Nanofluidics

Nanofluidics

Nanofluidics Patrick Abgrall Nam-Trung Nguyen

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ISBN-13:  978-1-59693-350-7 Cover design by Yekaterina Ratner © 2009 ARTECH HOUSE 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. 10 9 8 7 6 5 4 3 2 1

Contents   Chapter 1  Introduction 1.1  1.2  1.3  1.4 

Classical Areas Related to Nanofluidics Technology Developments Leading to Nanofluidics New Phenomena and Applications of Nanofluidics Organization of the Book

  Chapter 2   Fundamentals of Mass Transport at the Nanoscale 2.1  Fluid Flow at the Microscale 2.1.1  Scaling Laws 2.1.2  Continuum Model 2.1.3  Pressure-Driven Flow 2.1.4  Capillary Flow 2.1.5  Electrokinetic Flow 2.1.6  Mixing in Microscale 2.2  From Microscale to Nanoscale 2.2.1  Continuum Assumption 2.2.2  Kinetic Theory 2.2.3  Molecular Dynamics and Direct Simulation      Monte Carlo (DSMC) 2.3  Diffusion in Nanochannels 2.3.1  Knudsen Diffusion in Nanochannels 2.3.2  Hindered Diffusion in Nanochannels 2.4  Electrokinetics in Nanochannels 2.5  Water in Nanochannels 2.6  Capillary Filling in a Nanochannel 2.6.1  Modeling of the Filling Process 2.6.2  Asymptotic Solutions 2.7  Nanofilters 2.7.1  Electrostatic Effect 2.7.2  Reptation Effect 2.7.3  Steric Effect 2.7.4  Entropic Effect References Selected Bibliography

1 1 3 4 6

9 9 9 11 13 19 22 34 41 41 43 46 48 48 49 49 51 53 53 59 63 64 65 66 66 67 69



vi

Contents

  Chapter 3   Fabrication Technologies of Nanochannels

71

3.1  Basics About Micro- and Nanofabrication 3.1.1  Silicon/Glass Micro- and Nanofabrication 3.1.2  Polymer Micro- and Nanofabrication 3.2  Application to Micro- and Nanofluidic Systems 3.2.1  Microfluidic Devices 3.2.2  Nanofluidic Devices References

71 72 103 109 109 117 127

  Chapter 4   Applications of Micro- and Nanofluidics

141

4.1  Microfluidics and Lab-on-Chips 4.1.1  Biomolecular Analyses 4.1.2  Microfluidics Beyond Biomolecular Analyses 4.2  Handling Ions by Electrokinetic Effects in Nanochannels 4.2.1  Nanofluidic Resistors as Surface Charge Sensors 4.2.2  Semipermeability and Preconcentration 4.2.3  Routing Ions by Nanofluidic Electronics 4.3  Separations in Nanofluidic Devices 4.3.1  Batch Separations 4.3.2  Continuous-Flow Separations 4.4  Linear Analysis of Biomolecules 4.4.1  Why Stretching DNA? 4.4.2  Sizing Confined DNA 4.4.3  Nanopore Sequencing References

142 142 155 159 159 163 167 170 170 174 178 178 179 180 181

  Chapter 5   Conclusion

197

About the Authors Index

199 201

Chapter 1

Introduction According to the International Union of Pure and Applied Chemistry (IUPAC), pores are classified into three categories: micropore, mesopore, and macropore, with pore sizes less than 2 nm, between 2 and 50 nm, and larger than 50 nm, respectively. The nanoscale of the microelectromechanical system (MEMS)/nanoelectromechanical system (NEMS) community does not fall into this categorization. In this field, the prefixes micro, submicro, and nano usually correspond to objects with dimensions in a range 1–100 mm, 100 nm–1 mm and 1–100 nm, respectively. Nanofluidics encompasses science and technologies that involve a fluid flowing in a system with a least one dimension in the nanoscale, usually defined in this community by the range 1–100 nm. Both internal and external flows inside or around nanoscale objects can be considered as topics of nanofluidics. Although the transport phenomena of fluids at the nanoscale have already been studied in the past, the terminology of nanofluidics has been recently crystallized from the field of microfluidics. Both microfluidics and subsequently nanofluidics have emerged from the miniaturization technologies pioneered by the field of microelectronics. The relevant scales of nanofluidics and microfluidics are illustrated in Figure 1.1. Figure 1.1 shows that although nanofluidics introduces new challenges in terms of fabrication technologies, it also offers tremendous opportunities to directly interact with objects with dimensions close to the nanoscale, such as ions, DNA strands, or proteins.

1.1  Classical Areas Related to Nanofluidics The interest in shrinking dimensions below 100 nm originates from the remarkable properties appearing at the nanoscale and the associated phenomena. Materials such as gel matrices, ion-track–etched membranes and many other porous materials have been investigated and used in applications such as separation, filtration, and catalysis. Classical science and engineering disciplines already provide a wide, well-established base of knowledge for the understanding of these phenomena. Examples of areas that deal intensively with nanoscale phenomena include tribology, surface sciences, and colloid sciences (see Figure 1.2). Tribology is the science and technology of interactions between moving surfaces that are in contact. Tribology investigates fundamental phenomena and their applications related to friction, lubrication, and wear. Tribology originally found applications in designing bearing systems, but it has been extended to any system with two rubbing surfaces. For instance, the reliability of a micro electromechanical system (MEMS) is significantly affected by the tribological interactions of its surfaces. Together with the development of micro- and nanotechnologies, it has been 



Introduction

Figure 1.1  Length scales and volume scales of nanofluidics, microfluidics, common microtechnologies, and common objects.

established as an important research field. Tribology helps explain the lubrication effects of thin films, whose typical dimensions are in the nanoscale. This knowledge from tribology clearly contributes to the field of nanofluidics. Another classical research field that previously dealt with nanofluidic phenomena is surface science, which studies phenomena occurring at the interface of two phases such as solid-liquid interfaces, solid-gas interfaces, and liquid-gas interfaces.

Figure 1.2  Classical areas of science and engineering related to nanofluidics.

1.2  Technology Developments Leading to Nanofluidics



Surface chemistry deals with chemical reactions at interfaces. One application of surface chemistry is surface engineering, which has a great impact in nanofluidics due to the surface-dominated nature of the nanoscale. The chemical composition of a surface can be finely tuned using selected elements or functional groups that lead to the desired properties. Surface science also encompasses electrokinetics, which is one of the main transport phenomena in micro- and nanofluidics. A number of natural systems provide pores with sizes in the nanometer scale. For instance, aquaporins are proteins that naturally form pores in the bilipid membrane of cells, regulating molecules going in and out. Zeolites, a range of naturally occurring minerals, have been used as porous catalysts in the petroleum industry for decades. Activated carbon is another example of a porous material that has been used for a long time in chemical reactions due to its very large surface-to-volume ratio. Artificial gel matrices have been widely applied to bioanalysis, with pore sizes typically ranging from 200 nm to 500 nm for agarose, and from 5 nm to 100 nm for polyacrylamide. Nuclear-track–etched membranes are also common in the lab for filtration applications.

1.2  Technology Developments Leading to Nanofluidics Recent advances in micro- and nanotechnologies have allowed the creation of pores with precise control over the geometrical parameters as well as the modification of the physical and chemical properties of the substrate material. The miniaturization of electronic devices, or microelectronics, can be considered as one the main achievements of humanity in the last century. Microelectronics has been the enabling technology for almost all engineering and scientific progress, including the strides that led to the development of the Internet. Since its introduction during the 1990s, the miniaturization in microelectronics has followed Moore’s law, which predicts a twofold miniaturization every 18 months. Recently, this pace has slowed down to 24–36 months due to the limit of photolithography, which has been the workhorse of this industry for the last few decades. Following the trend in electronic devices, mechanical devices have also been miniaturized using microelectronics technology. Silicon micromachining has led to the development of MEMS, and to the more general concept of Microsystems, including not only mechanical and electrical components, but also optical and fluidic functions. Indeed, the integration of fluidic components into a microsystem led to as the development of the microfluidics field. While silicon-based components such as microflow sensors, micropumps, and microvalves dominated the early development stage of microfluidics, applications for analytical chemistry and life sciences became the key drivers at the later stage. Microfluidic devices have been found in a wide spectrum of applications including biological analysis, chemical synthesis, electronic cooling, and fuel cells. Microchannels and related microfluidic structures may have sizes on the order of millimeters to centimeters. Thus, instead of having a footprint of a few square millimeters as in microelectronics, a microfluidic system can be as large as several square centimeters. The large footprint makes the fabrication of silicon-based microfluidic devices cost-ineffective. As a result, the last 10 years has witnessed a shift from silicon to polymeric micromachining as a



Introduction

low-cost alternative for the fabrication of microfluidic devices. Elastomers such as polydimethylsiloxane (PDMS) and thermoplastics such as polycarbonate (PC) or polymethylmethacrylate (PMMA) have progressively appeared in every microfluidics lab. The advantages associated with microfluidics’ use in chemical analysis and life sciences include a minimized consumption of reagent, faster operation, better performance, automation, integration, and reduced manufacturing costs. Since most microfluidic systems work with a length scale on the order of several tens of micrometers to several millimeters, the breakdown of continuum assumption actually does not occur in microfluidics. Standard tools and approaches well-known in the macroscale can still be used for designing and investigating microfluidic devices. On the other hand, precautions have to be taken with the continuum assumption and the nonslip boundary conditions as the dimensions are getting closer to the nanometer. As mentioned above, materials such as zeolites naturally provide a random network of pores, but new fabrication techniques have allowed the fabrication of well-defined, deterministic networks of nanochannels. Experimental studies using these deterministic nanostructures have given a new insight on nanofluidic phenomena, which were previously only accessible through macroscopic experiments and models. The very high level of control on the geometry given by the progresses in micro- and nanotechnologies has allowed for the implementation of a smart, custom-designed array of nanostructures. This unique feature not only results in improved performances in applications where random, isotropic porous media were traditionally employed, but it has also led to new applications previously inaccessible. Beyond the geometrical aspect, planar integration has also allowed the coupling of these nanostructures with other functions into a complete system (e.g., microchannels and electrodes). In summary, planar micro- and nanofabrication techniques are clearly a key enabler of nanofluidics. New strategies have been developed to reach the nanoscale. While serial nanolithography techniques such as electron beam lithography, focused ion beam lithography, and scanning probe lithography are highly flexible in term of design, they are still expensive and slow methods. More economical solutions are replication techniques, including nanoimprint lithography, where the pattern of a template (often fabricated using a serial fabrication method) is transferred into a resist at the surface of the substrate using a mechanical contact. Many alternative lithography methods have recently emerged to further simplify the patterning at the nanoscale. These methods can usually be applied to simple patterns of nanostructures, such as arrays of nanodots or nanolines, which are common in nanofluidics. Chapter 3 presents these techniques, classified according to both the geometry of the channel and the network.

1.3  New Phenomena and Applications of Nanofluidics New phenomena at the nanoscale bring forward a wide range of new applications. The dimensionless channel size kh can be used to determine the mode of electrokinetic transport inside a channel, where h is the characteristic channel dimension such as the diameter or the height and k is the reserve Debye length or the thickness

1.3  New Phenomena and Applications of Nanofluidics



of the electric double layer (EDL) that is determined by the electrolyte concentration and the zeta potential of the channel wall. With a typical Debye length from 1 to 100 nm, the channel size in microfluidics is much larger than the thickness of the electric double layer (kh>>1). In contrast, the channel size in nanofluidics is of the same order as the thickness of the electric double layer (kh<1). As the EDLs overlap, the profile of the electro-osmosis velocity becomes parabolic. This effect has been applied to a separation method close to field flow fractionation. In contrast to microfluidics, surface charges affect the whole fluid volume inside a nanochannel. The excess charges on the channel wall result in permselectivity. The nanochannel rejects ions of the same charge (co-ions) while letting ions of the opposite charge (counter-ion) go through. This ion-selective phenomenon can be used to integrate filtration or concentration functions, which are frequently required pretreatment steps in the analysis of proteins. As the surface charge determines the ion transport in nanochannels, tailoring the surface chemistry allows for direct control of ionic transport characteristics through what can be called a nanofluidic resistor. On the other hand, monitoring the ion conductivity can be used as a method to detect adsorption of molecules onto the walls of the nanochannel. By patterning the surface chemistry of nanochannels, a barrier similar to a p-n-junction in microelectronics can be realized. Nanofluidic diodes could be one of the key components for potential new applications. Using the analogy of a field effect transistor, surface charges can also be controlled by a gate electrode that makes the realization of a nanofluidic transistor possible. The concepts of nanofluidic resistor, diode, and transistor open new opportunities in terms of ionic manipulation, paving the way to nanofluidic ionic integrated circuits. The interactions between nanopores and large molecules have been widely investigated in order to get a better picture of DNA electrophoresis, which plays a central role in standard sequencing methods. Deterministic, custom-designed nanofluidic arrays comprise an ideal media for the direct observation of sieving mechanisms and other fundamental experiments in biophysics. The high level of control on the geometry affords the optimization of the porous media, resulting in better separation performance, and the implementation of smart designs that make possible the exploration of new separation methods. Information such as the size of the strand (for restriction mapping) and the positions of fluorescent tags (which can be attached to a hybridization probe or to a transcription factor) can be directly obtained by stretching a DNA molecule through a nanochannel. The extreme confinement allows the isolation of a single molecule. By fabricating just below the nanochannel, near-field optics can be integrated leading to very high resolution. Nanofluidics can also be applied to the fields of water purification and alternative energy sources to meet the growing global need for clean water and clean energy. Nanofluidic structures, which match the typical pore size of the ultrafiltration designed to filter macromolecules, can filter small particles such as viruses. Nanopores in proton-conducting membranes are the key elements for microfuel cells, which can convert chemical energy directly into electricity. Current protonconducting membranes are made of nanoporous polymers such as Nafion. A number of current problems in membrane-based fuel cells, such as fuel crossover and water management, can be solved with fabricated nanostructures where the pore



Introduction

size, substrate material, and surface properties can be controlled. More details on existing and potential applications of nanofluidics are discussed in Chapter 4.

1.4  Organization of the Book This book is divided into five chapters. This chapter introduces the field of nanofluidics by reviewing classical research areas and material systems that dealt with nanofluidics in the past. This chapter also discusses the historical technology development leading to the establishment of the field of nanofluidics. In addition, this chapter briefly reviews recent applications of and perspectives on the potentials of nanofluidics. Chapter 2 discusses the fundamentals of mass transport on the microscale and nanoscale. Nanofluidic devices usually integrate both micro- and nanofluidic channels. Thus, the first part of Chapter 2 gives an overview of fluid flow management in microfluidics, describing the origin of nanofluidics and highlighting the similarities and differences between flows at the microscale and the nanoscale. While the nonslip condition and continuum assumption can comfortably be applied in microfluidics, they have to be carefully considered at the nanoscale. Next, Chapter 2 discusses the fundamentals of electrokinetics, which is a key transport effect in nanochannels. Sieving effects, such as the electrostatic effect, steric effect,and entropic effect, are treated in a separate section due to their importance in separation and concentration applications. Finally, Chapter 2 discusses the theoretical aspects of capillary filling, which is a simple but practical mode for liquid transport in nanochannels. Next, Chapter 3 discusses techniques of fabrication, first reviewing the fundamentals of micro- and nanofabrication. As nanochannels are usually integrated with microchannels and the manufacture of microfluidic and nanofluidic devices presents many similarities, the second part of Chapter 3 is dedicated to microfluidic fabrication methods. Although the most direct method to reach the nanoscale makes use of slow and expensive serial nanolithography tools, Chapter 3 shows how alternative methods can be applied to nanofluidic devices, depending on the geometry of the channels and the network. Starting with planar nanochannels, Chapter 3 guides readers through the different technologies for fabricating square and high-aspect-ratio nanochannels. Finally, Chapter 3 discusses technologies for fabricating solid-state nanopores and high-aspect ratio nanochannels. Subsequently, Chapter 4 describes the main applications of nanofluidics. First, Chapter 4 introduces the concept of lab-on-chip and provides an overview of microfluidic applications. Second, Chapter 4 details the application of nanofluidics to a range of techniques of manipulation for ions and biomolecules, including injection, routing, preconcentration, and batch or continuous flow separations. Sensing applications such as conductivity sensors and single molecule analysis are further practical applications of nanofluidics. Finally, Chapter 4 describes nanofluidic diodes and transistors and concludes by discussing further promising applications in DNA linear analysis. Chapter 5 summarizes the important topics covered by nanofluidics and gives an outlook on future development.

1.4  Organization of the Book



At the time we started writing this book, materials on nanofluidics were scattered throughout different books and technical papers. This book aims to serve as a reference on nanofluidics for readers working in academia and industry. Accordingly, we present here a complete review of current state-of-the-art nanofluidics technology. As a research field, nanofluidics is still in an early developmental stage. Many topics such as new fabrication technologies, the interactions of the different physics on the nanoscale, and applications of these new nanofluidic effects are still under intensive research. Thus, the ultimate goal of this book is to motivate researchers to move into this exciting field and to contribute more to our knowledge of fluid flow at the nanoscale.

Chapter 2

Fundamentals of Mass Transport at the Nanoscale

2.1  Fluid Flow at the Microscale 2.1.1  Scaling Laws

Microfluidic and nanofluidic applications benefit from the miniaturization of fluidic components such as flow channels. Miniaturization changes the ratio of impacts of different physical effects leading to new behaviors with new applications. The change of the spatial dimension or the size of the fluidic component of interest is characterized by the scaling factor S = l/l0, where l0 is a reference size. Miniaturization means that the scaling factor is smaller than unity (S < 1). For the entity of interest, a positive power of S means a decrease in its value. Next, we will consider the scaling law applied to dimensionless numbers. Dimensionless numbers are practical tools in fluid mechanics. The use of these numbers eliminates the dimension factor in an analysis. Independent of their size, systems with the same dimensionless numbers are considered to be similar. Many dimensionless numbers represent the ratio between a pair of forces and consequently their significance. The different forces occurring in the micro- and nanoscale can be generally categorized as surface forces and volume forces. Surface forces such as surface tension and viscous friction are proportional to the surface area, while volume forces such as weight, buoyancy force, or inertial force are proportional to the volume. The scaling of the ratio between surface force and volume force is formulated as:

Surface force 1 A L2 = = 3 = = S−1 Volume force V L L

(2.1)

where L, A, and V are the characteristic length, surface area, and volume, respectively. The above relation is also called the square-cube law, which implies that surface forces become dominant with miniaturization, while volume forces vanish. The ratio between inertial force and the viscous friction force is characterized by the Reynolds number:

Re =

r uL Inertial force = Viscous friction force m



(2.2)

where u is the average flow velocity, L is the characteristic length, m is the dynamic viscosity, and r is the density of the fluid. Under continuum assumption, the dynamic viscosity and the density remain constant with miniaturization. Thus, 

10

Fundamentals of Mass Transport at the Nanoscale

the Reynolds number is proportional to the characteristic length L. Keeping the timescale constant, the velocity is also proportional to the length scale. Thus, the Reynolds number is proportional to the square of the characteristic length as well as the square of the scaling factor:

Re ∝ S2



(2.3)

The Reynolds number decreases with two orders of magnitude of characteristic length. A small number indicates that the continuum flow in microscale is stable and laminar. The ratio between the inertial force and the surface tension force is characterized by the Weber number:

We =

Inertial force r u2 L = Surface tension force s



(2.4)

where s is the surface tension. Keeping the timescale constant, the Weber number is proportional to the cube of the length or the scaling factor:

We ∝ S3



(2.5)

The above relation implies that compared to the interfacial or surface tension force, the inertial force is negligible in microscale. Comparing the viscous friction force with the surface tension force results in the capillary number:

Ca =

mu Viscous force = Surface tension force s



(2.6)

Keeping the timescale, the capillary number is proportional to the velocity or the length scale. The scaling relation for capillary is then:

Ca ∝ S





(2.7)

Thus, it is expected that the surface tension force dominates over the viscous force in the micro- and nanoscale. For many transport effects, the ratio of the timescale can also be used to describe their physics. For instance, the ratio between the relaxation time of a nonNewtonian fluid and the characteristic time of the system is called the Weissenberg number:

Wi =

l lu Relaxation time = = Characteristic time L/u L

(2.8)

The term u/L is also the characteristic shear rate. Since relaxation time is assumed to be a material constant, the scaling relation of Weissenberg number is:

De ∝ S



(2.9)

2.1  Fluid Flow at the Microscale

11

Since both the Weissenberg number and the Reynolds number are proportional to the velocity u, the elasticity number is used to estimate the ratio between the elastic effects of non-Newtonian fluid and the inertial effect:

El =

l m0 Elastic force Wi = = Inertial force Re r L2

(2.10)

where m0 is the dynamic viscosity at zero shear. The scaling relation for the elasticity number is:

El ∝ S−2



(2.11)

The above scaling relation is important for the behavior of viscoelastic fluids in microscale. The viscoelastic force is dominant compared to the inertial force. Thus, turbulence and instability, which is caused by inertial force in macroscale, may still be possible with viscoelastic fluids in microscale. The ratio between the timescales of diffusion and convection is called the Peclet number:

Pe =

Diffusion time L2 /D Lu = = Convection time L/u D



(2.12)

where D is the diffusion coefficient. With D remaining constant with miniaturization, the scaling relation for the Peclet number is similar to the Reynolds number:

Pe ∝ S2



(2.13)

According to the above scaling relation, the diffusion time may approach the order of the convection time. This fact leads to the possibility of real-time measurement of reaction kinetics in the micro- and nanoscale. The above scaling relation shows that a number of interesting physical effects occur with miniaturization. Although the scaling laws are formulated based on continuum assumption, the general trend of the different effects is still valid in spatial dimensions on the order of 10–100 nm. Section 2.1.2 discusses the basic theories based on the continuum model of fluids. 2.1.2  Continuum Model 2.1.2.1  Governing Equations

In the case of liquid flow in nanochannels with a characteristic length of more than 10 nm, the flow of a Newtonian, isotropic, and incompressible fluid can be described respectively by the continuity equation, the Navier-Stokes equation, and the energy equation [1–4]. The continuity equation has the form:

Dr + r div∇ = 0 Dt

(2.14)

12

Fundamentals of Mass Transport at the Nanoscale

D where Ñ is the napla operator, ¾ is the total derivative operator: Dt

D ∂ ∂ ∂ ∂ ∂ = +u +v +w = + (v · ∇) Dt ∂t ∂x ∂y ∂z ∂t



(2.15)

and v = (u, v, w) is the velocity vector. The Navier-Stokes equation is derived from Newton’s second law:

r

Dv = r F − ∇p + m ∇2 v Dt



(2.16)

where r is the density of the fluid, and m is the dynamic viscosity of the fluid. The energy equation is formulated for the absolute temperature T as:

r cp

Dp DT = bT + div(k∇T) + F Dt Dt



(2.17)

� � 1 ∂p , k, and F are the specific heat at constant pressure, the where cp, β = − p ∂T p thermal expansion coefficient, the thermal conductivity and the dissipation function, respectively. The dissipation function for a Newtonian fluid is:



� � � � � � � �2 � � � ¶u 2 ¶v ¶w 2 ¶v ¶u 2 ¶w ¶v 2 F=m 2 +2 +2 + + + + ¶x ¶y ¶z ¶x ¶y ¶y ¶z �2 � �2 � � (2.18) 2 ¶u ¶w ¶u ¶v ¶w + m + + + + ¶z ¶x 3 ¶x ¶y ¶z 2.1.2.2  Slip Boundary Condition



The above set of partial differential equations can be solved for given boundary conditions. In macroscale and in most cases of microfluidics, the no-slip boundary condition is assumed at the solid-fluid interface, where the velocity at the wall is zero. While the no-slip boundary condition can be used for the Navier-Stokes equation, no-penetration boundary condition and no-temperature-jump boundary condition are used for the continuity equation and the energy equation, respectively. Navier proposed that the difference between the fluid velocity at the wall and the velocity of the wall itself is proportional to the velocity gradient at the wall or the shear stress. The proportional factor is called the slip length. The corresponding slip boundary condition is [5]: � ∂ u �� (2.19) Δu|wall = u|wall − uwall = Ls � ∂ y wall

where u is the velocity of the fluid and uwall is the velocity of the wall. The slip length can be interpreted as the distance between the channel wall and the virtual point where the velocity is zero.

2.1  Fluid Flow at the Microscale

13

For gas flow, the slip condition of the velocity and the jump condition of the temperature were derived by Maxwell [6] and Smoluchowski [7] as:

Δu|wall



� � 2 − σv ∂ u �� 3 η ∂ T �� =λ + σv ∂ y �wall 4 ρ Tgas ∂ x �wall

ΔT|wall =

(2.20)

� 2 − σT 2k λ ∂ T �� σT k + 1 Pr ∂ y �wall

(2.21)

where sv, sT, and Pr are the tangential momentum coefficient, the temperature accommodation coefficients, and the Prandt number, respectively:

sv =

cp h ti − trf dEi − dErf , sT = and Pr = ti − tre dEi − dEre k



(2.22)

where t, dE are the tangential momentum and the energy flux, respectively. The subscripts i, rf, and re stand for incoming, reflected, and reemitted, respectively. The tangential momentum coefficient and the temperature accommodation coefficient have a value between 0 and 1. Based on results from molecular dynamics simulation (see Section 2.2.3), liquid flow in channels larger than 10 fluid molecular diameters can still be considered as continuum [8]. And thus, the Navier-Stokes equation still can be used to describe them. However, an effective viscosity needs to be used in the case of smaller nanochannels. For a channel larger than 10 fluid molecular diameters, the effective viscosity is equal to the bulk viscosity. For a channel diameter less than 10 fluid molecular diameters, the effective viscosity in a nanochannel is larger than the bulk viscosity and increases significantly with a decreasing channel diameter. 2.1.3  Pressure-Driven Flow 2.1.3.1  Pressure-Driven Flow in Basic Channel Geometries

Figure 2.1 shows the basic geometries of a channel cross section. As mentioned above, the continuum assumption and the theoretical treatment from the mac-

Figure 2.1  Basic channel geometries: (a) circular cross-section; (b) parallel plate; and (c) rectangular cross-section.

14

Fundamentals of Mass Transport at the Nanoscale

roscale can still used for most cases of liquid flow in microchannels. The entrance length of a pressure-driven flow in a channel is estimated as [1–4]:

Lentrance = Dh

�

� 0.6 +0.056Re 1 + 0.035Re

(2.23)

where Dh is the hydraulic diameter of the microchannel, which is calculated from the cross-section area A and the perimeter U of the channel as:

Dh =

4A U

(2.24)

The typical small Reynolds number in microchannels leads to a negligible entrance length. Thus, a pressure-driven flow in a microchannel can be considered as laminar and fully developed. The pressure drop is proportional to the product of the Fanning friction factor and the Reynolds number fF Re as:

Dp = fF Re

2Lm D2h

u

(2.25)

where L is the length of the channel. Applying the Navier-Stokes equation to a channel with a circular cross-section of radius R, the velocity distribution and the average velocity are:









u(r) = −

� � dp 1 R2 − r2 dx 4m

u¯ =

dp R2 dx 8m



(2.26)

(2.27)

The product of the Fanning friction factor and the hydraulic diameter in this case are, fF Re = 16 (2.28) Dh = 2R Another simple case is a pressure-driven flow between two parallel plates that are 2H apart: � � dp 1 2 2 (2.29) y −H u(r) = − dx 2m

u¯ =

dp H2 dx 3 m

fF Re = 24 Dh = 2H





(2.30)

(2.31) (2.32)

2.1  Fluid Flow at the Microscale





15

The velocity distribution and the average velocity of a channel with rectangular cross-section H ´ W are [9]: � � � � n−1 np z dp 16W2 ∞ 1 cosh(np y/2W) 2 (2.33) cos 1− u(r) = − (−1) å dx mp 3 n=1,3,... n3 cosh(np H/2W) 2W

�� � � np H 192 W ∞ 1 dp W3 1− 5 u¯ = å n5 tanh 2W dx 3m p H n=1,3,...

(2.34)

The product of the Fanning friction factor and the Reynolds number is a function of the aspect ratio a = H/W [9]:

� � fF Re = 24 1 − 1.3553 α + 1.9467 α 2 − 1.7012 α 3 + 0.9564 α 4 − 0.2537 α 5 (2.35) At a small aspect ratio H<<W or a ® 0, the above product approaches that of a parallel plate. The hydraulic diameter of this channel type is determined as Dh = 4HW/(H + W ). 2.1.3.2  Macro Model for Complex Fluidic Networks

Pressure-driven flow in a complex microfluidic network with complex channel geometry requires the numerical solution of the governing equations. The complexity increases further for time-dependent analysis. Such simulation would need a huge computing capacity and computing time. A solution for this problem is reducing the complexity by increasing the abstraction level in modeling the flow in a microfluidic network. Components in the network are replaced by behavior-describing macro models. The physics of such a macro model can be analyzed using molecular-based or continuum-based or experimental approaches. The behavior-describing parameters are subsequently extracted and used in the macro model. This approach is similar to the modeling approach used in electronics, where components such as resistors, capacitors, inductors, diodes, and transistors are described by their behavior. The network of these components can be solved by well-known tools such as PSPICE. To utilize macro-modeling tools from electronics, an analogy between fluidics and electronics can be used to formulate basic components such as resistors, capacitors, and inductors (Figure 2.2). Electric current I, voltage V are analogs to mass flow rate m  ˙ and pressure drop Dp in fluidics. Thus, the fluidic resistance Rf, fluidic capacitance Cf, and fluidic inertance Lf are defined by the following basic relations [10]:

Dp ˙ m



Rf =



Dp = Lf



˙ = Cf m

(2.36)

˙ dm dt

(2.37)

dDp dt

(2.38)

16

Fundamentals of Mass Transport at the Nanoscale

Figure 2.2  The product of Fanning friction factor and the Reynolds number as a function of the aspect ratio a = H/W [9].

Source components such as pressure source and mass flow source can also be defined. Similar to electronics, more complex fluidic components such as micropumps and microvalves can be idealized and modeled by a network of the above basic fluidic components. Value tables can be used for complex components that cannot be described by a linear differential equation. The fluidic components are linked to a network governing their interactions. Their interactions are governed by the network. There are certain physical limits in exploiting the analogy between electronics and fluidics. For instance, an electric potential can have any value, while the absolute pressure can only be positive. The gauge pressure or the relative pressure to the atmospheric pressure p0 can only have a smallest value of -p0. While the propagation velocity of electric signals is approximately the light speed, the signals in a fluidic network propagate with a sound velocity that is 6 orders of magnitude slower. The light speed is the upper limit of an electromagnetic wave, while a velocity higher than the sound speed is still possible in fluidics. Applying the scaling relation to the wave frequency:

f=

us l

(2.39)

where us and l are the sound speed and the wavelength, respectively. Thus for the same wavelength, the propagation of a 100-Hz wave in microfluidics corresponds to that of a 100-MHz electromagnetic wave. The use of a network model with fluidic resistance, capacitance, and innertance is only valid if the channel length of the component is shorter than a critical length determined by the wavelength of the corresponding sound wave [10]:

L << lsound =

us f

(2.40)

Another limitation is the assumption that the mass flow rate and the pressure drop have the same phase in time-dependent processes. This assumption is only

2.1  Fluid Flow at the Microscale

17

correct if the velocity distribution in the channel has the same phase of the oscillating pressure. At high oscillation frequencies and large channel cross-sections, the distributions of pressure and velocity can be out of phase. To estimate the critical out-of-phase situation, the dynamic Reynolds number is introduced [11]:

Red =

Dh 2

�

wr m

(2.41)

where w = 2pf is the angular frequency and Dh is the hydraulic diameter of the channel. The dynamic Reynolds number is derived from the Laplace solution of the Navier-Stokes equation in a cylindrical channel:

ρ

� � ∂u dp 1 ∂ u ∂ 2u =− +u + 2 ∂t dx r ∂r ∂r

(2.42)

The Laplace solution of the velocity distribution in frequency domain is [11]

� � 1 dpˆ J0 (r*) −1 uˆ = sr dx J0 (R*)

(2.43)

where s is the Laplace transform and J0 is Bessel function of zero order and first kind. The complex arguments of the Bessel functions are:



� sr /m � R* = jDh /2 sr /m r* = jr

(2.44)

ˆ For a harmonic forcing function of the pressure d p/dx = Dp/Dx sin(w t), the Laplace transform s can be replaced by jw, and the velocity distribution can be written as: � � J (r*) 1 Dp (2.45) −1 uˆ = sin(w t) 0 jwr Dx J0 (R*) Figure 2.3 shows typical velocity profiles at different dynamic Reynolds numbers. Integrating the Laplace solution of the velocity distribution across the channel crosssection results in the relation between the pressure drop and the mass flow rate:



dpˆ 4s J0 (R*) ˆ˙ = m dx p D2 J2 (R*)

(2.46)

where J2 is Bessel function of second order and first kind. The series development of the Bessel quotient results as:

� � 128 m dpˆ 16 1 ˆ˙ m + jw =− dx 3p D2h p D4h r

(2.47)

18

Fundamentals of Mass Transport at the Nanoscale

Figure 2.3  (a–c) Velocity profile at different dynamic Reynolds numbers.

with the definition of fluidic resistance and fluidic inertial per unit channel length:

rf =

128 m

p D4h r

(2.48)

16 1 lf = 3p D2h



The relation between pressure drop and mass flow rate is:

dpˆ ˆ˙ = −(rf + jw lf )m dx



(2.49)

The critical frequency can be determined as:

wcr =

1 r 24 m = f = t lf r D2

(2.50)

With the above critical frequency, the critical dynamic Reynolds number is:

Red,cr =

√

24/2 ≈ 2.45

(2.51)

Thus, the condition for the use of network model (see Figure 2.4) is:

Red ≤ Red,cr ≈ 2.45



(2.52)

The continuity equation in the frequency domain is formulated as:

ˆ˙ ∂m = −jω (ρ Aγ + celastic )pˆ ∂x

(2.53)

and in the time domain as:

ˆ˙ ∂m ∂p = −(ρ Aγ + celastic ) ∂x ∂t

(2.54)

2.1  Fluid Flow at the Microscale

19

Figure 2.4  Equivalent network model of a microchannel.

where g is the compressibility of the fluid. The above equation indicates the two components of the fluidic capacitance: the compressibility of the fluid and the compliance of the channel wall dA/dp. The elastic component of the fluidic capacitance is:

celastic = r





dp dA

(2.55)

The effective fluidic capacitance consists of both compressibility and elastic compliance: � � dp (2.56) cf = r Ag + celastic = r Ag + dA Figure 2.4 shows the equivalent network model for a microchannel. According to the above model, a cylindrical channel with a diameter of D and a length of L and filled with a fluid of dynamic viscosity m and density r can be described with the following basic components:

128 m L pr D4 16m L Lf = 3p D2 � � p D2 Cf = rg + celastic L 4 Rf =



(2.57)

2.1.4  Capillary Flow 2.1.4.1  Surface Tension

Capillary flow is the motion of a fluid driven by surface tension or by interfacial tension. The surface tension at an interface is defined as the amount of Gibbs free energy per area at a given pressure and temperature. Therefore, surface tension has a unit of energy per unit area (J/m2) or force per unit length (N/m). Surface tension is caused by the intermolecular forces of fluid molecules. In the bulk, a fluid molecule is surrounded and attracted by other molecules. Taking a

20

Fundamentals of Mass Transport at the Nanoscale

simple cube model, a molecule experiences six chemical bonds to other molecules. Each molecule is pulled equally in all direction resulting a zero net force. At the surface, one bond is missing making the molecule to be pulled inward and leading to a higher free energy of the molecules at the surface. The zero net force is only warranted due to the resistance to compression of the fluid. That means, energy is required to form a surface, and a free surface tends to minimize its energy by retaining a minimum surface area. 2.1.4.2  Young-Laplace Equation



The existence of pressure difference across a surface and its surface tension leads to a curved shape. The pressure difference across a surface leads to a force acting normal to the surface. To achieve a force equilibrium, the surface should be curved so that the normal force balances with the surface tension forces [4]: � � 1 1 (2.58) Dp = s + r1 r2 where Dp is the pressure difference, also called the capillary pressure, and r1 and r2 are radii of curvature in the axes parallel to the surface, as shown in Figure 2.5. The terms 1/r1 and 1/r2 are called the curvatures of the surface. Equation (2.58) leads to the following scaling relation:

Dp ∝ S−1



(2.59)

That means that the capillary pressure increases with miniaturization and can be used as an actuation source for delivering liquids. 2.1.4.3  Contact Angle

In a system with three phases such as gas, liquid, and solid, the contact between these phases forms a line. The surface tension forces between the phase pairs should balance on this line, where the contact angle is defined as the angle between the solid-liquid and liquid-gas interfaces. This balance equation is called the Young equation [1–4]:

ssg − ssl = slg cosq



Figure 2.5  Capillary pressure.

