonia, while the decrease of a with increasing [Br"] seems to be due to the additional adsorption of bromide ions. Since both Yioo 3nd y^^^ must be reduced by adsorption of NH3 and Br", the drop in a seems to be due to the more pronounced reduction of y^^^. The extremely small change of a in the pBr range below ca. 1.8 may be explained by the limited adsorption of negatively charged Br" ions and bromide complexes of silver ions (see section 1.1.3). 3,2,2, Steady Forms It is expected that, if particles grow under a constant supersaturation with a constant composition in the solution phase, the particles may assume a constant growth form. We may call such a constant growth form a "steady form."' a) Steady Forms in the Reaction-Controlled Growth Mode Figure 3.4(b) is a transmission electron micrograph of a carbon replica of AgBr particles aged from the octahedral particles in Fig. 3.4(a) at pBr = 1.40 under the same conditions used for attaining the equilibrium forms in Fig. 3.3. Figure 3.4(c) shows particles grown from those in Fig. 3.4(b) by subsequent simultaneous addition of AgN03 and KBr solutions under otherwise identical conditions of pBr, [NH3], [NH4NO3], pH, and temperature for the formation of the particles in Fig. 3.4(b) from those in Fig. 3.4(a). Obviously the growth form in Fig. 3.4(c) is different from the equilibrium form in Fig. 3.4(b). If a particle is in a steady state of growth, it is necessary for the/? value of the particle that p = 0.

(3.2.3)

On the other hand, the differentiation ofp with regard to time yields

126

FUNDAMENTALS

1 ^m Fig. 3.4. TEMs of replicas of AgBr particles in a reaction-controlled steady form (a, c) and in an equilibrium form (b) at the same pBr. They were all formed in a system containing 0.5 mol dm"^ NH3 and 1.0 mol dm'^ NH4NO3 at 50 °C and pBr 1.40, but (a) original octahedra grown by the CDJ method, (b) transformed ones after aging of the original octahedra for 30 min, and (c) sharp-cornered octahedra resumed after CDJ processing of the particles in (b). (From Ref. 4.)

P=-

100

'111

'100

'100/

(3.2.4)

It follows from Eqs. (3.2.3) and (3.2.4) that P=-

111

(3.2.5)

'100

Designating the steady state with an asterisk, Eq. (3.2.5) can be written as •

^iii(C

Cjii)

P =

:

(3.2.6)

^ioo(^ ^loo) unless S^oo = 0 or S^^^ = 0, where k^^^ and k^^^ are the growth rate constants of the {100} and {111} faces, respectively, on the assumption of reaction-

127

3. HABIT FORMATION

0

0.2

0.4 0.6

a)

o(^l//J-

0.8

1.0

(DCl/2)

1.2 1.4

1.6

1.8

0.6

0.8 1, 0

1.2 1.4

1.6

1.8

0.6

0.8

1.0

1.2 1.4

1.6

1.8

Fig. 3.5. The graphic method for the determination of the steady forms. The arrows indicate the directions of the driving forces. (From Ref. 4.)

controlled growth, C the solute concentration in the bulk solution, and C\^^ and C*ioo the solubilities of the {111} and {100} faces of the steady form, respectively. The solutions of Eq. (3.2.6) for/?* are given graphically by intersections of the following two functions ofp: (3.2.7)

V^lll and C-C,,, 3^2 =

C-C 100

CIC^-cjq?{n.JRT)

(3.2.8)

C/C^-ei^(fi,JRT)

where C„ is the solubility of the bulk solid - see Eq. (3.1.12). The broken lines in Fig. 3.5 are the straight lines ofy^ passing through the origins. The solid lines are the curves of ^2 calculated for the three ranges of a: a :^ 1/V3, 1/V3 < a < V3, and a ^ V3 represented by a = 1/2, 1, and 2, respectively, where we assume that C/C^ = 1.3, v = 1 \AnP, and y^oo = 1^^ mJ m'^.

128

FUNDAMENTALS

In Fig. 3.5, the intersections of P^ and P2 satisfy Eq. (3.2.3). However, the following condition is required additionally for a particle to be stable: ^<0. dp

(3.2.9)

This relationship means that if/? is slightly displaced from/^*, the particle is subject to a force of pulling back to /?*. Thus, only P^ out of P^ and P2 satisfies Eq. (3.2.9). Furthermore, P3, where/? = 1/V3 or V3, satisfies Eqs. (3.2.3) and (3.2.9) independently of Eq. (3.2.6). The F3's in Fig. 3.3 correspond to the equilibrium forms in the respective a ranges. Hence, there is only one steady form in the case of 1/V3 < a < V3, while there are two steady forms in some ranges of ^ni/^ioo ^^ ^^e cases where a ^ 1/V3 and a ^ V3, suggesting the possibility of simultaneous formation of two dif^" rent steady forms in these a ranges. The/? values of the steady forms are ^ )wn in Fig. 3.6 as functions of ^ni/^ioo ^^^ ^he respective a ranges. The approximate p* values are listed in Table 3.4 for the special cas of 2yVJrRT « 1 and 1.2 < C/C^ < 1.5. For 1/V3 < a < V3, all particles in a growth system have a single p* of P^ during their growth [see Figs. 3.5(b) and 3.6(b)], and /?* is given approximately by p =— v3

for

= 'hll ^100

^100

^— , v3

for±<^<^, v^

^•loo

^100

Figure 3.7 shows the pBr-dependence of the growth rates of the {100} and {111} faces of AgBr microcrystals, obtained from the growth rates of cubic particles 11^^^, (T^QQ = 0.462 \im) or octahedral particles 11^^^ {r^^^ = 0.376 [xm) by dissolution of a large amount of much smaller cubic particles Ub (''loo = 0.115 fim) under the conditions of [NH3] = 0.50 mol dm"\ [NH4NO3] = 1.0 mol dm•^ pH = 8.48, [gelatin] = 1 wt%, and temperature = 50 °C. It is obvious from this figure that the growth rates of the {100} and {111} faces show opposite pBr dependences in this system, and that the change of the solubility of the bulk solid with pBr is not proportional to the

3. HABIT FORMATION

129

Fig. 3.6. Steady forms as functions of A:,ii/A,oo for the three ranges of a. (From Ref. 4.) Table 3.4. Approximate p* values of the steady forms in a case of lyVJrRT « 1 and 1.2 < C/C^ < 1,5 {Source: Ref. 4) ^iiAoo scale 0

a <. l/>/5

\l>fh
^ ^


y/k\ - CJC)

i/>/5(i - c^C)

J3

iM

V3

1/^

\/S \/J3

v5

l/>/5

kiii/kioo

y/3

^Ill/^IOO y/3

1

^ ^

changes of the growth rates of both faces. This means that the growth of these faces is in the reaction-controlled mode (see section 2.6.6). Since r^^Q

130

FUNDAMENTALS —q 10

p

\

h

\

4 ^

p

J

a

F^

n

H -|

[r

\(100)

H H

• « - / ' " "

P

H

y

r r

J

J /

1

> -J

/

4n

1 ^'^

10

1

1

1

Solubility

Li.. 1 1 1 1 1 i 1

1 1.5

2.0 PBr

2.5

Fig. 3.7. The pBr dependence of the growth rates of the {100} and {111} faces of AgBr microcTystals in the presence of 0.5 mol dm"^ NH3 plus 1.0 mol dm"^ NH4NO3 at 50 °C, where the particles were grown at the expense of much smaller coexisting particles. (From Ref. 4.)

and f^^^ were obtained from the growth rates of the sufficiently large cubic and octahedral particles, respectively, the solubilities of these {100} and {111} faces are approximately equal to the solubility of the bulk solid, and thus one may consider that rin/^'ioo * ^m/^ioo- Hence, the broken line of ^111/^100 i^ Fig. 3.8 was obtained from the ratio of r^^^ to T^QQ ^^ ^ function of pBr in Fig. 3.7. On the other hand, within the range of 1/V3 < ^m/^ioo < V3, the/7 value (= r^^^lr^^ of the steady form, /?*, must be nearly equal to A:in/^ioo> while p* must be almost equal to 1/V3 or V3 for k^x\l^ws ^ 1/^^

3. HABIT FORMATION

131

Cubic Range

'

(kiii/kiooi»^)

i^^--

kin/kioo

vj

1.5 h

1.0

0.5

Tetradecahedral Range

/ ^ Steady

(l//3
j

\

\

rm/rioo _\

,'^—

Equilibrium

X\\\IX\Q^

o /o/

L

/ Octahedral Range

/

(kin/kiooil/ZJ)

/

// / / L...