(2.60)

2.1  Fluid Flow at the Microscale

21

Figure 2.6  Balance of interfacial tensions in a gas-liquid-solid system.

where ssg, ssl, and slg are the interfacial tensions of the solid-gas, solid-liquid, and liquid-gas interfaces and q is the contact angle. Figure 2.6 illustrates the balance of the three components at the contact line, schematically showing the shape of a liquid plug in a capillary. A concave meniscus has a contact angle less than 90°, while a convex meniscus has contact angle of greater than 90°. The contact angle represents the property of the surface. A contact angle less than 90° indicates a hydrophilic surface, while a contact angle greater than 90° represents a hydrophobic surface. 2.1.4.4  Capillary Flow in a Horizontal Channel

A liquid column is driven in a cylindrical capillary by the surface tension and the meniscus (see Figure 2.7). If the capillary has an inner radius of R, the balance equation between the inertial force, friction force, and the driving surface tension force results in:

Finertial + Ffriction = Fsurface tension

(2.61)

� � d dx dp +p R2 x rp R2 x = 2p Rslg cosq dt dt dx

(2.62)

or

where x is the position of the meniscus. The friction force or the pressure gradient can be estimated with the Hagen-Poiseuille model:

dp 8 m dx = 2 dx R dt



Figure 2.7  Capillary flow in a horizontal cylindrical channel.

(2.63)

22

Fundamentals of Mass Transport at the Nanoscale

Substituting the friction term into the force balance results in the governing equation of capillary flow:

� � 2p Rslg cosq 1 d2 x 1 dx 2 8m dx + + 2 − =0 2 dt x dt rR x R r dt



(2.64)

with the initial conditions:

x|t=0 = 0 � dx �� =0 � dt t=0



(2.65)

� � 1 dx 2 is small and can be neglected. Substituting y = x2 In (2.64), the term x dt into (2.64) results in the ordinary differential equation:

y¨ + a y˙ = b

(2.66)

with

8m

a=



2

R r

b =

4slg cosq

rR



(2.67)

The solutions of the meniscus position and meniscus velocity are:

�

b b b exp(−a t) + t − 2 a2 a a b [1 − exp(−a t)] u= 2a x x=



√

y=

(2.68)

For the starting period t ® 0, the above solution has the form:

x=

�

b t= a

�

s cosq t 2m R

(2.69)

which is known as the Washburn equation. 2.1.5  Electrokinetic Flow 2.1.5.1  Electric Double Layer and the Electric Field

Electrokinetic effects are caused by surface charges at the interface between a solid and a liquid or between two liquids. Electrokinetic effects are categorized into four basic groups [12, 13]: ·

Electro-osmosis is the flow of an electrolyte in an electric field relative to a charged surface;

2.1  Fluid Flow at the Microscale

23

Figure 2.8  The electric double layer: (a) the concept of Stern layer and Gouy-Chapman layer and (b) the potential distribution.

· · ·



Electrophoresis is the motion of charged particles relative to the surrounding liquid under an electric field; Streaming potential is the opposite effect of electro-osmosis; an electrolyte moving relative to a charged surface generates an electric field; Sedimentation potential is the opposite effect of electrophoresis; moving charged particles relative to a charge surface generates an electric field.

The preceding electrokinetic effects are caused by an electric double layer at the solid-liquid interface. The electric double layer is the result of the interaction between an electrolyte and a charged surface (Figure 2.8). Counter ions from the electrolyte are attracted by the surface and form a thin charge layer. This charge layer is also called the Stern layer. The Stern layer is immobilized on the surface by electrostatic force between opposite charges. The Stern layer builds up a thicker charge layer inside the electrolyte. The second layer is called the diffuse layer or the Gouy-Chapman layer. Under an electric field, the second layer can move relative to the solid surface. The immobile Stern layer and the mobile Gouy-Chapman layer form the so-called electric double layer (EDL). The interface between these two layers is called the shear layer. The electric potential of the wall and of the shear layer are called the wall potential and the zeta potential, respectively. The existence of ions in a liquid leads to the electric potential distribution in the channel. The electrostatic forces for transport in micro- and nanochannels are directly determined by this electric field. The electric field is given by the PoissonBoltzmann equation: � � re e zi eY 2 0 (2.70) ∇ Y = − = − å zi ni exp − e e i kB T where Y is the electric potential; re is the charge density; e = 1.602 ´ 10-19 C is the elementary charge; e = er e0 is the relative permittivity of the liquid, with e0 = 8.854 ´ 10-12 F/m the dielectric constant of vacuum and er the relative permittivity; zi is the valence number of the ion species and n0i is its bulk concentration; and kB = 1.381 ´ 10-12 J/K is the Boltzmann constant. In the case of a binary solution with cations (z1) and anions (-z2), the Poisson-Boltzmann equation becomes:



� 2 � 1/lD ∇ Y* = − exp(−z1 Y*) − exp(−z2 Y*) z1 + z2 2

(2.71)

24

Fundamentals of Mass Transport at the Nanoscale

eY is the dimensionless electric potential and lD is the Debye length kB T or the thickness of the electric double layer: where Y* =

� � � lD = � �



e kB T 2



(2.72)

e2 å z2i n0i i=1

For a symmetric electrolyte (z1 = z2 = z), the Poisson-Boltzmann equation can be simplified as:

∇2 Y* =



1 sinh(zY*) lD2 z



(2.73)

The equation can be simplified further for the case of low potential, where the hyperbolic sine term becomes linearized [sinh(zY*) » zY*]:

∇2 Y* =

1 Y* (zY*) = 2 2 lD z lD

(2.74)

The above equation is linear and can be solved analytically. With the wall condition Y*= z*, and the infinity condition Y*|y=∞ = 0 , the solution of (2.44) is:

Y* = z *exp(−y/lD )

(2.75)

ez is the dimensionless zeta potential. kB T In microfluidics, the channel diameter is usually much larger than the thickness of the electric double layer. Considering a zero-pressure gradient and steady-state condition, the Navier-Stokes equation has the simplified form:

where z * =



−u

d2 u d2 Y = e E dy2 dy2

(2.76)

where e is the dielectric constant and the dielectric constant of vacuum. Solving (2.76) results in the electro-osmotic velocity or the Smoluchowski velocity:

ueo = −

e Ez m

(2.77)

At a constant viscosity and a constant zeta potential, the Smoluchowski velocity is proportional to the applied electric field E (Figure 2.9). The proportional factor is called the electro-osmotic mobility meo:

meo = −

ez m

(2.78)

2.1  Fluid Flow at the Microscale

25

Figure 2.9  The concept of electro-osmotic flow.

2.1.5.2  Electrokinetic Flow Between Parallel Plates

In the case of a one-dimensional micro/nanochannel, which is modeled as a gap between two parallel plates (Figure 2.10), the channel height h is the width of the gap. The velocity profile u(y) of the electro-osmotic flow between the two parallel plates can be derived from the one-dimensional form of the Navier-Stokes equation (2.14). For further simplification, a function U(y) is introduced [14]:

U(y) = u(y) −

ueo Y(y) z

(2.79)

where ueo is the Smoluchowski velocity. The Navier-Stokes equation then has the homogenous form:

d2 U(y) =0 dy2



(2.80)

with the boundary conditions:



y=h

The solution for (2.76) is:

� dU(y) �� =0 dy �y=0 � U� = −ueo

u(y) = ueo

�

(2.81)

� Y(y) −1 z

Figure 2.10  Model of electro-osmotic flow between two parallel plates.



(2.82)

26

Fundamentals of Mass Transport at the Nanoscale

Introducing the dimensionless velocity, potential, zeta potential, and coordinate:

u* = u/ueo ; Y* = zeY/kB T; z * =

zez and y* = y/h kB T

(2.83)

The velocity distribution and the Poisson-Boltzmann equation have the dimensionless form:

u*(y*) =



dY* = dy*

Y*(y*) −1 z* �

h lD

�2

(2.84)

Y*

(2.85)

The boundary conditions for (2.85) are:

Y*|y*=1

� dY* �� = z *, =0 dy* �y*=0

(2.86)

The solution of (2.85) is then:

Y* = z *

cosh(hy*/lD ) cosh(h/lD )



(2.87)

Substituting (2.87) into (2.84) results in the dimensionless velocity distribution of an electro-osmotic flow between two parallel plates (Figure 2.11):

u*(y*) =

cosh(hy*/lD ) −1 cosh(h/lD )



Figure 2.11  (a, b) Typical solution of potential and velocity between two parallel plates.

(2.88)

2.1  Fluid Flow at the Microscale

27

2.1.5.3  Electrokinetic Flow in a Cylindrical Capillary

The Navier-Stokes equation and the Poisson-Boltzmann equation are formulated in the cylindrical coordinate system as (Figure 2.12):



� � � � 1 d du dY d r = eE r −m r dr dr dr dr

(2.89)

� � � � ze 1 d dY 2zen∞ r = sinh Y r dr dr e kB T

(2.90)

introducing the dimensionless variables

u* = u/ueo ; Y* = zeY/kB T; z * =

zez and r* = r/R kB T

(2.91)

where R is the radius of the cylindrical capillary. Under the linearized assumption of low potential, the Navier-Stokes equation and the Poisson-Boltzmann equation have then the dimensionless forms:



� � 1 d du* 1 r* = − Y* r* dr* dr* z* d2 Y* dr*2

1 dY* + = r* dr*

�

R lD

�2

sY*



(2.92)



(2.93)

The boundary conditions of the Navier-Stokes equations are:

u*|r*=1 The solution of (2.92) is:



u*(r*) = 2

� du* �� = 0, =0 dr* �r*=0

(R/lD )2 ∞ Cn J0 (ln r*) å l 2 J2 (l ) z * n=1 n 1 n

Figure 2.12  Model of electro-osmotic flow in a cylindrical capillary.

(2.94)

(2.95)

28

Fundamentals of Mass Transport at the Nanoscale

where J0 and J1 are the Bessel function of the first kind and of the zero and first order, respectively. ln is the nth zero point of the Bessel function of the first kind, zero order J0(ln) = 0. Cn is the function of the dimensionless potential Y*:

Cn =

�1

xJ0 (ln x)Y*(x)dx

(2.96)

x=0

The boundary conditions of the Poisson-Boltzmann equation are:

� � dY* �� � Y* r*=1 = z *, =0 dr* �r*=0

(2.97)

Solving (2.93) with the above boundary equation results in

� I0 (Rr*/lD ) Y*�(r*) = z * I0 (R/lD )



(2.98)

where I0(x)=i-nJ0(ix) is the modified Bessel function of the first kind and zero order. Substituting (2.98) into (2.96) results in:

Cn =

z* J1 (ln )I1 (R/lD ) I0 (R/lD ) ln [1 + (ln )2 ]

(2.99)

The dimensionless velocity distribution in a cylindrical micro/nanochannel is then (Figure 2.13):

� �2 R z* J1 (ln )I1 (R/lD ) u*(r*) = 2 lD I0 (R/lD ) ln [1 + (ln )]2

Figure 2.13  (a, b) Typical solution of potential and velocity in a cylindrical capillary.

(2.100)

2.1  Fluid Flow at the Microscale

29

2.1.5.4  Electrokinetic Flow in a Rectangular Channel

The Navier-Stokes equation and the Poisson-Boltzmann equation are formulated in the Cartesian coordinate system as (Figure 2.14):



�

� 2 � ¶ Y ¶ 2Y = eE + ¶ 2z ¶ 2y

(2.101)

� � ze ∂ 2Ψ ∂ 2Ψ 2zen∞ sinh + 2 = Ψ ∂ 2z ∂ y ε kB T

(2.102)

−m

¶ 2u ¶ 2u + 2 ¶ 2z ¶ y

�

Dimensionless equations are achieved by introducing the following dimensionless variables

u* = u/ueo ; Y* = zeY/kB T; z * =

zez , y* = y/Dh and z* = z/Dh (2.103) kB T

with the hydraulic diameter Dh = 4WH/(W + H) as the characteristic length. The linear assumption of the hyperbolic function sinh(x) = x is only correct for channel larger then the Debye length, which may not be true in the case of nanochannels. However, for simplicity we still consider this assumption here. The dimensionless forms of Navier-Stokes equation and Poisson Boltzmann equation are:





∂ 2 u* ∂ 2 u* + 2 = −ΓΨ* ∂ 2 z* ∂ z* ∂ 2 Ψ* ∂ 2 Ψ* + 2 = ∂ 2 z* ∂ y*

�

Dh λD

�2

Ψ*

(2.104)

(2.105)

� � 1 Dh 2 represents the ratio between the The dimensionless number G = z * lD electrostatic force and the friction of the flow and is called the electroviscous number. With the boundary conditions: � � � � � � ∂ u* ∂ u* � � u*�z*=W/D = 0, u*�y*=H/D = 0, = 0, = 0 (2.106) � h h ∂ z* z*=0 ∂ y* �y*=0

Figure 2.14  Model of electro-osmotic flow in a rectangular channel.

30

Fundamentals of Mass Transport at the Nanoscale

The solution of the velocity distribution in the rectangular channel is:

u* =

×

4GD2h ∞ ∞ cos(an z*)cos(bm y*) å å WH n=1 m=1 an2 + bm2 (2.107)

W/D � h H/D � h

cos(an z*)cos(bn y*)Y*dz*dy*

0

0



where

(2n − 1)p Dh for n = 1, 2, 3 . . . 2W



(2.108)

(2m − 1)p Dh for m = 1, 2, 3 . . . 2H



(2.109)



an =



b =



The potential distribution is calculated based on the Poisson-Boltzmann equation. With the potential boundary conditions: � � � � ∂ Ψ* �� ∂ Ψ* �� � � (2.110) Ψ* z*=W/D = Ψ* y*=H/D = ζ *, = h h ∂ z* �z*=0 ∂ y* �y*=0

The distribution of the dimensionless potential is (Figure 2.15) [10]: ⎧ �� � ⎪ (2n − 1)2 p 2 D2h Dh ⎪ n+1 ⎪ 1+ y* ⎪ � � ⎨ ∞ (−1) cosh 4(W/lD )2 lD (2n − 1)p Dh � Y* = 4z * å z* � cos � ⎪ 2 p 2 D2 2W ⎪ n=1 − 1) (2n H ⎪ h ⎪ (2n − 1)p cosh 1+ ⎩ 4(W/lD )2 lD

⎫ (2.111) � 2 p 2 D2 ⎪ − 1) (2n D ⎪ h h 1+ z* (−1)n+1 cosh ⎪ � �⎪ ⎬ 2 4(W/lD ) lD (2n − 1)p Dh � cos y* + � � ⎪ 2H ⎪ (2n − 1)2 p 2 D2h W ⎪ ⎪ (2n − 1)p cosh 1+ ⎭ 2 4(W/lD ) lD ��

Figure 2.15  (a, b) Typical solution of potential and velocity in a rectangular channel.

2.1  Fluid Flow at the Microscale

31

Figure 2.16  Motion of a charged spherical particle in an electric field.

2.1.5.5  Electrophoresis

Electrophoresis is the movement of a charged surface relative to a stationary liquid under an electric field. Considering a spherical particle with radius R and a surface charge of qs moving under the influence of an electric field E in a stationary liquid with viscosity m and a low electric conductivity, the balance between the driving electrophoretic force and the friction force is [15] (Figure 2.16):

qs E = 6p m Ruep

(2.112)

where uep is the electrophoretic velocity of the particle. The Stokes drag model was taken for the friction force. We assume a relatively slow velocity and a small particle size. Thus, the Reynolds number based on the particle diameter is small, and the Stokes flow regime with a Reynolds number less then unity can be assumed. Rearranging the above equation leads to:

qs E 6p m R

uep =



(2.113)

The electrophoretic velocity is proportional to the applied electric field. The proportional factor is called the electrophoretic mobility:

uep =

qs 6p m R



(2.114)

The dependence of electrophoretic mobility on the surface charge qs and the size R leads to applications in capillary electrophoresis separation. A mixture of molecules with different sizes and different charges can be separated using an electric field. The smallest molecule with the largest surface charge will arrive first due its high electrophoretic mobility. The surface charge density of a particle can be estimated based on the zeta potential z, the Debye length lD, and the permittivity e as [15]:

q��s

�

∂Φ = −ε ∂r

�

r=R

� � R εζ 1+ = R λD

(2.115)

For small molecules, the size of the molecule is much small than the Debye length, which is on the order of 100 nm. With R << lD or R /lD ® 0. The electroosmotic velocity of charged molecules or small nanoparticles can be estimated as:

32



Fundamentals of Mass Transport at the Nanoscale

uep =

qs 4p R2 ez /R 2 ez E E= E= 6p m R 6p m R 3 m

(2.116)

For particles larger than the Debye length, the electrophoretic velocity approaches the Smoluchowski velocity for flat surfaces:

ueo =

ez E m

(2.117)

2.1.5.6  Electrowetting

Electrowetting is also called the electrocapillary effect, where the force balance at the interface or at the contact line is manipulated by electrostatic forces. The three electrocapillary effects that find applications in microfluidics and nanofluidics are continuous electrowetting, direct electrowetting, and electrowetting on dielectric. Continuous electrowetting occurs at the interface between two electrically conducting liquids. If two immiscible conducting liquids such as mercury and an aqueous electrolyte are in contact, an electrical double layer exists at their interface. With this electrical double layer, a mercury droplet in an aqueous electrolyte can be considered as a charged particle. Exposing this droplet to an electric field causes the droplet to move, similar to the case of electrophoresis. For a droplet with a length L in a flat channel with a height H (Figure 2.17), the balance between the driving electrostatic force and the friction force is:

qs E =

6 m uCEW H

(2.118)

where qs is the surface charge density of the droplet. The friction is estimated here based on the Hagen-Poiseuille model. If the voltage across the droplet DV is known, the electric field over the droplet is estimated as DV/E. The droplet velocity caused by continuous electrowetting is then:

uCEW =

q2 H DV 6m L



(2.119)

Direct electrowetting is another electrocapillary effect at the contact line of a gas-liquid-solid system (Figure 2.18). The solid phase is made of an electrically

Figure 2.17  Continuous electrowetting: (a) formation of an electric double layer at the interface and (b) the droplet moves in an electric field.

2.1  Fluid Flow at the Microscale

33

Figure 2.18  Direct electrowetting: (a) formation of an electric double layer at the interface and (b) an applied voltage changes the contact angle.

conducting material. An electrical double layer exists at the interface between the liquid and the solid. The electrical double layer represents an electric capacitance per unit surface of:

cEDL =

e lD

(2.120)

where e is the permittivity of the electrolyte. The interfacial energy between the liquid and solid is reduced by the amount energy stored in the electric double layer. The relation between the solid-liquid interfacial tension ssl and the applied voltage DV is described by the Lippmann equation [16]:

ssl = ss10 −

cEDL DV2 2

(2.121)

where ssl0 is the initial interfacial tension at DV=0 V. Combining the Lippmann equation with the Young equation, the contact angle can be determined as a function of the applied voltage:

� � 1 cEDL DV2 q = arcos cosq0 + slg 2

(2.122)

where q0 is the initial contact angle without the applied voltage. To cause an originally hydrophobic surface to become hydrophilic, the contact angle should be less than 90°; the corresponding condition for the applied voltage is:

DV >

�

−2slg cosq0 cEDL

(2.123)

The major drawback of direct electrowetting is that the capacitance per unit surface is determined by the thickness of the electric double layer. Coating the electrode with a hydrophobic dielectric material such as Teflon, the capacitance per unit surface can be adjusted by the film thickness of the dielectric d (Figure 2.19):

cEDL =

e d

(2.124)

34

Fundamentals of Mass Transport at the Nanoscale

Figure 2.19  Electrowetting on dielectric: (a) the hydrophobic dielectric layer acts as a capacitor and (b) an applied voltage changes the contact angle.

The relation between the contact angle and the applied voltage in the case of electrowetting on dielectric can then be described with the Lippmann equation as in the case of direct electrowetting. 2.1.6  Mixing in Microscale 2.1.6.1  Stokes-Einstein Model of Diffusion and Fick’s Law

The final phase of any mixing process is molecular diffusion. Due to the small size involved in microfluidics and nanofluidics, diffusion processes can occur much faster than in microscale. We can take the same model of a spherical particle used previously for electrokinetics to estimate the diffusion coefficient of a particle with a radius of R. In this case, the driving force for the particle is the random force F(t) coming from the liquid molecules. The force balance is then:

m

du = 6p m Ru + F(t) dt

(2.125)

where m is the mass of the particle. This force F(t) is random; thus its auto-correlation function represents the Dirac delta function:

�F(t)F(t )� = Ad (t − t )



(2.126)

The solution of the force balance equation is [17]:

� � � �t � � � 6p m Rt 6p m Rt 6p m Rt + exp − dt (2.127) u(t) = u0 exp − exp − m m m 0

The variance of the displacement x(t) is:

dx2 = 2x(t)u(t) = 2 dt

�t 0

2

u(t)u(t )dt = 2�u �

�t 0

� −6p m Rt dt exp m �

(2.128)

For a timescale larger than m/6p mR:

�x2 (t)� = 2Dt

(2.129)

2.1  Fluid Flow at the Microscale

35

The diffusion coefficient of the particle can then be determined as:

D = �u2 �

m 6p m R

(2.130)

The kinetic energy of the particle is proportional to the absolute temperature:

1 1 kB T m�u2 � = kB T or �u2 � = 2 2 m

(2.131)

where kB is the Bolzmann constant. Substituting (2.131) into (2.130) results in the diffusion coefficient:

D=

kB T 6p m R



(2.132)

The unit of the diffusion coefficient is square meters per second, the same as the kinematic viscosity and the temperature diffusivity, which are respectively diffusion coefficients for momentum and temperature. The flux of diffusion j is related to the diffusion coefficient by the Fick’s law:

j = −D

dc dx

(2.133)

where D is the diffusion coefficient and c is the species concentration. 2.1.6.2  Taylor-Aris Dispersion

Figure 2.20 shows the concept of Taylor-Aris dispersion. Taylor-Aris dispersion is caused by the distributed velocity profile in a microchannel. If the velocity distribution is plug-like as in the case of electrokinetic flow [Figure 2.20(a)], all fluid layers move with the same velocity, and the species are dispersed with the molecular diffusion. If the velocity distribution is parabolic as in the case of pressure-driven flow, different fluid layers move with different velocity. Diffusion between the fluid layers leads to a higher apparent diffusion coefficient in the flow direction. For instance, the apparent diffusion coefficient in the flow direction in a cylindrical capillary is [17]:

Figure 2.20  The impact of velocity distribution on the dispersion of a species: (a) plug-like flow and (b) pressure-driven flow.

36





Fundamentals of Mass Transport at the Nanoscale

� � 1 2 D* = D 1 + Pe 48

(2.134)

where D is the molecular diffusion coefficient and Pe is the Peclet number based on the capillary diameter. In the case of a flow between two parallel plates, the apparent diffusion coefficient is: � � 1 2 (2.135) D* = D 1 + Pe 210 where the Peclet number is evaluated based on the gap between the two plates. Because diffusion is a slow process, the basic mixing concept is to divide the mixing domain into smaller segments and to reduce the diffusion length. In the macro scale, mixing is improved by creating turbulence. Since the Reynolds numbers in microfluidic and nanofluidic devices are often less then unity, inertial effects are negligible, and turbulence based on inertial/viscous competition is not achievable. Mixing devices are categorized as active and passive micromixers according to the way the mixing domain is segregated. The following sections discuss the basic passive concepts for decreasing the mixing path such as parallel lamination, sequential lamination, sequential segmentation, and chaotic advection. 2.1.6.3  Parallel Lamination



Parallel lamination divides the mixing stream into n substreams. As a result, mixing time is reduced to 1/n2 of the original time. For the simple model of a flat channel with two inlets as shown in Figure 2.21, the concentration is governed by the transport equation: � � 2 ∂c ∂ c ∂ 2c (2.136) u + =D ∂x ∂ x 2 ∂ y2

Figure 2.21  (a, b) Model of mixing based on parallel lamination.

2.1  Fluid Flow at the Microscale

37

Figure 2.22  Concentration distribution with parallel lamination: (a) two streams [18]; and (b) multiple streams [19].

The flow rates and the concentration of the inlets are Q1, c0 and Q2, 0, respectively. Assuming the same viscosity and density, the interface position is r = Q1/(Q1 + Q2), which is the same as the final mixing ratio a. Normalizing spatial parameters by the channel width W and the concentration by c0, the transport equation has the dimensionless form:

Pe

∂ c* ∂ 2 c* ∂ 2 c* + = ∂ x* ∂ x*2 ∂ y*2

(2.137)

where Pe = uW/D is the Peclet number. The star indicates the dimensionless variables. The inlet conditions are c*(x* = 0, 0 £ y* < r) = 1 and c*(x* = 0, r £ y* < 1) = 0. Full mixing is assumed at an infinite outlet ¶ c*(x*®¥, y*)/¶ x* = 0. The channel wall is impermeable and thus ¶ c*/¶ y* = 0. Solving the above equation and the corresponding boundary conditions using the method of separation of variables results in the concentration distribution [Figure 2.22(a)] [18]: � � 2 ∞ sinap n 2n2 p 2 � c*(x*, y*) = a + å x* (2.138) cos(np y*)exp − p n=1 n Pe + Pe2 + 4n2 p 2 A solution for mixing of multiple streams can be achieved with the same method. Figure 2.22(b) shows the typical concentration of parallel lamination of multiple streams [19].

2.1.6.4  Sequential Lamination

Sequential lamination segregates the joined stream into two channels and rejoins them in the next stage. Using n such splitting stages in serial, 2n layers can be created. The mixing path is reduced to 1/2n of the original distance. Consequently, the mixing time is reduce to 1/4(n−1) of the original mixing time. Figure 2.23 illustrates this mixing concept. This mixing concept is also called the split-and-recombine (SAR) concept. 2.1.6.5  Sequential Segmentation

Sequential segmentation divides the mixed fluids in segments along the flow direction. In this way, the effective diffusion coefficient caused by Taylor dispersion leads

38

Fundamentals of Mass Transport at the Nanoscale

Figure 2.23  (a–c) Sequential lamination.





to a shorter mixing time. The segments are injected using active valves at the inlets of the mixer. The mixing ratio is adjusted by the opening time of each valve. Figure 2.24 shows a simple one-dimensional model to demonstrate the concept of sequential segmentation [20]. Pressure-driven flow in microchannels with a low aspect ratio W >> H can be reduced to the model of two parallel plates. The model assumes a flat velocity profile in the channel width direction (z-axis) and a Poiseuille velocity profile in the channel height. Thus, the effective diffusion coefficient can be taken from the model of Taylor-Aris dispersion in a flow between two parallel plates. The concentration distribution across the channel width and height is assumed to be uniform. Thus the problem can be reduced to a single dimension. At a flow velocity u and a switching period T, the characteristic segment length is defined as L = uT. The general transport equation is reduced to the transient onedimensional form: ∂c ∂c ∂ 2c (2.139) +u = D* 2 ∂t ∂x ∂x where D* is the effective diffusion coefficient taken from the Taylor-Aris dispersion model for parallel plates. The periodic boundary condition at the inlet is: ⎧ ⎪ c 0 ≤ t ≤ a T/2 ⎪ ⎨ 0 (2.140) c(t, 0) = 0 a T/2 < t ≤ T − a T/2 ⎪ ⎪ ⎩c T − a T/2 < t ≤ T 0

2.2  From Microscale to Nanoscale

39

Figure 2.24  Sequential segmentation: (a) concept, (b) boundary condition, (c) concentration distribution.

where c0, T, and a are the initial concentration of the solute, the period of switching, and the mixing ratio, respectively. Figure 2.24(a, b) illustrates this model. Normalizing the concentration, the spatial variable, and the time by c0, L, and T, respectively, leads the dimensionless form of the transport equation:







∂ c* ∂ c* , 1 ∂ 2 c* − = ∂ t* Pe ∂ x*2 ∂ x*

(2.141)

The Peclet numbers are defined based on the segment length L and the effective diffusion coefficient D*. The dimensionless boundary condition is [Figure 2.24(b)]: ⎧ ⎪ 1 0 ≤ t* ≤ a /2 ⎪ ⎨ (2.142) c*(t*, 0) = 0 a /2 < t* ≤ 1 − a /2 ⎪ ⎪ ⎩ 1 1 − a /2 < t ≤ 1 Solving the above equation leads to the time-dependent dimensionless concentration distribution [20]: � � � � � � ∞ 2sin(ap n) � 1 exp Pe − Pe2 + 8p nPei x* c*(x*, t*) = R a + å pn 2 1 �� (2.143) × exp(2p t*i)

40

Fundamentals of Mass Transport at the Nanoscale

where i is the imaginary unit and R indicates the real component of a complex number. The typical solutions are shown in Figure 2.24(c). 2.1.6.6  Chaotic Advection

Advection is the transport of a particle by the flow. Chaotic advection is the phenomenon where a simple Eulerian velocity field leads to a chaotic response in the distribution of a Lagrangian particle. Thus, a laminar and deterministic flow can lead to a chaotic motion of particles positioned in this field. Chaotic advection can be generated either in a simple two-dimensional flow with time-dependent disturbance or in a three-dimensional flow. Since the flow leading to chaotic advection is deterministic, this effect is not turbulence. For a flow system without disturbance, the velocity components of chaotic advection at a point in space remain constant over time, while the velocity components of turbulence is random. The streamlines of the steady chaotic advection flow cross each other, causing the particles to change their paths. Under chaotic advection, the particles diverge exponentially and enhance mixing between the solvent and solute flows. In experiments, the flow is visualized by trajectories of fluorescent particles. If the particles are idealized so that they are small enough not to disturb the flow, but large enough so that molecular diffusion is neglected, they can move passively with the flow. The particle transport is described by the advection equations [17]:




dx    dt = u(x, y, z, t)   

dy = v(x, y, z, t) dt dz = w(x, y, z, t) dt

(2.144)

The trajectories of particles are obtained by solving the above equation system with numerical integration methods such as the Runge-Kutta method and initial conditions for t, x, y, and z. The advection equations represent a system of coupled ordinary differential equations (ODEs). Similar dynamical systems in engineering and physics have shown a strong chaotic behavior. Poiseuille flow in a straight microchannel is considered as a one-dimensional incompressible flow at low Reynolds number. The dynamics of this flow are simple and nonchaotic. In the case of a two-dimensional flow, the dynamic behavior of the flow is more interesting. The two-dimensional continuity equation:

du dv + =0 dx dy

(2.145)

is fulfilled by the stream function y:




∂Ψ dx   =  dt ∂y  dy ∂ Ψ   = dt ∂x

(2.146)

2.2  From Microscale to Nanoscale

41

The above equation system has the same form of the Hamilton equation of motion, where the stream function y is the Hamiltonian. Thus, a steady twodimensional incompressible flow results in deterministic dynamics. Adding one more dimension to the system makes the equations nonintegrable leading to chaotic dynamics. The advection equations can be solved explicitly for the fluid particle position (x, y, z) at a given time t. The solution of (x, y, z) is then used for describing the motion of fluid particles in a region R such as a channel cross-section, and thus can be called a mapping function S [21]. The region R can be transformed into a new region using S. This transformation or mapping can mathematically described as S(R). Each transformation is called an advection cycle, which corresponds to a mixing element in different micromixer designs, such as sequential lamination. Repeating these mixing elements n times means repeated application of S to R, or Sn(R). With the discrete number of advection cycles, the transformation S is understood as a discrete operation and not continuous as in the case of the time function. Since the volume (in a three-dimensional case) or the area (in a two-dimensional case) is preserved for incompressible fluids after each transformation, the above mapping function S is called a volume-preserving transformation or an area-preserving transformation. The trajectories of a chaotic three-dimensional flow are complicated. The threedimensional positions of fluid particles are reduced into a two-dimensional map called the Poincaré section. In a time-periodic system, a Poincaré section is a collection of intersection of a trajectory with a plane. The continuous trajectory becomes discrete points of the transformations PnPn+1. The time needed between the two points Pn and Pn+1 does not need to be the period of the system. In threedimensional space-periodic systems, the plane is taken at the same position of the repeated spatial structure. A trajectory will intersect all these periodic planes at several points. The collection of these points forms the Poincaré section. In this case, the transformation PnPn+1 is the advection cycle.

2.2  From Microscale to Nanoscale 2.2.1  Continuum Assumption

In classical fluid mechanics and in most cases of microfluidics, the fluid is generally assumed to be a continuum. As discussed later in this section, this assumption may still be valid for channel structures with characteristic sizes on the order of few nanometers. In a continuum, all quantities of interest such as density, velocity, viscosity, pressure, and diffusivity are assumed to vary continuously in space. These quantities are defined as a spatial function and can be described by a set of partial differential equations. Depending on their characteristics, these continuum quantities are categorized as point quantities and transport quantities [15]. A point quantity should satisfy the statistic requirement that its values remain constant in the given sampled volume. Point quantities are kinematic quantities of the fluid such as velocity as well as thermodynamic properties such as pressure and density. Due to the discrete nature of fluid molecules, if the sampled volume is too small, the values of point quantities will be affected if one molecule enters or

42

Fundamentals of Mass Transport at the Nanoscale

escapes the sampled volume. Thus, the sampled volume should be large enough so that the statistical variation of its point quantity is less then 1% [15]. According to random process theory, a number of molecules of N = 104 should allow this small variation. The molecular number density n at the standard condition of liquid and gas can be taken approximately as 2 ´ 1025 m-3 and 3 ´ 1025 m-3, respectively. Assuming that the sampled volume has a form of a sphere of diameter D, the relation between the number of molecules, number density, and sampled volume is:



V=

p D3 N = 6 n



(2.147)

Rearranging the above relation leads to the critical length dimension: � 3 6N (2.148) D= pn Using the corresponding number densities leads to the critical dimension of 10 nm and 90 nm for point quantities of liquid and gas, respectively. The point quality is related to the density distribution of a liquid. The structure of a liquid is represented by the radial distribution function (RDF) g(r), which describes the probability density of finding a particle at a distance r from a given molecule at r = 0 [22]. Figure 2.25 shows the typical RDF of gas and liquid. For a liquid, the value at a short distance is zero because of the repelling force. The first peak in the distribution is caused by the attractive force of the molecule at r = 0. The distribution value drops significantly due to the repelling forces of the molecule at the first peak. This behavior repeats with a further distance. In a bulk solution, the distribution is averaged over many particles and is constant. In a nanochannel or near the channel wall, the molecule of the wall is not mobile and can cause a similar fluctuation in distribution or density. Transport quantities such as viscosity and diffusivity are affected by interactions between fluid molecules. In a confined volume, the fluid molecules should interact more frequently with each other than with the volume boundary. In order to maintain the continuum condition, the diameter of the sampled volume should be approximately 10 times larger than the length scale of the interactions. For gases, this length scale of interactions is the mean free path where the molecule can freely move before colliding with the next molecule. The mean free path of gas molecules at the standard condition is approximately 100 nm. For liquids, the length scale of interactions can be taken as the distance between the neighboring molecules. Based on the number density of 2 ´ 1025 m–3, the distance between two liquid molecules is approximately 0.4 nm. Thus, the critical dimensions for transport quantities for liquid and gas are 4 nm and 1 mm, respectively. Considering both point quantities and transport quantities and taking the larger critical dimension into account, the critical dimensions for continuum assumption of liquids and gases are 10 nm and 1 mm, respectively. It is apparent that gases cannot be handled in nanofluidics as a continuum. However, for most current applications in nanofluidics with dimensions between 10 nm and 100 nm, a continuum model with or without slip conditions can be assumed. In nanofluidics, the main theory for gas flow is kinetics theory. The corre-

2.2  From Microscale to Nanoscale

43

Figure 2.25  Molecular distribution of (a) a gas such as nitrogen and (b) a liquid such as water.

sponding techniques for modeling gas flow would be molecular dynamics or Monte Carlo simulation, where individual molecules or a group of them are modeled as hard spheres. The position and velocity of the spheres are modeled and tracked using Newton’s second law under consideration of molecular forces. For liquids in nanofluidics, the continuum flow can still be described by a set of partial differential equations with their corresponding boundary equations. 2.2.2  Kinetic Theory

The most important parameter for transport quantities of a gas is the mean free path length l, the average velocity ua, the effective velocity urms, the most probable velocity up, the sound velocity us, and the kinematic viscosity n. For an ideal gas whose molecules are modeled as a hard sphere the main free path length is calculated as [23]:

1 l = √ 2ps 2 n



(2.149)

44

Fundamentals of Mass Transport at the Nanoscale

where s is the diameter of the molecule, and n is the number density. The probability P(u) of the velocity u follows the Maxwell-Boltzmann distribution:

�

1 P(u) = 4π 2π RT

�3/2

� � u2 u exp − 2RT 2



(2.150)

where R is the gas constant of the gas and T is the absolute temperature. The average velocity of a molecule can be determined from this distribution as:

ua =

0



√ 2 2√ uP(u)du = √ RT p

�∞

The effective velocity of the gas molecule is determined as [24]: � �∞ �� √ � urms = � u2 P(u)du = 3RT

(2.151)

(2.152)

0

The effective velocity is representative for the kinetic energy KE of the molecule, which is determined as:

KE =

1 3 Mu2rms = kB T 2 2

(2.153)

where M is the molecular mass and kB is the Boltzmann constant. The most probable velocity is the value at the peak of the distribution (2.150):

√

up =

2RT

(2.154)

The sound speed can be calculated from the ratio of the specific heats k=cp/cv

us =

√ kRT

(2.155)

Thus, the velocity value of a gas molecule is on the order of the sound speed. The velocity of a gas is often nondimensionalized by the sound speed, the resulting dimensionless number is called the Mach number:

Ma = u/us

(2.156)

A flow with a Mach number less than 0.3 can be considered as incompressible. The kinematic viscosity n of a gas can be calculated from the main free path length l and the effective velocity urms as:

n=

1 l urms 2

(2.157)

As mentioned in Section 2.1.1, the ratio between inertial forces represented by the velocity u and the friction forces represented by the kinematic viscosity n is called the Reynolds number:

2.2  From Microscale to Nanoscale



45

Re =

uL n

(2.158)

where L is the characteristic length scale of the flow such as the channel diameter. The ratio between the mean free path and the characteristic length scale is called the Knudsen number, which in turn can be evaluated from the Mach number and the Reynolds number as:

l Kn = = L

�

kp Ma 2 Re

(2.159)

The Knudsen number is one of the criteria for determining a suitable model to describe fluid flow in nanofluidics. For a small Knudsen number (Kn < 10–3), the fluid is considered as continuum with nonslip boundary conditions. For a larger Knudsen number between 10–3 and 10–1, continuum model with slip boundary conditions should be applied. For a Knudsen number between 10–1 and 10, the flow is in transition model and still can be described with equations modified from the continuum model. For a large Knudsen number (Kn > 10), molecular dynamics can be used to describe the free molecular flow (Figure 2.26). In the free molecular regime, also called Knudsen diffusion, the channel dimension is smaller than the mean free path. The collisions between molecules are negligible. Transport takes place through free molecular motions. The motion direction changes after each collision with the channel wall.

Figure 2.26  Flow regimes for gases and relationship between the Knudsen number, characteristic length L, density n, molecular diameter s, and molecular distance d.

46

Fundamentals of Mass Transport at the Nanoscale

2.2.3  Molecular Dynamics and Direct Simulation Monte Carlo (DSMC)



Molecular dynamics (MD) is a simulation method for explicit calculation of the motion of many particles in a system. As mentioned above, the interaction between the molecules in a system can be described by classical mechanics (e.g., Newton’s second law). The simplest model of a molecule is a hard sphere with a mass m. The interaction between two neutral molecules can be described by the Lennard-Jones potential [25]: � � �−12 � �−6 � r r (2.160) Yij (r) = 4e cij − dij s s

where r is the distance between the two molecules, cij and dij are the coefficients for the interacting molecules. The diameter of the molecule s is taken as the characteristic length. The characteristic energy of the molecule is e, which represents the strength of interaction between molecules. The term with the power of 12 describes the repulsion between the molecule, if they are to near to each other. The term with the power of -6 describes the attraction between them. From (2.160) the interaction force between two molecules can be described as follows [8, 26, 27]:

� � � � � �−13 dYij (r) dij r −7 r 48e cij Fij (r) = − − = dr s s 2 s



(2.161)

The characteristic time of the oscillation between the attraction and repulsion of the two molecules is calculated as:

t=s

�

m/e



(2.162)

From (2.161) the forces between molecules i and j are of the same magnitude but opposite signs:

Fij (r) = −Fji (r)



(2.163)

If the position vectors of the two molecules are ri and rj, the distance r is determined as r = ½ri-rj½. The relation between the interaction and the position of the molecule can be described by Newton’s second law:

d2 rj 1 N = å Fij dt2 m j=1, j�=i



(2.164)

where N is the total number of molecules considered in the simulation. The basic steps of molecular dynamics simulation are listed as follows: · · · ·

Determining the initial conditions and geometrical parameters; Determining the interaction forces; Integrating the equation of motion (2.164) for the next position (or state) of the molecules; Repeating over the required number of time steps.

2.2  From Microscale to Nanoscale

47

The fluid model used in MD with Lennard-Jones interactions is called a “simple fluid” or Lennard-Jones fluid. The “simple fluid” model offers a cost-effective model for describing liquid flow in the nanochannel. MD simulation of liquid indicates that the continuum model with Navier-Stokes equations breaks down in a nanochannel smaller than 4 molecular diameters, or approximately 1 nm. This result is very near to the limit of 10 nm derived from the approach based on the properties of a continuum discussed in Section 2.1.1. A continuum model with Eyring fluid can be used to describe the flow in a nanochannel such as a carbon nanotube (CNT) [28]. The Eyring fluid model was used to describe the non-Newtonian flow of lubricants. Eyring fluid has a shearthinning non-Newtonian fluid in which the viscosity is shear stress-dependent. The results reveal that the Eyring fluid flows like a Newtonian fluid at a small pressure gradient and a plug-like fluid at a high pressure gradient. The flow rate of the Eyring fluid in the nanotube at a high pressure gradient with the Navier slip condition is several orders of magnitude higher than the Hagen-Poiseuille result for the flow of Newtonian fluids in the nanotube. This trend is similar to the experimental results reported in the literature, suggesting that the Eyring fluid can be used to model nanoscale fluid flow. Experimental data can be reproduced by introducing a slip length on the order of the radius of the CNT as a nanochannel. The interaction between fluid molecules and channel wall molecules can be well described by the Lennard-Jones potential. The fluid density fluctuates near the wall due to this interaction. The higher the interaction energy e, the more apparent is the fluctuation. Thus, simple fluids are inhomogeneous in nanochannels. Due to the interactions with wall molecules, a simple fluid exists as layers in a nanochannel. This layering effect is determined by the molecular structure of the wall, the fluid-wall interaction, and the channel width. The stronger the interaction energy between the wall molecules and the liquid molecules, the more apparent is the layering effect [29–31]. For a channel width less than 10 molecular diameters, the density distribution or the position of the fluid layer depends on the width of the channel (Figure 2.27). For larger channels, the position of the fluid layer does not depend on

Figure 2.27  Density distribution in a nanochannel with a width of 11 atomic diameter.

48

Fundamentals of Mass Transport at the Nanoscale

the width and the density in the middle of the channel approaching the bulk value [32]. At the nanometer scale, a flow in a nanochannel has a negligible effect on the density distribution near the channel wall [33]. The layering effect explains the increased effective viscosity in channels with a diameter less than 10 molecular diameters. The actual flow is confined in layers smaller than the channel itself; thus the viscosity appears to be higher. Molecular dynamics is a deterministic method. The computational expense is therefore very high. Molecular dynamics is suitable for complex problems with liquids. Problems with gases can be solved by combining statistical methods and the deterministic method of particle dynamics. This later approach is called DSMC. In DSMC, instead of tracking the position of each single molecules, many molecules are grouped together as a particle. While the simulation of the particles is deterministic, the interaction of the molecules in the particle is modeled statistically. Each step of DSMC consists of three substeps: modeling of particle motion, indexing and cross-referencing of the particles, simulation collisions and probing macroscopic properties. Similar to MD, the new position of the particle is determined by integrating the equation of motion over a time step Dt. This time step should be less than the characteristic time of the particle, which is the time of the oscillation between the attraction and repulsion of the two particles. Thus, a particle can first move without colliding on another particle. After each time step, some particles are selected for collision. The selection is realized by indexing and cross-referencing the particles. If some particles exit the calculation domain, they must be considered by proper boundary conditions. For instance, to maintain the same number of particles in the simulation domain, particles exiting at one boundary should reenter at another boundary. The selection for collision is realized by dividing the simulation domain into cells. Only particles in the same cell can collide. The size of the cell should be less then three times of the main free path. The probability of collision is then calculated for every particle pair. This calculation requires a large computational resource. Thus, the acceptance-rejection scheme is often used for selecting the collision pair. The scheme selects two arbitrary particles and calculates their relative velocity. The pair is selected as collision pair if their relative velocity is higher than a certain threshold. If the particle pair is selected for collision, the collision is simulated and new velocities are updated. The velocities of the particles after the collision are calculated based on the conservation of impulses and of kinetic energy. For the macroscopic properties, the average value over all particles is calculated. The temperature can be evaluated from the kinetic energy of the particles. The viscosity can be evaluated according to theories such as the Chapmann-Enskog theory [23].

2.3  Diffusion in Nanochannels 2.3.1  Knudsen Diffusion in Nanochannels

In a nanochannel, the simplest mode of diffusive transport is Knudsen diffusion, where the channel size is smaller than the mean free path (Kn > 1). The diffusion process is dominated by collision with the channel wall and not by collision with the surrounding molecules as estimated with the Stokes-Einstein’s theory (Section

2.4  Electrokinetics in Nanochannels

49

2.1.6.1). For a liquid, the distance between molecules is on the order of the molecular diameter. Thus Knudsen diffusion is not significant. At a standard condition, the mean free path of gas molecules is on the order of 100 nm. Therefore, the diffusive transport of a gas through a nanochannel is mainly determined by Knudsen diffusion. The diffusion coefficient can be estimated based on the kinetic theory



DKn =

1 Lup 3

(2.165)

√ where L is the characteristic length such as channel diameter and up = 2RT is the most probable molecular velocity (see Section 2.2.2). The Knudsen diffusion coefficient can be determined as: 1 √ (2.166) DKn = L RT 3 where R = k/M is the gas constant with the molecular weight M. 2.3.2  Hindered Diffusion in Nanochannels





In the case of a molecule size on the order of the channel diameter, the diffusion coefficient is estimated as [34]: � � 9 −1 (2.167) Dhindered = D 1 − l lnl − 1.539l + O(l ) 8 where D is the diffusion coefficient evaluated with the Stokes-Einstein theory, (2.132), l = s/L is the ratio between the molecule diameter and the channel diameter. The polynomial form of the coefficient of hindered diffusion is the Renkin equation [8, 9]: � � (2.168) Dhindered = D 1 − 2.104 l + 2.09l 3 − 0.95 l 5 Equations (2.167) and (2.168) are useful for large molecules in a liquid-filled nanochannel.

2.4  Electrokinetics in Nanochannels



Surface charge on the channel wall is caused by dissociation of nonelectric adsorption of ion in the solution to the surface [35]. The surface charge can be positive or negative, depending on the pH value of the solution. At a specific pH value, the net charge can be zero. For instance, the surface charge of glass is zero at a pH value of approximately 2 [36]. The amount of charge per unit area is called the surface charge density: å zi e (2.169) ss = i A where zi is the valence of ion i, e is the elementary charge, and A is the surface area. In nanochannels, counterions accumulate on the surface charge and form the Stern

50

Fundamentals of Mass Transport at the Nanoscale

layer. These counterions extend into the channel through the Gouy-Chapman layer. Across the electric double layer, co-ions are repelled. From (2.72), the thickness of the EDL, the Debye � length, is inversely proportional to the square of the bulk con� � centration lD ∼ 1/ n0i . Thus, if the EDL is small compared to the channel, both ion types are transported through the channel, and the conductance is proportional to the ion concentration. If the EDL from channel walls overlap and cover the entire nanochannel, co-ions will be repelled out of the nanochannel. Only counterions can be transported. The concentration of counter-ions is determined by the distribution in the EDL and thus only depends on the surface charge. In case of overlapping EDL, the ion conductance is expected to remain constant, independent of the ion concentration. In a nanochannel, the surface charge density should balance the charge density in the solution:

ss = −

�∞

re dx



(2.170)

d2 Y dx2



(2.171)

0

where the charge density is:

re = −ee0

Using the Poisson-Boltzmann equation, the relation between the surface charge density ss and the surface potential Ys can be derived as [12]:

ss =

�

� � � � −zi eys 2ee0 kT å n0i exp −1 kT i

(2.172)

For a 1:1 electrolyte such as KCl (mK+ = mCl- = mi, z = 1), the part contributed by bulk conductance is:

Gep =

WH 2WH mi nKCl e (mK+ + mCl− )nKCl e = L L



(2.173)

where W, H, and L are the width, height, and the length of the nanochannel, respectively. At low concentration, the EDL overlaps, co-ions are excluded out of the nanochannel, and only the counter-ion remains to conduct:

Geo =

2W 2W m + ss = mi ss L K L

(2.174)

Thus, the simplified model for the conductance of a nanochannel for all concentration is [37]:

Geo =

2WH 2W mi nKCl e + mi ss L L

(2.175)

Figure 2.28 shows the relation between the conductance of a nanochannel and the different parameters such as height, width, length, and surface charge.

2.5  Water in Nanochannels

51

Figure 2.28  Typical behavior of the conductance of a nanochannel: (a) dependency on channel width W, (b) dependency on channel length L, (c) dependency on channel height H, and (d) depen­ dency on conductance.

The change in width and length follows the rule of an ohmic conductor (e.g., the conductance is proportional to the width and inversely proportional to the length) [Figure 2.28(a, b)]. The change in height only follows the Ohm’s rule if the nanochannel is in the bulk conductance regime [Figure 2.28(c)]. At low concentrations, the conductance is independent on the channel height, but is determined by the surface charge. The higher the surface charge, the higher is the conductance.

2.5  Water in Nanochannels Due its unique properties and the many nanofluidic applications associated with water, the behavior of water in nanochannel is discussed here in detail. A water molecule consist of two hydrogen atoms and one oxygen atom, Figure 2.29(a). The hydrogen atoms are linked to the oxygen atom through two bonds with a length of 0.970Å each. The two bonds form an angle of 106°. This molecular asymmetry makes a water molecule act as an electric dipole, where the oxygen site appears to be negatively charged and the hydrogen site positively charged. The symmetry axis of the water molecule can be considered as the dipole axis. The dipole moment of a water molecule in liquid state at 300K was determined experimentally as 1.95±0.2 Debye (1 Debye = 3.336 ´ 10–30 Cm) [38].

52

Fundamentals of Mass Transport at the Nanoscale

Figure 2.29  Models of a water molecule for MD simulation: (a) a single molecule and (b) two molecules with a hydrogen bond.

MD simulation involving water is more complex than that with gases or simple liquids as discussed previously. The main challenge is the accurate description of the potential of interactions between molecules. There are for instance 40 different models for the simulation of water. In addition to the Lennard-Jones potential, electrostatic interactions of the dipole are considered for simulating the interaction between the molecules. Each model is optimized to fit a critical parameter, but none can be applied in general. Water as a unique liquid is even more difficult to simulate and to understand. One of the key issues of research on water in the nanoscale is the nature of the hydrogen bond (HB) network. The strength of approximately 21 kJ/mol of a hydrogen bond is between that of a covalent chemical bond (~420 kJ/ mol) and that of the weak van der Waals interaction (~1.25 kJ/mol). Energetic and geometric conditions are often used to determine the formation of a HB. The energetic condition is based on the interaction energy. If the interaction energy between two molecules is less than a threshold energy (typically -10 kJ/mol), a HB can be formed. The geometric condition is based on the distance and the angle between the bonds of the atoms [Figure 2.29(b)]. If the distance between two oxygen atoms (the first coordination shell) and the angle between the O-O and O-H bonds (30°) are less then a respective threshold, an HB can be formed. Since HBs determine the properties of bulk water, the influence of nanochannels on HB determines the type of its transport. Hummer et al. reported a highly oriented HB inside a carbon nanotube. An HB is much more stable inside the channel. The average lifetime of an HB inside a carbon nanotube is 5.6 ps, while it is 1.0 ps in bulk water [39]. At a critical channel diameter of 8.6Å, water molecules are immobilized in the nanotube as a stable HB network [40]. This so-called water wire allows protons to hop from one molecule to the other leading to selective protonconducting properties of the nanotube.

2.6  Capillary Filling in a Nanochannel

53

The transport of the “water wire” in a nanochannel is as follows [41]. A water molecule enters the channel with the hydrogen atom first. Water fills the channel in a chain form with the same orientation of water molecules. Depending on the interaction to the channel, the chain can be broken, causing a burst of molecule transport. MD is suitable for studying transport phenomena in subnanometer channels such as the carbon nanotube. Gordillo and Marti [42] found with an MD simulation that inside a CNT with a diameter of 0.8 nm, the number of hydrogen bonds of water decreases compared with bulk water in larger channels. Water molecules align in a one-dimensional line and can move more quickly through the CNT. In contrast to flows in larger channels, water transport through subnanometer channels are semi-frictionless and nearly independent of the channel length. MD simulation can be combined with the continuum approach to avoid the huge computational expense. For instance, the near wall region can be simulated with MD while the bulk area is solved with the continuum model. This approach is suitable for a relatively large nanochannel on the order of 10–100 nm. Yen et al. [43] applied this concept to Couette and Poiseuille flow where the near wall region should be at least 12 molecular diameters in size. The continuum domain and the MD domain should overlap by at least 10 molecular diameters.

2.6  Capillary Filling in a Nanochannel 2.6.1  Modeling of the Filling Process

Capillary filling in a nanochannel is the best example demonstrating the interaction between different phenomena in the nanoscale. The following modeling approach was reported by Phan et al. [44]. Figure 2.30 shows the geometry for the model of capillary filling in a planar nanochannel with a height of h and a width w. The channel width is much larger than the channel height (w/h >> 1). Thus the filling process can be described as a two-dimensional flow between two parallel plates. The channel wall is assumed to be positively charged. The shaded areas in Figure 2.30 represent the EDL consisting of a compact layer attached to the channel wall and a mobile layer. When the liquid fills the nanochannel, the absorption of positive charge into the wall causes negative ions to accumulate at the meniscus. Charge imbalance between the meniscus area and the channel inlet induces a streaming voltage. This streaming voltage in turn generates a conductive current. The capillary filling phenomenon in a nanochannel assumes the liquid to be a continuum. The filling process is driven by the surface tension of the fluid and the contact angle between the fluid and the channel wall. With the continuum assumption, the flow in the nanochannel has the typical parabolic velocity profile. When the liquid fills the nanochannel, the zeta potential z of the channel causes the redistribution of ions to form the channel EDL, as shown in Figure 2.30. If the channel wall is positively charged, it induces apparent accumulation of negative charge ions in the meniscus region. The total electric charge in this region, termed here as the accumulated charge qa, creates an electric field ES that in turn generates a conductive current. Voltage drop along this electric field is the streaming voltage US. The charge balance considers the total charge of the liquid column, the total charge on the channel’s wall, the accumulated charge, the streaming current,

54

Fundamentals of Mass Transport at the Nanoscale

Figure 2.30  Capillary filling in a nanochannel: Ion distribution and movement inside a liquid column moving in a nanochannel due to capillary filling.



and the conductive current. The charge density distribution in the nanochannel is governed by the Poisson-Boltzmann (PB) equation. For low zeta potential z, the Debye-Hückel approximation can be used to linearize the PB equation. Electrostatic interaction between the induced streaming potential and the charge density of the liquid column slows down the capillary filling effect. Electrostatic interaction is governed by Gauss’s theorem and Lorentz’s force law. The effect of all the above forces on the dynamics of the fluid column is governed by Newton’s second law. According to the Young-Laplace equation, the pressure difference across the meniscus of the moving liquid column is: � � 1 1 Dp = s + (2.176) r1 r2 where Dp is the pressure difference, s is the surface tension, and r1 and r2 are the two principal radii of curvature. Assuming w >> h, the capillary force can be described as the sum of the two forces at the two contact lines (Figure 2.30):



FS = 2s wcosq

(2.177)

where q is the contact angle between the fluid wall surface. To simplify the problem, the variation of surface energy due to accumulation of charged particles near the meniscus, as described in Gibbs-Duhem equation, is neglected. The liquid slip at the solid-liquid interface [45] and interaction with electrokinetic effect [46] may lead to the modification of the effective contact angle. However, the shear stress and streaming voltage diminish quickly during the filling process. Thus, the change in effective contact angle is also negligible. Capillary filling can be considered as a pressure-driven flow, where the velocity at the meniscus has a plug distribution. Similar to the entrance effect at the inlet,

2.6  Capillary Filling in a Nanochannel





55

the velocity changes from its parabolic profile to the plug-like profile in the exit region. This exit condition is important at the initial stage of the filling process, especially when the filling length is comparable to the exit length. The exit effect introduces an additional flow resistance. However, when the filling length is much longer than the exit length, the important of the entrance effect diminishes. In this investigation, because the asymptotic solution is considered, the entrance effect is assumed to be negligible due to the low Reynolds number here. The velocity profile of a one-dimensional laminar pressure-driven flow between two infinite parallel plates is given as � 2 � y 3 h h (2.178) u = u¯ 6 2 − ,− < y < 2 2 2 h where u is the fluid velocity, u– is the average velocity across the channel height, and y is the coordinate across the channel height. The average velocity u– can be proven to be equal to the velocity of the meniscus due to its plug-like velocity distribution. – represents both front-surface velocity and average velocity across the The variable u channel height. With the above assumptions, the viscous force acting on the entire liquid column can be determined as � � � � du �� du �� 12 m u¯ wx = − FV = m � −m � wx (2.179) h h dy y= dy y=− h 2



2

where FV is the viscosity force acting on the fluid, and m is the dynamic viscosity of the fluid. The charge density distribution determines the electrostatic force in the capillary filling process. In a nanochannel, the potential distribution and consequently the charge density distribution is governed by the Poisson-Boltzmann equation. Because the filling length is considered much longer than the channel height, it is appropriate to assume that the potential and charge density distribution across the channel is influenced by the zeta potential z only and not by the streaming potential: � � ρq e zi eΨ 0 (2.180) ∇2 Ψ = − =− z n exp − i i εε0 εε0 ∑ kT i

where Y is the electrostatic potential, rq is the charge density, e = 1.6021 ´ 10–19 C is the elementary charge, e is the relative permittivity of the fluid, e 0 = 8.854 ´ 10–12 CV–1m–1 is the permittivity of vacuum, zi is the charge number of ionic species i, ni0 is the bulk concentration of ionic species i, k = 1.381 ´ 10–23 JK–1 is Boltzmann’s constant, and T is the temperature. For a symmetric binary electrolyte (i.e., z1 = z2 = z), (2.180) can be written as

� = ∇2 Y

� � k2 � sinh zY z

(2.181)

� = eY is the dimensionless potential, and k is the Debye parameter, where Y kT 2 2 0 2e z n k2 = with the bulk concentration of the solute n0. According to the ee0 kT Debye-Hückel approximation for low potential, the distribution (2.181) can be linearized to

56

Fundamentals of Mass Transport at the Nanoscale



� = k 2Y � ∇2 Y

(2.182)

At room temperature, this approximation is only valid for z potential below 26 mV [23]. The solution of the dimensionless potential is then (Section 2.1.5.2):

ez cosh(−k y) � Y(y) = kT cosh(k h/2)

Therefore,

Y(y) = z



cosh(−k y) cosh(k h/2)



(2.183)



(2.184)

The charge density distribution can then be determined as

rq = −ee0

d2 Y dy

= −ee0 z k 2

2

cosh(k y) cosh(k h/2)



(2.185)

The conductivity of an electrolyte in the nanochannel can be calculated as

� � l = å li = å �zi �eni ni



i

(2.186)

i

where l is the total conductivity of the electrolyte, l i is the individual contribution of ionic species i to the conductivity, ni is the electrical mobility of ionic species i, and ni is the concentration of ionic species i. Substituting the Poisson-Boltzmann equation into (2.186) results in

� � � � � l = n0 å �zi �eni exp −zY



(2.187)

i

The average conductivity across the channel height can be calculated as h



1 l¯ = h

�2

− 2h

h

1 l dy = n0 e h

�2 h 2

� �

å�zi �ni exp i

� � � dy −zi Y



(2.188)

The average conductivity (2.188) can be expressed with the Debye-Hückel approximation as h



1 l¯ = n0 e h

�2

− 2h



� � � � � dy 1 − zi Y

å�zi � ni i

(2.189)

If a monovalent electrolyte such as an NaCl solution (z+ = –z- = 1) is considered, (2.189) can be rewritten as ⎞ ⎛ h

1 n− − n+ ⎜ l¯ = n0 e(n + + n− ) ⎝1 + h n+ + n−

�2

− 2h

� ⎟ Ydy ⎠

(2.190)

Substituting the solution of the potential distribution (2.184) into (2.190) and performing the integration results in

2.6  Capillary Filling in a Nanochannel





57

� �⎞ kh 2tanh ⎜ z e n− − n + 2 ⎟ ⎟ l¯ = n0 e(n + + n − ) ⎜ 1 + ⎠ ⎝ kT n+ + n− kh ⎛

(2.191)

Defining the molar conductivity of a monovalent electrolyte Lm = å eNAni , we i can obtain the expression from the average conductivity as � �⎞ ⎛ kh 2tanh Lm ⎜ 2 ⎟ ⎟ ⎜1 + z e n− − n+ (2.192) l¯ = n0 ⎠ ⎝ NA kT n + + n− kh

Unlike in pressure-driven flow or electro-omosis flow, where the ion absorption at the channel wall is saturated, in capillary filling, ion absorption takes place continuously during the filling process, creating a charged surface attached to the wall. Due to ion absorption, an accumulated charge exists in the fluid. Because the channel wall far away from the meniscus is quickly saturated, the absorption only takes place near the meniscus. Hence, the charge accumulation is assumed to be at the meniscus only. The development of the charged surface is assumed to be as fast as the speed of the meniscus, and the net charge in the nanoscale volume must neutralize the surface charge. The contribution of ion absorption to the development of accumulated charge can be expressed by the streaming current IS h



IS =

�

urq dA =



urq wdy

(2.193)

h

AC



�2

−2

The accumulated charge induces streaming voltage US , which in turn causes the conductive current US l¯ AC (2.194) IC = − x where AC = wh is the cross-section of the channel. If the accumulated charge at the meniscus of the liquid column is qa, the charge balance for the liquid column requires d (2.195) Ic + Is = qa dt Substituting (2.193) and (2.194) into (2.195) results in the charge balance equation h



�2 Us l¯ wh d − urq wdy = qa + x dt −

(2.196)

h 2

It was shown in [46] that the convective transport of ions under the influence of streaming field is usually ignored in the literature, leading to inaccuracy in the estimation of streaming potential. In (2.196), the streaming current is expressed using the general velocity field u, without explicit relation to the causes of this velocity; hence the error can be avoided.

58

Fundamentals of Mass Transport at the Nanoscale

The force balance on the moving liquid column is expressed according to Newton’s second law

FS + FV + Fe =

d ¯ (rm whxu) dt



(2.197)

where Fe is the electric force due to interaction between the induced streaming potential and the EDL charge density of the channel wall, and rm is the mass density of the fluid. The electric charge accumulated at the front of the fluid column produces an electric field, namely the streaming potential field. Such an electric field, in turn, causes an electrostatic force acting on the entire liquid column. Assuming a constant electric field along the channel, the electric force is h

Fe = −x



�

AC

Us rq dA = −xw x

�2

− 2h

Us rq dy x

(2.198)

Applying Gauss’s theorem to the meniscus area, where the accumulated charge is located, the streaming electric field strength is

−Es =



Us qa qa = = x AC ee0 whee0

(2.199)

Substituting (2.199) into (2.198) gives an expression for the electric force h



Fe = −

qa x hee0

�2

rq dy

(2.200)

−h 2

Also, substituting (2.199) into (2.196) results in h

�2 qa l¯ d − + urq wdy = qa ee0 dt



(2.201)

− 2h

The force balance and the charge balance that form the equation system for the capillary filling effect in a nanochannel are expressed as


 2s wcosq − 12m u¯ wx − q x  h hee a

0



 − q l¯ + u¯ w  ee a

0

h �2

−

h 2

�

h

�2

−

rq dy =

h 2

� 3 y2 d − 6 2 rq dy = qa 2 h dt

d ¯ (rm whxu) dt (2.202)

2.6  Capillary Filling in a Nanochannel

59

2.6.2  Asymptotic Solutions

The system of nonlinear ordinary differential equations (ODE) (2.202) cannot be solved analytically. However, an asymptotic solution can be found. Introducing × c = xx, × and x = q x/w - = x, u a With

h

12m 1 Fˆ s = 2s cosq , A = ,B = h hee0

�2

−

rq dy =

h 2

� � kh 2z k l¯ tanh , C = rm h , D = h 2 ee0

and h

E = −ee0 z k 2

�2 �

−

h 2

3 y2 −6 2 2 h

�

cosh(k y) 12ee0 [k h − 2 tanh(k h/2)] dy = cosh(k h/2) (k h)2

the equation system (2.202) can be written as










Fˆ s − A c − Bx = C c˙   

x −Dx + Ec + u¯ 2 = x˙ c    �c˙ − u¯ 2 � u¯ = u˙¯ c

(2.203)

Applying algebraic transformation to the above equation system leads to the solution:
Fˆs A B  c˙ = − c− x  C C C  x x˙ = −Dx + E c + u¯ 2 (2.204) c  � �ˆ  3  u˙¯ = Fs − A c − B x u¯ − u¯ C C C c c × × By setting c× = 0, x = 0, –u = 0, it is straightforward to find the solution of (2.204)
as DFˆsˆ  c=   BE + AD EFˆ s (2.205) x=   BE + AD  u¯ = 0

Because the velocity approaches zero u– ® 0 when the time goes to infinity t ® ¥, the critical point described by (2.205) is a stable point. Substituting c = xx× back into the first equation of (2.205) and performing the integration with the initial con­ dition x(0) = 0 results in the position of the meniscus as function of time � �1/2 2DFˆs (2.206) x= t BE + AD

60

Fundamentals of Mass Transport at the Nanoscale

Equation (2.206) presents an asymptotic solution of the displacement as t ® ¥ with consideration of electrokinetic effects. This equation is qualitatively similar × to the Washburn equation, which states that x µ t 1/2, and therefore, –u = x µ t  –1/2. The results lead to a singularity that u– ® ¥ when t ® 0. However, this singularity is solved with consideration of the entrance effect at the initial stage of filling. For comparison, the asymptotic solution of the displacement without the EDL effect can be expressed as

x=

� ˆ �1/2 2Fs t A

(2.207)

Substituting back Fˆ s = 2scosq and A = 12m/h into the two solutions results in

�



� �1/2 � 2DFˆ s s cosq h 1/2 = t t BE + AD 3 ma � ˆ �1/2 � � 2Fs s cosq h 1/2 = t t A 3m



(2.208)

(2.209)

where ma is the apparent viscosity, which is higher than the real viscosity m due to the electrokinetic effects. The ratio between these two viscosities can be obtained as

� � ma BE BE + AD 2Fˆs = = 1+ m A AD 2DFˆs



(2.210)

The relative change of the viscosity is expressed as



� �� � �� kh kh tanh k h − 2 tanh Dm ma − m BE 2 2 � = = = h� z e n− − n+ 2 tanh(k h/2) m m AD 2 (k h) 1+ kT n− + n+ kh

(2.211)

2e 2 e02 z 2k 2 4ee0 NA z 2e2 ≈ is a dimensionless parameter. m Lm kT m l¯ Under normal conditions (i.e., z < 100 mV, T » 300K) the dimensionless paze rameter has a value on the order of unity. For many common binary solutions, kT n− − n+ reduces to zero. In where the mobility of ion species is the same, the ratio n− + n+ Dm n− − n+ < 1 . This ratio does not change the trend of the extreme cases, −1 < m n− + n+ versus k h significantly (Figure 2.31). Therefore, (2.211) can be approximated by where h =



Dm =h m

tanh

�

kh 2

�� � �� kh k h − 2 tanh 2 2 (k h)

(2.212)

2.6  Capillary Filling in a Nanochannel

61

Figure 2.31  Dependence of the ratio Dm/m on the dimensionless channel height kh for two different cases (with the dashed line indicating that the mobility of two species of ion is almost the same and the solid line indicating that the mobility of anion is much higher than that of cation).

The dimensionless channel height kh is determined by both the channel height and the electrolyte concentration. At room temperature, Debye length k -1 ranges from several nanometers to a theoretical maximum of submicron for pure water. Therefore, in order to observe the electroviscous effect, it is necessary to fabricate channels with height in nanoscale. With a specific channel height, there is one optimum concentration at which the electroviscous effect is most significant. A diluted or concentrated solution (relative to such optimum concentration) gives a lower increase in viscosity, as illustrated in Figure 2.32. Figure 2.33 shows that if the channel height is on the order of tens to hundreds of nanometers, the solution must be very dilute (i.e., 10–6M) to reach the maximum apparent viscosity. At a higher concentration, the viscosity simply increases with decreasing channel height. Setting the first order derivative of the relative change of viscosity (2.212) to Dm zero shows that the maximum of is reached when kh » 4. Figure 2.34 shows the m Dm with respect to normalized channel height k h for various values of h. In ratio m Dm Figure 2.35, the ratio with respect to normalized channel height kh is plotted mh in comparison with the corresponding ratio reported previously [47], the results showed a close agreement. The apparent viscosity observed in reported experiments still seems to be higher than that predicted by the present model (Figure 2.34). The deviation between current theories on the electroviscous effect and experimental results on the reduction in filling speed suggests that the electroviscous effect is not the only cause of the speed reduction. There are reports on the formation of air bubbles during the filling process [50, 51]. The negative pressure across the meniscus in the nanochannel

62

Fundamentals of Mass Transport at the Nanoscale

Figure 2.32  Ratio Dm/m versus concentration at different channel heights.

may release the gas diluted in the fluid in form of bubbles. A fraction of energy is stored in terms of surface energy of the air bubbles; hence, less energy transfers to the kinetic energy of the fluid column, which leads to a reduction in filling speed. Therefore, in order to observe the electroviscous effect, it is essential to eliminate, or quantitatively evaluate, other factors contributing to filling speed reduction, such as bubbles formation.

Figure 2.33  Ratio Dm/m versus channel height at different concentrations.

2.7  Nanofilters

63

Figure 2.34  Ratio Dm/m versus normalized channel height with different h. The white circles show experimental results from [48]. The black dots show experimental results from [49].

2.7  Nanofilters One of the key applications of nanochannels is their use as nanofilters, which have a wide range of applications from biochemical separation to water purification and

Figure 2.35  Ratio Dm/(mh) versus normalized channel height. The solid line shows results derived from (2.231). The dashed line shows the corresponding results from [47].