1

1

-fc---1--

1 . 1

1

1

1

1

1

1

L . „ . l ..J

L

pBr

Fig. 3.8. The /:n#ioo ratio, steady r^^^lr^^ ratio, and equilibrium rm/rloo ratio of AgBr particles, as functions of pBr, in the system at 50 °C containing 0.5 mol dm"^ NH3 plus 1.0 mol dm"^ NH4NO3. The particles were grown at the expense of much smaller coexisting particles for the ^ni/^ioo and steady r^^-Jr^^^ ratios, while the data for the equilibrium r^^-^lr^^^ ratio are from Fig. 3.3. (From Ref. 4.) or ^111/^100 ^ ^3, respectively, since a is in the range of 1/V3 < a < V3. The solid line in Fig. 3.8 was obtained from this theoretical prediction. The circles in this figure are data of/?* values actually determined by the dyeadsorption method for AgBr particles obtained by simultaneous introduction of 500 cm"^ of 1.2 mol dm"^ AgNOg and almost the same volume of 1.2 mol dm"^ KBr solution containing 1.0 mol NH3 and 2.0 mol dm"^ NH4NO3

132

FUNDAMENTALS

Fig. 3.9. AgBr particles prepared at pBr 1.40, 1.90, and 2.30 in the presence of 0.5 mol dm"^ NH3 plus 1.0 mol dm"^ NH4NO3 at 50 °C by the CDJ method. (From Ref. 4.)

into 1000 cm^ of an agitated aqueous solution containing 2 wt% gelatin, 0.50 mol NH3, and 1.0 mol dm'^ NH4NO3 (pH 8.48) without seed particles at 50 "^C. Thus, the concentration of ammonia and the temperature in the reaction vessel are the same as those for the growth of the AgBr particles in the closed systems for Fig. 3.7. In this experiment, the addition rate of AgN03 was kept constant, and that of the KBr solution was regulated potentiometricaUy with a silver electrode to keep the pBr of the dispersion constant. This method is referred to as "controlled double jet (CDJ) technique."^"^^ Their excellent agreement with the solid line verifies the theoretical prediction for the steady form in the case of 1/V3 < a < V3 in Fig. 3.6. The pBr dependence of the/? value of the equilibrium form in Fig. 3.3 is also shown again by the dotted line for reference. Figure 3.9 shows transmission electron micrographs of carbon replicas of octahedral, tetradecahedral, and cubic particles of AgBr prepared by the simultaneous addition of AgNOj and KBr solutions at pBr 1.40, 1.90, and 2.30 under the same conditions as for the growth of AgBr particles represented by the circles in Fig. 3.8. The shapes of these AgBr particles in Fig. 3.9 are typical steady forms in the reaction-controlled growth mode. Stranski and Kaischew^^ suggested that the activation energy for the formation of the two-dimensional nuclei on a {111} face is almost zero in the absence of adsorption, whereas it is high enough on a {100} face. Thus {100} faces are developed to form a cubic shape. However, if some molecules are adsorbed to {100} and {111} faces, the peripheral surface energy of the two-dimensional nuclei on the {100} faces is lowered by the

3. HABIT FORMATION

133

preferential adsorption of the molecules to their peripheral steps whereas that of the {111} faces is, on the contrary, raised by the lower adsorption to the peripheries of the two-dimensional nuclei than to the rest of the {111} planes.^^'^"* Thus, any adsorption to {100} faces and/or {111} faces favors the development of {111} faces to give an octahedral shape. Moisar and Klein^^ used the Stranski theory to explain the morphological change of AgBr microcrystals, from cubes to octahedra, in terms of the adsorption of bromide ions to both {100} and {111} faces on the basis of their experiments on the adsorption isotherms of bromide ions to these faces. This may be a reasonable explanation for the opposite pBr dependences of the growth rate constants of the {100] and {111} faces shown in Fig. 3.7. If only one steady form is found in a reaction-controlled growth system of polyhedral particles bound by more than one kind of faces, there may be the following general relationship from the above discussion: r,:r,:r,...:r,...:r„=k,:k,:k,...:k,...:k„,

(3.2.10)

where r. and k. are, respectively, the central distance to and the growth rate constant of the faces of the zth kind. Here it should be noted that, if two kinds of steady forms are found in a system at the same time, or if the polyhedral particles are bound by only one kind of faces, this relationship does not necessarily hold. b) Steady Forms in the Diffusion-Controlled Growth Mode If a polyhedral microcrystal grows under ideal diffusion control where both growth and dissolution at the interface are sufficiently fast compared to the diffusion of solute toward the interface, the concentration of solute at interface must be virtually in equilibrium with each face of the microcrystal. Accordingly, the steady form of a polyhedral microcrystal which grows under ideal diffusion control must be the same as the equilibrium form.^^ In other words. Curie's hypothesis, referred to earlier, is valid for a fine microcrystal which grows under the ideal diffusion control mechanism. This model for a tetradecahedral particle in 1/V3 < a < V3 is illustrated in Fig. 3.10. Even if the solubility levels of the two kinds of faces a and b, C^ and Q, are initially different, as shown in the left-hand diagram, both of them will finally become equal to the concentration level, C-, of the solute at the solid/liquid interface after recrystallization by dissolution of face a and growth of face fo, as shown in the right-hand diagram. Therefore, p * for a tetradecahedral microcrystal bound by {100} and/or {111} faces in the

134

FUNDAMENTALS

diffusion-controlled growth mode is written as: . 1 p =—

. 1 for OS — ,

= Illl for±
= v/3

V3

for

a^y/3.

Ca I

Ca. Cb

Cfe

1^

non-steady state

steady state

Fig. 3.10. Concentration levels of solute for diffusion-controlled growth of a tetradecahedral particle in a non-steady state and in the steady state for 1/V3 < a < V3. The two kinds of faces are designated by a and b. The arrows indicate the directions of mass transfer of solute. (From Ref. 15.)

Figure 3.11 shows the critical growth rates of the {100} and {111} faces as functions of pBr, obtained by the CDJ technique with simultaneous introduction of AgNOj and KBr solutions into 1000 cm^ of an agitated solution, containing 1 wt% inert gelatin (pH 5.80) and 2.35x10"^ mol dm'^ monodisperse cubic AgBr particles (T^QQ = 0.255 \im) or octahedral AgBr particles (r^^^ = 0.221 \mi), at 60 °C in the absence of ammonia. Each critical growth rate, corresponding to the maximum growth rate under the critical supersaturation for nucleation, was determined by varying the addition rate of AgNOj at each pBr, in the presence of the seed particles, to detect the threshold of renucleation with an electron microscope.^^ Since the spontaneous nucleation occurs at a certain level of the supersaturation ratio irrespective of pBr, the supersaturation ratio is common for all pBr values.

135

3. HABIT FORMATION 20 ^1

^\ \ ^ \ >- \ h \ \ \ k

1 h

-

\

\ \

10

,-, ^ c S 8 o ^

\ ^\

6

4-*

\\

± S

"^^

\

\\

/ //

/^

/

X

• •

*

N

^

\

1

A

1

\

1

/ / •

, 1.5

2.0

_

iJc

.

.

If ^

^

Cub

1

1

2.5

1

1

4.0

pBr

Fig. 3.11. Critical growth rates of the {100} and {111} faces of AgBr as functions of pBr in an open system at 60 °C containing no ammonia. (From Ref. 15.) The remarkably different pBr dependence in the growth rate of the {100} and {111} faces in Fig. 3.7 is mainly due to the absence of anraionia that reduces the growth rate constants of each face by its adsorption.^^ Figure 3.12 shows the pBr dependences of the ratio of the apparent growth rate constants, K^^^IK^^^, obtained from r^i and r^^^ in Fig. 3.11 and/?* (= steady ^iii/^ioo) obtained from the dye-adsorption method of the particles actually formed by the CDJ technique in the absence of the seed particles. Since /?* changes from 1/V3 to V3 while K^-^^IK^,^^ stays close to unity, it is obvious

FUNDAMENTALS

136

Cubic Range

(kiii>kioo>v'3')

/T

^ ic

L—steady

rui/rioo

/ / 0/ //

*

/ /

0.5

/

/

/ •''

1

^ 1/T

/

/ /

/ /

1

1 Octahedral Range

I

1

1

(KUI/KIOO ^ 1 / ^ )

1

1

1

Fig. 3.12. ^111/^100 ^"^ ''in/^'ioo ratios as functions of pBr for AgBr particles prepared in the absence of ammonia at 60 °C. (From Ref. 15.) that the steady form is not a consequence of the reaction-controlled growth. Particularly, in the pBr range from 2.6 to 3.5 in which Kxw^^m = 1' ^^'^ of the {100} and {111} faces are in the diffusion-controlled growth mode. Hence, the morphological change from octahedron to cube is due to the change of Ym/Yioo fr^i" 1/^3 to V3 with increasing pBr. The growth rate constant of the surface reaction of the {100} face must increase with decreasing pBr, so that the critical growth rate of the {100} face in the pBr range below 2.6 in Fig. 3.11 must be the diffusion-controlled growth rate. At the same time, the critical growth rate of the {111} face in this pBr range gradually deviated downwards with reducing pBr, due to the increasing

137

3. HABIT FORMATION

Fig. 3.13. AgBr particles prepared by the CDJ method at pBr 2.0, 2.8, and 4.0 in the absence of ammonia at 75 ^C. (From Ref. 17.)

contribution of the reaction-controlled mode. Similarly, the critical growth rate of the {111} face in the pBr range above 3.5 corresponds to the diffusion-controlled growth rate, while that of the {100} face deviates downwards with increasing pBr by the increasing contribution of the reaction-controlled mode. Figure 3.13 shows transmission electron micrographs of carbon replicas of AgBr particles prepared by the CDJ technique, in the absence of anunonia at 75 °C at different pBr's.^^ Although the octahedral particles prepared at pBr 2.0 and the cubic particles at pBr 4.0 involve some contribution of the reaction-controlled mode in the growth of their respective {111} and {100} faces, the tetradecahedral particles prepared at pBr 2.8 are of a typical diffusion-controlled steady form equivalent to the equilibrium form. The steady form of a general polyhedral particle bound by more than one kind of faces after a typical diffusion-controlled growth can be written as r,:r^:r^...:r..