64

Fundamentals of Mass Transport at the Nanoscale

desalination. The channel size in nanometer scale allows the filtration of molecules by a various effects. In separation science, deterministic nanochannels and nanostructures could replace the randomly distributed nanopores in materials such as gels or ion-track-etched membranes. The following sections discuss the main effects for filtration applications. For further information on nanofilters, readers may refer to the review paper by Han et al. [52]. 2.7.1  Electrostatic Effect

As discussed in Section 2.5, if the ion concentration is low and the channel is small, the EDL is on the same order as the channel size. The overlapped EDL prevents co-ions that have the same polarity as the surface charge from entering the nanochannel. This behavior makes the nanochannel charge-selective. The exclusion of co-ions from the nanochannel and the enrichment of counterions inside the nanochannel is called the exclusion-enrichment effect (EEE) [53]. The permeability of a nanochannel is defined as the ratio between the difference in concentration Dn across the nanochannel and the instantaneous flux f:

P=

f Dn

(2.213)

With a diffusion coefficient of D, the permeability can be estimated from the channel cross-section A and channel length L as:

P=

A D L

(2.214)

If the permeability and the concentration distribution along the x-axis of the channel without electrostatic effects of the EDL are P0 and c0(x), the exclusionenrichment effect is characterized by the factor:



n (x) b = eff = n0 (x)

1 A

�

c(x)dA

A

c*(x)

=

Peff P*

(2.215)

where c(x) is the concentration distribution in the cross-sectional area A, ceff(x) is the averaged concentration across the channel cross-section, and Peff is the effective permeability resulting from the EEE. Enrichment and exclusion are characterized by b > 1 and b < 1, respectively. Considering the nanochannel (Figure 2.10), which is a gap of 2h, the potential distribution is:

Y=z

cosh(y/lD ) cosh(h/lD )

(2.216)

The above distribution overestimates the actual potential due to the DebyeHückel approximation but can be used for estimating the exclusion-enrichment coefficient. The ion concentration in a nanochannel according to the Boltzmann relation is:

2.7  Nanofilters

65

� � zeY n = n0 exp − kT



(2.217)

Thus, the exclusion-enrichment coefficient can be determined as:



1 2h

b =

�h

ndy

−h n0

1 = 2h

�h

−h

� zez cosh(y/lD ) dy exp − kT cosh(h/lD ) �

(2.218)

From the above equation, the coefficient b can be adjusted by the zeta potential. This coefficient is exponentially dependent of the charge q = ze. Thus, filtering based on electrostatic effect is suitable for biomolecules, which has a relatively large net charge. 2.7.2  Reptation Effect

Transport effects of large molecules, such as DNA, in a nanochannel also depend on the shape of the molecules. For instance a DNA molecule in a relaxed state has a spherical shape. The radius of the sphere is called radius of gyration. If the radius of gyration is larger than the pore size, the transport regime of the molecule is called reptation. Under an electric field, the large molecule slides or reptates through a “tube” whose contours are defined by the immobile phase such as gel polymers or a matrix of nanostructures as shown in Figure 2.36. The motion of the polymer chain transverse to the contour defined by the nanostructures is restricted. The reptation effect can be used for separation of the molecules. However, if the molecules are too long, it will be stretched and move along the field direction, making separation impossible [Figure 2.36(a)]. Bakajin et al. used a pulsed field technique to alternately switch the field direction [54]. The larger molecules get trapped at the nanopillars, while the smaller molecules still can move forward. Thus separation in this mode is possible.

Figure 2.36  Reptation effect. (a) Large and small DNA molecules slide along the gap between the nanopillars in the field direction. No separation is possible. (b) Repeated switching of field direction makes long DNA being hooked on the nanopillars, while smaller DNA molecules can still move forward. Separation based on molecule size is possible.

66

Fundamentals of Mass Transport at the Nanoscale

2.7.3  Steric Effect



Steric effects arise from the repulsion force of molecules. When a large molecule moves through a nanochannel that is larger than its diameter in a relaxed state, steric repulsions from the channel wall create a size-dependent configurational energy barrier at the entrance of the nanochannel. This concept is also called Ogston sieving [55]. In contrast to the entropic effect, the steric effect causes longer molecules to move more slowly. The calculation of the mobility of a molecule in Ogston sieving is based on the Ogston-Morris-Rodbard-Chrambach (OMRC) model, which describes the filtering phenomena as a partitioning process. The effective mobility of a molecule m* is set to be equal the partition coefficient f, which warrants an equilibrium between the nanopore and the open space. The partition coefficient in gel electrophoresis is represented by the volume fraction, which is the ratio between the free volume and the total gel volume [56]. Ogston estimated this volume fraction as a function of gel fiber length per unit volume l¢, gel fiber radius r, the molecule radius rs, and the total gel concentration c0: � � (2.219) f ∝ exp − p l� (r + rs )2 c The above estimation is based on the fact that the pore size is randomly distributed. For nanochannels with deterministic height h, the partition coefficient is



f ∝ 1 − rs /2h



(2.220)

2.7.4  Entropic Effect



If the molecule is forced through a nanochannel smaller then its diameter, it will deform and take a shape with a higher energy than that of the relaxed state. The energy cost for stretching the polymer is called the entropic trap (Figure 2.37). At the entrance, large molecules are more likely to be stretched and pass through the nanochannel, while smaller molecules are trapped. This filtering regime is considered to be the transition between the reputation regime and the steric regime discussed in Section 2.5. The trapping lifetime characterizes the entropic effect: � � � � Demax a (2.221) = t0 exp t = t0 exp kB T EkB T

Figure 2.37  Entropic effect. Small molecules are trapped at the thick region. Large molecules can find the entrance to the thin region more easily and move faster.

2.7  Nanofilters

67

where Demax = a /E is the magnitude of entropic trap that is inversely proportional to the driving electric field strength E, and a is a constant.

References   [1] Fox, R. W., and A. T. McDonald, Introduction to Fluid Mechanics, New York: John Wiley & Sons, 1999.   [2] White, F. M., Viscous Fluid Flow, New York: McGraw-Hill, 1991.   [3] Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge, U.K.: Cambridge University Press, 2000.   [4] Panton, R. L., Incompressible Flow, New York: John Wiley & Sons, 1996.   [5] Navier, C. L. M. H, “Memoire sur les lois du mouvement des fluides,” Mem. Acad. Roy. Sci. Inst. France, Vol. 1, 1823, pp. 389–440.   [6] Maxwell, J. C., “On Stresses in Rarefied Gases Arising from Inequalities of Temperature,” Philosophical Transactions of the Royal Society Part 1, Vol. 170, 1879, pp. 231–256.   [7] Smoluchowski, von M., “Über Wärmeleitung in verdünnten Gasen,” Annalen der Physik und Chemie, Vol. 64, 1898, pp. 101–130.   [8] Karniadakis, G., A. Beskok, and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation (Interdisciplinary Applied Mathematics), 2nd ed., New York: Springer-Verlag, 2005.   [9] Shah, R. K., and A. L. London, Laminar Flow Forced Convection in Ducts, New York: Academic Press, 1978. [10] Nguyen, N. T., Mikrofluidik, Stuttgart, Germany: Teubner Verlag, 2004. [11] Richter, M., Modellierung und experimetelle Charakterisierung von Mikrofluidsystemen und anderen Komponeneten, Ph.D. thesis, Universität der Bundeswehr München, 1998. [12] Probstein, R. F., Physicochemical Hydrodynamics: An Introduction, 2nd ed., New York: John Wiley & Sons, 1994. [13] Hunter, R. J., Zeta Potential in Colloid Science, New York: Academic Press, 1981. [14] Yang, C., C. B. Ng, and V. Chan, “Transient Analysis of Electro-Osmotic Flow in a Slit Microchannel,” Journal of Colloid and Interface Science, Vol. 248, 2002, pp. 524– 527. [15] Nguyen, N. T., and S. T. Wereley, Fundamentals and Applications of Microfluidics, 2nd ed., Norwood, MA: Artech House, 2006. [16] Lippmann, M. G., “Relations Entre Lésenomèn Électriques et Capillares,” Ann. Chim. Phys., Vol. 5, 1875, pp. 494–549. [17] Nguyen, N. T., Micromixers: Fundamentals, Design, and Fabrication, Norwich, NY: William Andrew Publishing, 2008. [18] Wu, Z., N. T. Nguyen, and X. Y. Huang, “Nonlinear Diffusive Mixing in Microchannels: Theory and Experiments,” Journal of Micromechanics and Microengineering, Vol. 14, 2004, pp. 604–611. [19] Wu, Z., and N. T. Nguyen, “Convective-Diffusive Transport in Parallel Lamination Micromixers,” Microfluidics and Nanofluidics, Vol. 1, 2005, pp. 208–217. [20] Nguyen, N. T., and X. Y. Huang, “An Analytical Model for Mixing Based on Time-Interleaved Sequential Segmentation,” Microfluidics and Nanofluidics, Vol. 1, 2005, pp. 373–375. [21] Ottino, J. M., The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge, U.K.: Cambridge University Press, J, 1989. [22] Prakash, S., et al., “Nanofluidics: Systems and Applications,” IEEE Sensors Journal, Vol. 8, 2008, pp. 441–450. [23] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford, U.K.: Clarendon Press, 1994.

68

Fundamentals of Mass Transport at the Nanoscale [24] Vincenti, W., and C. Kruger, Introduction to Physical Gas Dynamics, Huntington, NY: Robert E. Krieger Publishing Company, 1977. [25] Israelachvili, J., Intermolecular & Surface Forces, New York: Academic Press, 1992. [26] Koplik, P. J., and J. R. Banavar, “Continuum Deductions from Molecular Hydrodynamics,” Annu. Rev. Fluid Mech., Vol. 27, 1995, pp. 257–292. [27] Gad-el-Hak, M., “Flow Physics,” in M. Gad-el-Hak, (ed.), The MEMS Handbook, Boca Raton, FL: CRC Press, 2002. [28] Guillot, B., “A Reappraisal of What We Have Learnt During Three Decades of Computer Simulations of Water,” J. Mol. Liquid, Vol. 101, 2002, pp. 219–260. [29] Thompson, P. A., and M. O. Robbins, “Shear Flow Near Solids: Epitaxial Order and Flow Boundary Conditions,” Phys. Rev. A, Vol. 41, pp. 6830–6837. [30] Travis, K. P., and K. E. Gubbins, “Poiseuille Flow of Lennard-Jones Fluids in Narrow Slit Pores,” J. Chem. Phys., Vol. 112, 2000, pp. 1984–1994. [31] Sokhan, V. P., D. Nicholson, and N. Quirke, “Fluid Flow in Nanopores: An Examination of Hydrodynamic Boundary Conditions,” J. Chem. Phys., Vol. 115, 2001, pp. 3878–3887. [32] Somers, S. A., and H. T. Davis, “Microscopic Dynamics of Fluids Confined Between Smooth and Atomically Structured Solid Surfaces,” J. Chem. Phys., Vol. 96, 1992, pp. 5389–5407. [33] Bitsanis, I., et al. “Molecular Dynamics of Flow in Micropores,” J. Chem. Phys., Vol. 87, 1987, pp. 1733–1750. [34] Gaydos, L. J., and H. Brenner, “Field-Flow Fractionation: Extensions to Non-Spherical Particles and Wall Effects,” Separation Science and Technology, Vol. 13, 1978, pp. 215– 240. [35] Behrens, S. H., and D. G. Grier, J. Chem. Phys., Vol. 115, 2001, p. 6716. [36] Iler, R. K., The Chemistry of Silica, New York: John Wiley & Sons, 1979. [37] Schoch, R. B., J. Han, and P. Renaud, “Transport Phenomena in Nanofluidics,” Review of Modern Physics, Vol. 80, 2008, pp. 839–883. [38] Gubskaya, A. V., and P. G. Kusalik, “The Total Molecular Dipole Moment for Liquid Water,” J. Chem. Phys., Vol. 117, 2002, pp. 5290–5302. [39] Hummer, G., J. C. Rasaiah, and J. P. Noworyta, “Water Conduction Through the Hydrophobic Channel of a Carbon Nanotube,” Nature, Vol. 414, 2001, pp. 188–190. [40] Mashl, R. J., et al., “Anomalously Immobilized Water: A New Water Phase Induced by Confinement in Nanotubes,” Nano Letter, 2003, Vol. 3, pp. 589–592. [41] Waghe, A., J. C. Rasaiah, and G. Hummer, “Filling and Emptying Kinetics of Carbon Nanotubes in Water,” J. Chem. Phys., 2002, Vol. 117, pp. 10789–10795. [42] Gordillo, M. C., and J. Marti, “Hydrogen Bond Structure of Liquid Water Confined in Nanotubes,” Chem Phys Letter, Vol. 329, 2000, pp. 341–345. [43] Yen, T. H., C. Y. Soong, and P. Y. Tzeng, “Hybrid Molecular Dynamics Continuum Simulation for Nano/Mesoscale Channel Flows,” Microfluidics Nanofluidics, Vol. 3, 2007, pp. 665–675. [44] Phan, V. N., C. Yang, and N. T. Nguyen, “Analysis of Capillary Filling in Nanochannels with Electroviscous Effects,” Microfluidics Nanofluidics, January 29, 2009. [45] Yang J., F. Lu, and D. Y. Kwok, “Dynamic Interfacial Effect of Electro-Osmotic Slip Flow with a Moving Capillary Front in Hydrophobic Circular Microchannels,” Journal of Chemical Physics, Vol. 121, 2004, pp. 7443–7448. [46] Chakraborty, S., and S. Das, “Streaming-Field-Induced Convective Transport and Its Influence on the Electroviscous Effects in Narrow Fluidic Confinement Beyond the DebyeHückel Limit,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 77, 2008. [47] Mortensen, N. A., and A. Kristensen, “Electroviscous Effects in Capillary Filling of Nanochannels,” Applied Physics Letters, Vol. 92, 2008.

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[48] Tas, N. R., et al., “Capillary Filling of Sub-10 nm Nanochannels,” Journal of Applied Physics, Vol. 104, 2008, p. 014309. [49] Persson, F., et al., “Double Thermal Oxidation Scheme for the Fabrication of SiO2 Nanochannels,” Nanotechnology, Vol. 18, 2007. [50] Han, A., et al., “Filling Kinetics of Liquids in Nanochannels as Narrow as 27 nm by Capillary Force,” Journal of Colloid and Interface Science, Vol. 293, 2006, pp. 151–157. [51] Thamdrup, L. H., et al., “Experimental Investigation of Bubble Formation During Capillary Filling of SiO2 Nanoslits,” Applied Physics Letters, Vol. 91, 2007. [52] Han, J. Y., J. P. Fu, and R. B. Schoch, “Molecular Sieving Using Nanofilters: Past, Present and Future,” Lab on a Chip, Vol. 8, 2008, pp. 23–33. [53] Plecis, A., R. B. Schoch, and P. Renaud, “Ionic Transport Phenomena in Nanofluidic: Experimental and Theoretical Study of the Exclusion-Enrichment Effect on a Chip,” Nano Letter, Vol. 5, 2005, pp. 1147–1155. [54] Bakajin, O., et al., “Separation of 100-Kilobase DNA Molecules in 10 Seconds,” Analytical Chemistry, Vol. 73, 2001, pp. 6053–6056. [55] Fu, J., P. Mao, and J. Han, “Nanofilter Array Chip for Fast Gel-Free Biomolecule Separation,” Appl. Phys. Lett., Vol. 87, 2005. [56] Ogston, A. G., “The Spaces in a Uniform Random Suspension of Fibers,” Trans. Faraday Soc., Vol. 54, 1958, p. 1754.

Selected Bibliography Conlisk, A. T., “The Debye-Hückel Approximation: Its Use in Describing Electro-Osmotic Flow in Micro- and Nanochannels,” Electrophoresis, Vol. 26, 2005, pp. 1896–1912. Preisig, P. A., and C. A. Berry,“Evidence for Transcellular Osmotic Water Flow in Rat Proximal Tubule,” Am. J. Physiol. Renal Fluid Electrolyte Physiol., Vol. 249, 1985, pp. F124–F131.

Chapter 3

Fabrication Technologies of Nanochannels

3.1  Basics About Micro- and Nanofabrication Actual micro- and nanofabrication techniques mainly originated from the microelectronics industry. Since the invention of the planar transistor by J. A. Hoerni of the Fairchild Corporation at the end of the 1950s, microfabrication has continuously improved, always resulting in faster, smaller, and better chips. The ubiquitous presence of silicon as a semiconductor material is mainly due to the outstanding properties of its oxide as a masking or insulating layer. The idea of using the microelectronics toolbox to fabricate micromechanical parts emerged in the 1950s after the discovery of the piezoresistive effect in silicon and germanium at Bell Laboratories [1]. Known as the microelectromechanical system (MEMS) in the United States and as microsystem technologies (MST) in Europe, silicon-based MEMS was originally an expensive and exotic technology, but now its progress has been reflected in many commercial successes. Examples include pressure and inertial sensors in the automotive and medical industry, but also ink-jet nozzles, read-write head positioners in hard drives, and micromirror arrays in projectors. MEMS technologies take advantage of the benefits associated with mass fabrication that drastically reduce manufacturing costs. Owing to the integration, microsystems are smaller and more reliable than their macro counterparts. Miniaturization also provides a means of performing new functions that were previously not possible, an example being the direct switching of optical signals by micromirrors. More recently there has been a growing interest in microsystems involving fluids with applications in chemistry and in the biomedical field. In the beginning of the 1990s, Manz introduced the concept of mTAS (miniaturized total chemical analysis system), which integrates a chemical sensor with a pretreatment step in a microsystem, improving the overall selectivity [2]. This was extended later to the concept of lab-on-a-chip, aiming to integrate and automate all the functions of a macroscopic lab on a microchip. Over the last 15 years, this popularity has been a driving force for the development of new types of microsystems combining electrical, mechanical, and microfluidic functions (e.g., wells, channels, valves, and pumps), promoting the development of new processes and materials in microtechnologies. In the following, we will introduce all the basic techniques commonly used for the fabrication of micro- and nanofluidic devices.

71

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Fabrication Technologies of Nanochannels

3.1.1  Silicon/Glass Micro- and Nanofabrication

Due to the 50 years experience in silicon machining, a microfabrication process commonly starts with a disc-like silicon wafer. The extensive use of silicon wafers in the microelectronics industry made them fairly cheap considering their excellent characteristics (e.g., monocrystalline, very low roughness, and high level of purity). Wafers have typical thicknesses in a range 200–800 mm and diameters below 300 mm (commonly 100–150 mm in academic laboratories). Table 3.1 lists the basic properties of selected materials commonly used in microfabrication. Ingots of single-crystalline silicon are grown from a seed crystal either by the float zone or the Czochralski method. In the Czochralsi method, the seed crystal is pulled away from a crucible in which polycrystalline silicon (or polysilicon) is molten. The float zone method uses a circular electrical heater that crystallizes a polysilicon ingot. The orientation is given by the seed located at one end of the ingot. In both cases, the diameter of the ingot depends on the rotation speed of the seed. Though Czochralski methods are 10–20% cheaper, fusion zone results in a lower oxygen and carbon contamination. Flats are cut in the ingots to provide information about the crystallographic direction and the dopant type, according to the semiconductors equipment and materials (SEMI) standard. Wafers are then cut using a diamond rotating saw or a wire saw, and polished on one or both sides. 3.1.1.1  Thin Films

Starting from this silicon wafer, the most general scheme for fabricating a microcomponent consists of stacking and patterning thin films. Many methods have been developed to grow or deposit thin films. Properties such as uniformity, conformality, quality, homogeneity, adhesion, and stress, among others, are commonly used to characterize a film and its deposition process. The uniformity relates the variation of the thickness across a wafer, or from wafer to wafer. The conformality of Table 3.1  Properties of Selected MEMS Materials

Material Silicon Corning Pyrex 7740 Quartz SiO2 Si3N4 SiC Diamond PMMA Polyimide Parylene SU-8 PDMS

Dielectric Constant

Thermal Conductivity [W.m–1.K–1]

Coefficient of Thermal Melting Expansion Temperature [10–6K–1] [°C]

0.22 0.2

11.7   4.6

157 1.35

2.6 3.3

1,415 821

0.16 0.17 0.25 0.14 0.1 0.35-0.4 0.35 0.4 0.22 0.5

3.75 3.9 4-8 9.7 5.7 2.6 2.95-3.15 2.65-3.15 3 2.3-2.8

1.4 1.4 19 500 990-2,000 0.2 0.12 0.082 0.2 0.15

0.55 0.55 2.8 4.2 1 70-80 20 35-69 52 310

1,610 1,700 1,800 1,800* 3,652* 110** 360-410** 80-100** 210** −120**

Young’s Density Modulus [g.cm–3] [GPa]

Poisson’s Ratio

2.4 2.2

160 63

2.65 2.2 3.1 3.2 3.5 1.19 1.42 1.1-1.4 1.19 1.03

107 73 323 450 1,035 3.2 2.5 2-5 2-4 360−870 ´ 10–6

Source: [3, 4]. * Sublimates before melting. ** Glass transition temperatures.

3.1  Basics About Micro- and Nanofabrication

73

a coating refers to its ability to conform to various shapes. Figure 3.1 illustrates this notion. The conformality can be quantified by the step coverage, often defined as the minimum thickness deposited on the side of a step divided by the thickness deposited on the top horizontal surface. The aspect ratio of a step is defined by its height divided by its width. The conformality of a coating usually decreases as the aspect ratio increases (i.e., it is difficult to cover deep and shallow trenches with a high conformality). Depending on its role, the quality of the film may refer to its composition, the level of contamination, the density of defects, or mechanical and electrical properties. The homogeneity of the process is defined by the variation of these properties over the wafer and from wafer to wafer. The control of the adhesion is also important. Though it is usually preferred to have a coating that sticks to the underlying surface, low-adhesion surfaces may be preferred in particular processes (e.g., layers that have to be released). Thermal stress originates from differences in thermal expansion between the substrate and the film, while intrinsic stress results from structural inhomogeneities (e.g., voids). Though the resulting bending of the stack is highly dependent on the film thickness, the value of the stress itself is independent of the thickness of the film. When the film thickness is negligible compared to the substrate thickness, the mechanical stress s can be deduced from the curvature of the bilayer using the Stoney equation:



sStoney =

1 Es t2s 1 6 1 − vs tf R



where E is the Young modulus, n is the Poisson coefficient, R is the curvature radius, d is the thickness and the subscripts f and s refer, respectively, to the film and substrate. In the case of a thick film, a better approximation is given by:

sthick = KsStoney

Figure 3.1  Illustration of the conformality of a deposition process over a trench.

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Fabrication Technologies of Nanochannels

where K is a correction factor [5]:



K=

1+

Ef /(1−vf ) Es /(1−vs )

� d �3 f

ds

1 + df /ds



3.1.1.2  Oxidation

Silicon wafers are naturally covered with a native layer of silicon dioxide (SiO2) with a thickness close to the nanometer. As stated above, the ability to grow SiO2 films is at the origin of the popularity of silicon in the semiconductors industry. Oxide films can be grown under exposure to oxygen by heating the silicon substrate at a temperature typically in a range 900–1,200°C. The reaction consumes some silicon from the substrate to form SiO2, so the silicon thickness decreases. However the total thickness of the Si/SiO2 stack increases, as depicted on the Figure 3.2. The most common model for the growth kinetics was given by B. E. Deal and A. S. Grove in the 1960s by considering the flux of oxidant molecules traveling from the vapor phase trough the oxide to react at the oxide/silicon interface [6]. According to this model, the linear-parabolic correlation between the thickness dSiO2 of the oxidation layer and the oxidation time t is:



dSiO2

A = 2

��

2

1 + (2B/A )(t + t ) − 1

�



with

t = (d20 + Ad0 )/B



where B/A is the linear constant, B is the parabolic constant, and d0 is the initial oxide thickness. The following relation is valid for short oxidation times (up to ~200–300 nm):

Figure 3.2  Consummation of silicon during the oxidation process.

3.1  Basics About Micro- and Nanofabrication



dSiO2 =

75

B (t + t ) A



while for long oxidation times, it can be approximated by:

d2SiO2 = Bt



Wafers are usually stacked vertically in a three-zone furnace having three independent heating zones. A typical setup is depicted in Figure 3.3. The control of these zones by independent thermocouples results in a uniform temperature essential for a homogeneous growth kinetics. Oxidation takes place in a so-called “dry” or “wet” environment, depending whether silicon react with oxygen or with water molecules, respectively. Dry oxidation is slower (typically films are not thicker than a few hundreds of nanometers) but leads to a better quality oxide with a low density of electrical defects. On the other hand, thicker (up to a few microns) SiO2 films can be grown by wet oxidation, for instance, for masking purposes or to provide an electrical insulation between metal layers. 3.1.1.3  Chemical Vapor Deposition

In chemical vapor deposition (CVD), films are formed by the chemical reaction between gases and a substrate [7, 8]. Typically, reactants diffuse from a gas stream to the wafer surface. Following adsorption, reactive species can migrate along the surface to reaction sites. After reaction, byproducts are desorbed from the surface and diffused back to the main gas stream. Considering competing effects between mass transfer and reaction kinetics, a treatment similar to that used for oxidation allows identifying two limit cases for the deposition kinetics. At a high temperature, the deposition rate is limited by the diffusion from the main stream to the surface. This regime is relatively constant with temperature. However, the design of the reactor in a mass transfer–controlled process is critical to ensure a homogeneous gas flow, thus a homogenous deposition. Typically wafers are placed horizontally and edge to edge in the chamber.

Figure 3.3  Schematics of a horizontal oxidation furnace.

76

Fabrication Technologies of Nanochannels Table 3.2  Chemical Reactions Used in Some CVD Processes Material

Chemical Reactions

Silicon

SiH4 ® Si + 2H2 � SiH2Cl2 ® SiCl2 + 2H2 � SiCl2 + H2 ® Si + 2HCl � SiH4 ® Si + 2H2 � (@630°C, 60Pa) SiH4 + O2 ® SiO2 + 2H2 � (@430°C, 1bar) SiH4 + O2 ® SiO2 + 2H2 � (@430°C, 40Pa) Si(OC2H5)4 ® SiO2 + gas � (@700°C, 40Pa) Si(OC2H5)4 + O2 ® SiO2 + gas � (@400°C, 0.5bar) SiH2Cl2 + 2N2O ® SiO2 + gas � (@900°C, 40Pa) SiH4 + 4N2O ® SiO2 + gas � (@350°C, plasma, 40Pa) SiH2Cl2 + 4NH3 ® Si3N4 + gas � (@750°C, 30Pa) 3SiH4 + 4NH3 ® Si3N4 + gas � (@700°C, plasma, 30Pa) 3Si + 4NH3 ® Si3N4 + 6H2 � (@300°C, plasma, 30Pa) 4SiH4 + 2WF6 ® 2WSi2 + 12HF �+ 2H2 (@400°C, 30Pa) 4SiH2Cl2 + 2TaCl5 ® 2TaSi2 + 18HCl � (@600°C, 60Pa) 2SiH4 + TiCl4 ® 2TiSi2 + 4HCl + 2H2 � (@450°C, plasma, 30Pa)

Polysilicon Silicon dioxide

Silicon nitride Silicide

Source: [9].

At lower temperatures, the kinetics are limited by the surface reaction. In contrast with the mass transfer–limited regime, this process is highly sensitive to variations in temperature, but has fewer restrictions in terms of reactor design and wafer placement. A reactor similar to the oxidation furnace described in the previous section may be used, where wafers are stacked vertically leading to higher throughput, without affecting the wafer-to-wafer uniformity. However, as the temperature is decreased, the deposition rate is reduced and the quality of the film is affected. Lowering the pressure promotes the transport of reactive species, extending the range of temperatures of the reaction-limited deposition regime. Longer migration paths also result in a higher conformality. Low pressure chemical vapor deposition (LPCVD) usually takes place in a reactor similar to an oxidation furnace (see Figure 3.3), coupled to a pump, at a pressure in the 0.1–2.0 torr range. Deposition temperatures commonly range from 300°C to 900°C. Low-pressure chemical vapor deposition leads to high-quality, uniform, homogeneous, and conformal films. Plasma-enhanced chemical vapor deposition (PECVD) is another popular chemical vapor deposition method. In this process, plasma source is used to bring the energy required for the chemical reactions to occur. Though room temperature deposition is possible using PECVD (an example is the deposition of parylene), typical processing temperatures range from 200°C to 400°C. The highly reactive species created in the plasma leads to a high deposition rate at low temperatures. This allows for deposition on a wider range of materials, including metals such as aluminum with a low melting temperature. However, the high-reaction kinetics associated with plasma leads to shorter migration, which in turn leads to a lower conformality. Table 3.2 lists common examples of chemical reactions used in CVD. 3.1.1.4  Physical Vapor Deposition

Other popular deposition methods rely on physical rather than chemical principles. Physical vapor deposition (PVD) processes include evaporation and sputtering. The first technique consists of evaporating a material that condenses on the cold

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77

substrate. A high vacuum (typically less than 1.10–5 torr) is required to avoid the material to condense between the source and the wafer [i.e., (1) for the vapor to reach the substrate, and (2) to avoid the formation of particles]. This results in relatively straight paths from the heated source to the substrate. The energy is usually provided either by injecting a current though a crucible containing the material to deposit (i.e., Joule heating) or by locally heating using an electron beam (E-beam) focused by a magnetic field. E-beam evaporation is usually preferred because higher temperatures can be reached allowing the deposition of a wider range of materials. Furthermore E-beam evaporation induces less contamination since only the source is heated and not the crucible. The fairly straight emission from a punctual source results in a relatively low conformality, explaining that this method is often exploited for patterning by the lift-off process that will be described in Section 3.1.1.11. Because of the differences in the vapor pressures of different elements, evaporating alloys or compounds is particularly challenging. In sputtering, a target of the material to be deposited is bombarded with ions generated in an inert gas plasma (usually argon). Atoms are knocked from the target and travel through the plasma to the substrate where they condense. A first advantage of sputtering is its ability to deposit thin films of alloys with a stoechiometry identical to the target. The vacuum requirements are also less severe than for evaporation (from 1 to 100 mtorr). Finally, atoms are emitted from the whole surface of the source: Using large enough targets, this leads to highly conformal coatings. A magnetic field is often introduced close to the target to increase the electron path length (spiral motion) resulting in more ionization events. The so-called magnetron sputter deposition leads to better efficiencies, including higher deposition rates (up to 1 mm.min–1, or 10–100 times faster than without magnetron). The thin films of metals are commonly deposited using PVD techniques. Table 3.3 lists some metals commonly used in microfabrication. 3.1.1.5  Electroplating

While CVD and PVD allow the deposition of thin films, typically less than a few microns, electroplating is better suited for making films of conductive materials up

Table 3.3  Properties of Selected Metals Used in Microfabrication Resistivity r [10–6 W.cm] Aluminium (Al) Copper (Cu) Titanium (Ti) Chromium (Cr) Gold (Au)

2.7 1.7 42 12.9 2.4

Platinum (Pt)

10.6

Silver (Ag) Nickel (Ni) Indium-tin oxide (ITO) Source: [3, 10].

1.58 6.8 300–3,000

Use Electrical interconnect Low-resistivity electrical interconnect Adhesion layer Conductor, electrochemistry, bonding layer for Si/Au eutetic bonding Electrochemistry, often used as electrode for biological applications Electrochemistry Ferromagnetic, often used to make electroplated templates for molding in plastics Transparent conductor

78

Fabrication Technologies of Nanochannels

to a few hundreds of microns. Examples of applications include the fabrication of copper vias in multilevel metallization processes and the realization of hard nickel templates for replication in plastics (e.g., the LIGA process). The most common setup in MEMS fabrication consists of two electrodes dipped in a solution of metal salt. The wafer to be coated is placed at the cathode. The counter electrode is usually made out of platinum. When an electrical potential is applied, a redox reaction takes place resulting in a deposition at the cathode. Basically, electrons are released at the anode and react with the metallic ions at the cathode to form a coating on the wafer. It should be noted that an electrical contact (i.e., a thin metal layer deposited by other means such as CVD or PVD) is necessary to apply the required potential in the areas to be covered. Though the deposition kinetics are relatively high, precautions have to be taken to ensure an electric field as homogeneous as possible to avoid inhomogeneities in the film thickness. An alternative to this process is electroless deposition where no external electrical potential is applied. Instead, the solution contains a reducing agent catalyzing the deposition. Compared to more conventional electroplating, the deposition kinetics are slower and the chemistry involved is more complex. 3.1.1.6  Spin- and Spray-Coating

Thin films of fluid resins such as photoresists are often applied to the substrate by spin-coating. After dispensing a sufficient amount of a solution of polymer, the wafer is rotated at high speed, homogeneously spreading the resin over the surface. Most of the solvents usually evaporate during this step. A subsequent baking is often necessary to complete the drying. The thickness t of the coating mainly depends on the dynamic viscosity h and the highest rotation speed w, but is also influenced by the spinning time and the exhaust rate. It is usually expressed by the empirical expression:

h t≈k √ r w where k is a constant and r is the density. The acceleration towards the final speed affects the homogeneity and multiple steps processes are common to improve the quality of the deposition. Thicknesses from 10 nm up to a few hundreds of microns can be achieved using this method. This technique is also often used for planarization purposes. Spraying a resin is another method capable of producing a highly uniform coating with thicknesses ranging from 100 nm up to 100 mm. The method results in a conformal deposition, which is especially interesting for coating nonplanar substrates. However, spraying equipment is still more expensive and not as widespread as spin-coating systems. Other methods of resist deposition include dry lamination, dip coating, screen printing, and electrodeposition. 3.1.1.7  Scanning Beam Lithography

Drawing structures in a substrate or in a thin film is usually performed in two steps: (1) A pattern is transferred into a resist coated on the substrate/film; next (2) the

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79

same pattern is transferred into the substrate/film during the etching step using the resist as a mask. A first approach is to directly write into the resist as we are writing on a sheet of paper with a pen. A “pen” with a tip as small as possible is then essential to get a good resolution. This is typically achieved by scanning a beam of particles over the surface of the wafer, giving the name scanning beam lithography (SBL) to this set of techniques. Practically, beams of photons (i.e., laser light), ions, or electrons are employed. Laser Machining

Laser machining is not only used to pattern resists, but also for local deposition, ablation, or etching. Masks for photolithography are usually patterned with a laser or by electron beam lithography. Virtually all types of materials, such as metals, ceramics, plastics, and wood can be machined by laser ablation. This technique can remove minute amounts of materials with only a small heat-affected zone. Depending on the energy, the material can be evaporated, sublimated, or converted into a plasma. Laser machining is a convenient method for the rapid prototyping of microfluidic devices [11]. A drawback of laser ablation may be the redeposition of the substrate material, making sometimes the quality control of machined surfaces difficult. Indeed, it is also possible to use laser ablation in a PVD process called pulsed laser deposition (PLD). The laser is then focused on a rotating target of the material to deposit. The material ejected from this target is deposited onto the facing substrate. For micromachining purposes, the resolution is limited by the spot size. The theoretical resolution dmin (mm) is given by:

dmin =

lf p d0

where: l (mm) is the wavelength of the laser; d0 (mm) is the diameter of the beam at the focusing lens; f is the focal length of the lens. Thus, the main way of increasing the resolution is by reducing the wavelength. Common lasers for micromachining applications include the following: · · ·

Excimer lasers with ultraviolet wavelengths (351, 308, 248, and 193 nm); Nd:YAG lasers with near infrared (1,067 nm), visible (533 nm), and UV wavelengths (355 nm, 266 nm); CO2 lasers with a deep infrared wavelength (10.6 mm).