^ = Yi:Y2-Y3-*Yr-*Vn'

(3.2.11)

where r. and y^. are, respectively, the central distance to and the specific surface energy of the faces of the zth kind. This relationship agrees with the Wulff theorem in Eq. (3.1.1) for the equilibrium form. Here, it should be noted that if the particle is bound by only one kind of faces this relationship does not necessarily hold.

138

FUNDAMENTALS

References 1. J. W. Gibbs, "Collected Works." Longmans, New York, 1928. 2. G. Wulff, Z. Krist. 34, 449, (1901). 3. P. Curie, Bull. Soc. Fr. Mineral. 8, 4 (1885). 4. T. Sugimoto, J. Colloid Interface Sci. 91, 51 (1983). 5. A. H. Herz and J. O. Helling, Kolloid-Z. Z. Polym. 218, 157 (1967). 6. W. L. Gardner, D. P. Wrathall, and A. H. Herz, Photogr. Sci. Eng. 21, 325 (1977). 7. T. Tani, "181st ACS Nat. Meet., Atlanta 1981." Am. Chem. Soc, 1981. 8. C. R. Berry, J. Opt. Soc. Am. 52, 888 (1962). 9. C. R. Berry, S. J. Marino, and C. F. Oster, Jr., Photogr. Sci. Eng. 5, 332 (1961). 10. C. R. Berry and D. C. Skillman, Photogr. Sci. Eng. 6, 159 (1962). 11. E. Moisar and E. Klein, Ber. Bunseges. Phys. Chem. 67, 949 (1963). 12. I. N. Stranski and R. Kaischew, Phys. Z. 36, 393 (1935). 13. O. Knacke and I. N. Stranski, Z. Elektrochem. 60, 816 (1956). 14. I. N. Stranski, VDI-Berichte 20, 5 (1957). 15. T. Sugimoto, J. Colloid Interface Sci. 93, 461 (1983). 16. J. S. Wey and R. W. Strong, Photogr. Sci. Eng. 21, 14 (1977). 17. T. Sugimoto, MRS Bull. 14(12), 23 (1989).

CHAPTER 4 RECRYSTALLIZATION 4.1. Phase Transformation Let us consider a solid species suspended in a liquid. If a more stable species is known, it is at least thermodynamically possible for the less stable species to be transformed into the more stable one. The stability of a solid species in a liquid may be measured by its solubility, Le., the higher solubility corresponds to the lower stability. For example, amorphous ferric hydroxide has a higher solubility than goethite (a-FeOOH) and akaganeite O^-FeOOH), and the latter have higher solubilities than hematite ( a F^a^a)-^'^ In this case, ferric hydroxide is the least stable and hematite is the most stable among them. The stability also depends on the composition of the solution phase and on the temperature as well. In any case, when two different solid species of different solubilities coexist in a liquid system, the species of higher solubility will be dissolved and the other one of lower solubility will grow by deposition of the solute released from the former. In general, the concentration level of the solute in the solution phase is located somewhere between the two solubilities. If the dissolution rate constant of the higher-solubility species is much greater than the growth rate constant of the other of lower solubility, the steady concentration level is close to the higher solubility level. In this case, the rate of the total recrystallization process is limited by the deposition process (depositioncontrolled growth). In contrast, if the growth rate constant is much greater than the dissolution rate constant, the concentration level is close to the lower solubility level, and thus the rate-determining step is the dissolution process (dissolution-controlled growth). These relations are illustrated in Fig. 4.1. The phase transformation through this kind of recrystallization is of great importance for the formation of monodisperse particles in which some unstable vsolid species is used as a reservoir of the monomers of the final stable product, as will be described in detail in Part 2. Also, the

FUNDAMENTALS

140

distinction between deposition control and dissolution control is essential for the regulation of the supersaturation during the growth of monodisperse particles.

Dissolving Solid

Solution Phase

Growing Solid

Fig. 4.1. The concentration levels of solute in the deposition-controlled and dissolution-controlled growth modes.

4.2. Ostwald Ripening The Ostwald ripening is nonnally known as a growth process of particles by the dissolution of coexisting smaller particles, as illustrated in Fig. 4.2. The phenomenon was first described by Ostwald,^ and afterwards named "Ostwald ripening" by Liesegang."* This process is caused by the solubility difference between large particles and the smaller particles of a higher solubility due to the Gibbs-Thomson relation in Eq. (2.6.1). Since it is important not only for the fundamental study of particle growth but also for industrial applications, there are a number of theoretical^"^"^ and experimental^^"^^ studies. Practically, it can be used for the preparation of uniform

4. RECRYSTALLIZATION

141

Fig. 4.2. Concept of the Ostwald ripening.

particles in a closed system in which relatively large seed particles grow at the expense of a great amount of small particles. In fact, the supersaturation during the particle growth can be kept constant without renucleation in this simple closed system, and the supersaturation level can be controlled by changing the size of the smaller particles. These advantages were fully used for fundamental studies of habit formation of polyhedral microcrystals^^'^^ and the growth mechanism of tabular grains of silver bromide.^^"^^ In addition, the preparation of silver halide particles in open systems, widely used in the photographic industry, is essentially based on the Ostwald ripening process, in which exceedingly fine embryos - instantly generated by continuous introduction of silver and halide ions - act as a source of silver and halide ions, through dissolution, for the growth of the stable nuclei."^^'-^^ The fundamental theory of Ostwald ripening in closed systems was established by Lifshitz and Slyozov^ and by Wagner^. It is often referred to as "LSW theory." In this section, only the essence of this theory will be described from Wagner's paper.^ 4.2.1. Diniision-Controlled Ostwald Ripening For simplicity, both the dissolution and growth are assumed to be controlled by the diffusion of the solute. In this case, the size distribution function in the steady state of dissolution and growth is given by

142

FUNDAMENTALS /

/(r,0 = const {Utlz'j,)'^

3 yi/3

2

,7/3

13 +p

3

:;:"P

U J

(

exp -P 3 r"P

\

(4.2.1)

U i

where p s rlr"*", r is the particle radius, r* the particle radius in equilibrium with the solute, t the time, and T'p a time constant given by / ^ 9ro*'/?r

(4.2.2)

8Y/)C.^^^

where r^* is r* at r = 0. This equation holds for p ^ 3/2, while /(r,0 = 0 for p ^ 3/2. If the arithmetic mean radius of the particle system is denoted by r, it is given by (4.2.3)

r^r . Also, r*{t) is given by r\t) = Tl{Utlx'j,yi' 1/3

\l/3

^yDcjj

(4.2.4)

9RT

Hence r* or r is proportional to the cube root of t. 4.2.2. Reaction-Controlled Ostwald Ripening On the assumption of the first-order reaction control with an identical reaction constant, the size distribution function for p ^ 2 was derived as

fint)

where

const (1^^/t^/

(l^Hl^t).

,4.2.5)

143

4. RECRYSTALLIZATION

Reaction Control

Diffusion Control

Fig. 4.3. The steady forms of the size-distribution functions of diffusion-controlled and reaction-controlled Ostwald ripening. (From Ref 6.)

/ _

'•o'^^

"= R =

(4.2.6)

ykcX

and k is the growth rate constant in dr/dt = kVJC-CJ. 0. The relationship between f and r* is given by - 8 . r=—r . 9

For p a 2, f(r,t)

(4.2.7)

Also, r*(t) is given as a function of time by: r-(r) = ro(l+r/x'^)i/2 \l/2

v^'.;

- vi^ ykcj^^t

(4.2.8)

RT

so that r* or (9/8)f is proportional to the square root of t. Figure 4.3

144

FUNDAMENTALS

illustrates the size distribution functions of the diffusion-controlled and reaction-controlled Ostwald ripening. Although the LSW theory has been formulated on the simple assumption that both the dissolution and growth of particles are either diffusioncontrolled or reaction-controlled in a first-order reaction with the same reaction constants, the Ostwald ripening processes in real systems are not necessarily so simple. For example, one of the dissolution and growth processes is diffusion controlled and the other is reaction controlled in a higher-order reaction mode. Some theoretical studies^^'^^ deal with such cases.

4.3. Self-Recrystallization As has been discussed in chapter 3, if the surface chemical potential, or the solubility, of one kind of faces of a polyhedral particle bound by two kinds of faces is higher than the other kind in a closed system, the former will be dissolved and the latter will be grown by the deposition of the solute released from the former, in order to approach the equilibrium form of the particle. The particles in Fig. 3.4(b) were thus formed from those in Fig. 3.4(a). This intra-particle recrystallization may specifically be called the ''Self-Recrystallization,'' Figure 4.4 shows the equilibrium forms reached from cubic AgBr particles at pBr 1.40 (a) and from octahedral ones at pBr 2.48 (b), by self-recrystallization for 30 min at 50 °C in the presence of 0.5 mol dm'^ NH3 plus 1.0 mol dm'^ NH4NO3, similar to the transformation of the octahedra in Fig. 3.4(a) to the particles in 3.4(b).^^ If the size distribution is broad, the Ostwald ripening, a kind of inter-particle recrystallization, will occur at the same time. In this case, a perfect equilibrium form will not be attained in a reaction-controlled growth system.

4.4. Reversed Ostwald Ripening If monodispersed small cubic AgBr particles (I^J ^ ^ present with monodispersed larger octahedral particles (11^^^) in a closed reactioncontrolled system, in which 1/V3 < Ym/Yioo < ^^ and the pBr is sufficiently low for ^in/^ioo to be less than 1/V3, the smaller cubic particles I^^^ will grow at expense of the larger octahedral ones 11^^, since the surface chemical potential of the {100} faces at the comers of the octahedral

145

4. RECRYSTALLIZATION

30 min later * ^

1 |i m Fig. 4.4. TEM images of replicas of the equilibrium fomis reached from cubic AgBr particles at pBr 1.40 (a) and from octahedral ones at pBr 2.48 (b), by selfrecrystallization for 30 min at 50 °C in the presence of 0.5 mol dm'^ NH3 plus 1.0 mol dm-^ NH4NO3. (From Ref. 27.)

FUNDAMENTALS

146

particles is higher than that of the {100} faces of the cubic particles (case 1).^^ In this case, while the cubic particles 1^^^, will be subject to the selfrecrystallization owing to the high surface chemical potential of the {111} faces at their own corners, it will not contribute to the increase in their particle volume. Also, the growth of the {111} faces of the octahedral particles 11^,^^ will be negligible due to ^m/^ioo < 1/^3. Similarly, if small octahedral particles coexist with larger cubic particles at a sufficiently high pBr to attain k^n/k^oo > ^^^ ^^^ smaller octahedral particles will grow at the expense of the larger cubic particles (case 2). Since smaller particles are grown at the expense of the larger ones in these systems, one may call this inter-particle recrystallization the ''Reversed Ostwald Ripening.'' If the size difference between the two kinds of particles is sufficiently small in these systems, the final shape of all particles at the end of the

so b)

B

40

^'

*

30 •

20

-

10

.

I

i 1

[0.6

[Oct]

0 .8

1

1

1.0

1

1

1.2

[M]

1

r

1

1.4

p " p'

1

1.6

Fig. 4.5. (a) Diagram for the change of the / values of 1(100), 1(111), 11(100), and 11(111) with the progress of the reversed Ostwald ripening in case 1; (b) Diagram for the corresponding change of each particle volume of I and II. Here, 1(100), 1(111), 11(100), and 11(111) indicate the respective face indices, {100} and {HI}, for I^^ and Iloct,- p is equal to r^u/r^QQ-, the prime designates the quantities of 11^ in distinction from those of ' [Cub] Icub- (From Ref. 33.) 1

1

4. RECRYSTALLIZATION

147

reversed Ostwald ripening will be close to the equilibrium form, proper to the given system. Since the driving force for the growth of the smaller particles will disappear at this moment, the normal reaction-controlled Ostwald ripening for the growth of the larger particles will follow at the expense of the smaller particles. Figure 4.5 illustrates the change of the reduced surface chemical potential (f) and that of the particle volumes of 1^^^ and IIQ^ in case 1 for the reversed Ostwald ripening and succeeding normal Ostwald ripening processes, where the reduced surface chemical potential / for I^.yb or IIQ^^ is a surface chemical potential of a facet of I^^^, or II^^^ divided by the initial surface chemical potential of the {100} faces of a cubic particle at the start of the reversed Ostwald ripening (see Fig. 3.2). The initial surface chemical potential, ([iioo)o' ^^ ^^^ {100} faces of the cubic particles is given by (|iioo)o = ^Yioo^y^ioo* where YIOO is the specific surface energy of the {100} faces, V^ is the molar volume of the solid, and U^QQ is the radius of the inscribed sphere of the cube. ^..^

I fim

^

Fig. 4.6. TEM images of carbon replicas showing the evidence of the reversed Ostwald ripening in case 1: (a) original mixture; (b) resulting particles after aging at 50 °C for 60 min in the presence of 0.5 mol dm"^ NH3 plus 1.0 mol dm"^ NH4NO3 and 1 wt % gelatin. (From Ref. 33.)

148

FUNDAMENTALS

50

-

40

,

g 30 u L. 0)

o.

(

E 20 3

> 10 0

^ 0.2

,

^

—1

0.4 0.6 0.8 1.0 2.0 diameter of the equivalent sphere [pin]

i—

4.0

Fig. 4.7. Histograms of the size distributions of the particles in Fig. 4.6(a) and (b), corresponding to the broken and solid lines, respectively, obtained with a Coulter counter, where the equivalent sphere is a sphere having the same volume as a nonspherical particle. (From Ref. 33.)

The TEM images of carbon replicas of the AgBr particles in Fig. 4.6 show an example of the reversed Ostwald ripening in case 1. The corresponding histograms of the size distributions, obtained with a Coulter counter are shown in Fig. 4.7. One may find there considerable narrowing of the size distribution by the reversed Ostwald ripening. More detailed theoretical treatment is given in ref. 33.

4.5. Contact Recrystallization When silver halide particles are in contact with each other in an aqueous solution by coagulation, centrifugation, or filtration, they are found to undergo an extremely rapid recrystallization, which is completed within a few seconds at room temperature.^"**^'^ This recrystallization, as well as the normal or reversed Ostwald ripening, can be inhibited by some additives, such as 4-hydroxy-6-methyl-l,3,3a,7-tetraazaindene or l,l'-diethyl-2,2'cyanine. For example. Fig. 4.8 shows the degrees of the exchange of chloride ions between AgCl particles and the solution phase, r, changing with the progress of the Ostwald ripening in a suspension of AgCl particles

4. RECRYSTALLIZATION

149

i.
A

o| o

o •!

^y

ael uV

•

/

\i2Vf 0°

1°

2° 3° 4° 5° Excbaneint; time (hr)

6

Fig. 4.8. Effect of Aza added before filtering on the curve of the degree of ion exchange. Curve A is that of a AgCl system plotted with samples filtered without Aza, while curve B is that with Aza added immediately after every sampling. (From Ref. 24.)

at 25 °C, which was determined radiochemically, with radioactive ^^Cr ions, by following the change of the radioactivity of the ^^Cl" ions in the solid, after the addition of the radioactive ^^Cl" to a suspension of AgCl particles.^"* As defined by Kolthoff and Rosenblum,^^ the degree of the ion exchange for chloride ions between the solid and liquid phases in a suspension of AgCl particles is given by (4.5.1)

where AQ is the total radioactivity of the suspension; A^ is the radioactivity of the solution phase at time t; M^ and M^ are the mole numbers of the nonradioactive chloride ions in the solid and solution phases, respectively. The separation of the solid phase from the liquid phase is normally performed by filtration. The curve A in Fig. 4.8 was obtained by filtering each sample by suction immediately after sampling, while the curve B was obtained in a similar manner, but each sample was once mixed with a solution of 4~ hydroxy-6-methyl-l,3,3a,7-tetraazaindene (Aza) to halt the Ostwald ripening immediately after sampling and filtered 5 min later. The great discrepancy between the two curves was due to the dramatic effect of Aza, as shown by the TEM images of the replicas of AgCl particles in Fig. 4.9, in which the original cubic AgCl particles, without mixing with Aza before filtration, were subject to a drastic recrystallization into a continuous

FUNDAMENTALS

150

iJL Fig. 4.9. TEM images of the carbon replicas of AgCi particles: (a) original particles after centrifugal separation with Aza; (b) a sample on afilterafterfiltrationwithout Aza added; (c) a sample on a filter after filtration with Aza added. (From Ref. 24.) structure consisting of large flat grains on the filter (b), whilst the particles treated with Aza beforehand remained intact (a). It is evident from the rapid exchange of chloride ions between the solid and solution phases that the dramatic change of the particles in Fig, 4.9(b) is due to a fast recrystaUization via the solution phase. Since the rapid recrystallization was caused by the contact of particles, it was named ''Contact Recrystallization,'' The same phenomenon was observed whenever any kinds of silver halide particles, including AgCl, AgBr, Agl, and their mixture, were in contact by flocculation on addition of electrolyte such as KNO3, or by centrifugation. Figure 4.10 shows the instantaneous halide-ion exchange by addition of KNO3 to a suspension of AgCl or AgBr at 25 °C. Figure 4.11 demonstrates AgCl