Another major parameter of laser micromachining is the power. The choice of the power depends on the desired structure size and the ablation rate. Lasers can be run continuously or pulsed. Each pulse removes only a very thin layer of material; thus the ablation is controlled with a high accuracy by the number of pulses. As an example, using a Nd:YAG laser, a pulse with a duration of a few tens of nanoseconds

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Fabrication Technologies of Nanochannels Table 3.4  Typical Ablation Depths per Pulse of Different Material (Nanosecond Laser) Materials

Depth per Pulse [µm]

Polymers Ceramics and glass Diamond Metals

0.3-0.7 0.1-0.2 0.05-0.1 0.1-1.0

Source: [12].

will typically vaporize the surface material to a depth of 0.1–1 mm (see Table 3.4). Beyond this, short pulses decrease heat transfer issues. Local deposition is also possible using a CVD process (laser CVD or LCVD). The laser is focused at the deposition location in the presence of the suitable reactive gases, providing the energy for the deposition to occur. In a similar fashion, the process called microstereolithography consists of polymerizing a liquid photoresist layer by layer. Resolution as low as 70 nm has been demonstrated using a laser-assisted etching technique [13]. Electron Beam Lithography

In electron beam lithography (EBL), the pattern is written using a beam of electrons focused by electrostatic lenses under vacuum. Typically, EBL is used for direct writing on a wafer coated with a thin layer of resist, or used for the fabrication of high-resolution photomasks. Due to the very small wavelengths of high-energy electrons, EBL is not limited by diffraction; thus very high resolutions can be attained. Features with lateral dimensions below 4 nm have been reported using this method (see Table 3.5) [14]. However, other factors than the probe size determine the resolution. The lithography performance is also greatly affected by both the electron-resist system and the

Table 3.5  Comparison of Different Lithography Techniques Current Capabilities Technique

Minimum Feature a

Half Pitch b

32 nm

Photolithography [15]

10 nm

Scanning beam lithography c [14, 16]

3–4 nm

13.5 nm

Molding, embossing, and printing [17, 18] Scanning probe lithography [19, 20]

1 nmd

6 nm

<1 nm

1 nm

Edge lithography [18]

4 nm

6 nm

Self-assembly

>1 nm

>1 nm

Pattern Parallel generation of arbitrary patterns Serial writing of arbitrary patterns Parallel formation of arbitrary patterns Serial positioning of atoms in arbitrary patterns Parallel generation of noncrossing features Parallel assembly of regular, repeating structures

Source: [21]. The minimum feature corresponds to the lateral dimension of a single structure, while the half pitch refers to a periodic array of lines. a Structure obtained using extreme UV lithography. b Resolution demonstrated by Intel in 2008 using 193-nm immersion scanners. c Obtained with an electron beam. d Replication of single wall carbon nanotubes.

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underlying substrate. Electrons are scattered as they penetrate the resist layer. Figure 3.4(a) depicts exposed areas in the case of a resist thicker than the penetration depth. The size of these lobes depends on the energy of the electrons and the atomic mass of the resist molecules. Considering the straight shape at the top of the lobe, a first solution to avoid the loss of resolution induced by the forward scattering of electrons is to use a layer of resist thin enough for a given energy. However, proximity effects occur [Figure 3.4(a)], also limiting the maximal EBL resolution. Indeed some of the electrons, after going through the thin layer of resist, are backscattered by the substrate, accidentally exposing neighboring areas. Parameters influencing the backscattering include the energy of the electrons, the substrate material (i.e., the effect is smaller for materials with lower atomic weight), and the resist properties and its thickness. Using very sensitive resists with a high contrast is also essential in order to lower the required dose, and thus the exposure time. Other important characteristics of resist include the thermal stability and the etching resistance (from which the name “resist” originates). Many formulations have been developed for this purpose. Polymethylmetacrylate (PMMA) is an example of an inexpensive resist with a high resolution capability. As with many polymers, bombardment of a PMMA layer by electrons induces bond breakage. After exposure, the layer is usually rinsed with a chemical, called the developer, selectively etching the exposed areas. In the case of PMMA, the developer is simply IPA (Isopropyl alcohol). PMMA is a positive resist for EBL, because the exposed areas are dissolved in the developer. In the case of a negative tone photoresist, the unexposed areas are dissolved in the developer. Examples of negative resists will be given in the next sections about photolithography. Ion Beam Lithography

The principle of ion beam lithography [often named “focused ion beam” (FIB)] is very similar to EBL. Ions being heavier than electrons, they are less prone to suffer from front- and backscattering effects, leading, in theory, to better resolutions than EBL. The smallest beam size is in the range 4–6 nm. Compared to photons or elec-

Figure 3.4  (a) “Pear-shaped” scattering in a thick resist for electrons with different energies, and (b) backscattered electrons after collision with the substrate. (After: [22].)

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trons, ions are able to chemically react with the substrate. As a result the focused ion beam is a very versatile tool that can be used for patterning resists, but that also offers the ability to locally etch, deposit, or implant dopant species. More precisely, most common applications of FIBs include the repair of photomasks and integrated circuits (e.g., add or cut electrical connections) and maskless implantation. As the penetration depth of slow and heavy ions is low (typically 30–500 nm for heavy ions with energies below 1 MeV), only thin layers can be patterned. Despite the higher sensitivity of resists to ions, FIB is also serial and is performed under vacuum. Heavy ions being more difficult to deflect than electrons, this technique is even slower than EBL, and hence mainly employed for research purposes. An attractive alternative is the proton beam writing technique [23]. Using protons at higher energies (typically at 2 MeV) results in a much larger penetration depth. Very high aspect ratio micro- and nanostructures can be fabricated using this technique. 3.1.1.8  Scanning Probe Lithography

Scanning probe lithography (SPL) represents a powerful tool for even smaller dimensions [24] (see Figure 3.5). An ultimate example is the manipulation of individual iron atoms on a substrate using a scanning tunneling microscopy (STM) tip [19]. Dip pen nanolithography (DPN) perfectly illustrates the concept of the extremely small writing nib [25]. Actually the pen is an atomic force microscope (AFM) tip (i.e., a cantilever supporting an extremely small tip) also produced using silicon technologies. This tip is dipped in a solution of the material to be transferred on the surface. Then, the tip is brought in contact with the substrate, depositing the adsorbed ink along the pattern. Features as small as 50 nm can be achieved using this method. Self-assembled monolayer (SAM) precursors have commonly been employed as an ink for DPN. SAMs are surfaces consisting of one molecular layer.

Figure 3.5  Illustrations of some scanning probe lithography techniques: (a) handling of individual Fe atoms using a scanning tunneling microscope (STM) tip [19]; (b) elimination of a SAM using an AFM tip [27]; (c) local oxidation of a silicon substrate with a carbon nanotube attached to an AFM tip [28]; and (d) deposition of a SAM using an AFM tip (DPN) [25]. (After: [21, 24].)

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During the deposition in liquid or vapor phase, the molecules organize themselves on the surface to form a monolayer. A typical example is the self-assembly of alkylsilanes [e.g., octadecyltrichlorosilane (OTS)] on a silicon dioxide surface, or the self-assembly of alkane thiol on gold. Features with desired properties are obtained by selecting the appropriate functional headgroup. SAMs may also be used as a resist during a later etching step. This has been demonstrated for the wet etching of metals (i.e. Au, Pd, and and Ag) of nanostructures with critical dimensions as small as 12 nm [26]. It is still not clear whether the transfer of the molecules is mediated by the liquid or results from tip-surface interactions. Various parameters such as the humidity, the reactivity of the precursor with the substrate, the radius of curvature of the probe, and the linear velocity of the probe affect the spreading of the ink [21]. Instead of adding material, the tip may be used to locally scratch away a SAM [27]. A sufficient load has to be applied on the atomic force microscopy tip to selectively remove the film. Using the same technique, it is also possible to substitute the removed material with other molecules [29, 30]. Selective oxidation of a silicon substrate has been demonstrated by applying an electrical potential between the substrate and a carbon nanotube mounted at the end of an atomic force microscopy tip and dipping in water [28]. The 10-nm-wide lines have been patterned using this method. Another example of chemical modification using SPL is the photochemical oxidation of SAMs of mercaptoundecanoic acid using a near-field scanning optical microscope (NSOM) [31]. This process was employed to etch 55-nm-wide trenches in gold. The availability of scanning tunneling, atomic force, or near-field scanning optical microscope makes them convenient tools for writing nanostructures. SPL is a serial technique and exhibits an even lower throughput than scanning beam lithography methods, restricting its use to research or mask fabrication repair. Parallelization was demonstrated to overcome this issue [32]. The “millipede” developed at IBM for high-density data storage applications is an AFM with an array of 32 by 32 (1,024) cantilevers [33]. However, the reproducibility using this concept is challenging. Indeed the substrate topography may vary from one array to another, and the shape of the tip also varies with time and use. 3.1.1.9  Photolithography

Photolithography has been the key enabling technology of the microelectronics industry. Even though there have been calls to replace it with alternative technologies—as the technological node has been shrunk far beyond the wavelength of the UV sources—photolithography has always kept the pace imposed by the Moore’s law. Opposite to the SBL techniques that were described in Section 3.1.1.7, photolithography is a parallel patterning technique with a very high throughput. Figure 3.6 illustrates the basic principles of this patterning method. The process starts with a thorough cleaning step. The procedure depends on the nature of the substrate and the deposited films. A classic example for silicon wafers is the standard RCA procedure developed by Werner Kern in 1965 while working for the Radio Corporation of America, and detailed in Table 3.6. However in academic labs, for MEMS applications, it is common to clean wafers only with

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Figure 3.6  Basic principles of photolithography: (a) a film of photoresist is deposited onto the substrate of the film to be patterned; (b) after a baking step, the substrate is exposed to UV light through a photomask; (c) next the resist is developed, leaving on the substrate exposed or unexposed areas depending on the tone; and finally, (d) the substrate or the thin film is etched, and (e) the residual resist is removed.

the piranha solution (step 1) and skip the other steps. After cleaning, a dehydration step in an oven is necessary to remove water traces and improve photoresist adhesion. In addition, the surface is often treated with adhesion promoters such as hexamethyldisilazane (HMDS). Table 3.6  Standard RCA Cleaning of Silicon Wafers Step

Solution

Temperature [°C]

Duration

Effect

1 2 3

H2SO4 + H2O2 (4:1)a DIW NH4OH (29%) + H2O2 (30%) + DIW (1:1:5)b DIW HCl (37%) + H2O2 (30%) + DI–H2O (1:1:6)c DI–H2O HF + H2O (1:50) DI–H2O

120 Ambient 75–80

10 minutes 1 minute 10 minutes

Organic substances Rinsing Particles

Ambient 75–80

1 minute 10 minutes

Rinsing Metal ions

Ambient Ambient Ambient

1 minute 15 seconds 1 minute

Rinsing Native oxide etching Rinsing

4 5 6 7 8

Source: [34]. a Piranha solution; b RCA SC1 solution; c RCA SC2 solution; SC standard clean; DIW de-ionized water.

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Photoresists

After this priming step, a photoresist is deposited onto the wafer, usually by spincoating. A photoresist is composed of a polymer (the base resin), solvents making the resist liquid, a photoactive compound (PAC) giving the photochemical properties, and eventually additives to ease the spreading during the spin-coating. Most commercially available i-line resists (sensitive at a wavelength of 365 nm) belong to the DQN family composed of phenolic novolak resin (N) as a base and diazoquinone ester (DQ) as a photoactive sensitizer. An important quality of the photoresist is the difference in dissolution rate between the unexposed and exposed photoresist polymer. While novolak resins are typically soluble in alkali solutions, the addition of diazoquinone inhibits the dissolution as long as the resist is not exposed. The exposure to UV locally increases the dissolution rates (say, with two to three orders of magnitude). Beyond this, other requirements for a photoresist include high sensitivity, good resolution, good etching resistance, easy processing, high purity, long shelf life, and low cost. A high glass transition temperature Tg is also an advantage to avoid a reflow, which may lead to a modification of the pattern during the eventual baking steps following the exposure (i.e., post-exposure bake and hard bake). New photosensitive materials have been developed as the photolithography pro­ cess has evolved. As an example, at the smaller wavelengths required for patterning smaller features (243 and 193 nm), DQN resists had to be replaced because of their high absorbance in the deep UV (DUV). Due to the lower output power of early 248-nm laser sources compared to the mercury arc lamp, chemically-amplified resists with a higher sensitivity were introduced with poly-hydroxy-styrene (PHS) as a resin base [35]. The concept of chemical amplification has been kept for UV sources at 193 nm but the polymer backbone has been changed from PHS to mainly acrylate chemistry in order to keep a sufficient transparency at this wavelength [36]. Techniques such as immersion also required new resist formulation. Photomasks

The fast evaporation of the solvent during the spin-coating step leads to an expansed polymer film, with a high content of free volume. Thus, after spin-coating, the film of resist is densified by a first baking step (i.e., soft bake). Next, the photoresist is exposed to UV through a photomask (or “reticles”) with opaque and transparent areas defining the pattern [Figure 3.6(b)]. Photomasks have usually been made of fused silica, but the material can also be borosilicate glass for less critical layers. Chromium has traditionally been used as an absorbing layer, sometimes covered with an antireflective coating (chromium oxide or oxynitride) to reduce flare in the imaging system. The patterning of the mask is done by direct writing (see dedicated sections) either using a laser or an e-beam writer. With photomasks becoming increasingly complex and expensive, industrial masks are inspected using dedicated software and optical systems. Errors are corrected using a laser or a FIB (local etching or deposition). As the resolution gets closer to the theoretical limit, new techniques have been developed to improve photomasks. Aberrations are evaluated and compensated for during the design of the mask using dedicated software [optical proximity correction (OPC) techniques], resulting in a final image on the wafer different from the original image on the mask. Others resolutionenhancement techniques include off-axis illumination and phase-shifting masks

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Fabrication Technologies of Nanochannels

[37, 38]. The principle of a phase-shift mask is described in Figure 3.7. By introducing a transparent spacer with a well-defined thickness, it is possible to reverse the phase of the incoming light on the wafer, giving rise to destructive interference at the edge of the step. The depth D of this spacer is given by:

D=k

l 2(n − 1)



where k is an integer, l is the wavelength, and n is the index of refraction of the mask. Though features below 10 nm have been demonstrated using this method, photomasks are more expensive and the design is limited to noncrossing lines. Soft polymeric phase-shifting mask made from polydimethylsiloxane (PDMS) are promising low-cost alternatives to quartz phase-shifting masks [39, 40]. Contact, Proximity, and Projection Aligners

The early days (1970–1980) of photolithography were dominated by contact and proximity aligners. In the contact mode, photomasks and wafers are brought together in contact. This intimate contact between the absorber and the photoresist limits the loss of resolution induced by the diffraction. However, the contact also greatly reduces the durability of the mask because of the mechanical wear and induces a high density of defects due to the contamination between the mask and the resist. Furthermore, distortions may occur due to the possible bending of either the mask or the substrate. Proximity printing overcomes these drawbacks by allowing a gap between the mask and the substrate. On the other hand, the diffraction of the transmitted light through this spacing reduces the resolution. For shadow printing (i.e., contact or proximity), the theoretical half-pitch bmin (half the distance between the centers of adjacent features in periodic structures) is given by:

bmin

� � � �� 1 z =3 l s+ 2



Figure 3.7  (a) Conventional and (b) phase-shifting photomasks. Introducing a phase-shifting layer improves the contrast by locally reversing the phase of the incoming light. (After: [10].)

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87

where s stands for the gap between the mask and the photoresist surface, l stands for the wavelength of the exposing radiation, and z stands for the photoresist thickness. Practically, the minimum resolution reachable using shadow printing is above or close to the micrometer. The diffraction issue of shadow printing was overcome by introducing projection systems in the early 1970s. The image of the pattern is projected onto the photoresist-covered wafer using a high-resolution lens, avoiding any contact. Initially the pattern on the mask remained at the same size as the pattern printed on the wafer (1´ projection system), although only part of the image was exposed simultaneously (the “exposure slit”) for better control of the complex optic. 1´ projection aligners have been used for features between 3 and 0.7 mm. The half-pitch bmin of projection systems is usually evaluated by the Rayleigh equation:

bmin = k1

l NA



where k1 is a process-dependent factor mainly determined by the illumination conditions, the materials of the photomask, the quality of the imaging optics, and the performance of the photoresist, and NA the numerical aperture. It should be noted that the Rayleigh limit is on the pattern pitch not on how small an individual pattern can be printed [i.e., the critical dimension (CD)]. Thus, the CD can be much smaller than the half-pitch (or technological node). The trends in optical lithography are to reduce the wavelength l (deep and extreme UV) and to increase the numerical aperture. Mercury arc lamps (at 405 nm and 365 nm), KrF lasers (at 248 nm) and ArF lasers (at 193 nm) have been successively used as UV sources. The transition to molecular fluorine lasers at 157 nm or extreme UV sources (EUV or soft X-ray) should allow reducing these dimensions further in the future, but there are still many technical challenges [41]. The numerical aperture is proportional to the minimum index of refraction of the imaging medium, final lens element, or resist. Increasing the refraction index has led to immersion solutions where the objective is immersed in a liquid such as water instead of air. In the 1980s, step and repeat projection systems (or “steppers”) with a reduction optic (4´, 5´) were introduced. The mask contains the pattern of only one or a couple of dies. This elementary pattern is reduced and repeated all over the surface of the wafer. This has greatly reduced the complexity of the mask. Additionally, the reduction optic has allowed the mask to be magnified compared to the image on the wafer, simplifying its fabrication, as well as its inspection and repair. Steppers have been employed for critical dimensions between 0.7 and 0.25 mm. Finally, step and scan systems (or “scanners”) were introduced. Similar to 1´ projection aligners, only a part of the image is exposed through an exposure slit. After the whole pattern has been scanned, the operation is repeated on the next position to complete the whole mask image. This has allowed further reducing and improving the optics, leading to an actual resolution of 32 nm using 193-nm immersion scanners. These projection systems have a very high throughput, but their very high prices (equipment and photomasks) prevent their use in academic research or in the MEMS industry, where contact and proximity aligners are still preferred.

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Fabrication Technologies of Nanochannels

Development, Etching, and Removal

After the photoresist has been exposed to UV and after the eventual baking step, the resist is developed [Figure 3.6(c)], leaving on the substrate exposed or unexposed areas depending whether the photoresist is negative or positive, respectively. Developers for positive photoresists are alkaline aqueous solutions while organic solvents such as xylene are typically used for negative photoresists. Thus positive resists are often preferred in the industry because of their smaller environmental impact, although negative resists are substantially less expensive due to the reduced content of sensitizers. After development, organic residues are eventually etched away using an oxygen plasma reactor (descumming process). Next, the pattern is transfer in the substrate or the thin film using etching technique introduced in the next section [Figure 3.6(d)]. Finally, resists are removed in an organic solvent or in an oxygen plasma reactor. 3.1.1.10  Unconventional Patterning Techniques

SBL, SPL, and photolithography are patterning techniques with the ability to write a random, two-dimensional structure. Recently, many simple techniques have emerged, making possible patterning at the nanoscale of simple structures such as array of lines or dots. Although these techniques are limited in term of design, their simplicity and relatively low cost makes them particularly attractive. Edge Lithography

Edge lithography is a set of techniques taking advantage of small features in the vertical direction (e.g., a step) to generate small features in the lateral direction. Such a method was described in a patent from Texas Instruments in 1982 (Figure 3.8).

Figure 3.8  1) Step lithography. Starting from a layer of polysilicon deposited on a step etched in SiO2 (a), a selective and anisotropic etching was performed in the polysilicon layer (b), and next in the SiO2 layer [43]. 2) Etching at an angle. A layer of chrome was deposited on a step etched in a glass substrate (a) The anisotropic etching of chrome at an angle resulted in the formation of a triangular nanowire (b) [44]. 3) Deposition at an angle. The directional deposition of metal on a step (a) led to the formation of a nanowire (b). Finally, the residual chrome layer is etched away (c) [45].

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First, a vertical step was etched in a layer of SiO2 using an anisotropic etching process (i.e., a process etching only in the vertical direction). Next, a layer of polysilicon with a thickness t was deposited and etched down to the SiO2 layer using an anisotropic process. This resulted in a line of polysilicon with a height and a width roughly equal to t. Finally, the layer of SiO2 was anisotropically etched down to the silicon substrate to form the gate of a MOS transistor. The inventor claimed the fabrication of lines as narrow as 50 nm. Critical dimensions below 10 nm have been demonstrated using a similar process [42]. Similar results have been obtained by taking advantage of anisotropic etching or deposition processes at an angle [Figure 3.9(2, 3)]. Prober et al. used a thin layer of gold-palladium deposited on a glass step [44]. Etching this layer at an angle resulted in a nearly triangular wire having dimensions as small as 30 nm. A subsequent anisotropic evaporation step in the direction parallel to the substrate led to a square nanowire. By directly evaporating chromium at an angle on steps vertically etched on a substrate, Flanders and White obtained lines as small as 10 nm [45].

Figure 3.9  1) Controlled under-etching. A patterned photoresist (a) was used as a mask to overetch a metal layer (b). Then a second metal layer was deposited (c) and the resist was lifted off the substrate (d) [46]. 2) Trench refilling. Starting from amorphous silicon lines on a silicon nitride layer (a), silicon dioxide was grown, defining the width of the patterns (b). Next, a second layer of amorphous silicon was deposited (c) and etched down to the silicon dioxide by chemical mechanical polishing (CMP) (d), before etching the silicon dioxide in a buffered oxide etch (BOE) solution (e) [47].

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Fabrication Technologies of Nanochannels

Love et al. introduced a method of patterning by over-etching a metal layer [Figure 3.9(1)]. A photoresist was patterned on a metal substrate. The over-etching distance was controlled by the duration of the subsequent isotropic wet etching step. After etching, a second metal layer was deposited, and then lifted off with the photoresist, letting on the substrate features with dimensions defined by the undercutting. Critical dimensions of 50 nm were obtained [46]. Lee et al. made use of a trench refilling technique depicted in Figure 3.9(2) to pattern 10-mm-long lines as narrow as 25 nm. The authors grew a silicon dioxide layer with a thickness t on amorphous silicon (a-Si) lines patterned on a thin film of silicon nitride (SiNx). Silicon dioxide does not grow on silicon nitride during the thermal oxidation process, leading to a nonconformal coating (no oxide at the bottom of the trench). A second deposition step of amorphous silicon filled the open trenches. After planarization by chemical mechanical polishing (CMP), the silicon dioxide was etched away, converting the thickness t in a width [Figure 3.9(e)]. The 7-nm pores were fabricated using a similar method [48]. The atomic step edges of a single crystal surface often have properties that differ from those of the bulk material. More precisely, increased reaction kinetics can be observed at exposed edges. The step-decoration method takes advantage of this property to selectively deposit or grow metallic nanowires at atomic step edges. As an example, metallic wires of molybdenum with diameters ranging from 15 nm to 1 mm were obtained by electrodeposition at the exposed edges of a graphite layer [49]. It was then possible to transfer these nanowires by casting polystyrene on the substrate. However, the orientation and spacing of these structures is determined by the step edges of the graphite. The order of SAMs is perturbed at the edge of steps. This was observed on silver features patterned either by lift-off or stenciling on a silver surface [50]. The resulting local disorder of an alkanethiol layer induced preferential growth location of calcite crystals at the edges of the steps. A highest etching kinetics was observed at the interface, making possible the fabrication of metallic features as small as 50 nm. Phase-shift lithography, already introduced in the section dedicated to photolithography (Figure 3.7) is also a kind of edge lithography [38]. A step with adequate dimensions locally results in a destructive interference, equivalent to a narrow opaque area on a photomask. The dimensions of this area depends on multiple parameters, such as the periodicity of the features, the wavelength of the exposure light or the depth of the step [51]. While quartz phase-shifting photomasks are relatively expensive, soft phase-shifting mask replicated in PDMS from a silicon master make this process suitable for academic structures [39]. Features as small as 30 nm have been obtained using a stamp made with a modified, harder PDMS [40]. Superlattices are stack of different thin films alternating periodically, usually obtained by molecular beam epitaxy (MBE). Such heterostructures have applications in microelectronics or in photonics. The thickness of these layers is very well controlled, down to 1 nm or less. Using a GaAs/Al0.8Ga0.2As superlattice, a technique for fabricating nanowires was demonstrated [52]. Voids were created at the edge of the superlattice by selectively etching the AlGaAs roughly 20–30 nm deep using a dilute mixture of buffered hydrofluoric acid. Next, metal nanowires were evaporated at an angle of 36°. Eventually the wires were transferred on an oxidized

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silicon substrate using a 10-nm-thick intermediate layer of epoxy adhesive; 2–3-mmlong wires 8 nm in diameter, at a pitch of 16 nm were fabricated using this method. Similar to this process, Xu et al. exposed the edge of thin films embedded in an epoxy matrix by cutting it with the glass knife of a microtome [53]. Interference Lithography

Interferences between two coherent light beams results in periodic series of fringes. Interference lithography (IL) consists of exposing a photoresist using the interference pattern resulting from two or more laser beams [54]. The fringe-to-fringe period T of the intensity pattern is given by:

T=

l q sin 2 2

where l is the wavelength and q is the angle between the two light beams. With immersion techniques, the smallest half-pitch is given by:

bmin =

l 8n

where n is the refractive index of the immersion material. Using an excimer laser at 193 nm and water immersion, the minimum half-pitch is around 18 nm. IL is fast and does not require any mask, but is limited to periodic array of dots or lines. It has been often used to test photoresists at new wavelengths. Controlled Wrinkling and Cracking in Thin Films

The formation of wrinkles in thin films occurs as a response to a stress. As an example, wrinkles may form in a film deposited on a substrate to relieve the thermal stress accumulated over thermal cycles due to the mismatch in coefficient of thermal expansions. Different strategies have been proposed to order the resulting structures. Bowden et al. used a micropatterned PDMS slab coated with 50 nm of gold to create ordered patterned over large areas [55]. The process was later simplified by replacing gold by the stiff silicate layer created during the exposure of a PDMS slab to an oxygen plasma [56]. The control of the buckling has also been demonstrated by applying a patterned PDMS mold during the thermal cycle, by mechanically stretching the substrate, or by a selective adhesion resulting from the patterning of SAMs [57–59]. The stress can also be chemical or induced by the UV curing of polymers [60, 61]. The resulting structures have lateral dimensions in the micrometer range and height down to the nanoscale, making them useful for the fabrication of planar nanochannels. The formation of cracks with widths in the range 120–3,200 nm was demonstrated by applying a mechanical strain on a oxidized PDMS slab [62]. Self-Assembly

Self-assembly is defined as the spontaneous organization of two (or more) components into larger aggregates using covalent and/or noncovalent bonds [63]. SAMs are a common example of this process. The single layer of molecule results from the

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Fabrication Technologies of Nanochannels

specific interactions between the molecules and the substrate. Block copolymers consist of at least two different, immiscible polymeric chains (block) that are covalently linked (see Figure 3.10). Block copolymer lithography is based on the self-assembly of blocks into periodic microdomains, on the 10–100-nm scale (set by the block’s chain length). Depending on the relative length of the two blocks, diblock copolymers assemble in various geometries (e.g., lamella, pillars, or spheres) [64]. When confined in a thin film, microdomains tend to align with a particular orientation to the substrate. Regular arrays of dots or lines are obtained after selective etching of one of the block. However, though the domains formed by diblock copolymers are locally extremely well ordered, it is difficult to control their order on a large scale. Strategies to order diblock copolymer patterns on a large scale are pretty similar to those used to induce anisotropy during the wrinkling or cracking processes just introduced in the same section. This includes the chemical patterning of surfaces using SAMs [65, 66], microfabricated grooves [67, 68], electric fields [69] or by applying a shear [70]. An effective combination of those methods consists of imprinting the film using a patterned mold. Deng et al. demonstrated the alignment of microdomains by flowing block copolymers solution through a PDMS microfluidic network reversibly bonded to the substrate [71]. 3.1.1.11  Etching

The next step after the lithography is usually the etching of the pattern directly in the substrate or in the thin film covering it. The patterned resist is used as a mask,

Figure 3.10  (a) Schematic representation of a diblock copolymer AB and the process of selfassembly into lamellar domains. (b) In the thin film state, domains have a particular orientation to the substrate surface. A selective etching step results in the formation of regular patterns.

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93

as depicted in Figure 3.6(d). During the etching step, the material is removed at an etch rate R defined by:

R=

etching depth etching time



Figure 3.11 illustrates the notion of isotropic and anisotropic or directional etching. An etchant with an etching rate identical in all directions is called isotropic. Etchants with an etchant rate dependent on the direction are called anisotropic. The anisotropy A can be evaluated by:

A= 1−

RL RV

where RL is the lateral etch rate and RV is the vertical etch rate as specified in Figure 3.11. In the case of an isotropic etching, RL = RV so A = 0. This results in an underetching. Eventually, this may lead to the etching of the gap between two close features (dashed lines in the isotropic etching case, Figure 3.11), so the loss of the original pattern. For high resolutions, an isotropic or directional process is then preferred (ideally RL = 0 so A = 1). Another important parameter is the selectivity of the etching. Considering the Figure 3.11, the etch rate of the resist and the substrate should be ideally much lower than the etch rate of the layer to etch. The selectivity S of the process is defined by:

S=

RV RV0

where RV is the vertical etch rate of the material to etch while RV0 is the vertical etch rate of another material. A low selectivity results in an over-etching of the resist or the substrate as shown in Figure 3.11. For a selectivity large enough, only the target layer is etched down to the interface with another material.

Figure 3.11  Illustration of concepts of isotropy and selectivity in an etching process.

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Isotropic Wet Etching

Historically, silicon etching was first performed in liquid phase in a mixture of nitric acid (HNO3), hydrofluoric acid (HF), and acetic acid (CH3COOH), also called HNA. Silicon is oxidized in HNO3. Simultaneously, the resulting silicon dioxide is etched by HF. Wet etching usually provides a higher selectivity than dry etching techniques and higher etch rates (a few tens of microns per minute for isotropic wet etching, around 1 mm/min for anisotropic wet etching and 0.1 mm/min for dry etching). Wet etching technologies are also simpler and easier to implement than plasma-based methods. Depending if the process is transport- or reaction-limited, the etch rate is a function of the solution composition and concentration, the temperature and the stirring method. If the etch solution is well stirred, the isotropic etch front has almost a spherical shape (considering a transport limited process), while a poor agitation results in a profile with a flat bottom. Glass etching in HF is common in the fabrication of microfluidic devices. However, as explained in the previous section, isotropic etching leads to under-etching issues, making the method inappropriate for high resolutions. Common etchants and corresponding etch rates for a range of materials can be found in the literature [72, 73]. Anisotropic Wet Etching

Later, it was discovered than some alkaline solutions have different etching rates for different orientation of a monocrystalline silicon substrate. As examples, aqueous solutions of KOH, NaOH, LiOH, CsOH, and RbOH are anisotropic silicon etchants. Etching is faster for the (100) planes than for the (110) planes and much faster than for the (111) planes. Figure 3.12 shows some example of possible geometries in (100) and (110) silicon wafers. Figure 3.12 also introduces the concept of geometric etch stop, where the etching process stop by itself as a pyramid is fully formed, the depth being defined by the width of mask aperture. However this method is limited in term of design and does not allow the fabrication of membranes. The etch-stop can also be controlled by introducing a protective layer. Anisotropic silicon etchants usually demonstrate good selectivity against silicon dioxide, silicon nitride, or silicon heavily doped with boron (because electrons are involved in the etching reaction). The formation of a buried layer of these materials at the required depth is possible by the implantation of O+, N+, or boron ions respectively. Another common etch-stop method is based on an electrochemical setup and involves a wafer with a p-n junction. Briefly, the p-n junction is first reverse-biased and no current flows. The p-silicon is then etched down to the p-n interface. As the p-n junction is

Figure 3.12  Anisotropically etched grooves in a (100) and in a (110) silicon wafer.

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destroyed and the n-silicon is exposed, the potential at the interface is automatically increased to the passivation potential. In this electrochemical regime, a silicon dioxide layer forms, ending the etching process. CMOS circuits have been fabricated on membranes or on cantilevers using this method, allowing the thermal decoupling of the components from the rest of the chip [74]. Potassium hydroxide KOH is one of the most common anisotropic etchants and demonstrates a relatively high etching rate (Table 3.7). The most reliable etch mask is a LPCVD silicon nitride layer. The etch rate of PECVD silicon nitride is also very low, but usually results in a higher pinholes density due to the lower quality of the deposited layer, and a larger under-etching due to the lower adhesion. Silicon dioxide may also be used as a mask for shorter etching times. Aluminium, often used as a material for electrical interconnects, is attacked violently by KOH with etch rates larger than 1 mm/min at 95°C. Even a 2-mm-thick protective layer of PECVD SiO2 is not sufficient to protect aluminum, KOH penetrating through pinholes to destroy the structures. This problem, combined with ionic contamination issues in semiconductor, make KOH anisotropic etching hardly compatible with CMOS processes. Ethylene diamine pyrocathecol EDP demonstrates a better compatibility with integrated circuit fabrication and a good selectivity against aluminium. However, EDP solutions are toxic and corrosive. Furthermore, EDP solutions age fast and oxidize in contact with air. Though its higher selectivity against more lightly p-doped silicon layers gave EDP an advantage, the introduction of electrochemical etch-stop techniques made it less popular. On the other hand, tetramethyl ammonium hydroxide (TMAH) solutions demonstrate lower etching rates, but are easier to handle and have a high selectivity with SiO2, Si3N4, and aluminium making them compatible with CMOS technologies. Electrochemical Etching

Electrochemical etching methods often demonstrate very high etching rates, making them promising for the surface micromachining of nanofluidic channels. Compared to other wet etching methods, electrochemical micromachining usually offers Table 3.7  Characteristics of Different Anisotropic Wet Etchants

Solution (weight %) Temperature [°C] <100> etch rate [mm/min] Etch rate ratio (100)/(111) Masking layer (etch rate) Etch rate ratio (100) p-Si/(100) Si Remarks

Source: [10, 34].

KOH Potassium hydroxide (24 %)

EDP Ethylene diamine pyrocathecol with pirazine additive

TMAH Tetramethyl ammonium hydroxide (8 %)

85 1.7

115 1.25

90 0.9

400

35

40

SiO2 thermal (0.43 mm/min), SiO2 PECVD (0.7 mm/min), Si3N4 (not attacked) 1:20 (boron, >1020 cm–3)

SiO2, Si3N4, Au, Cr, Ag, Cu

IC incompatible, strong formation of H2 bubbles, etches oxide fast

IC compatible, toxic, ages fast, reacts with oxygen

SiO2 (4 orders of magnitude less than <100> Si), Si3N4 1:40 (boron, > 4.1020 cm–3) IC compatible, easy handling, smooth surface finish

1:50 (boron, >5.1019 cm–3)

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better control and flexibility, requires very little monitoring, and has minimum safety and environmental concern. Anodic metal dissolution is a typical example with industrial applications in machining and polishing [75]. Basically, the work piece is placed at the anode of an electrochemical cell. Oxidation takes place and the metal is removed at a rate controlled by an external current. Most of the thin films of metals and alloys that are of interest in the microelectronics industry can be anodically dissolved in neutral salt electrolytes (e.g., sodium nitrate, sulfate, or chloride). In these electrolytes, the dissolved metal ions form hydroxide precipitates that remain in suspended form in solution and can be easily filtered, thus significantly minimizing problems of safety and waste disposal. While wet etching is isotropic, the metal removal rate in the lateral direction may be significantly reduced through proper consideration of mass transport and current distribution, leading to a better slope control [76]. An example of anodic dissolution is the etching of aluminium in a neutral sodium chloride solution [77]. Electrochemical dissolution exhibits a potential dependence, making possible the selective etching of only one metal in the presence of other metallic layers. Anodic dissolution of silicon in fluorhydric acid leads to electropolishing or to the formation of porous silicon, depending on both the anodic current density and the HF concentration. The porosity varies with the current density. Pores with diameters ranging from 2 nm to 10 mm have been achieved. Due to its very large specific area, porous silicon is highly reactive, and oxidizes and etches at a very high rate. Depending on the concentration and the nature of the dopant species, injection of holes by illumination (photo injection) may be necessary in order to achieve a significant holes concentration [78, 79]. Galvanic corrosion may occur when two different metals in electrical contact are immersed in an electrolyte. The less noble metal corrodes preferentially. For a given environment, the relative nobility of metals is estimated by their position in a galvanic series. Galvanic corrosion has usually to be avoided, for instance in construction where materials far apart in the galvanic series should not be in contact. An example is the corrosion of the iron armature of the Statue of Liberty because of its contact with the copper skin. However, it is also possible to take advantage of galvanic corrosion to speed up etching. It was demonstrated that coupling chromium with a layer of copper results in an etching speed an order of magnitude faster than that of a pure copper layer [80]. Using a similar technique, the selective etching of a chromium layer was shown to be greatly improved by the presence of gold electrodes [81]. Dry Etching

In dry etching techniques, the substrate is exposed to a gas or a plasma (ionized gas) of the etchant species. Two phenomena are usually considered and are illustrated in Figure 3.13. The chemical component involves a reaction between ions or radicals and the surface. Similar to PECVD, reactive species diffuse to the surface where they are adsorbed and migrate along the surface to a reaction site. After reaction, byproducts are transported away from the substrate. Actually there is often a competition between deposition and etching. The chemical etching process provides a means of achieving faster rates and has a good selectivity, but is usually isotropic. Usually the chemical etch rate is promoted by increasing the plasma power (leading

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Figure 3.13  Chemical and physical components of a dry etching process. The chemical process is isotropic and has a higher etching rate and a higher selectivity, while the physical process is anisotropic and has a lower etching rate and demonstrates a poor selectivity.

to higher dissociation efficiency), by increasing the pressure, or by increasing the gas flows (larger supply of reactants). The physical component is similar to the sputtering process or, at the atomic scale, to sandblasting in which the sand is replaced by inert species such as argon, helium, or neon ions. Inert ions are accelerated towards the surface where they knock out target atoms. An important feature of physical etching is the angle dependence of the etch rate. In fact, removing an atom from the substrate results from the momentum transfer between the incident ion and the substrate. After a perpendicular incidence, the impulse vector has to be rotated by 180°, while with smaller incident angles a smaller directional change of the impulse is necessary. This effect results in a maximum etch rate for an angle of incidence around 60°, thus a slight tapering at the edge of the structures (Figure 3.14). As a consequence, absolutely perpendicular walls cannot be produced. Physical etching is usually poorly selective

Figure 3.14  Ideal and realistic profile after physical etching. The tapered profile of the structure results from the angle-dependence of the etching rate.