151

4. RECRYSTALLIZATION

10 20 30 40 50 Exchanging time (min)

Fig. 4.10. Rapid halide-ion exchange on addition of KNO3 in a AgCl or a AgBr system at 25 °C, where [KNO3] = 0.2 mol dm'l (From Ref. 24).

particles after flocculation or centrifugation.^^ When a mixed suspension of AgCl and AgBr or that of AgBr and Agl was filtered by suction, they instantly formed the respective solid solutions.^"* When the halide-ion concentration is in excess of the silver-ion concentration in the solution phase of a suspension of silver halide particles, the driving force of the Ostwald ripening, in terms of the chemical potential difference between the monomers in a small particle of radius r^ and in a larger particle of radius r^, (Ajx)^^, is given from the Gibbs-Thomson equation by

(1

(4.5.2) '2)

where [Ag^J^ and [Ag*]2 are the concentrations of silver ions in equilibrium with the particles of radii r^ and r^, respectively. If two spherical particles of radius r^ are in contact and in the progress of contact recrystallization in a monodisperse system, the driving force of the contact recrystallization, (A[iXanr inay be written as

152

FUNDAMENTALS

Fig. 4.11. TEM images of carbon replicas of AgCl particles showing the evidence of the contact recrystallization by flocculation or centrifugation: (a) original particles, (b) sample after flocculation of the particles in (a) with KNO3 shown in Fig. 4.10; (a') original particles (4 h old), (b') sample after centrifugation for 30 min with the particles in (a'), started at the age of 3.5 h. (From Ref. 34.)

(ALL)

=/?7Tn——i= yV

(4.5.3)

where r2a is the longitudinal negative radius of the neck in the boundary region of the particles in contact, and r2i, is the latitudinal positive radius, as illustrated in Fig. 4.12.^ In general, the absolute value of r2a is very small, especially in the early stage of contact recrystallization. This model is reminiscent of the sintering theory of Clark and White.^^ The solute preferentially deposits onto the necks, owing to the total negativity of the term, {llr2a + V^ih)^ i^ Eq. (4.5.3), which induces the dissolution of the surfaces of the particles of radius r^ by reducing the concentration of the

153

4. RECRYSTALLIZATION

D

Fig. 4.12. Model of the contact recrystallization. (From Ref. 34.) solute. If the r^ values m Eqs. (4.5.2) and (4.5.3) are equal, the driving force of the contact recrystallization, (Api^^^ is much greater than (Ajx^^) for the normal Ostwald ripening, owing to the small negative value of r2a' This seems to be the reason why the rate of contact recrystaUization is much greater than that of Ostwald ripening. Although the typical contact recrystallization, which readily occurs even at room temperature, was first found for silver halide microcrystals, it may not be limited to silver halides. Probably, it may be common to the colloidal particles whose rate constant of the surface reaction for their growth is sufficiently high as to make the growth mode a diffusioncontrolled one, or very close to it. Moreover, even if the drastic contact recrystallization is not observed, it is not rare to find that the contact points of aggregated particles act as active sites for the preferential deposition of solute in ordinary growth systems, leading to rigid agglomerates or junctioned particles.

References 1. C. M. Flynn, Jr., Chem, Rev. 84, 31 (1984). 2. M. A. Blesa and E. Matijevic, Adv. Colloid Interface Sci. 29, 173 (1989). 3. W. Ostwald, Z Phys. Chem. 34, 495 (1900). 4. R. Liesegang, Z Phys. Chem. 70, 374 (1910).

154

FUNDAMENTALS

5. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 6. C. Wagner, Z Elektrochem. 65, 581 (1961). 7. L. Kampmann and M. Kahlweit, Ber. Bunsenges. Phys. Chem. 71, 78 (1967). 8. M. Kahlweit, Ber. Bunsenges. Phys. Chem. 78, 997 (1974). 9. M. Kahlweit, Adv. Colloid Interface Sci. 5, 1 (1975). 10. D. B. Dadyburior and E. Ruckenstein, J. Crystal Growth 40, 279 (1977). 11. T. Sugimoto, J. Colloid Interface Sci. 63, 16 (1978). 12. T. Sugimoto, J. Colloid Interface Sci. 63, 369 (1978). 13. J. A. Marqusee and J. Ross, J. Chem. Phys. 79, 373 (1983). 14. J. Schmelzer, Z. Phys. Chem. (Leipzig) 266, 1057 (1985). 15. C. W. Davies and G. H. NancoUas, Trans. Faraday Soc. 51, 818 (1955). 16. E. J. Meehan and W. H. Beattie, J. Phys. Chem. 65, 1522 (1961). 17. C. R. Berry and D. C. Skillman, J. Phys. Chem. 70, 1871 (1966). 18. C. R. Berry and D. C. Skillman, Photogr. Sci. Eng. 13, 69 (1969). 19. H. Mager and K. H. Lieser, Z. Phys. Chem. (N. F.) 79, 288 (1972). 20. E. Klein, E. Moisar, and E. Roche, /. Photogr. Sci. 19, 55 (1971). 21. F. H. Claes and H. Borginon, J. Photogr. Sci. 21, 155 (1973). 22. L. E. Oppenheimer, T. H. James, and A. H. Herz, in "Particle Growth in Suspensions," (A. L. Smith, Ed.), p. 159. Academic Press, New York, 1973. 23. T. G. Bogg, M. J. Harding, and D. N. Skinner, J. Photogr. Sci. 24, 81 (1976). 24. T. Sugimoto and G. Yamaguchi, J. Phys. Chem. 80, 1579 (1976). 25. T. Sugimoto and H. Yanagida, Nippon Shashin Gakkaishi 40,15 (1977). 26. J. S. Wey and M. J. Schad, J. Imaging Sci. 30, 193 (1986). 27. T. Sugimoto, J. Colloid Interface Sci. 91, 51 (1983). 28. T. Sugimoto, J. Colloid Interface Sci. 93, 461 (1983). 29. T. Sugimoto, Photogr. Sci. Eng 28, 137 (1984). 30. T. Sugimoto, J. Imaging Sci. 33, 203 (1989). 31. C .R. Berry, Photogr. Sci. Eng 20, 1 (1976). 32. T. Sugimoto, J. Colloid Interface Sci. 150, 208 (1992). 33. T. Sugimoto, J. Soc. Photogr. Sci. Technol. Jpn. 46, 306 (1983). 34. T. Sugimoto and G. Yamaguchi, J. Crystal Growth 34, 253 (1976). 35. I. M. Kolthoff and C. Rosenblum, J. Am. Chem. Soc. 56, 1658 (1934). 36. P. W. Clark and J. W. White, Trans. Brit. Ceram. Soc. 49, 305 (1950).

CHAPTER 5 SOLID-SOLUTION FORMATION Functional particles used as advanced materials are, in many cases, composite particles consisting of more than one component. Such multicomponent particles often exhibit unexpected properties entirely different from those of the individual constituents due to their complex composition and internal structures. For controlling the structures which determine their properties, some fundamental knowledge of thermodynamics and kinetics relevant to the formation process of mixed particles are required. In this chapter, the thermodynamic aspects of this process and the kinetics of the conversion process for the formation of mixed particles will be described with concrete examples of well-defined silver halide systems.

5.1. Equilibrium Compositions 5.1.1. Theory Let us consider the solid mixing of two kinds of sparingly soluble monovalent salts which differ in either the cations or anions. Since the treatment is the same for either case, we consider only the case of mixing two kinds of salts, AB^ and AB2, having identical cations A and different anions, B^ and 82^ In a system of mixed particles of A{Bi,Bo) in the form of a solid-solution, suspended in a solution of their solute, the activity of the monomers oiAB^ in the liquid phase in equilibrium with the monomers of AB^ in the mixed particles may be approximated by their mole fraction in the liquid phase, jc^, from the assumption of a sparingly soluble solid. In this case, the chemical potential, fx^, of 1 mole of monomers AB^ is given by Hi = | i i + / f n n x i ,

(5.1.1)

where m® is the standard chemical potential of 1 mole of monomers AB-^ in

156

FUNDAMENTALS

the solution phase. If the particles are pure AB^, the equilibrium chemical potential of the monomers in the liquid phase, \I^Q, is given by

where X^Q is the activity of the monomers AB^ in this case. Hence, m can be rewritten as |jii=|jii,+/?nn-^. •^1,0

(5.1.3)

On the other hand, the chemical potential, ^^^ of the monomers AB^ in the solid-solution is given by

where /^ is the activity coefficient of the monomers AB^ in the solidsolution, and x^^ is their mole fraction in the solid-solution. Since \X^Q = \i^^, and since at equilibrium it holds that \ir-\^l

(5.1.5)

one obtains, on an assumption of a constant ionic strength, that

^,,;.A.I1TW,

(5.1.6)

where \A^] and [B^~] are the concentrations of ^"^ and 5 f in the liquid phase, respectively; and Kj^^ is the solubility product of the pure solid AB^, since the ion product [^^][^r] is proportional to Xp irrespective of the composition of the solid. The corresponding relation for component 2 is

'2.0 •*'»*'

As a consequence, one obtains

-»™ sp

157

5. SOLID-SOLUTION FORMATION

(5.1.8)

If JC2VJC1* and [B2']/[B^'] are measured analytically, f^ and /2 can be determined as a fimction of jc/ or X2 from Eq. (5.1.8) and the following Gibbs-Duhem equation:

(5.1.9)

XidWi*4dhif2 = 0

If some appropriate electrode is available, it is possible to determine these activity coefficients directly from the electrochemical measurement. For example, if AB^ and AB2 are silver halides of different halide ions, the /i and /2 can be determined readily by setting up the following cell with silver electrodes, where Ag(Xi,X2) is sufficiently large particles of a mixed silver halide consisting of AgX^ and AgX2; AgX^ is sufficiently large pure AgX^ particles whose solubility product Kj^^ is much greater than that of AgX2, KJ"^^; KX^ and KX2 are the potassium sahs of the respective halide ions. The concentrations of KX^ in both half cells are set to be equal and sufficiently high. KXi

Ag

KX2 Ag(X„X2)

Ag AgX,

The concentration of silver ions in the solution of the left-hand half cell, [Ag*]xi^. and that of the right-hand half cell, [Ag*]xi, are given from Eq.(5.1.7) as [Ag*]X [ ^

and

_

aiP

m

(5.1.10)

158

FUNDAMENTALS

(5.1.11)

[^8\ =

Hence, the potential difference, E, between the left- and right-hand electrodes is given by

£= M]j,^lL}^

= ^lnf,xl.

(5112)

[Agl where E is always zero or negative. From the measurement of E, one can directly obtain the value of/p and thus/2 as well, from the Gibbs-Duhem relation in Eq. (5.1.9). The free energy of formation, AG, of one mole of a solid-solution consisting of AgX^ of mole fraction x^^ and AgX2 of mole fraction X2 from each pure substance of AgX^ and AgX2 is given as AG = xli\i\-\i\) =

+x^i\il- ji^

RT(x^f,x',^x;}nf^) (5.1.13)

= RT

= FE+x^RThi

K^m

In some cases, the crystal system of a solid-solution is different from that of one of the pure starting solids. If pure AgX2 is different in crystal system from the solid-solution of Ag(XpX2), AG consists of AG^ for the transition from the pure AgX2 to pure AgX2 of the new crystal system of the solid-solution and AG2 for mixing with AgX^ to form the solid-solution. If the pure AgXj of the new crystal system of the solid-solution is indicated by an asterisk (*), AG^ and AG2 are given by

5. SOLID-SOLUTION FORMATION

159

and AG^ = FE^j4RThii^,

(5.1.15)

respectively. After all, AG (= AG^ + AG2) agrees with Eq. (5.1.13), so that Eq. (5.1.13) holds even if the solid solution is different in crystal system from the starting crystal of AgX2. As a matter of course, the situation is the same even if AgX^ or both of AgX^ and AgX2, are transformed into a solid-solution of a different crystal system. 5.1.2. Ag(Cl,Br) Systems The equilibrium between the mixed crystals of AgCl and AgBr and their halide ions in the solution phase was first studied by Kiister\ and has been investigated by many researchers since then.^"^ Figure 5.1 shows the relationship between the mole fraction of AgBr in the solid-solution of Ag(Cl,Br) and [Br"]/[Cr] in the solution phase at equilibrium, given by Pouradier et al} by averaging the data of Chateau,^ Flood,"^ Yutzy and Kolthoff,^'^ and Eastman and Milner.^ From this figure, it is obvious that the composition of the solid-solution of Ag(Cl,Br) continuously changes from jc^gj,/ = 0 to 1 with the change in the [Br"]/[Cr] ratio in the solution phase, and the [Br-]/[Cl-] ratio at jc^^^/ = 0.5 is 2.7x10"^ at 25 °C. Flood,^ particularly, defined the following parameter Kj as

K,^^En^,

(5.1.16)

and determined Kj as a function of the mole fraction of AgBr in the sohdsolution, Jc^gfl/, and the absolute temperature, T as follows:

FUNDAMENTALS

160

0,4

0,6

0,8

1,0

Fig. 5.1. [Br-]/[Cl-] vs x^^^. (From Ref. 8.)

890+180xLa,

(5.1.17)

If we use this parameter, it is possible to calculate the relationship between [Br"]/[Cr] and x^^^ at an arbitrary temperature. For the activity coefficients /^ and / j , Flood and Bruun^ and Chateau^ gave the following relationships:

where parameter ^ is a function of temperature and is 0.239 at 25 °C. Pouradier et a/, ^determined the relation between [Br"]/[Cr] and E using a cell of Ag(Cl,Br)-AgCl system similar to the one described in the foregoing section (AgX^ = AgCl; AgX2 = AgBr). They found an excellent agreement

161

5. SOLID-SOLUTION FORMATION

with the same relationship calculated from Eq. (5.1.12) with the data of [Br ]/[Cl-] v^ jc^g/ shown in Fig. 5.1 and available f^^a- ^ ^c^' Pouradier et al} used a cell with silver electrodes coated with a thin layer of AgCl or AgBr by electrolysis, instead of Ag(Cl,Br) and AgCl particles, and found the electromotive forces identical, irrespective of the kinds of electrodes. Also, X-ray diffraction data on the overall compositions of the silver halide layer were not in accord with those expected from the electrochemical measurement. Thus, they suggested that only the top layer of the silver halide coating was in equilibrium with the halide ions in the solution phase. 5.1.3. Ag(Br,I) Systems In contrast to the Ag(Cl,Br) system, there is an upper limit to the mole fraction of Agl component in mixing with the face-centered cubic (f.c.c.) AgBr crystal, Le., practically, around 0.40 in mole fraction. When the ratio of [r]/[Br"] in the solution exceeds a level in equilibrium with this maximum mole fraction in the f.c.c. solid phase, a new solid phase of hexagonal Ag(Br,I) crystals containing only a small amount of AgBr, less than ca, 5 mol %, appears. This is because iodide ions, having larger dimensions than bromide ions, raise the lattice strain energy with their increasing content in the f.c.c. crystals of Ag(Br,I) until, finally, the enthalpy

20

40

60

80

100

Mole percent iodide Average composition of solid

Fig. 5.2. Equilibrium Agl contents in f.c.c. Ag(Br,I) phase and in P-type Ag(Br,I) phase as a function of average iodide content in solid. (From Ref. 9.)

FUNDAMENTALS

162

increase by this effect exceeds the increase of the mixing entropy term. Figure 5.2 shows the equilibrium composition in the solid phases as a function of average composition of solid.^ Such a phase separation is actually observed when a mixed solution of soluble salts of bromide and iodide is added to a solution of a soluble silver salt (reverse mixing), or simultaneously added with the silver salt solution to an inert aqueous solution (double jet mixing) (see section 8.4.1). However, there seems to be no theoretical reason for the reduction of the iodide content of the cubic Ag(Br,I) phase with the increasing average iodide content of solid above ca, 40 mol % in Fig. 5.2. Chateau et al}^ studied the equilibrium of the Ag(Br,I) system. Figure 5.3 is one of their results for the linear relationship between [r]/[Br"] and x^J at 25 °C and 70 "C ([r]/[Br-] = 6WxlO-^xJ at 25 °C, and [r]/[Br-] = 2.07xl0-^x^/ at 70 °C). Figure 5.4 is their result for £ vs x^J using Ag electrodes coated with AgBr, KBr for KX^ and Kl for KX2, but for the data at x^J = 0.52 they used some supersaturated mixed crystals prepared from a mixed melt.

0,10

0,20

0,30

X^gi

Fig. 5.3. [r]/[Br-] vs x^J at 25 °C and 70 °C. (From Ref. 10.)

163

5. SOLID-SOLUTION FORMATION

0,10

0,20

0,30

0,40

QSO XAgi

Fig. 5.4. E vs x^J. (From Ref. 10.)

The thermodynamically stable form of pure Agl below 137 °C at 1 atm is a hexagonal crystal (P-type), while the Agl component in a f.c.c. solidsolution of Ag(Br,I) is in the form of f.c.c. crystal. On the other hand, the pure f.c.c. Agl crystal (8-type) is stable only above 3020 atm (= 306 MPa). Hence, for the formation of a f.c.c. solid solution of Ag(Br,I) near the room temperature, it is reasonable to consider that the P-AgI is once transformed into 8-AgI and then incorporated into the f.c.c. solid solution.^° Thus, Eq. (5.1.8) for a f.c.c. solid solution may be written as (5.1.19)

[Br-] Kl^Y,x\ where the components 1 and 2 correspond to AgBr and Agl components, respectively; the superscript 6 indicates the parameters for the f.c.c. crystals. On the other hand, from Eq. (5.1.6),

If a hexagonal solid-solution coexists, an analogous relation holds for it as

where the superscript p indicates the parameters for the hexagonal crystals.