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and demonstrates very low etching rates (some 10 nm/min). In a physicochemical etching process, the physical component is favored by decreasing the pressure, leading to a longer mean free path of the ions. Many different setups have been used; they can be divided into two main categories. In glow discharge techniques (diode setup), the plasma is generated in the chamber where the wafer is located. In ion-beam techniques (triode setup), the plasma is generated in a separate location. The resulting ions are then directed towards the substrate. A diode setup similar to the one employed for deposition by sputtering can be used for etching (sputtering or ion-etching technique) by replacing the target of the material to deposit by the substrate to etch. However, this configuration results in considerable redeposition and eventually damage from ion bombardment. In the case of physical etching methods, triode configuration helps by decoupling the plasma source from the substrate. Ion-beam etching (IBE) or milling is a purely physical etching process where ions are generated in a plasma from a neutral gas and accelerated towards the substrate (triode setup). The angle-dependent etch rate may lead to slight tapering of the structures (faceting). The collision of ions at a glancing angle on these tapered structures may locally increase the etch rate, leading to the formation of a trench at the edge of the structure (erosion or ditching) becoming noticeable for long etching times. Sputtering involatile species away from the substrate may also result in redeposition, and eventually micromasking leading to the formation of needles or spikes on the surface (e.g., black silicon or silicon grass). Though this method is capable of very high resolutions, it suffers from a relatively low etching rate and a poor selectivity. The etching rate can be enhanced by increasing the density of the plasma. Applying a magnetic field with the correct orientation gives electrons a helical trajectory, increasing their ionization efficiency (magnetically enhanced ion etching, MIE). The selectivity and etching rates can be improved by adding a chemical component to these physical etching methods. Examples of such ion-beam techniques include reactive IBE (RIBE) and chemically assisted IBE (CAIBE). While both methods involve reactive species in a triode setup, in CAIBE the reactive gas and the inert ions source are independent while in a RIBE reactor the reactive species are used to generate the plasma itself. The difference between the two methods lies in the transport of the reactive species (ions or neutral particles) from the source to the substrate (i.e., bombardment or diffusion). The most common setup for etching is reactive ion etching (RIE) usually performed in a reaction chamber with parallel plate electrodes (diode setup). In this configuration, the plasma is generated from a mixture of inert and reactive gases, resulting in the presence of reactive neutral radicals, reactive ions, and inert ions. Also neutral particles are always present. Examples of gas mixtures and corresponding target materials are listed in Table 3.8. High-aspect ratio etching is of paramount importance in MEMS and micro- and nanofluidics technologies. Though physicochemical methods greatly improve etching rates and selectivity, improvements are required for high aspect ratio etching. As discussed above, the angle dependence of the etch rate results in oblique sidewalls, limiting the maximal depth. Thus lateral removal has to be prevented to resolve deep and narrow structures. Cryogenic etching involves the cooling of the substrate at a temperature as low as –120°C. HBr, S2F2, or SF6 are typical process gas for this technique. At low temperatures, HBr or S2F2 condensate on the cool substrate

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Table 3.8  Selected Gas Mixtures for Dry Etching Material

Etchant Gases

Selective to

Si

BCl3/Cl2, BCl3/CF4, BCl3/CHF3, Cl2/CF4, Cl2/He, Cl2/CHF3, HBr, HBr/Cl2/He/O2, HBr/NF3/He/O2 CF4/H2, C2F6, C3F8, CHF3, CHF3/O2, CHF3/CF4, (CF4/O2) CF4/H2, (CF4/CHF3/He, CHF3, C2F6) BCl3, BCl3/Cl2, BCl3/Cl2/He, BCl3/Cl2/CHF3/O2, HBr, HBr/Cl2, SiCl4, SiCl/Cl2, Cl2/He O2, O2/CF4, O2/SF6

SiO2

SiO2 Si3N4 Al Organics

Si (Al) Si (SiO2) SiO2 –

Source: [9].

and forms a homogeneous passivation layer of SiBrx or sulphur compounds respectively. The passivation layer act as an etch barrier on the side walls, while the floor layer is constantly removed by physical etching. Using SF6, the low temperature of the substrate promotes the etching process on the floor of the structure. The deep reactive ion etching process (DRIE), also called the advanced silicon etch process (ASE) or the Bosch process (because it was invented and patented by Robert Bosch, GmbH, in Reutlingen, Germany) also involves passivation layers on the sidewall. Opposite to cryogenic etching, anisotropic etching and homogeneous passivation are not run simultaneously but in sequence. First the silicon surface is etched with SF6. Next, the wafer is homogeneously coated with a Teflon-like fluoropolymer layer using a PECVD with CF4 as a precursor. Finally, another RIE step with SF6 etches preferentially the floor layer. This cyclic process leads to a lateral roughness characteristic of DRIE (Figure 3.15). Though etching rates as high as 50 mm/min

Figure 3.15  A DRIE or Bosch process consists of alternating deposition and etching steps. After a homogeneous coating of the wafer with a passivation layer, an anisotropic etching step results in the preferential etching of the floor layer.

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have been recently demonstrated on a commercial DRIE source, values around 5 mm/min are more usual, making possible the etching of holes through a 500-mmthick wafer within 2 hours. XeF2 is a gas used for the etching of silicon without plasma excitation. The process is purely chemical, thus nearly isotropic and is extremely selective to SiO2, Si3N4, Al, and many other materials including numerous polymers. Etching rates as high as 10 mm/min can be achieved. XeF2 etching is simple and promising as a backend process to release MEMS or micro- and nanofluidic structures integrated with CMOS elements. Lift-Off

Metals such as platinum or tungsten may be hard to remove using common etching methods. Furthermore, substrate materials (e.g., polymers) may be incompatible with some chemical etchants. The lift-off process depicted in Figure 3.16 allows for skipping the etching step and converts it into a step of resist dissolution. Opposite to other patterning process, the resist is deposited and patterned before deposition and not after. After patterning of the resist, a layer of the material to structure is deposited. Finally the resist is dissolved and the top metal layer is lifted off the substrate. The deposition of the metal should be directed perpendicular to the wafer. The deposited layer has to be thinner than the resist layer. An overcut of the resist profile is preferred to ease the lift-off step. This is realized by using a high exposure dose, so that the profile is dominated by the adsorption of radiation, and the photons are reflected at the substrate-resist interface. It is also possible to use dedicated lift-off resists (e.g., Clariant AZ5214E image reversal photoresist or Microchem LOR bilayer processes). Lift-off methods are often used to pattern metal very small dimensions structures, as wet etching is not capable of high resolution due to its isotropy and simple dry etching processes of metals are not always available. 3.1.1.12  Wafer Bonding

Many techniques have been presented to pattern open structures. Starting from this point, a method to form micro- and nanochannels consists of sealing this struc-

Figure 3.16  Patterning by lift-off. Before the deposition of the material to pattern, (a) a resist is deposited and structured. Next (b) the metal is deposited, and (c) the resist is dissolved.

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ture with another substrate. Many wafer-bonding techniques have been developed. Main applications include the fabrication of silicon-on-insulator wafers and the packaging of MEMS components (e.g., pressure sensors). Table 3.9 lists and compares the most common wafer bonding techniques. Interested readers can refer to the review article from Niklaus et al. [82]. Wafer direct bonding, also called fusion bonding, involves the contact between two mirror-polished and thoroughly cleaned wafers adhering at room temperature without using outside forces or any kind of adhesives [83, 84]. After starting the bonding process by applying a slight pressure at some point, the bonded area spreads laterally over the entire wafer area, typically in a few seconds. The bonding achieved at room temperature is usually relatively weak, and, thus, for many applications, bonded wafers have to undergo an additional heat treatment to strengthen the bonds across the interface. Typical annealing temperatures range between 600°C and 1,200°C. Direct bonding usually leads to strong bonds and is widely used in the fabrication of silicon-on-insulator (SOI) wafers, as described in Figure 3.17. The mechanism of direct bonding is believed to rely on a chemical reaction between hydroxide groups (—OH) present in the interface of the wafers according to the following reaction: Si—OH + OH—Si ® Si—O—Si + H2O



in which water is formed and released under heating, causing the hydrogen bonds between the silanol group (Si—OH) at the surface of an oxide layer to turn into covalent siloxane bonds (Si—O—Si). This oxide layer can either be native or chemically grown by hydration in a piranha solution (mixture of H2O2 and H2SO4), thermally grown in a furnace, or may result from an oxygen plasma treatment. Direct bonding can be applied to silicon, glass or quartz substrates.

Table 3.9  Comparison of Common Wafer Bonding Techniques Wafer Bonding Technique

Typical Bonding Conditions

Direct bonding Si—Si glass – glass quartz—quartz

600–1,200°C room-temperature schemes have been reported small or no bond pressure

Anodic bonding Si – glass glass—glass and Si—Si with intermediate layers

150–500°C 200–1,500V Small or no bond pressure

Adhesive bonding between almost any substrate

Room temperature up to 400°C Low to moderate bond pressure

Source: [82].

Advantages and Disadvantages

Application Areas

+ High bond strength + Hermetic + Resistant to high temperatures – High surface flatness required – High bond temperatures not always compatible with electronic wafers + High bond strength + Hermetic + Resistant to high temperature – Bond temperatures in combination with voltage not always compatible with electronic wafers + High bond strength + Low bond temperature + Works practically with any substrate material including electronic wafers – No hermetic bonds – Limited temperature stability

SOI wafer fabrication, glass microfluidic devices

sensor packaging

MEMS, sensor packaging, 3D-ICs, temporary bonds

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Figure 3.17  Fabrication of SOI wafers using the smart-cut process. First (a) light ions (e.g., hydrogen or helium) are implanted below the wafer surface. After (b) direct wafer bonding, (c) the annealing of the implanted wafer leads to splitting parallel to the wafer surface due to agglomeration of implanted ions and thereby crack formation. Finally (d) the split wafer surface is planarized by polishing or etching.

Anodic bonding between glass and silicon is one of the most common methods for packaging and protection of MEMS (see Figure 3.18). Typically, a glass substrate is brought into intimate contact with a silicon wafer. The stack is heated at 400°C and is supplied with a voltage of about 1,000V. Under this electric field, Na+ ions (part of the composition of the glass) migrate at the silicon-glass interface while O– ions migrate towards the cathode. This induces a strong electrostatic pressure between the two substrates. The presence of oxygen ions at the interface results in an oxidation reaction, and eventually a bond between the two wafers. In order to avoid thermal stress issues, glass and silicon wafers should have a matching thermal coefficient of expansion. An example of suitable glass for this application is the

Figure 3.18  Anodic bonding of silicon and glass substrates. Heating (~400°C) and applying a voltage (~ 1,000V) lead to the migration of oxygen ions at the silicon-glass interface. The induced electrostatic pressure and oxidation process leads to a strong bonding between the wafers.

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pyrex Corning 7740. Many variants of these methods have been reported using intermediate layers (silicon, silicon dioxide, silicon nitride, metals) to bond siliconglass, silicon-silicon and glass-glass substrates [85–89]. In adhesive wafer bonding, a third material is introduced to glue the two substrates. Examples of intermediate gluing materials include UV epoxies, PMMA, PEEK, fluoropolymers, PDMS, benzocyclobutene (BCB), parylene, polyimide, positive or negative photoresists, glass frits with low melting temperatures, and sodium silicate solutions [82, 90]. This intermediate layer is often spin-coated, but can also be laminated, sprayed, screen-printed, sputtered, or stamped. Adhesive wafer bonding is based on the fact that atoms and molecules adhere to each other when they are brought in sufficiently close contact. Adhesives deform to create a conformal and intimate contact between each interface (i.e., substrate A-adhesive and adhesivesubstrate B). Thus a good wetting of the surface of both substrates is of paramount importance. Usually, after this close contact has been established, the adhesive is hardened to make the bond mechanically stable. For instance, this may be done by cross-linking a thermoset or a UV polymer, heating a solution of polymers in order to evaporate the solvents, or cooling down a thermoplastic. Advantages of adhesive wafer bonding include the low temperature, required by many sensitive devices, its lower sensitivity to contamination compared to direct or anodic bonding (the adhesive conforms to the substrate and the eventual dusts or particles), and its ability to bond all kinds of materials. 3.1.2  Polymer Micro- and Nanofabrication

Though silicon fabrication technologies have been intensively developed, they are not always adapted to the requirements of micro- and nanofluidics. Lab-on-chips usually have a relatively large footprint, and thus a rather large price for a component that is often disposable because of chemical or biological contamination issues. The introduction of polymeric material in micro- and nanotechnologies has proven to be one of the recent key developments of the field. Mass production molding techniques for thermoplastics such as injection molding, hot embossing, and thermoforming have been successfully adapted to the fabrication of microcomponents [91]. The availability of a wide range of materials with tailored properties is crucial for applications in research fields such as microfluidics, where the control of the material properties (e.g., wettability, autofluorescence, transparency, dielectric strength, and biocompatibility) is of paramount importance. The possibilities of bulk and surface modification promise to further extend the control over the structural material properties [92–94]. 3.1.2.1  Polymers

Polymers are very large natural or synthetic molecules, also called macromolecules, composed of up to millions of repeating units, or monomers, linked by covalent bonds. An easy way to represent polymers is to compare them to a plate of spaghetti, with each strand of spaghetti being able to slide along the others (Figure 3.19). One of the most important parameters for polymer molding is the glass transition temperature Tg. Below the Tg, the polymer is said to be in the glass region.

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Figure 3.19  Schematic representation of spaghetti-like polymers looped into each other. Crosslinking results in the formation of bonds between the polymer chains.

The macromolecules are frozen in. At the macroscopic level, the material is hard and brittle like glass. As the temperature increases above the glass transition, the energy of motion of parts of the polymer chain becomes large enough to overcome intramolecular friction. The long chains are then free to slide along each other, and the material becomes soft and flexible (viscous state). Examples of thermoplastics include polymethyl methacrylate (PMMA), polycarbonate (PC), or polyetheretherketone (PEEK). Thermoplastics can also be softened by dissolution. After shaping, the material is hardened by evaporating the solvent or by other methods of phase separation [95, 96]. Thermoplastics are usually high-molecular-weight polymers that can be reversibly switched from the glassy state to the viscous state. Chains are linked only through weak interactions. They are usually molded at a temperature above the glass transition, in the viscous state. Once the shape has been given, the polymer is cooled down, so the plastic part retains its shape. On the other hand, thermosetting plastics (or thermosets) are polymers that irreversibly cure from a soft to a hard form under the influence of heat, chemicals, or exposure to radiation such as UV light. Typical examples are the negative photoresist SU-8 or PDMS. In fact, thermosets are usually polymers in solution or with a Tg (before curing) below the glass transition temperature. Thus they are malleable and can consequently be processed before curing. The addition of curing agents results, under adequate conditions (i.e., heat or illumination), in the formation of covalent bonds between the polymer molecules, limiting the movements of the chains. This reaction is also called crosslinking (Figure 3.19). Elastomers are usually thermosets with a glass transition after curing still below the glass transition temperature, giving their characteristic elasticity. Table 3.10 lists the properties of selected polymers. 3.1.2.2  Replication Techniques

Invaluable experience has been accumulated in silicon fabrication from both the academic research and the microelectronics industry over the last 60 years. Therefore the idea is obviously not to give up silicon, but to fabricate a master using well-established silicon technologies, then copy this structure in a polymer as many times as desired using molding techniques.

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Table 3.10  Properties of Selected Polymers Thermal Expansion Coeff Resistant [ppm/K] Against

Name

Density [g/cm3] Tg [°C]

Young’s Modulus [MPa]

PMMA

1.19

110

3,200

70–80

PC

1.19– 1.24

148

2,200– 2,400

70

PP

0.9

0–10

1,450

100–200

PS

0.9– 1.24

100

2,300– 4,100

30–210

PE (LD/ HD) COC COP PEEK

0.91/ 0.967 1.02 1.01 1.3

110/140 200/1,000 170/200 78 138 143

2,600 2,400 3,700

70 70 17

Parylene 1.1–1.4 80–100

2,000– 5,000

35–69

PDMS

1.03

–120

310

SU-8

1.19

210

PI

1.42

360– 410

0.360– 0.870 2,000– 4,000 2,500

52 20

Not Resistant Against

Organic Solvent Stability

Attacked by most solvents (e.g., acetone, benzene, dichloromethane) Attacked by most solvents (e.g., acetone, methylene chloride) Petrol, benzole, Xylol, tetraline, Acids, bases, alcohol, organic hydrocarbons decaline solvents, fats Attacked by most Bases, alcohols Conc. acids, solvents (e.g., ether, hydrocarbons acetone, benzene, dichloromethane) Acids, bases, Hydrocarbons Trichlorobenzole, alcohols, oil xylole, hexane Acids, bases

Alcohols, Acids, bases (medium conc.), acetone, benzole, UV oil, petrol radiation Alcohols, acids Hydrocarbons, ketones, KOH

Most organic and inorganic

Conc. nitric acid, sulfuric acid, UV light

Acids, most organic and inorganic Weak acids and Strong acids, bases hydrocarbons Acids, bases, most solvents Acids, bases, solvents

No known solvents

Note: PMMA: polymethyl methacrylate; PC: polycarbonate; PP: polypropylene; PS: polystyrene; PE (LD/HD): polyethylene (low density/high density); COP/COC: cycloolefin polymer/copolymer; PEEK: polyetheretherketone; PDMS: polydimethylsiloxane; PI: polyimide. Source: [4].

Molding Lithography

A number of sophisticated direct writing techniques have been developed to pattern features with resolutions down to the atomic scale (see Sections 3.1.1.7 and 3.1.1.9). However, as the critical dimensions shrink, the time required to write over large areas has become prohibitive. Photolithography has made possible the fast replication of a stencil (the photomask) fabricated using slow direct writing methods. However, as the cost and the complexity of photolithography equipment get higher and higher, alternative replication methods have been actively investigated [21, 97, 98]. Imprinting is one of the most promising patterning processes among these emerging techniques. Imprinting relies on a physical contact between a template and a malleable resist. Figure 3.20 illustrates three different approaches. The first paper about nanoimprint lithography (NIL) was published in 1995 demonstrating already critical dimensions as small as 25 nm. The technique consists in pressing a template against a resist (e.g., PMMA) heated above its glass transition temperature. After the imprinting, the resist is cooled down and the template removed (Figure 3.20). Usually, a thin layer of resist remains at the bottom of the

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Fabrication Technologies of Nanochannels

Figure 3.20  Molding lithography strategies. 1) Nanoimprint lithography (NIL): a thermoplastic solution is spin-coated (a), then heated above its glass temperature Tg. Next a template is pressed against the resist (b) and finally the template is released after cooling down below Tg. 2) Step-andflash imprint lithography (SFIL): a UV polymer is deposited (a) and exposed to UV through a template pressed against it (b). The template is released after completion of the curing (c). 3) Reversal imprint: first a template is coated with a resist (a), next the resist is transferred on a substrate (b). The template is released after hardening of the resist (c). In every situation, after patterning of the resist, the residual layer is etched away in oxygen plasma (d) before the pattern is transferred into the substrate by etching (e). Finally the resist is removed.

features. This residual layer is etched away in oxygen plasma. Molds are made in silicon, in metals or in polymers with a high enough Young’s modulus. Usually metallic or polymeric templates are replicated from a silicon master. Metallic templates are harder and more durable than silicon templates. However, a high pressure is necessary to obtain a conformal contact over a large area. Using flexible templates results in a better homogeneity for a reduced applied pressure. Furthermore soft polymeric templates are easier to fabricate and cheaper and make the release process easier. However, the achievable resolution usually decreases with the hardness of the template. Rigiflex molds, with hard features patterned on a soft substrate, combine the advantage of both approaches [99]. Another important consideration in NIL is the coefficient of thermal expansion. A mismatch between the mold and the substrate would result in pattern distortion or stress built up during the thermal cycle, affecting the resolution and the registration accuracy.

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Step-and-flash imprint lithography (SFIL) is an alternative to NIL at low working pressure and ambient temperature, thus overcoming homogeneity and thermal issues [100]. In SFIL, the thermoplastic solution is replaced by a low-viscosity UV polymer. The template is pressed against a few droplets of the photopolymer homogeneously deposited on the substrate and is released after curing by exposure to UV (Figure 3.20). Patterning features at different scales (e.g., micro- and nanoscales) may result in homogeneity issues that can be overcome by combining photolithography and SFIL [101, 102]. Another technique to solve NIL homogeneity and thermal issues is to spin-coat the resist not on the substrate but onto the template (Figure 3.20) [103]. The resist is next transferred to the substrate and the template is released. Three-dimensional layer-by-layer assembly is possible using the reverse imprint technique. Many variants have been developed using different methods of deposition or transfer: polymer bonding [104], microtransfer molding (mTM) [105], and nanotransfer printing [106]. In solvent-assisted micromolding (SAMIM), the resist is locally swollen or dissolved by an elastomeric stamp previously dipped in an adequate solvent. The resist is imprinted, then hardened as the solvent evaporates. Again, the method avoids thermal cycling and the use of a soft template results in a conformal contact with the substrate, thus a good homogeneity at low pressure [107]. Micromolding in capillaries (MIMIC) is another example of a soft lithography approach to pattern a resist [108]. In this technique, a PDMS template is first placed into contact with the substrate, forming a reversibly bonded microfluidic device. Next, the cavities between the substrate and the mold are filled by capillary effect with a low-viscosity UV polymer, before curing and releasing the template. Only isolated microstructures can be fabricated using this method. Structural Molding

Most of the techniques introduced above are lithography techniques. Their initial aim is to pattern a substrate using a sacrificial resist layer. On the other hand, micromolding techniques such as casting, embossing, injection molding, or thermoforming have been developed to shape plastic microparts [4, 91]. The most common and straightforward is the casting of polydimethylsiloxane (PDMS). After mixing with a curing agent and outgassing, PDMS is simply poured on the mold. After curing (typically 4 hours at 65°C or 1 hour at 100°C), the replica is peeled off the master. The smallest feature sizes mainly depend on the resolution of the mold and can be below 100 nm [109]. This technique is often used in research laboratories for fast prototyping. Since its introduction in the 1990s, the method has been widely adopted by the microfluidic community as a low-cost alternative to silicon-glass technologies [110, 111]. Hot embossing is the most common micromolding method to process thermoplastics such as PMMA or PC. Briefly, the mold is pressed against a heated polymer sheet then cooled down and released, producing an inverted replica of the mold. Opposite to NIL, embossing is not dedicated to lithography. The structural material itself is molded, not a sacrificial resist. Typical cycle times range between 5 and 15 minutes. In injection molding, the molten polymer is injected in a cavity containing the mold. This method is by far the most widespread for plastic molding in the

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macroworld. Shorter cycle times are achievable (30 seconds to 5 minutes, depending on the surface-to-volume ratio of the part to mold) [4]. In the LIGA process (from the German lithographie, galvanoformung und abformung, or lithography, electroplating, and molding), a metallic mold is replicated in plastic by injection molding. The metallic mold is replicated by electroplating from a thick PMMA layer patterned by X-ray lithography. The chemical structure of PMMA is altered by irradiation with X-rays, and it can then be removed with an appropriate solvent. X-rays are very penetrating radiation with a small wavelength, allowing the patterning of very high-aspect ratio structures. However, X-ray lithography involves expensive mask fabrication (e.g., diamond membranes with gold absorbers) and a synchrotron source, making it prohibitive for most of the potential users. At the end of the 1990s, SU-8, a photoresist with the ability to be patterned with very high aspect ratios, was introduced, making possible the fabrication of metallic molds at a significantly lower cost (“UV-LIGA” or “poor man’s LIGA”) [112, 113]. Compact discs are manufactured using a combination of hot-embossing and liquid injection, allowing for further reduction of cycle times down to about 5–10 seconds. In this process, called injection compression molding, the molten polymer is injected on the template inside the semi-closed cavity, and then pressed into the microstructures by closing the tool. The injection is faster since the cavity is closed after introduction of the polymer, while in injection molding the polymer has to enter a small, closed cavity. In thermoforming, a thin film of plastic is heated slightly above its Tg and is blown by a pressurized gas against the mold. The method is widely used at the macroscale for the production of thin-walled plastic products such as bottles or yogurt cups. Although the resolution is very poor compared to hot embossing or injection molding, the simplicity and the speed of the process make it promising for some applications. Table 3.11 compares characteristics of these different micromolding techniques. 3.1.2.3  Direct Writing Laser Ablation

Other methods are available to structure polymers directly, without any mask or template. These methods are usually dedicated to prototyping. CO2 laser machining is an example of a fast prototyping method able to draw microstructures down to with lateral features down to 100 mm (see also Section 3.1.1.7) [11, 115]. The pat-

Table 3.11  Comparison of Replication Methods Process Casting Hot embossing Injection molding Injection compression molding (CD manufacturing) Source: [4, 114].

Tool/Infrastructure Costs

Cycle Times

Design Flexibility

Production Automation

Low Medium High High

Long (min-hours) Medium (min) Short (s-min) Very short (s)

High (3D) Medium (2D) High (3D) Limited

Not existing Medium High Very high

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tern is first drawn using a two-dimensional computer-aided design (CAD) program. Next, the file is sent to the laser writing system, similar to a printer. The position, the speed, and the number of passes of the writing head, as well as the laser intensity, are controlled by the system, according to the design. Simple microstructures can be written within a few seconds. Photolithography

Photoresists have been traditionally employed as a temporary mask to transfer a pattern during a lithography process, but have also been recently introduced as a structural material in micro- and nanofabrication. A typical example is SU-8, a negative thick-film photoresist patented by IBM in 1989. This resist is an EPON SU-8 resin (from Shell Chemicals) photosensitized with triaryl sulfonium salts. SU-8 is particularly well suited for thick-film applications since it can be dissolved at high concentrations [116]. As a direct consequence, layers of a few hundred of microns can be easily spin-coated and patterned with conventional UV exposure systems. Its very low absorbance in the near-UV range enables the fabrication of structures with very high aspect ratios. The good chemical compatibility and biocompatibility makes SU-8 a material of choice for microfluidic devices [117, 118]. Electrical [119], magnetic [120], optical [121, 122], or mechanical properties [123, 124] of the free-standing part can be tuned by mixing the resist with functional materials. Using photoresists as a structural material also represents new opportunities in terms of system integration [113, 125].

3.2  Application to Micro- and Nanofluidic Systems The fast growth of nanofluidics in recent years has been driven by the progress in micro- and nanotechnologies. This has made possible the fabrication of a deterministic network of nanochannels and their integration in a chip with other components. Practical nanofluidic devices integrate not only nanochannels but also microchannels, and eventually actuation (e.g., electrodes and valves, pumps) or sensing elements (e.g., electrodes, optical fibers, and photodiodes). This section first introduces some classic examples of fabrication of microfluidic devices (i.e., channels with at least one dimension in the range 1–100 mm, and all dimensions larger than 1 mm), before describing the fabrication of nanochannels (i.e., channels with at least one dimension in the range 1–100 nm) and their integration in a chip. 3.2.1  Microfluidic Devices 3.2.1.1  Silicon Bulk Micromachining Silicon-Glass

The strategies to fabricate silicon or glass micro- and nanochannels fall into two categories: bulk micromachining and surface micromachining. Bulk micromachining refers to methods where the structures are patterned into the substrate itself. The structures must then be sealed using wafer bonding methods. A remarkable example of bulk micromachining is the silicon gas chromatograph developed by Terry and colleagues at Stanford in the 1970s, more than 10 years before the concept of

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microTAS [2, 126]. This microsystem integrated not only microchannels, but also a valve dedicated to gas injection and a hot wire detector (thermal conductivity detector). Figure 3.21 depicts this process. Microchannels with a width of 200 mm and a depth of 30 mm were defined by standard photolithography and isotropic wet etching in HNA. As the photoresist was not able to withstand long exposure to the wet etchant, a 1-mm-thick silicon dioxide layer was used for masking during the etching of the microchannels. To limit their footprints, access holes were defined from the basckside and etched by anisotropic wet etching in a KOH solution. Finally, after stripping of the oxide, anodic bonding was used to seal the microchannels with a glass wafer. Glass allows a visual inspection during fluidic experiments. This process combines the advantages of the silicon (well-mastered fabrication techniques) and the transparency of glass. The technique is highly reliable and allows for the fabrication of well-defined, airtight microchannels able to withstand high pressure. It has been widely used in the microfluidic community, albeit with a few improvements. Etching the channels in silicon by isotropic wet etching results in rounded profile with low aspect ratios and poor resolution. Anisotropic wet etching is possible but is limited in term of design by the crystalline orientation of the silicon substrate [127, 128]. DRIE avoids faceting effects and allows for the fabrication of both microchannels and through holes [129]. Using DRIE, highaspect ratio structures are achievable with high resolution. Masking layers are usually photoresists with a thickness depending on the depth of the structure to etch.

Figure 3.21  Silicon/glass fabrication process of the Stanford gas chromatograph. After (a) the growth of a 1-mm-thick silicon dioxide film on a (100) silicon substrate, (b) microchannels were defined in a SiO2 masking layer using a photolithography process and a wet etching. Next, (c) microchannels were etched in silicon using an isotropic etchant solution. After (d) a second oxidation, (e) access holes were defined in the oxide on the backside of the wafer, then (f ) etched in silicon with an anisotropic etchant. Finally, (g) the oxide was completely stripped away and a glass wafer was anodically bonded to seal the microstructure.

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Electrokinetic-driven flows and separations are common in micro- and nanofluidics. Electric fields in a range 100–500 V/cm are typical for this purpose. For analytical applications, channels often have lengths of a few centimeters; thus voltages often exceed 1,000V. Since silicon is a semiconductor, an insulating layer is necessary between the substrate and the channel. A breakdown voltage of 720V was reported in an early article using a 1-mm-thick layer of good-quality, pinholefree silicon dioxide [128]. Using thicker layers would result in prohibitive growth times or a lower film quality, and bonding issues. Therefore silicon-glass technologies have generally been limited to devices with a pressure-driven actuation. Glass-Glass

All glass fabrication quickly emerged in the 1990s to overcome the electrical limitations of silicon devices [130]. Glass etching is not as advanced as silicon etching. Glass is usually isotropically etched in fluorhydric acid HF [131]. Depending on the stirring method, the section of the channel is circular or rounded with a flat bottom. Deep structures with high aspect ratios cannot be achieved using this method. Photoresists quickly deteriorate in HF, and thus cannot be used as a masking layer but for very shallow structures. Chromium/gold masking layers are most common, chromium being used as an adhesion layer for gold [132, 133]. Polysilicon and amorphous silicon may also be used [134, 135]. RIE methods of glass exist but are not as common as for silicon and may lead to contamination issues when reactors are employed for multiple purposes. A liquid chromatography device was etched in glass by RIE [136]. The stationary phase was an array of 5 ´ 5 ´ 10 mm posts separated by 1.5-mm-wide and 10-mm-deep channels. DRIE of glass and quartz has also been demonstrated [137–139]. Microchannels can also be etched in glass using powder blasting [140, 141] or ultrasonic machining [142]. A photopatternable glass, FOTURAN, has also been investigated [142]. Glass-glass bonding is usually a direct bonding process, but anodic bonding using intermediate layers, or adhesive bonding is also possible (see Section 3.1.1.12). Access holes are machined though the glass substrate by mechanical or ultrasonic drilling, or powder blasting. 3.2.1.2  Silicon and Polymer Surface Micromachining

Surface micromachining refers to a set of technique producing structures from thin films deposited above the substrate. In contrast with bulk micromachining, surface micromachining leaves the underlying substrate intact. Surface micromachining uses the sacrificial layer method. Figure 3.22 depicts a simple example. Using this method, the inner part of the channel is first defined in a sacrificial material, and then coated with a structural material. Finally, the sacrificial material is etched away. This technique has been often used to integrate micromechanical elements (e.g., cantilevers, bridges, micromirrors, and micromotors). It offers many advantages compared to bulk micromachining: Wafer bonding is avoided, access holes are easier to fabricate, and system integration is straightforward (above IC). The resulting channels have a thin capping layer, which is an advantage for optical measurements. On the other hand, thin capping layers can make these devices delicate to handle. As stated above, multiple centimeter–long channels are often involved in lab-on-chips. Removing a sacrificial material with micro- or nanodimensions

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Figure 3.22  Sacrificial layer method: (a) a sacrificial material is deposited and patterned; (b) the structural material is deposited; (c) access holes are defined; and (d) the sacrificial layer is etched away.

over such a distance may require a very long etching time in aggressive chemicals. Therefore an excellent selectivity is of paramount importance. Polysilicon has often been used as a structural material, in combination with silicon dioxide as a sacrificial material. Mechanical and electrical properties of polysilicon can be tuned by doping. The control of the stress in the deposited layer is an important issue, limiting the number of polysilicon layer stacked on the substrate (up to five for Sandia National Laboratory’s SUMMiT V process), as well as the thickness of the structure. While polysilicon and other thin film materials have been used for the fabrication of nanochannels, thicker layers of sacrificial material (1–100 mm) are necessary for the realization of microfluidic devices. Electroplated Metal

Thick layers of metal can be deposited by electroplating. A layer of copper with a thickness up to 40 mm was used as a sacrificial layer to realize ink-jet devices. Diamond was used as a structural material. One day of etching in a 30% nitric acid solution was necessary to release the structure [143, 144]. Leichle and colleagues simply used the copper layer of a printed circuit board as a sacrificial layer [145]. An epoxy photoresist (i.e., SU-8) was used as a structural material. Silicon-on-Insulator

SOI wafers are used in a sacrificial process commercialized by Tronics microsystems. Briefly, silicon is etched down to the buried oxide, which is used as a sacrificial layer. The method allows the fabrication of monocrystalline silicon mechanical structures. Björkman et al. used SOI wafers with a 46-mm-thick device layer as a sacrificial material to fabricate an all-diamond microfluidic liquid chromatograph [146]. Channels were defined in the silicon layer, and then encapsulated with diamond deposited by hot filament chemical vapor deposition (HFCVD). Finally, silicon was etched away in a HF:HNO3 mixture at 80°C. Two hours were required to

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etch 2 cm, 350 hours to etch 3 cm. The slowdown of sacrificial etching over time is a common issue in the fabrication of microfluidic structures. It is believed that byproducts progressively agglomerate at the etching front, diffusion being the only way to escape from the microchannels. The progressive saturation of the solution in byproducts locally reduces the etching kinetics. Photoresists

Using photoresists as a sacrificial material further simplifies the patterning process. Furthermore, high etching rates of photoresists have been reported in organic solvents [72, 73]. The low temperature processing allows a convenient integration above IC. Webster et al. combined AZ9620 from Clariant as a sacrificial layer and parylene-C as a structural layer to fabricate an electrophoresis device integrated on top of silicon photodiodes [147]. Parylene is a polymer that can be deposited by PECVD at ambient temperature (i.e., no thermal stress) in a highly conformal way. The film is hydrophobic and exhibits excellent barrier properties, a high chemical resistance, and high dielectric strength. Typically, a parylene coating is dedicated to protect materials in rough environment (for example, electronics in spatial environment). Twenty hours of etching in acetone were necessary to release a 1.3-cm-long channel with a width of 200 mm and a depth of 20 mm. Uncrosslinked SU-8

Deep structures with high aspect ratios can be easily patterned by standard lithography using SU-8, resulting in a higher geometrical flexibility compared to more conventional photoresists. Many processes have been developed to use uncrosslinked SU-8 as a sacrificial layer, with cross-linked SU-8 as a structural material. The two strategies that have been followed are illustrated in Figure 3.23 [113]. A first way consists of embedding a metal layer to block the cross-linking of the sacrificial SU-8 defining the microchannels during the exposure of the cover layer. Aluminum and magnesium have been used as UV blocking layer [148–152]. The following principles were used to minimize the exposure to UV during the process: (1) deposition by evaporation of metals presenting a sufficient vapor pressure rate at a temperature low enough to prevent significant UV radiation is preferred, (2) the amount of heating of the substrate has to be kept small, and (3) metal must be removed by wet etching to avoid an irradiation during a RIE. An alternative solution consists in printing the metal layer using a soft PDMS stamp [153]. In the partial exposure process, a single layer of SU-8 is exposed layer-by-layer (Figure 3.23). This process was demonstrated using UV illumination [154, 155]. An antireflection coating may be used to avoid back-illumination by reflection on the substrate [154]. A positive resist on the top of the SU-8 has also been employed to partially absorb UV illumination [122]. In a recent publication, Ceyssens et al. used two different wavelengths, taking advantage of the higher absorption of UV light in SU-8 for a wavelength below 350 nm [152]. Partial exposure was successfully implemented using a wavelength of 313 nm, while full exposure was performed at 365 nm. Partial exposure was also demonstrated using laser [156, 157], electron beam [158], and ion beam [159] sources. SU-8 as a sacrificial layer has proven to be suitable for the fabrication of MEMS structures, but the long dissolution time may be prohibitive for the fabrication of microchannels.