164

FUNDAMENTALS

Hence,

and

[n_CV«

CV

[Br-] ^^V-f

0,\^

(5.1.23)

since /2^ « 1. This is because even the maximum mole fraction of the AgBr component in the hexagonal solid-solution is only around 0.01, which is achieved when the equilibrium mole fraction of the Agl component in the coexisting f.c.c. solid-solution reaches saturation (JC2* = 0.312 at 25 °C).^^'^^ When f.c.c. and hexagonal solid-solutions coexist, the potential difference E is generally given by

E= ^ I n / i V - —In F

F

^^—^

.

(5-1-24)

i£j>^([/-]/[Br-])

When x^^ = 0.312, [rMBr] = 1.89x10"^ and thus E = -2.92 mV at 25 °C. This E value is in good agreement with -3.2 mV obtained by the electrochemical measurement for Fig. 5.4. If C2 and VQ denote the molarity of the monomers AB2 and the molar volume of the solvent, respectively, the mole fraction of AB2 in the liquid phase is generally given by x, = c,V,.

(5.1.25)

Also, if K2 denotes the stability constant of AB2, K^

Hence,

^ - .

(5.1.26)

5. SOLID-SOLUTION FORMATION

165

x^ = IC,V,[A^][B^].

(5.1.27)

From Eq. (5.1.2), 1^2= 1^2.0= \xl*RTlnx,,= vil^RTlnK^V.^RThiK^K

(5.1-28)

Thus, from Eq. (5.1.4), one obtains

If we apply this formula to the f.c.c. solid-solution of ^4(5^^2)' nf = iil^RmK,V,^RTinKS\4 •

(^'^'^O)

For the hexagonal solid-solution, fxf = ,.^/^^lnJ^,K,.i^7^n4^>V.^^ If the two solid-solutions coexist, Kj'^%%^ = Kj^^h^ so that u

=11^ =11^

from

(5.1.31) Eq. (5.1.22)

(5.1.32)

where \i^^j, [x^^/, and \i^J are the chemical potentials of the Agl monomers in the liquid phase, 8-type solid-solution, and p-type solid-solution, respectively. Similarly, for the AgBr monomers, u >^ AgBr

=11^ =11^ y^ AgBr

(5.1.33)

V'AgBr'

^

The free energy of formation of 1 mole of a single f.c.c.-type solidsolution is given from Eq. (5.1.13) as AG = F£+Jc^^/?nn

{KT^' -sp j^sl^ sp

[/-]!

(5.1.34)

[5r-]

If the solid-solution of the same composition is formed from 1 mole of pure AgBr, the free energy of formation, AG', is given by

FUNDAMENTALS

166

AG

y "pJB

^S

Q5O

OJO

Q3O

Fig. 5.5. AG and AG' (= AG/(1^AJ)) as a function of x^^J, (From Ref. 10.)

X '

AG/ =

AG

(5.1.35)

\-x Azi The AG' corresponds to the change of free energy when 1 mole of AgBr is mixed with varied amounts of Agl, and hence as long as it is lowered with the increasing amount of Agl, the mixing is promoted. However, if it reaches a minimum and starts to increase with further incorporation of Agl, the system will rather stay near the minimum AG' by retaining the excess pure P-AgI unchanged. Figure 5.5 shows the changes of AG and AG' for the formation of the f.c.c. solid-solution of Ag(Br,I) at 25 °C as a function of x^^f}^ The minimum AG' gives the mole fraction of Agl in the solidsolution thermodynamically saturated with Agl (JC^^/ = 0.312). In other words, while the curve of AG in Fig. 5.5 corresponds to the free energy of formation of 1 mole of f.c.c. Ag(Br,I) from pure AgBr and P-AgI, as a function of the mole fraction of Agl, it is equivalent to the minimum free energy of formation at each overall mole fraction of Agl in the system up

5. SOLID-SOLUTION FORMATION

167

to 0.312. However, when the overall mole fraction of Agl exceeds 0.312, the total free energy of formation at each overall mole fraction of Agl is rather lowered below AG by increasing the proportion of unreacted pure p Agl at the expense of the f.c.c. Ag(Br,I) with a Agl mole fraction kept close to 0.312. The uniformly mixed particles of f.c.c. Ag(Br,I) can be prepared by the controlled double jet (CDJ) technique (see section 1.5.2) using a AgNOg solution and a mixed solution of KI and KBr with a mole fraction of KI equal to or less than 0.312. However, even if the mole fraction of BCI in the mixed solution of alkali halides exceeds 0.312 to some extent, we can obtain mixed particles of uniform f.c.c. solid-solution, because the increase of AG' with the increasing JC^^/ above 0.312 is rather small, and because some supersaturation in the liquid phase is needed for the nucleation and growth of P-AgI particles. The smaU reduction in the iodide content in the f.c.c. mixed crystals after the peak with the increasing total iodide m Fig. 5.2 may reflect the lower supersaturation for the growth of P-Agi than is needed for its nucleation. Normally, the maximum mole fraction of mixed Ag(Br,I) particles of f.c.c. solid-solution prepared by the CDJ technique is around 0.345 at 25 ""C, and the mole percent of the maximum Agl content increases with increasing temperature in degrees centigrade as^^ xlgjimol%) = 34.5 +0.165(f-25).

(5.1.36)

5.2. Conversion by Intra-Particle Recrystallization The conversion technique for transforming monodisperse particles into others, different in composition and/or structure, is one of the most important procedures for producing new kinds of particulate materials. In particular, conversion processes by intra-particle recrystallization (recrystallization within each particle) are of interest not only from a thermodynamic viewpoint but also have great practical significance for the production of hetero-junctioned particles, new types of particles containing a high density of crystal defects, etc. This section will be devoted to describing the fundamental aspects of this process, focusing on a well-defined conversion process of silver chloride particles by bromide ions.^"*'^^ 5.2.1. Conversion of AgCI Particles by Br" ions

168

FUNDAMENTALS

Fig. 5.6. Morphological change of the AgCl particles with time at 25 °C after introduction of a solution of KBr corresponding to 10 mol % of the AgCl: (a) 3, (b) 10, (c) 20, and (d) 50 s. Initial composition: [AgCl]o = 10"' M, [KCl]o = 10'^ M, [KBrJo = 10-2 ^^ ^^^ [gelatin] = 1 wt %. (From Ref. 14.)

5. SOLID-SOLUTION FORMATION

169

AgCI{220}

50" 20" 10"

39*

40'

4r

42'

DIFFRACTION ANGLE (20)

Fig. 5.7. Change of the corresponding XRD pattern. (From Ref. 14.)

If a different kind of soluble anions or cations is added to a suspension of sparingly soluble particles, and if the added anions or cations can form a more stable solid with the cations or anions of the suspended particles, the latter may be converted, partly or totally, into the more stable particles by recrystallization via the solution phase. Figure 5.6 shows transmission electron micrographs of carbon replicas of cubic AgCl particles of mean edge length 0.41 ^mi in the progress of a partial conversion at 25 °C after instantaneous addition of a KBr solution containing 10 mol % of KBr to AgCl (initial composition immediately after the addition of KBr: [AgCl]o = 10"^ M, [KCl]o = 10"^ M, [KBr]o = 10"" M, and [gelatin] = 1 wt %)}'^ The conversion reaction was stopped instantly by mixing each sample with a merocyanine dye for the electron microscopy. Figure 5.7 shows the corresponding XRD patterns. Figures 5.8 and 5.9 exhibit the change of the silver-ion concentration from potentiometry with a silver electrode, and the change of [Br'] calculated from the potentiometry for the silver-ion concentration (solid line) and from XRD on annealed solid samples (closed circles), respectively. The particles sampled at the end of

FUNDAMENTALS

170

0

10

20

30 40 t/sec

50

60

70

Fig. 5.8. Change of the corresponding silver-ion concentration. (From Ref. 14.)

b

Fig. 5.9. Change of the corresponding bromide-ion concentration. (From Ref. 14.) the initial sharp drop of [Ag^] revealed no appreciable change in morphology (Fig. 5.6(a)) or composition (Fig. 5.7). Thus, in this first step, the reaction was limited to the consumption of silver ions originally equilibrated with AgCl along with a surface conversion of AgCl particles by ionic exchange with bromide ions. The temporary stability in [Ag"^] at the end of the initial drop implies the establishment of the surface equilibrium. Soon after the first step very small verrucae were found to emerge on the comers and edges of the cubic AgCl particles (Fig. 5.6(b)) and then the ones on the comers grew, while the others on the edges appeared to be dissolved together with the open areas of each AgCl particle. The new solid parts consisted of nearly pure AgBr microcrystals, as demonstrated by XRD in Fig. 5.7. Figure 5.9, for the change of [Br"], clearly shows that almost all

171

5. SOLID-SOLUTION FORMATION

b) ;^: •*?;.•

''^'€^:Ps^ ^

"7- $ .