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Figure 3.23  Two strategies to use uncross-linked SU-8 as a sacrificial layer with SU-8 as a structural material. 1) UV-blocking process: (a) a first layer of SU-8 is spin-coated on the substrate and (b) a metal layer is deposited into which microchannels are defined. (c) The device is coated with second layer of SU-8 layer and (d) exposed to UV through the mask defining access holes. Finally, (e) the uncross-linked SU-8 is dissolved. 2) Partial exposure: (a) a SU-8 layer is spin-coated into which (b) the cover layer is defined by a partial irradiation. (c) The walls of the microchannels are then defined by a complete irradiation down to the substrate before (d) the dissolution of the uncross-linked SU-8 releasing the microstructure.

Thermodegradable Polymers

An original solution to overcome etching-related issues is to use a phase change to remove the sacrificial material. For instance thermodegradable polymers have been used for this purpose [160–162]. After completion of the device, the sacrificial layer is decomposed by heating. Complete decomposition of 1–12-mm-thick polynorbonene (PNB) films was observed after a 2-hour-long thermal treatment at 425°C [160]. SiO2, SiNx, and various polyimides were used as structural materials. Later, the process was simplified by making the PNB photosensitive [163]. Improvements in the chemistry of the sacrificial polymer resulted in a decomposition temperature as low as 180°C applied during only 1 hour [164].

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3.2.1.3  Photoresist Lamination

Lamination is another approach to integrate a microfluidic network in an above IC scheme. Following this method, microchannels were fabricated on the top of a substrate using low temperature processes [113, 118, 125, 165–167]. Figure 3.24 depicts the general procedure. Compared to the numerous fabrication techniques that have been proposed using SU-8 wafer bonding techniques [168–183], the lamination method offers many advantages. The flexibility of the film allows a conformal contact, resulting in a fast and reliable bonding at low pressure and temperature with simple equipment (i.e., a roll laminator). Beyond this, the use of a conventional photolithography aligner allows patterning with a high resolution and an excellent level-to-level registration. Other photoresist deposition methods (e.g., spin-coating or screen-printing) may be employed for planarization purposes before the layer-bylayer construction of the microfluidic network above the substrate.

Figure 3.24  Above IC microfluidic integration using the lamination of photosensitive films: (a) a substrate (silicon, glass, printed circuit board, PCB) embedding components (e.g., integrated circuits and sensors) is selectively planarized [e.g., by (b) spin-coating or (c) screen-printing]; (d) a dry film photoresist is laminated on the substrate and (e) patterned by photolithography. A microfluidic structure is formed by repeating (c) and (d).

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3.2.1.4  Polymer Replication PDMS Casting

PDMS has many disadvantages, including its tendency to swell in most of the organic solvents or its low Young modulus resulting in a bulging of the structures for applied pressure differences larger than 1 bar. Still, soft lithography techniques have been a key enabler of microfluidics. The most common fabrication process of microchannels by soft-lithography is described in Figure 3.25 [111, 184]. The main reason why PDMS casting is so popular, compared to other molding techniques, is probably the flexibility of the material. This property: (1) eases the unmolding step; (2) allows a highly conformal contact with another substrate, thus an easy bonding; (3) facilitates fluidic interfacing (simply using needles with the right diameter, with the elastomer acting as a gasket); and (4) allows for the integration of active elements (e.g., valves) [185]. Thermoplastic Molding

While PDMS has been widely used in microfluidics, its low Young’s modulus unfortunately restricts its use in nanofluidic application. Sealing such small structures without collapsing is very challenging. Even with an adequate bonding process, soft structures may collapse due to capillary forces or van der Waals interactions. Similar issues have been encountered previously for microcontact printing (mCP) [186–188] and for stiction, which has been identified as one of the major sources of failure in MEMS and hard drives [189–191]. The few examples of PDMS “nano”-channels reported have a smallest dimension larger than 150 nm, or have serious geometrical limitations [192–195]. Harder thermoplastics such as PMMA, PC, or PEEK seem to

Figure 3.25  PDMS casting of a microfluidic device: (a) the process starts with a negative template, and PDMS is (b) mixed with a curing agent, (c) outgassed, and (d) poured onto the substrate after a curing period in an oven (1 hour at 90°C, 4 hours at 65°C).

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be more adapted to nanofluidics [196]. Hot-embossing of thermoplastics has been widely used in the fabrication of microfluidic devices [4, 197, 198]. An example of this process will be given in Section 3.2.2. 3.2.2  Nanofluidic Devices

Nanochannels are channels with at least one dimension in the nanoscale (i.e., 1–100 nm in the MEMS/NEMS community) [199, 200]. As it will be seen, nanofluidic devices can be fabricated in silicon using techniques very similar to the bulk micromachining employed in microfluidics. However, depending on geometrical characteristics, alternative techniques can be used to simplify the fabrication process. Figure 3.26 introduces two important geometrical parameters, namely the aspect ratio (AR) between the height and the width of the cross-section and the dimensionality of the channel network. Most of the nanofluidic devices integrate planar nanochannels (width >> height). Keeping the width at the microscale, (1) allows the use of standard photolithography to define the network, and (2) offers higher flow rates than square nanochannels (width » height) for an equivalent

Figure 3.26  Geometrical definitions: one-dimensional and two-dimensional networks of planar, square, and HAR nanochannels.Other parameters have to be taken into account beyond the aspect ratio and the dimensionality. Wafer bonding techniques may be a better option for the fabrication of long nanochannels, where standard sacrificial methods result in excessive etching times. As for the structural materials, silicon is well-established and benefits from the long experience gained in microelectronics over the last 60 years. Silicon dioxide channels are naturally hydrophilic and therefore can be easily filled by capillarity. In addition, surface chemistry of silica is well-known and can be conveniently controlled using alkylsilane SAMs [94]. Polymers represent an attractive alternative. They come in various forms and shapes, with a very wide range of properties (e.g., optical, mechanical, chemical resistance, and biocompatibility). The lower cost associated with polymer molding techniques along with the low price of common plastics are also advantages for the eventual commercialization of nanofluidic devices.

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pressure. On the other hand, the fabrication of square nanochannels involves nanopatterning methods where the dimensionality of the network plays an important role. While SBL, combined or not with replication techniques, is usually involved in the fabrication of a two-dimensional network, alternatives like edge lithography or diblock copolymer lithography have to be considered for the realization of straight, parallel nanochannels. A dense array of high-aspect ratio (HAR) (height >> width) nanochannels is even more challenging to fabricate but would result in an even higher throughput. 3.2.2.1  The Simplest Case: Planar Nanochannels (AR << 1) Bulk Micromachining

Because the lateral dimension can be defined by conventional photolithography, patterning one-dimensional or two-dimensional networks of planar nanochannels in silicon is relatively straightforward. The challenge lies in: (1) the uniformity of the depth over a wafer, and (2) avoiding the collapse of very wide and shallow structures. The most common and earliest method to fabricate nanochannels is bulk micromachining (Figure 3.27). Nanochannels are defined by photolithography and

Figure 3.27  Fabrication of planar nanochannels using wafer bonding. (1) Bulk micromachining starting from (a) substrate, (b) nanochannels, (c) microchannels, and (d) access holes are etched in the substrate and then sealed with (e) a second substrate. (2) Spacer machining: (a) a thin layer is deposited, defining (b) the depth of the nanochannels. Using a selective etching method, nanochannels are patterned in the spacer. Next (c) microchannels and (d) access holes are etched in the substrate and then sealed with (e) a second substrate.

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etched into the substrate. Eventually, microchannels and access holes are also machined in the substrate before sealing the structures with a second substrate. As early as 1992, Volkmuth and colleagues reported the fabrication of a nanofluidic device dedicated to study the electrophoretic movement of a polymer chain in an ideal porous matrix. An array of 150-nm-high, 1-mm-diameter pillars separated by a 1-mm gap was etched in a 0.5-mm-thick silicon dioxide layer grown on silicon. The structures were capped with a pyrex wafer by anodic bonding. A similar technique was employed by Han and Craighead in 1999 to fabricate 20-mm-wide channels with alternating deep (around 0.65–1.6-mm) and shallow (90-nm) regions (“entropic traps”) with 4-, 10-, 20-, and 40-mm periods [201, 202]. Channels were etched by RIE. Access holes were opened using anisotropic wet etching. An insulating silica layer was grown before sealing the device with a pyrex substrate by anodic bonding. Using this approach, channels with ARs as low as 0.004 and 0.0005 were reported using silicon-glass and glass-glass technologies respectively [203]. Stein et al. used an adhesive bonding method to seal 70-nm-deep, 50-mm-wide nanochannels [90]. The cover plate was coated with a 20-nm film of sodium silicate using a 2% solution and then pressed against the structures for 2 hours at 90°C to seal them. A plasma-oxygen bonding method similar to the PDMS bonding technique was recently reported [204]. A thin film of polysilsesquioxane was used as an intermediate later between a silicon and a glass wafer. Channels with depth below 10 nm and ARs below 4.10–5 were reported. Using dry etching techniques, well-controlled and well-maintained plasma reactors are required to obtain channels with good depth uniformity and a low roughness. The double oxidation scheme illustrated in Figure 3.28 takes advantage of a

Figure 3.28  Double oxidation scheme for etching nanochannels: (a) a thick layer of silicon dioxide is grown on silicon (wet oxidation) into which (b) the width of the nanochannels is first defined. In the next step (c), the depth is defined by a dry oxidation (~ half of the oxide thickness). (d) Finally the silicon dioxide is etched in buffered HF.

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very uniform growth process, thermal oxidation, to define the depth of nanochannels [205]. First, the width is defined by photolithography and wet etching into a thick layer of silicon dioxide grown by wet oxidation. As the thick oxide layer does not evolve anymore, a second dry oxidation results in the growth of oxide layer only at the bottom of the trench. This process consumes silicon. Stripping off the oxide in a buffered HF solution leads to the formation of trenches with a depth roughly equal to half of the second oxide thickness. High uniformity and roughness (Ra) below the nanometer were measured using this method. Anisotropic wet etching of nanochannels was also investigated with excellent results [206]. Various solutions were tested. A commercial photoresist developer (Olin OPD 4262, an aqueous solution of TMAH at 2.5 wt% with some surfactant additives) was found to give the best results. The etching rate (3.7 nm/hour at room temperature) was low enough to allow a convenient control of the etching depth. The method resulted in very smooth channels (Ra below 0.5 nm) with straight sidewalls perpendicular to the bottom. An excellent uniformity was observed along the wafer and from wafer-to-wafer. The masking layer for the etching was simply the native oxide, making the process even easier. However, the design is limited by the crystalline plan of the silicon substrate. Spacer Micromachining

Spacer machining (Figure 3.27) is an alternative to precisely control the depth by taking advantage of well-mastered thin film deposition or growth technique, combined with a selective etching. Indeed the film is used as a spacer between two substrates. Amorphous silicon (aSi) has been often used for this purpose due to its high resistivity, the possibility to use it as an intermediate layer for anodic bonding between two glass substrates, and the achievable etching selectivity to glass. Kutchoukov et al. fabricated 2–100-mm-wide and 50-nm-deep channels by etching aSi using RIE (CF4/SF6/O2) and anodic bonding [207, 208]. Schoch and Renaud reported a similar process, in which the sputtered aSi layer was patterned by lift-off, preventing any over-etching issues [209]. Sacrificial Micromachining

Compared to bulk or spacer micromachining, sacrificial methods (Figure 3.22) avoid wafer bonding and allow an easier system-level integration. Thin capping layers result in a low light absorbance but are relatively fragile. Beyond this, integration of microchannels with surface-micromachined nanochannels is not straightforward. Paradoxically, this integration is often done using wafer bonding techniques [210]. The standard approach has been to etch the sacrificial layer in liquid phase. As early as 1992, Gajar and Geis reported the fabrication of a nanofluidic transistor using surface micromachining [211]. Using the same technique, chemical sensors embedding 4-mm-long nanochannels with a depth of 20 nm and widths ranging from 500 nm to 200 mm were fabricated [212]. The structural layer was silicon nitride and the sacrificial layer was aSi. The etchant was TMAH. Eighty hours were necessary to release a 1.5-mm-long channel with a cross-section of 50 nm by 1 mm. Polysilicon was also reported as a sacrificial layer, combined either with SiO2 [213] or Si3N4 [214] as a structural layer. Polyimide (PI) and SU-8 planar nanochannels were fabricated using metallic sacrificial layers [215, 216]. A few dry etching methods were

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also reported [210, 217–219]. Very long etching times associated with sacrificial methods result in a marked differential exposure time of the structural material to the etchant along the channel. The walls at the entrance of the channel are exposed a longer time to the etching solution than in the middle. Even with the excellent etching selectivity conferred by the couple SiO2/Si3N4, a tapering (32 nm/mm of channel) of the structure has been measured due to this differential exposure [220]. Many approaches have been developed to reduce etching times. A method consists of patterning small irrigation holes all along the channel to virtually reduce its length [221]. After etching of sacrificial layer, the access holes were sealed using a nonconformal deposition process (LPCVD of SiO2) as depicted in Figure 3.29. Many other examples of nonconformal depositions used to seal nanochannels have been reported. These include the LPCVD, PECVD, inclined evaporation or sputtering of SiO2 [47, 222–224], the PECVD of parylene [225], and the evaporation of gold [47]. Recently, a similar sealing technique was demonstrated using the wet oxidation of well-designed trenches [226]. The porous silicon layer was investigated as a sacrificial layer [227]. The large surface-to-volume ratio leads to a four-times increase of the etching kinetics. The roughness of the resulting channel was not given. Electrochemical methods demonstrate very attractive etching rates. As an example, Cheng and colleagues released a 100-nm-deep, 300-mm-wide, 2-cm-long nanochannel within only 4 minutes [228]. Copper was etched away using a solution of copper sulfate and an applied voltage of 6V. A drawback of the method is the need of an electrical contact with the metal lines during the etching. Galvanic corrosion demonstrates lower etching kinetics, but does not require any electrical connection. A tenfold increase in the etching kinetics of copper was measured by coupling it to a chromium layer [80]. A similar behavior was observed by coupling gold to a chromium sacrificial layer [81]. Polymer Replication

A low-cost, mass fabrication technology seems essential for an eventual commercialization of nanofluidic devices. The fabrication of plastic hybrid micro- and

Figure 3.29  Sealing of a HAR trench using a nonconformal deposition process.

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nanofluidic devices has been demonstrated using hot embossing and thermal bonding (Figure 3.30) [196]. Plastic microfluidic devices are usually sealed by adhesive bonding, solvent bonding, or thermal bonding [115, 229, 230]. In thermal bonding, the two pieces are brought into an intimate contact by heating at a temperature slightly above the glass transition and applying a sufficient pressure. The required pressure depends on the temperature, the quality of the surface, and the flexibility of the sheet. After contact, it is usually supposed that an entanglement of the polymer chains occurs at the interface leading to the bonding. In the case of planar nanochannels, it was observed that attempts toward thermal bonding at a temperature above the glass transition result in a collapse of the structure. The developed bonding method consists of: (1) an oxygen plasma treatment, and (2) a thermal bonding process at a temperature below the glass transition temperature. The bonding strength was estimated to be half as much as the bonding strength of a standard thermal process. The exact mechanism is still unclear, but it is supposed that bonding is due to both an entanglement of the polymer chains and a reaction between the hydroxyl groups generated at the surface by the oxygen plasma. 3.2.2.2 Nanopatterning of Two-Dimensional Networks of Square Nanochannels (AR~1) Serial Patterning Techniques

As the lateral dimensions shrink, the price of photolithography equipment quickly jumps to very high levels. Despite their very high throughput, the high cost of projection

Figure 3.30  Hot embossing and bonding of nanofluidic devices in PMMA: (a) access-holes are predrilled in a PMMA sheet with a CO2 laser; (b) PMMA is embossed using a two-level template (micro- and nanochannels); and (c) the mold is released. (d) Finally, the replica is sealed with a second PMMA sheet using a plasma-assisted thermal bonding process at a low temperature. (After: [196].)

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systems able to pattern below 100 nm prevents their use in academic research and small-scale production. SBL is the favorite method for prototyping devices with square nanochannels (i.e., with two dimensions in the nanoscale). Using femtosecond lasers, three-dimensional networks of channels have been drawn in glass and in PMMA [231, 232]. Minimum diameters were 400 and 700 nm, respectively, still above the nanofluidic range. Hibara and colleagues fabricated a micro- and nanofluidic device by combining EBL and photolithography [233]. Nanochannels were defined in pyrex by a dry etching technique, while microchannels were machined by powder blasting through a polyimide mask patterned with a laser. Finally the structures were sealed at room temperature with another pyrex wafer by introducing a drop of fluorhydric acid between the two substrates and pressing them together for 24 hours. Similar methods have been reported using EBL with dimensions down to 80 nm [234, 235]. Using FIB, nanofluidic channels with lateral dimensions as low as 20 nm were fabricated by milling glass or silicon substrate with gallium ions [236, 237]. Etching sacrificial layers in liquid phase becomes more and more difficult as the dimensions of the channel decrease. The introduction of irrigation holes drastically reduces the etching time and has been applied to the fabrication of square nanochannels, but it introduces a substantial additional number of fabrication steps [224]. Thermally degradable sacrificial layers allow for the release the structure at a rate independent of the length of the channel. Accordingly, 140-nm-deep, 400-nmwide channels were fabricated using polycarbonate (PC) as a sacrificial layer and silicon dioxide as a structural layer [161]. PC was patterned by EBL. PC was degraded by heating at a temperature of 300°C for 30–60 minutes. After spin-coating the PC sacrificial layer, the process had to be carried on at a temperature below 250°C to avoid accidental degradation. Polynorbonene (PNB) can be tailored to a wide range of degradation temperatures. A PNB with a degradation temperature of 425°C was used to fabricate nanochannels with cross-sections of 100 ´ 100 nm [162]. The highest degradation temperature allowed the authors to use a wider range of deposition processes. Replication Techniques

Serial techniques demonstrate a low throughput. A solution to overcome this issue is to replicate the high-resolution templates fabricated using this method. Replication has been used either: (1) to transfer a pattern in a resist used as a mask during a subsequent etching step, or (2) to directly imprint the structural material. Both bonding-based techniques [238, 239] and surface micromachining [162, 219] have been combined with NIL to defined square nanochannels. The templates have usually been made in silicon by EBL and RIE. Nonconformal deposition may be used to both: (1) reduce the dimensions, and (2) seal nanochannels defined by NIL [222] Channels with cross-sections of 10 ´ 50 nm were fabricated following this method. Guo et al. simply left the template in the resist after imprinting [240]. With an adequate resist thickness, gaps formed between the template and the PMMA layer. However, the template was lost. A direct imprint of thermoplastic substrates has been demonstrated for the fabrication of planar [196] and square [241] nanochannels. The technology is relatively straightforward and may be applied to the mass-fabrication of nanofluidic devices.

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On the other hand, there is a strong need for a rapid prototyping method equivalent to the PDMS casting widely spread in microfluidics. Due to the low Young modulus of the material, structures tend to collapse as the dimensions and the AR shrink. The smallest PDMS nanochannels reported in the literature have dimensions larger than 150 nm [192–194]. A notable exception is the recent fabrication of triangular nanochannels with a height of 80 nm and a base of 700 nm [195]. Channels were replicated from cracks thermally induced in an oxide layer formed by oxygen plasma on the top of a PDMS slab. The triangular cross-section prevents the channel from collapsing. Working with micro- or nanofluidic devices often requires preliminary surface treatments, often by flushing the channels with chemicals or introducing additives in the running solution. On the other hand, polymers offer a wide range of materials with tailor-made properties. An attractive alternative would be to directly use materials with adequate functionality as a structural material. Kim and colleagues demonstrated this concept by replicating channels using polyethylene glycol (PEG), a polymer with nonbiofouling properties, both as a structural and functional material. Figure 3.31 depicts the UV-molding process. Channels with typical dimensions ranging from 50 nm to 2 mm were replicated from a silicon template. Irreversible bonding was obtained by completing the crosslinking after contact with another substrate coated with PEG. Surface treatments were performed to ensure a good adhesion between PET or glass substrates and PEG layers. Similar to PDMS, PEG nanochannels can also be reversibly bonded to a substrate [242]. Beyond the functional aspect, this approach is promising as a fast prototyping method of nanofluidic devices.

Figure 3.31  Fast replication of square nanochannels in PEG. A few droplets of a UV-curable PEG are (a) dispensed on a silicon template and then (b) covered with a PET film and partially cured by UV during a few seconds. (c) After unmolding, the replica is brought into contact with another substrate (e.g., glass or PET) and coated with PEG. (d) Finally the UV curing is completed to irreversibly seal the channels.

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Reverse imprint (Figure 3.20) has been applied to the fabrication of gold nanochannels transferred on a GaAs substrate [243, 244]. This approach allows the transfer and stacking of multiple layers of nanostructures. The substrate was modified with an anchoring dithiol SAM. The resulting nanochannels have a crosssection smaller than 80 ´ 100 nm. Reverse imprint was demonstrated with a variety of other materials such as PMMA [245, 246], hydrogen silsequioxane (HSQ) [98], SU-8 [247, 248] and various UV adhesives [249]. 3.2.2.3  One-Dimensional Network of Square Nanochannels (AR~1) Self-Assembly

As the dimensionality of the fluidic network decreases, new solutions appear. Among others, self-assembly represents a powerful set of tools to pattern simple features at the nanoscale. For instance, microdomains of diblock copolymers can spontaneously arrange into cylinders with nanoscale diameters. Nanopipes can be formed by selective of the core material. As an example, Rzayev and Hillmyer reported the synthesis and the preparation of functional nanoporous monoliths using triblock copolymers polylactide-polydimethylacrylamide-polystyrene PLA-PDMA-PS [250]. The shear-aligned nanochannels were 5–50 mm long and had a diameter of 20 nm. Degradation of the polylactide was achieved in a 0.5 M NaOH (MeOH/water, 40:60 v/v) solution at 65°C. For such long channels, a period ranging from a few days to 14 days was necessary to release the structure, depending on the size of the PDMA block. PS was the structural material. The PDMA group allowed an easy functionalization after hydrolysis in a carboxylic acid group. Carbon Nanotubes

Another solution is to use carbon nanotubes. The challenge lies in their interconnection and their integration in a fluidic device. Nanofluidic experiments were performed on stand-alone nanotubes grown with an encapsulated fluid [251], on nanotubes vertically grown on a substrate [252, 253], on nanotubes mounted at the end of the tip of an AFM [254], and on nanotubes inserted across a polymer film [255]. However, a planar integration would be more convenient. A first solution was to deposit and pattern a silicon dioxide layer on nanotubes dispersed on a glass substrate [210]. Another reported solution consisted of trapping a carbon nanotube on a substrate between two electrodes by dielectrophoresis. Then a layer of SU-8 was spin-coated into which reservoirs were patterned [256]. Drawing Fibers

Optical fibers and glass capillaries are commonly drawn from bigger preforms, taking advantage of the Poisson effect. Following this idea, Sivanesan et al. shrank the diameter of a PC microchannel from 20–30 mm down to 400 nm by heating and pulling the whole device [257]. Electrospinning consists of drawing a material by the application of an electric field between a needle and a substrate. Using electrospinned nanofibers as a sacrificial material, channels with a diameter below 75 nm were fabricated [258, 259]. Coaxial fibers were extruded from a silica sol-gel and a motor oil [260]. Nanochannels with an inner diameter of 20 nm were formed after annealing and elimination of the sacrificial oil. Using a technique pioneered by Ev-

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ans and colleagues [261], nanochannels with an inner diameter of 50–150 nm were fabricated by pulling a micropipette away from a lipid vesicle [262–264]. Similar techniques have been demonstrated using a more robust cross-linkable polymer drawn with a micropipet or optical tweezers [265] or by pulling a sacrificial polymer with an AFM tip [266]. Edge Lithography

The step lithography technique depicted in Figure 3.8 allows for the fabrication of nanowire at the edge of step without nanolithography. This nanostructure can be used as a sacrificial layer. The fabrication of a nanochannel with a cross-section of 40 nm ´ 90 nm was reported using a sacrificial polysilicon nanowire and silicon nitride deposited by LPCVD as a structural layer [214]. Using a trench refilling technique (Figure 3.9) and a planarization by chemical-mechanical polishing (CMP), Lee and colleagues converted the thickness of a silicon dioxide film into the width of nanochannels sealed by a nonconformal deposition [47]. A well-controlled underetching (Figure 3.9) was used to fabricate 10-nm-high and 200-nm-wide nanochannels [223]. The Si3N4 surfaces were oxidized using an oxygen plasma before sealing to ensure homogeneous surface properties. 3.2.2.4  High-Aspect Ratio Nanochannels (AR >> 1) and Nanopores High-Aspect Ratio

Pushing a liquid at a reasonable flow rate in a long nanochannel usually leads to excessive pressure incompatible with most of the fabrication techniques. Scaling laws benefit more to electrokinetic- and capillary-driven flows, which are the most common techniques to run a liquid in a nanofluidic device. A dense array of HAR nanochannels would allow applications where pressure-driven flow is required. More generally, HAR nanofluidic devices would benefit from a higher throughput. However, only a few groups reported the fabrication of HAR nanochannels. O’Brien et al. reported the fabrication of 500-nm-high and 50-nm-wide silicon channels by combining interference lithography (IL), RIE, and anodic bonding [267, 268]. Also, 5-mm-high and 200-nm-wide channels were fabricated using proton beam writing in PMMA [269]. Other techniques employed for the fabrication of HAR trenches may be applied to nanofluidics. For example, 150-nm-wide trenches with an AR of 3 were reported using anisotropic wet etching [270]. The trench refilling technique introduced previously (Figure 3.9) is an edge lithography technique combining bulk and surface micromachining [271]. In a process developed by Ayazi and colleagues, a thin sacrificial layer was deposited in a trench etched by DRIE in silicon. Then the trench was filled, and the sacrificial layer was etched. Trenches with a width of 80 nm and a height of 20 mm were obtained [272, 273]. Using a similar technique, Martin et al. fabricated 7-nm-wide and 3-mm-high open channels corresponding to an AR of 430 [48]. A nanotrench with an AR of 125 was etched in silicon using a photoassisted electrochemical etching technique [274]. A 3-mm-high 200-nmwide gap was etched by combining EBL and an inductively coupled plasma (ICP) source [275].

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Nanopores

Nanopores can be defined as very short (i.e., 10 mm or less) nanochannels. Their diameters are usually on the order of the nanometer, tailored to stretch single-strand DNA for eventual sequencing applications [276]. First experiments were performed by taking advantage of biological structures. R-Hemeolysin is a transmembrane protein forming a pore with a diameter of 1.4 nm. These proteins spontaneously insert themselves through a lipid bilayer, similar to a cell membrane [277]. A few years ago, researchers started looking for new solutions to increase the stability of these structures and gain further control on their dimensions. Various alternative techniques have been developed, often based on the etching of insulating membranes. Nuclear track etching [278], ion beam etching [279], and electron beam etching [280] have been used to fabricate “solid-state” nanopores with diameters down to 0.8 nm [281]. The ability to finely shrink or enlarge their dimensions was demonstrated using an adequate ion or electron beam.

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Chapter 4

Applications of Micro- and Nanofluidics Nanofluidics has become popular over the last 15 years thanks to a continuous progress in micro- and nanofabrication. Indeed, while porous media like electrophoresis gels are characterized by a random distribution of both the position and the size of the pores, arrays of obstacles with well-determined shape, size, and diameter can be patterned by lithography. At the origin of nanofluidic integration, there was a will to speed up the human genome project via a shift from random porous media to regular, deterministic arrays of pores. The objective was: (1) to get a deeper insight into the biophysical mechanisms at the origin of the electrophoresis of polyelectrolytes, and (2) to get better, faster separations. Nanochannels have a characteristic size that makes possible a direct interaction with molecules such as DNA or proteins. Typical examples of functions include concentration, gating, and separation of these molecules, according to their charge or size. However, a complete process usually involves more than this. For instance, these biomolecules may first have to be extracted from a selected cell after growth in a controlled environment, mixed with a certain quantity of reactants, then concentrated, and separated before some of their properties can be measured. These operations involving different scales are usually performed using different pieces of equipment in a macroscopic lab. Miniaturization allows the integration of multiple steps on a single device. Practically, it means that not only do nanochannels have to be integrated on a chip, but so do microchannels, and eventually sensors or actuators (e.g., electrodes) for the most complex devices. In summary, micro- and nanofabrication techniques have both allowed: (1) a shift from a random, anisotropic porous media to a well-defined, customized array of pores, and (2) the integration of these nanochannels into a complex device. These technological advances offer tremendous applications in a wide range of areas, where a random array of holes can be advantageously replaced by a smart layout of nanopores integrated in a microsystem embedding microchannels, sensors, actuators, or integrated circuits. This chapter first introduces the concept of lab-on-chip and provides some examples of microfluidic chips, essentially focusing on biomolecular analytical devices due to their importance in nanofluidic applications. The second part of the chapter covers the application fields of nanofluidic technologies.

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4.1  Microfluidics and Lab-on-Chips 4.1.1  Biomolecular Analyses 4.1.1.1  Capillary Electrophoresis on Chip

At the end of the 1970s, Terry and colleagues from Stanford introduced a gas chromatograph micromachined in a 2-inch silicon wafer [1]. This early system integrated an injection valve, a 1.5-m-long spiral capillary column, and a thermal conductivity detector. The idea was to overcome the lack of sensibility of gas sensors by combining them with a separation step within a handheld format. Despite this early work, it was only in the beginning of the 1990s that the enlarged concept of micro-total chemical analysis system (mTAS) emerged. In his seminal paper, Andreas Manz took the example of blood analysis [2]. Ideally, a sensor dedicated to the detection of a compound present at a concentration of 10–5 moles per liter has to reject more than 100 compounds of higher concentration. Since sensors are usually not capable of such a high selectivity, a pretreatment step (e.g., mixing or separation) is often compulsory. Miniaturization and integration of the pretreatment and detection in a single chip yields numerous benefits, including a reduced volume of reagents, a possible lower cost (through mass fabrication), faster operation (due to reduced distance and automation), parallel analysis, and more sensitive detection. This was later extended to the concept of “lab-on-chip,” integrating and automating on a single component all the elementary steps involved in a macroscopic lab, including handling a certain quantity of a liquid, mixing, heating, diluting, concentrating, separating, and measuring. In the following years, researchers have been working on the development of building blocks for lab-on-chips. Figure 4.1 shows a now classical layout. The device, dedicated to capillary electrophoresis (CE), comprises four opened reservoirs connected by a network of microchannels. Fabrication of these devices has been demonstrated out of glass, silicon, or polymer [3–5]. Liquids are pumped using electrokinetic effects. Electric potentials are applied via external wires (usually in platinum to limit electrochemical reactions) dipped in the reservoirs or by electrodes integrated on the substrate. Using this simple, valve-free configuration, the direction of the flow is controlled by switching the four electric potentials. Two functions are integrated: electrokinetic injection and separation by CE. Compounds are commonly detected by an external fluorescence system or by electrochemical methods. The device is first filled with an electrolyte (Figure 4.1). Next, the sample is introduced into one of the reservoirs. A voltage is applied to fill the loading channel (segment S to SW). Next, the voltage is switched to inject the small volume geometrically defined by the intersection between the loading channel and the separation channel (segment B to BW). Under the action of the electric field, compounds with different electrophoretic mobilities are separated during their travel along the separation channel. At the end of this channel, compounds are detected, most often using fluorescence techniques. To this end, these compounds have to be tagged with fluorescent probes before injection. After separation, the excitation of these probes with a laser at the adequate wavelength results in a sequence of peaks of emission measured with a charge-coupled device (CCD) camera or a photomultiplier tube (PMT). The resulting trace of light intensity versus time is called an electrophoregram.

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Figure 4.1  CE on chip. The device comprises four reservoirs connected to electrodes. S, SW, B, and BW stand for sample, sample waste, buffer, and buffer waste respectively. After (a) filling the whole device with the buffer, (b) the sample is pushed from S to SW by applying adequate voltages. Once the sample has progressed beyond the intersection, (c) voltages are switched to inject the central volume of sample into the longer channel where (d) separation occurs by electrophoresis.

4.1.1.2  Electrokinetic Injection

Reliable injection of a controlled volume in the pico- or nanoliter range is critical to obtain reproducible, well-resolved separations. Thus multiple schemes of electrokinetic injections have been developed over the last 15 years [6]. Figure 4.2 depicts common methods. In T-type injection, the volume injected is controlled by the duration of the loading step [3, 7–9]. However, this method results in a continuous leakage of the sample during the separation step. Using the cross-geometry, the volume injected is defined by the volume of the intersection. This volume can be increased by using a double-T configuration, where the two loading channel arms are not aligned [10] to overcome sensitivity issues. During the separation step, the sample in the two loading channel arms is often pushed back towards their respective reservoirs (S and SW) to avoid a continuous leakage in the separation channel. Another issue may be the diffusion of the sample into the separation channel during the loading step. This can be avoided by pinching the sample flow during the loading step. The sample is focused by pushing the electrolyte from reservoirs B and BW to SW [11]. Another common method is gated injection, allowing control of the injection volume by the time interval (as for the T-type injection) but avoiding leakage issues. While pinched injection allows the dispensing of an extremely small volume defined by the layout of the device, gated injection gives more flexibility to deliver middle to large volumes independent of the geometry. Many variants to these methods have been developed; interested readers should refer to the review article from Blas and colleagues [6].

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Figure 4.2  Common injection modes for CE on a chip. T-type injection: (a) the sample is introduced in the separation channel during a given time interval and (b) the buffer is loaded during the separation step. Floating injection: (c) the sample is loaded in the loading channel and (d) the volume geometrically defined by the intersection of the channels is injected in the separation channel. Gated injection: (e) buffer and sample are coflowing in the device and (f) the potential of the reservoir B is left floating during a given time interval before (g) switching back the potentials to their initial states.