%K

-Q y%m

'1^ ' ^ I Pffil '"ftm^nr

•iiiiujrij,!.

Fig. 5.10. Electron micrographs of the composite particles with different [KBr]o/[AgCl]o molar ratios after aging at 25 °C for 50 s from the addition of KBr. [KBr]o/[AgCl]o: (a) 0.1, (b) 0.2, (c) 0.4, and (d) 1.0. (From Ref. 14.)

172

FUNDAMENTALS AgBr{220}

AgCI{220}

CKBr]^/[AgCI]o 1.0

3sr 4or 4r 42" DIFFRACTION ANGLE (26)

Fig. 5.11. XRD patterns corresponding to the particles in Fig. 5.10. (From Ref. 14.)

bromide ions were used up in 60 s for the formation of the AgBr particles joined to the comers of the AgCl particles. This period is defined as the second step. Figures 5.10 and 5.11 show electron micrographs and the corresponding XRD pattems for the particles after conversion for 50 s at 25 °C with a variety of initial molar ratios of [KBr]o/[AgCl]o. In any case, the AgBr parts finished forming in 60 s regardless of the initial molar ratio. Since the dissolution of the AgCl particles can readily follow the great increase in [KBrjg and finish the reaction at the same time, it is suggested that the the rate-determining step of the total reaction is not the dissolution process of the AgCl particles but the growth process of the AgBr particles in the second step. Interestingly, the original AgCl particles were almost totally decomposed into the nearly eight-fold number of AgBr particles when [KBr]o/[AgCl]o = 1 (Fig. 5.10(d)). After the end of the second step, [Ag^] still kept rising as shown in Fig. 5.8, when [KBr]o/[AgCl]o < 1. This implies the possibility of the third step, which appears, however, to be much slower than the earlier steps. In order to accelerate this slow process, the temperature was raised to 45 °C under the otherwise standard conditions. Figures 5.12 and 5.13 show electron micrographs and the corresponding XRD pattems of the particles in the course of conversion at 45 °C. Obviously, the particles at 1 and 3 min in

5. SOLID-SOLUTION FORMATION

173

Fig. 5.12 were already in a stage later than the second step, since they formed a new solid-solution of AgClo5Bro5 according to Fig. 5.13. The solid-solution must have grown from the borders of the guest/host joints and

Fig. 5.12. Morphological change of the composite particles of AgClo.9Bro.i with aging time at 45 °C: (a) 0.5, (b) 1.0, (c) 3.0, and (d) 10 min. (From Ref. 14.)

174

FUNDAMENTALS AgCI{220}

0.5' 39"

40'

4r

42^

DIFFRACTION ANGLE {2$)

Fig. 5.13. XRD corresponding to the particles in Fig. 5.12. (From Ref. 14.)

developed two-dimensionally toward the inside of the host surfaces, as clearly observed in Fig. 5.12(c). Here, it is noteworthy that the composition of AgClo 5610 5 was not altered, as long as the AgBr guests were present until 5 min as shown in Fig. 5.13. This stage, when the solid-solution of AgClo 5610 5 is grown by the simultaneous dissolution of the AgBr guests and AgCl hosts, is defined as the third step. The third step lasted for about 2 h at 25 °C. The third step is followed by still further steps to form a solid-solution having a higher chloride content, as shown by the XRD profile at 10 min in Fig. 5.13(d), corresponding to the irregularly shaped particles in Fig. 5.12(d). The solid-solution must have grown at the expense of the AgClo5Bro5 and the AgCl hosts. In this stage, however, there was no evidence of the definite stepwise progress. The reaction rate became slower by degrees as the chloride content of the solid-solution increased. When the initial molar ratio of [KBr]o/[AgCl]o was as low as 0.1, the reaction virtually stopped midway to yield double-structured particles consisting of a Ag(Cl,Br) shell and AgCl shell. For stabilizing the hetero-junctioned particles with eight AgBr guest particles at all comers of each AgCl host, some stabilizers, such as merocyanine dyes, cyanine dyes, mercaptotetra-

5. SOLID-SOLUTION FORMATION

175 3rd Step

2nd Step

A g *——^^ /<^^ , c r ^X /;rAg*' Ag As \ / \ Br--sU,Ag* '•V

Br"

fln*fK^"i^iii

Br"

' i l l " ' ""^fy^ Br" An"^

w. C r - i / ^ Ag* y

Vflr" Br"

A g * _ ^ \ ^ , - « ^ ^ A g * ^ ^ - ~ Ag*

Fig. 5.14. Schematic models of the first, second, and third steps of the partial conversion of AgCl particles with KBr. (From Ref. 14.) zoles, or tetraazaindenes, are used. Figure 5.14 illustrates the schematic models of the first, second, and third steps. The first step is an instantaneous step for the surface conversion of the AgCl microcrystals, within a few seconds after the introduction of bromide ions, to establish the equilibrium between the surfaces of the AgCl particles and the solution phase containing Br" and CI" ions. The second step is a rapid process of about 1 min at 25 ""C for the nucleation and epitaxial growth of nearly pure AgBr guest crystals on every comer and partly on the edges of the cubic AgCl hosts by the reaction of bromide ions with silver ions dissolved from the {100} faces of the AgCl particles. The third step is a much slower process of about 2 h at 25 °C (5 min at 45 °C) for the two-dimensional growth of the AgBro5Clo5 solid-solution on the AgCl host surfaces from the joints of guest/host by simultaneous dissolution of the AgBr guests and the open areas of the AgCl hosts. Finally, further steps form a solid-solution of a higher chloride content by the dissolution of the foregoing one formed in the preceding step together with the remaining host surfaces. From a thermodynamic viewpoint, the first and second steps are elementary processes for the reduction in enthalpy to form AgBr from AgCl (enthalpy processes), while the third and later steps are those for the increase in entropy to form AgCl^.^Br^ solid-solution from AgCl and AgBr (entropy processes). It is now obvious that the conversion reaction of this system proceeds by way of a stepwise intra-particle recrystallization, in which the nucleation and growth of the new solid phase initially occur at the comers and/or edges of each particle with the dissolution of other parts. The preferential nucleation of the guest particles on the comers and/or edges of the host particles seems to be due to the high

176

FUNDAMENTALS

density of bond-free sites on these parts. Then the AgBr guest crystals were found to be grown epitaxially on the AgCl host crystals, as revealed by electron diffractometry focused on a host crystal and one of its guest.^"^ Hence, generally, some active parts for the nucleation and growth of a new solid phase may be needed on the surfaces of the host particles for a wellregulated intra-particle recrystallization. Incidentally, there is no possibility of direct diffusion of bromide ions into the AgCl particles, as has been shown by a tracer technique^^'^^ and by the exceedingly low mobility of halide ions in silver halide crystals below 100 °C, as has been observed by Batra and Slifkin.^^ 5.2.2. Thermodynamics As has been shown in Fig. 5.8, the silver-ion concentration sharply dropped to reach a minimum in a few seconds after the introduction of Br" ions into the AgCl system, and a temporary equilibrium between the solution phase and the AgCl surfaces was established. The thickness of the surface layer rich in bromide content must correspond to one or more atomic layers, as is readily deduced from the radiochemical analyses, which indicate a very quick halide exchange between the the solution phase and silver halide surface layer.^^*^^ In this case, the atoms in the surface layer must be compressed with each other in order to be set epitaxially on the AgCl substrate of smaller lattice parameter (5.77 A/AgBr; 5.55 A/AgCl). On the other hand, if condensed CI" ions are introduced into a suspension of AgBr crystals with little excess of Br" ions in the solution phase, the surface layer of the AgBr crystals may turn into a solid-solution of silver chlorobromide of a high chloride content. In this case, the atomic bonds in the surface layer must be stretched laterally because of the greater lattice parameter of the AgBr substrate. The conceptual schemes of these situation are illustrated in Fig. 5.15. Thus, in both cases, some surface stress energy must be built up in the surface layer, and the chemical potentials of AgBr and AgCl components will be lifted by the surface stress energy (SSE), E^. For the AgBr component, the chemical potential, jx/, in the top atomic layer is given by

H^li^.E^./mnVJ.

(5-2.1)

where jx^^*^ is the standard chemical potential of the AgBr component in the top layer, /^ its activity coefficient, and JC/ its mole fraction. Thus one obtains from Eq. (5.2.1)

177

5. SOLID-SOLUTION FORMATION

K^ =

K^%x'^xpiEJRT),

(5.2.2)

where K^^ is the solubihty product of the AgBr component in the surface layer, and Kj^-^ is the solubihty product of pure AgBr. For the AgCl component. (5.2.3)

K^^K'^y^x'cOxpiEJRT).

The quantity E^ is a function of [Br']/[Cr] or x^/x^, and it will increase as jc/ becomes higher on the AgCl substrate. If £, = 0, then i C / = K^^^f^B so that K^^ is always equal to or smaUer than Kj^"