4.1.1.3  Separation of Small Molecules

Different separation methods have been integrated on microfluidic chips. Compared to more standard slab gel electrophoresis, performing electrophoresis in a microchannel leads to smaller values of the electric current for an equivalent electric field and much better heat dissipation. As a consequence, larger electric fields may be used in a microchannel. For a given compound, the “efficiency” P of a separation method is given by:

P=

16x2 w2

where x is the drift of the compound after time t, and w the band width [12]. Assuming that the band broadening is due only to diffusion, the efficiency P of CE is:

P=

8m E2 t D

where m is the mobility, E is the electric field, and D is the diffusivity of the compound. Thus, the larger electric fields accessible in CE result in a better efficiency. Though CE has traditionally been performed in glass capillaries, planar microtech-

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nologies have allowed the integration of complex layouts into lab-on-chips. The most common method is capillary zone electrophoresis (CZE) or free solution CE (FSCE) where no porous matrix is introduced into the separation channel. The total mobility m is the sum of the electrophoretic mobility m0 of the analyte itself and the electro-osmotic mobility meeo due to surface charges on the microchannel wall.

m = m0 + meeo



The plug-like electro-osmotic flow (EOF) induces a shift in the total mobility. For instance, the electro-osmosis mobility in a glass microchannel at neutral or alkaline pH is generally larger and opposite to the mobility of DNA, reversing the order of migration. One issue is the strong dependence of the EOF on the surface properties of the microchannel. In fact, many molecules (e.g., proteins) easily adsorb on surfaces, leading to an inhomogeneous electro-osmotic mobility along the channel, and thus a poor reproducibility of the migration times, and local recirculations enhancing dispersion [13]. Indeed patterning the chemistry or the topography of the surface of a microchannel is a common way to enhance mixing in microfluidics [14]. Many surface treatment strategies to avoid both adsorption of molecules and electro-osmosis have been developed [15]. One common solution is to attach a polymer “brush” of hydrophilic polymers thicker than the Debye layer in order to screen surface charges. The trend in microfluidics is to move from silicon or glass fabrication to cheaper polymer fabrication. While the surface chemistry of glass is well-mastered owing to the wide use of glass capillaries in analytical sciences, the development of surface modification strategies dedicated to polymeric devices is an important step towards efficient separations in low-cost microchips [16]. CE has been applied on chip to the separation of small molecules, DNA, proteins, and organelles [17]. Jacobson and colleagues demonstrated the separation of rhodamine B and dichlorofluorescein by CZE within 0.8 ms over 0.2 mm using an electric field of 30 kV/cm [18]. The Mars organic analyzer (MOA) is an instrument integrating a microchip with channels, valves, and pumps; pneumatic control; a high-voltage power supply; and fluorescence detection optics [19]. Analysis of soil extracts from the Atacama desert in Chile and the Panoche Valley in California detected amino acids at 70 parts per trillion to 100 parts per billion in jarosite, a sulfate-rich mineral associated with liquid water that was recently detected on Mars. These results were comparable or even superior to those of commercial chromatography systems. As standard CE can resolve only charged molecules, alternative modes have been integrated to separate neutral compounds. In micellar electrokinetic chromatography (MEKC), surfactants (e.g., sodium dodecyl sulfate SDS) are added in the running buffer at a concentration above the critical micelle concentration. In this condition, the mobility depends on the hydrophobic interaction between the charged micelles and the analyte. The stronger the interaction, the longer the solutes migrate with the charged micelles. Neutral molecules having no interactions with the micelles migrate at the electro-osmotic velocity. MEKC on chip was first demonstrated with the separation of three coumarin dyes [20]. MEKC has been widely applied to the separation of amino acids. On-chip separation of 19 tetramethylrhodaminelabeled amino acids was demonstrated within 165 seconds along a 22-cm-long

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spiral microchannel [21]. The separation of enantiomers has been made possible by adding chiral micelles or chiral selective agents [e.g., cyclodextrins (CDs)] to the background electrolyte. This was applied to the chiral MEKC of fluorescein isothiocyanate–labeled amino acids extracted from the Murchinson meteorite, in search for life signs in an extraterrestrial environment [22]. The separation of neutral or chiral compounds is also possible using chromatographic methods. Interactions between the analytes, a stationary phase (e.g., monoliths, packed columns, and walls of the microchannel) and a mobile phase (liquid, gas) lead to the separation. The flow is driven either by a difference of pressure [high-pressure liquid chromatography (HPLC) or gas chromatography (GC)] or by electro-osmosis [capillary electrochromatography (CEC)]. CEC has been more popular in the microchip format as HPLC requires large pressures, which are often incompatible with the limited bonding strength of microfluidic devices. Furthermore, the plug-like EOF results in a reduced dispersion. As early as 1994, Ramsey and coworkers reported the separation of coumarins by open-tubular CEC (OTCEC) [23]. In OTCEC, the stationary phase consists only of the walls of the channel coated with a selector (often a reverse-phase coating of octadecyl C18 groups). Harrison and colleagues demonstrated the separation of dyes using packedcolumn CEC [24]. Monoliths have also been integrated for the CEC of small compounds (e.g., alkylphenones and antidepressants) [25]. Interested readers can refer to review articles [26, 27] for a more complete overview of microchip CEC. 4.1.1.4  Separation of DNA



The need for faster, cheaper biomolecular analysis devices has never been greater than it is today. The ability to identify different biomolecules out of a complex biological sample will facilitate the shift from an essentially reactive health care scheme to predictive, preventative, and personalized medicine by following the molecular signature (proteins, mRNA) involved in cancers and other diseases [28, 29]. Mapping genetic polymorphisms for each individual will make possible a probabilistic statement about their disease likelihood. High-throughput analysis of the interaction between proteins (such as transcription factors) and DNA will permit one to understand regulatory networks and to detect pathogenic perturbations. In 2004, the National Institutes of Health (NIH) established a program to reduce the cost of sequencing from $3 billion (cost of the human genome project using the Sanger method) to $1,000 within 10 years. In 2008, the de novo sequencing of a single individual, James Watson, was completed within two months for a cost of $100,000 by pyrosequencing [30]. Separating long DNA strands is not as straightforward as separating smaller molecules. The electrophoretic mobility of a compound is usually equivalent to the ratio of charge Q to friction f: v0 Q m0 = = E f where v0 is the electrophoretic velocity. Long, uniformly charged polyelectrolytes such as DNA cannot be separated by CZE as their charge and friction scales linearly with the number of monomers (free draining property) [12, 31]. Stellwagen,

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Gelfi, and Righetti reported a free solution mobility m0 of 3.75± 0.04 cm2 V–1 s–1 in a Tris-Acetate-EDTA (TAE) buffer at 25°C, independent of DNA concentration, sample size, electric field strength, and capillary coating. The free solution mobility in Tris-Borate-EDTA (TBE) was found to be 4.5± 0.01 cm2 V–1 s–1, slightly higher presumably due to the formation of nonspecific borate-deoxyribose complexes. This mobility was independent of the molecular weight in a range from 48 kbp down to 400 bp, but decreased monotonically with decreasing molecular weight for smaller fragments. This balance between charge and friction can be broken by attaching a particle (“molecular parachute”) with a different mobility at the end of the polyelectrolyte to be separated. This method is called end-label free-solution electrophoresis (ELFSE) [32]. Sudor and Novotny reported the high-resolution separation of polysaccharides by attaching a neutral fluorescent label to their end [33]. ELSFE was also successfully applied to the separation of DNA [34]. Despite this remarkable work, the most common method to fractionate DNA by electrophoresis still involves the introduction of a sieving matrix inside the microchannels [capillary gel electrophoresis (CGE)]. As in standard slab gel electrophoresis, separation occurs because of the interactions between the matrix and the analytes. Cross-linked gels (e.g., polyacrylamide) were use in the beginning of CE, leading to unsurpassable efficiencies. However, the difficult task of loading a permanent gel into a microchannel gave rise to many issues in term of reproducibility and lifetime. The bonding of microfluidic chips usually does not withstand very large pressure (for instance, 30–50 psi for a PDMS device) [35]. Gels are sensitive to changes in temperature or pH and high voltages. Cracks and bubbles may appear during the polymerization of the matrix due to gel shrinkage. Thus, gels have been completely replaced by polymer solutions [36]. Un-crosslinked polymer solutions are less sensitive to changes in the physical environment, and microchannels can easily be refilled after each run leading to a better reproducibility. Using dilute, low-viscosity solutions makes the filling step easier. These solutions were applied to the fast separation of a large range of DNA sizes (up to 2 kb using constant field, up to Mb using pulsed-field) [37–41]. On the other hand, a higher polymer concentration results in better, more reproducible separations. Entangled polymer solutions remain by far the most common media for CE of DNA. One of the most common polymers for this purpose is linear polyacrylamide. However, due to its strong adsorption below 230 nm, which reduces the sensitivity of detection, other polymer solutions [e.g., Dextran, polyethylene oxide (PEO) and polyvinyl alcohol (PVA)] have been studied [36]. Automated systems for genomic and proteomic analysis integrating CGE microchips have been commercialized by companies such as Caliper (Labchip GX) or Agilent (bioanalyzer 2100). While increased concentrations lead to higher resolution, it also makes the loading of the polymer solution more troublesome. Doyle and colleagues proposed the use of superparamagnetic beads that self-assembled into a sieving matrix by applying a magnetic field [42]. The array returns to a liquid suspension upon field switchoff. Thermoresponsive solutions have also been investigated. Thermo-associating polymers have remarkable, counterintuitive properties: While their low viscosity at room temperature allows for easy filling, heating above a certain temperature leads to hydrophobic associations and thus higher viscosities, allowing separation

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with high resolutions. For instance, Sudor and colleagues demonstrated the application of poly(N-isopropylacrylamide) (pNIPAM) matrixes to DNA sequencing [43]. Thermothinning solutions demonstrate an opposite behavior: While they own a high viscosity at room temperature, heating above a critical temperature results in a drop in viscosity [44–46]. Another solution consists of patterning the sieving matrix into the microfluidic chip substrate itself using lithography techniques. The second part of this chapter details this method. 4.1.1.5  Separation of Peptides and Proteins

Protein separation has a wide range of applications including medical diagnostics, drug development in pharmaceutics, and quality control in the food industry [47– 49]. Opposite to DNA, proteins can be simply separated by CZE, without a sieving matrix [50]. Scientists from the Sandia National laboratory recently introduced mChemLab, a handheld device integrating a CE microchip [51]. Comparison between CZE and CGE of protein biotoxins were performed on this platform, demonstrating better reproducibility (run-to-run and chip-to-chip) of the CGE mode. Caliper, Agilent, and Bio-Rad have commercialized successful protein analysis systems integrating a CE microchip [52, 53]. CEC and HPLC have been widely applied to the analysis of proteins [26, 27]. In 2005, Agilent introduced a HPLC system based on a disposable microfluidic chip fabricated by lamination of polyimide sheets patterned by laser ablation [54]. The chip integrates an electrospray tip allowing easy coupling to a mass spectrometer. Amphoteric molecules carry both positive and negative charged functional groups. The charge of these groups is affected by the pH of the surrounding solution. For a given pH, a protein has a zero net charge, and therefore no electrophoretic mobility. This value of the pH is called the isoelectric point pI of the protein. In capillary isoelectric focusing (CIEF), also known as electrofocusing, a gradient of pH is generated along a microchannel [55, 56]. The pH gradient is usually generated by mixing carrier ampholytes with the sample prior to loading. An original solution proposed by Macounova and colleagues consists of taking advantage of water hydrolysis [57–59]. H3O+ and OH– ions generated at the electrodes mix along the channel leading to a pH gradient. Another method is to convert a temperature gradient into a pH gradient using the thermal pKa dependency of a tris-HCl buffer [60]. Under the action of an electric field, proteins introduced in this gradient migrate towards the region where the pH equals the pI. As the molecule tries to diffuse out of this region, a charge appears again moving it back. After focusing, proteins are detected by imaging the whole length of the microchannel or by mobilization (i.e., by moving the whole plug with the focused bands through a punctual detector). Hofmann and coworkers demonstrated CIEF on a microchip by separating a mixture of Cy-5 labeled peptides within 30 seconds [61]. The performance of analytical devices can be dramatically enhanced by coupling different separation principles [49, 56, 62]. The best performances are obtained by uncorrelated separation mechanisms (i.e., orthogonal separations). The possible integration of multiple functions without inducing dead volumes makes lab-on-chips an ideal platform for multidimensional separations. Technical challenges include the compatibility and the cohabitation of different buffers in a single device and the

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synchronization of the separations. Ramsey and colleagues pioneered the integration of two-dimensional separation on a glass microchip by coupling OTCEC as the first dimension and CE as the second dimension [63]. A critical point is the sampling rate of the analytes after separation in the first dimension. In this study, the buffer running in the OTCEC column was injected in the CE separation channel during 0.2 second every 3.2 seconds. After further optimization, impressive results were obtained by coupling MEKC and CE, resulting in a better orthogonality [64, 65]. Wang et al. designed a device coupling CIEF as a first dimension and CE as a second dimension [66]. The two separations were physically separated by PDMS valves. 4.1.1.6  Detection

While shrinking the dimensions is often associated with faster reactions or separations, the reduced amount of samples requires high-sensitivity detection methods. Laser-induced fluorescence (LIF) has been the most common detection method in microfluidics [67–70]. The principle of LIF is illustrated in Figure 4.3. Fluorescent compounds attached to the target compounds are excited by a laser. The emitted light is filtered and collected by a CCD camera or a PMT. Practically, the common setup in a microfluidics lab consists of a microscope including a set of fluorescence cubes (integrating an excitation filter, a dichroic mirror, and an emission filter). Mercury or xenon arc lamps are often utilized instead of lasers for less demanding applications, as they are less expensive and offer more flexibility in term of wavelength. Careful optimization has allowed very high sensitivity, and even single molecule detection [71–73]. Though ultraviolet (UV) absorbance is the most widely used detection method in CE and HPLC, it has seldom been employed in microfluidics due to the usually small depth of the microchannel resulting in a poor sensitivity. Harrison and coworkers enhanced detection by UV absorbance by using multiple reflections on aluminium mirrors patterned on the bottom and the top walls of the microchannel [74]. The general strategy to couple microsystems and UV absorbance has been to

Figure 4.3  Principle of the detection by fluorescence.

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extend the optical path length by integrating optical elements (e.g., lens) [75–80]. Detection methods based on chemiluminescence (CL) are highly attractive due to their high sensitivity, their low detection limits, and the fact that no external optical source is necessary. However, mixing the CL reagents with the analytes before detection involves a more complicated layout [81–85]. A variant is electrochemiluminescence (ECL) where the activation of the CL species is triggered by the application of an electric potential [84, 86, 87]. Mass spectrometry (MS) is a powerful, label-free alternative to optical detection techniques, especially for the detection of proteins. As the integration of the whole equipment on a chip is a very challenging task, most of the work has been focused on the coupling of microfluidic devices with MS. Within proteomics, sample preparation steps using conventional methods usually take much longer time than recording a mass spectrum. Therefore the idea is to integrate sample preparation (e.g., concentration and digestion) and separation on a chip to take advantage of the automation and speed associated with microsystems. This has been made easier by the comparable flow rates in microfluidics (from a few tens of nanoliters per minute to a few milliliters per minute) used in electrospray MS. The most common coupling scheme is electrospray ionization (ESI-MS), involving a high potential between a tip and the MS [88]. Important parameters include the flow rate and the sharpness of the tip. Outer walls have to be hydrophobic in order to avoid the solution from spreading out of the end of the microchannel and forming a big droplet. As stated above, the company Agilent commercialized a HPLC system with a microchip integrating the separation column and the nanospray emitter [54]. Both optical detection methods and MS involve bulky equipment that is difficult to fully integrate on a chip, even if significant progress has been done in this direction over the last few years [70, 89–91]. Due to their much easier integration, electrochemical detection (ECD) methods have attracted considerable interest and are promising candidates whenever portability is a requirement. Unlike LIF, ECD does not involve any labeling steps. Different modes have been integrated, including amperometry, conductivity, and potentiometry [92–95]. Though many groups have demonstrated a successful integration of ECD, a disadvantage of this method, as compared to LIF or MS, is its relatively poor sensitivity [96–98]. Micro- and nanoelectromechanical systems (MNEMS), such as suspended beams, are powerful detection tools [99]. Similar to atomic force microscopy (AFM), it is possible to weigh a very small quantity of analytes by monitoring the static deflection or the shift in the resonant frequency of a suspended structure upon binding. The method is versatile and highly sensitive and does not require any labeling procedures. While resonators offer better performancs in air, static deflection is usually better suited to operate in liquids as it is not subject to viscous damping. Instead of trying to integrate cantilevers inside a microchannel, Manalis and colleagues integrated a microchannel inside a cantilever, overcoming viscous damping issues [100, 101]. The device allows real-time, ultrasensitive detection of biological species via the successful coupling of resonant MNEMS and microfluidics. Surface-plasmon resonance (SPR) is another label-free detection method. SPR operates by monitoring the local change in the index of refraction near the region of a gold surface upon a binding event. Recently, Ly and coworkers integrated SPR with microchip CE, and reported the first results of protein separation integrated

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with real-time, label-free detection by SPR [102]. The need for a universal, sensitive detection method is still present, and many groups have reported the coupling of other detection techniques with microfluidic devices. Examples include field-effect transistors (FETs) [103–105], nanowire-based detection [106–108], hybridization and microarrays [109–111], nuclear magnetic resonance (NMR) [112], holographic refractive-index detection [113], backscattering interferometry [114, 115], and Raman spectroscopy [116–120]. Interested readers can refer to a recent review of unconventional detection methods in microfluidics [121]. 4.1.1.7  Pretreatment

As highlighted above, one of the key advantages of miniaturization is the possible integration of multiple functions on a single device. A simplistic example of DNA analysis would start with the capture of a cell of interest from a sample. After lysis, nucleic acids would be extracted and purified before selective amplification by polymerase chain reaction (PCR). Results would be obtained after immunoassay, CE, or hybridization on a DNA array. The same approach could be extended to RNA and transcriptome analysis by adding a reverse transcription step before amplification. PCR is a widely used technique able to create a large number of copies of a selected part of a DNA molecule. The principle of the method is illustrated in Figure 4.4. The idea consists of replicating a DNA template using DNA polymerase, an enzyme that catalyzes the polymerization of nucleotides at a temperature around 72°C. Before this, the double-stranded DNA (dsDNA) has to be broken into two single-stranded DNA (ssDNA). This is simply done by heating the molecule at 95°C, a process called denaturation. The target sequence is selected by designing and synthetizing primers, a short DNA strand with the desired sequence, which selectively binds to the template (annealing) at a temperature ranging from 50°C to 65°C. Thus, from a technical point of view, the amplification process consists of heating and cooling a solution, including the DNA template, some nucleotides, DNA polymerase, and the primers. The integration of PCR on microchip and its coupling with CE has received a lot of interest from the microfluidics community [122–124]. The reduced thermal mass and the integration of thin film heaters in microsystems allow very fast heating and cooling, dramatically reducing cycling times [125]. Basically, two approaches have been employed. The stationary method consists of introducing the reagents into a microwell integrated with a heating system. For instance, Northrup and colleagues demonstrated the coupling of PCR with CE by integrating polysilicon heaters over one of the reservoirs of their microchip [126]. The same group reported an instrument to perform real-time monitoring of the PCR using fluorescence techniques [127, 128]. The whole system fitted into a briefcase. Real-time quantification can be used: (1) to monitor the PCR efficiency before further processing, or (2) to directly detect the presence of the target sequence, which could be the signature of a pathogen. For instance, microfluidic PCR was used to amplify and detect a large variety of DNA and RNA targets, including the signature of human immunodeficiency virus (HIV) associated with acquired immune deficiency syndrome (AIDS). Marcus, Anderson, and Quake recently reported the parallel integration of reverse transcriptase-PCR (RT-PCR) in a PDMS microfluidic chip [129].

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Figure 4.4  Principle of the amplification by polymerase chain reaction (PCR). The objective is to isolate and create a large number of copies of a small, selected part of a long DNA molecule. First, the double strand is split into two single strands by heating (denaturation). At lower temperatures, primers (some short synthetic DNA strands) specifically bind to the ends of the target sequence (annealing). The last step of the cycle is extension, where the DNA template is replicated by a DNA polymerase at an intermediate temperature. Two copies of the specific fragment are obtained after three thermal cycles.

The second approach, continuous-flow PCR, consists of flowing the reagents though a microchannel running over heating zones at different temperatures (Figure 4.5) [130]. This principle avoids switching temperatures and allows an easier interface with continuous-flow separation methods. Using this design, an additional extension step is added between the denaturation and the annealing, which may result in an unintentional replication, and thus decreased efficiency. One way to overcome this issue is to use a circular design [131]. Quake and coworkers demonstrated a PDMS rotary device for PCR, integrating valves for isolation and peristaltic pumping, using only 12 nL of sample [132]. The membrane of a cell has to be open or broken to access its contents. One possible way to break the cellular membrane (in a process called “lysis”) is to induce an osmotic shock, simply by introducing the cell into a detergent (e.g., SDS or triton X-100) or simply deionized (DI) water [133]. Other methods include thermal

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Figure 4.5  Continuous-flow PCR on a microchip. The DNA template including the target sequence, nucleotides, DNA polymerase, and primers are introduced in the inlet and pushed through the serpentine microchannel, running through the different temperature zones.

treatment (4 minutes at 94°C), mechanical shear, electrochemical hydroxide generation, and electroporation [134–138]. Once the cell has been lysed, the genetic material is released along with other molecules, such as proteins and metal complexes, that would eventually bind to nucleic acids and affect the performance of the PCR. Purification is then essential to isolate DNA or RNA and wash away possible inhibitors. This step is known to be one of the most difficult and time-consuming of nucleic acid analysis. Christel et al. first demonstrated on-chip solid-phase extraction (SPE) using an array of 5,000 silicon pillars, about 200-mm-deep and 18 mm in diameter [139]. DNA was first introduced in a chaotropic salt solution allowing binding to the silicon dioxide surface and denaturation of proteins. Next, the device was flushed with trisEDTA or water in order to elute nucleic acids. Microchip SPE was also demonstrated using a microchannel packed with silica particles [140]. Cady et al. reported the integration of RT-PCR with purification by SPE [141, 142]. Mathies and coworkers integrated CE with a purification step using a microchamber filled with an affinity capture matrix (an acrylamide-copolymerized oligonucleotide) [143]. Another reported approach is to use dielectrophoretic forces to trap DNA [133]. DNA was trapped by an AC field while RNA, proteins, and membrane debris were pushed away by a small DC field. A preconcentration step is often required for the detection of low-abundance molecules [144]. For instance, there is no equivalent of PCR amplification for proteins. SPE is a common method for both purification and preconcentration [145]. Using a packed microchannel, Oleschuk and colleagues reported a 500-fold concentration by on-chip SPE integrated with a CEC [146]. Electrokinetic methods have been developed, making possible preconcentration without the need for integrating a porous matrix. For instance, IEF, previously introduced as a separation method, is a common method for preconcentrating proteins, using a gradient of pH. Alternative methods take advantage of an electric field gradient [electric field gradient focusing (EFGF)] or a temperature gradient [temperature gradient focusing (TGF)] to focus analytes [147, 148].

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In field-amplified stacking (FAS), a low-conductivity sample solution is injected into a high-conductivity background solution. With the electric current constant along the microchannel, the electric field in the sample solution plug is larger than in the background solution. The resulting difference of electrophoretic velocity results in an accumulation of the analytes at the interface between the sample and the background solution. Jacobson and Ramsey demonstrated the integration of on-chip FAS with CE [149]. Thousand-fold concentration has been reported by Santiago and colleagues [150]. In isotachophoresis (ITP), the sample solution is introduced between a leading electrolyte and a terminating electrolyte. The mobility of any analyte is lower than the mobility of leading ions and larger than the mobility of terminating ions. As an electric field is applied, analytes are first separated according to their electrophoretic mobilities; then they reach a steady state where every band travels at the same speed (isotachophoresis is derived from the greek “iso” meaning “equal,” “tachos” meaning “speed,” and “phoresis” meaning “to be carried”). Due to the electric field distribution, out-diffusing analytes are brought back into the band. The concentration in the band depends only on the concentration of the leading electrolyte. Recently on-chip ITP coupled with CE has been demonstrated for a million-fold concentration of fluorescent dyes within 2 minutes [151, 152]. While it may be more difficult to carry out, as a separation technique, an optimized ITP results in a performance that is superior to more conventional methods. In microfluidics, the self-sharpening of ITP has proved to be very useful, overcoming the “race-track” effect (dispersion induced by a turn) [153, 154]. Readers interested in a detailed overview of preconcentration techniques may refer to the recent article from Sueyoshi and colleagues [144]. Preconcentration schemes using nanofluidic devices will be detailed later in this chapter. 4.1.1.8  Integrated Systems

The company Medimate (a spinoff of the University of Twente, the Netherlands) proposes a handheld device able to measure the concentration of lithium ions in blood. The microfluidic chip integrates separation by CE and detection by conductivity detection. While standard analysis in a hospital lasts at least an hour, this system allows accurate measurements of lithium levels in less than 2 minutes virtually anywhere [155, 156]. However, most of the commercial microfluidic chips (e.g., Caliper and Agilent) integrate only a simple network of microchannels, sensors, and actuators being off the chip. On the other hand, many highly integrated devices have been developed in academic labs, demonstrating the full potential of lab-on-chips. As early as 1998, Burns and coworkers reported a DNA analysis device coupling metering and mixing of reagents, PCR, CE, and detection on-chip [157]. The chip integrated microchannels, heaters, temperature sensors, and fluorescence detectors. The same year, a device able to lyse cells, and then amplify, inject, and separate DNA was reported by the Ramsey’s group [134]. The same group introduced highly parallel capillary array electrophoresis devices integrating 96, then 384, radial lanes for high-throughput genetic analyses [158, 159]. Detection was performed via a dedicated rotary confocal fluorescence scanner. More recently, Easley and coworkers reported a system integrating lysis, purification by SPE, PCR, and CE [160]. The microfluidic system was able to drive a complete DNA analysis

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within less than 30 minutes starting from whole blood. Sanger sequencing was also integrated on-chip by coupling affinity-capture purification, PCR, and CE, resulting in a read length of 556 bp with an accuracy of 99% generated within 35 minutes [161]. Recently, Bontoux and colleagues from CNRS reported the integration of a whole transcriptome assay (i.e., cell capture, lysis, reverse transcription, and amplification) of a single cell on a PDMS chip, with impressive performances compared to more conventional methods [162]. As already mentioned, Fruetel, Renzi, and colleagues from Sandia national laboratories fabricated a handheld instrument (mchemlab) dedicated to the separation of proteins integrating a microchip CE, a high-voltage power supply, the pumping system, a LIF module and the associated electronics [51, 163]. Skelley and coworkers developed the Mars organic analyzer, an integrated CE instrument dedicated to extraterrestrial amino acid detection [19]. A system integrating a portable highvoltage power supply and a CE module with ECD was reported [164]. Another example of a portable system is the microfluidic platform described by Culbertson et al., which is able to separate amino acids in reduced-gravity and hypergravity environments [165]. Recently, isolation and lysis of a single cell, followed by labeling, injection, and CE separation of proteins was demonstrated on a PDMS microfluidic platform [166]. Single counting of the molecules was performed using an off-chip fluorescence system. The integration of immunoassays is also an important application field of microfluidics, with already a few commercial products. One example is the Gyrolab system, commercialized by the company Gyros, integrating an enzyme-linked immunosorbent assay (ELISA) on a CD format. Fluids are driven through the microchannels using centrifugal forces. The flow is controlled by hydrophobic patches patterned on the surface of the hydrophilic channels. The station also integrates a LIF unit for an accurate detection. 4.1.2  Microfluidics Beyond Biomolecular Analyses 4.1.2.1  Cellular Handling and Analysis

One unique feature of microsystems is their ability to interrogate individual cells. Biologists have essentially been working on large populations of cells. However, part of the information is lost as measured quantities are averaged over this large, often heterogeneous population. For instance, most of the cells within a tumor tissue might be healthy, while significant heterogeneity exists even among the cancerous cells [167, 168]. The shift to single-cell analysis avoids the loss of information associated with averaging. Of course, because of this heterogeneity, the objective is not to observe only one cell, but to probe individually each cell of a large population in order to obtain relevant statistics. The ability to pattern features at the cell scale added to the parallel, automated processing and the high level of control that make lab-on-chips attractive tools for single-cell analysis. Thus, a number of techniques have been developed to grow, select, handle, and probe individual cells in a controlled and automated fashion. In Section 4.1.1.7 we described how cells can be lysed to access their molecular content. Indeed it is becoming clear that more and more lab-on-chips will enable multiscale analysis, starting from the cell down to the biomolecular level.

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Reproducing in vitro the in vivo extracellular matrix (ECM) is a key to modern cell analysis. The devices illustrated in Figure 4.6 are remarkable examples of the level of control attainable using microfluidics. The first device [Figure 4.6(a)] takes advantage of the laminar flow in a microfluidic junction to stimulate two halves of a cell with different solutions, for instance media with different concentrations (chemical stress), or at different temperatures (thermal stress) [170, 171]. In the device shown in Figure 4.6(b), a linear gradient of concentration is generated by splitting and recombining. Using this technique, migration of human neutrophils by chemotaxis was observed in a gradient of interleukin-8 (IL-8) [169]. Using microfabrication techniques, a complex three-dimensional scaffold can be produced to mimic the complex in vivo environment and study the interaction between cells and ECM [172, 173]. Microfluidics also allows the control of the mechanical shear applied on adherent cells [174, 175]. Using a bed of flexible posts as a substrate, Tan and coworkers introduced a powerful though simple method to evaluate the distribution of forces exerted by cells on the ECM [176]. Their idea was to evaluate focal adhesion forces by measuring the deflection of these flexible posts fabricated in PDMS. After growth or stimulation, different techniques have been developed to integrate handling and sorting of cells on a chip. Dielectrophoretic forces have often been used for this purpose, taking advantage of the difference in the electrical properties of cells [177, 178]. Dielectrophoretic continuous-flow separation of viable and nonviable yeast cells has been demonstrated [179]. In addition, it is possible to separate cells using the affinity between the cells and a surface. Recently, the detection of rare circulating tumor cells from whole blood has been demonstrated using the interaction between cells and silicon microposts coated with antibody (EpCAM), allowing an early detection of cancer without the need of an invasive biopsy [180]. Fluorescence-activated cell sorters (FACSs) have been integrated on

Figure 4.6  Microfluidic control of the cell environment: (a) generation of a chemical stress and (b) a concentration gradient for the study of chemotaxis. (Adapted from [169].)

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chip, where cells are automatically sorted depending on their fluorescence properties [181, 182]. In flow cytometry, cells are aligned, often by hydrodynamic focusing, and flow through a detection system able to count and analyze one or multiple parameters [183]. One original example is the deformability-based flow cytometer developed by Lincoln and colleagues, which is able to measure the deformability of cells by stretching them as they flow through an optical trap [184]. While flow cytometry on-chip is still limited compared to state-of-the-art flow cytometers, especially in terms of the number of parameters analyzed, opportunities offered by system integration may lead in the future to highly miniaturized and automated devices. An excellent overview of microfluidics applications in cellomics can be found in recent articles [167, 168]. 4.1.2.2  Microreaction, Bubbles, and Droplets

Miniaturizing chemical proceses on a chip presents a number of advantages [185, 186]. Microstructured reactors allow a very efficient heat and mass transfer. While using large quantities of potentially dangerous chemicals often involves drastic security measures, small volumes associated with microreactors make the containment of an incident much easier and safer. On the other hand, parallel operation of multiple microreactors allows relatively large throughput, even to production scale. Because the requirements in microreaction are different from bioanalytical applications (i.e., large pressure, elevated temperatures, and strong chemicals), silicon, glass, ceramics, or metals are often preferred over polymers. German institutes like the Institut für Mikrotechnik Mainz (IMM) or the Forschungszentrum Karlsruke (FzK) (from where originates the LIGA fabrication technique) have been intensively working on industrial applications of microfluidic reactors. Losey and colleagues from MIT fabricated a device for heterogeneously catalyzed multiphase reactions [187]. The device, integrating heating, temperature measurement, and porous-silicon posts, was applied to the hydrogenation of cyclohexene with a mass transfer coefficient 100 times better than conventional laboratory reactors. Kobayashi et al. also demonstrated hydrogenation in a microchannel coated with palladium [188]. By carefully adjusting the flow rates, it was possible to flow hydrogen through the center of the tube while tetrahydrofuran (THF) was coflowing along the inner surface of the channel. A highly integrated PDMS device was reported for the synthesis of a radio-labeled imaging probe [189]. A process with sequential steps (i.e., processes—[18F] fluoride concentration, water evaporation, radiofluorination, solvent exchange, and hydrolytic deprotection) was implemented in an automated fashion, resulting in a higher yield, higher purity, and shorter synthesis time relative to more conventional approaches. Finding the correct parameters for the crystallization of proteins is a labor-intensive work in structural biology. Using a highly parallel architecture integrating 480 PDMS valves, Hansen and coworkers demonstrated the automated screening of 144 crystallization reactions, each of which using only 10 nl of the protein sample [190]. This technology is marketed by Fluidigm. Other applications of PDMS large-scale integration targeted by this company include genetic sequencing or gene expression analysis. Droplet microfluidics is another exciting branch of microfluidics [192, 193]. Instead of using continuous flow, the idea is to generate and handle droplets or bubbles

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acting as isolated microreactors. The discretization of the flow has also led to the term “digital microfluidics” by analogy with electronics. Droplet microfluidics can be divided into two subcategories, namely open microfluidic platforms and closed microfluidic platforms. In open microfluidic platforms, droplets are handled on an open substrate using, for example, electrowetting-on-dielectric (EWOD), dielectrophoresis (DEP), surface acoustic waves (SAW), vibrations, capillary, magnetic, or thermally-induced forces [194, 195]. In closed microfluidic platforms, droplets or bubbles are generated and handled inside microchannels. Figure 4.7 illustrates one example of a droplet generator [191]. In this example, droplets containing three different aqueous solutions are formed in oil. Opposite to continuous flow microfluidics, there is no dispersion as the reaction is restricted to the volume of the droplet. The ratios of the volumes of the different solutions inside the droplets are controlled by the relative flow rates. Using this configuration, with an inert divider (solution b) separating two reactants (solution a and c), the reaction can be considered to begin as soon as the droplet is formed since the content of such a small volume is rapidly mixed. As a consequence, this system allows excellent control of chemical reactions in time, which is simply a linear function of the mixing distance. Mixing can be further speeded up using winding channels. A direct application of this microfluidic platform is the measurement of reaction kinetics [196]. The kinetic activity of ribonuclease a (RNase a) was obtained with better than millisecond resolution using submicroliter volumes of solutions. Chan et al. reported a glass device for the production of CdSe nanocrystals in octadecene droplets at relatively high temperatures (240–300°C) using Long-chain perfluorinated polyethers (PFPE) as a carrier fluid [197]. Fast screening of protein crystallization condition was also demonstrated by varying the relative flow rates under computer control [198]. Raindance Technologies markets a droplet-based microfluidic platform (RDT 1000) dedicated to targeted sequence amplification by PCR. This company aims to apply their “Rainstorm” technology to many other aspects of DNA analysis, providing an alternative to more standard water-in-oil emulsions used, for instance, in pyrosequencing. Another straightforward application of droplet microfluidics is the synthesis of particles. Nisisako and colleagues demonstrated the synthesis of polymer microbeads with diameters in a range 30–120 mm by UV-curing droplets of acrylic monomer formed in water [199]. Janus particles (i.e., with two fused hemispheres made from two different materials) were also formed using two differently colored, coflowing monomers. A similar technique was reported by the Whitesides’ group to produce microparticles with various shapes, size, and composition, either by polymerization of the monomer or by lowering the temperature to reach a solid state

Figure 4.7  Microfluidic generation of droplets. a droplet is formed in oil from three aqueous solutions. Once the droplet is formed, rapid mixing occurs by recirculation. (Adapted from [191].)

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[200]. Dendukuri and colleagues recently introduced an elegant continuous-flow alternative [201]. Briefly, a photopolymer flowing into a PDMS microchannel is cured by exposition to UV through a transparency mask. The shape of the microparticle is then defined by the mask, polymerization close to the channel walls being inhibited by the oxygen diffusing through the PDMS.

4.2  Handling Ions by Electrokinetic Effects in Nanochannels The question that may arise is: What is so different in a nanochannel compared to a macro- or a microchannel? Readers familiar with microfluidics already know some specificities of the flow at the microscale, such as the laminar regime or the high surface-to-volume ratio improving heat and mass transfer and uncovering electrokinetic and capillary effects. Since Chapter 2 already covered nanofluidic theory, here we give only a phenomenological description, without the mathematical formalism, of the most important effects emerging at the nanoscale illustrated by recent experiments and highlighted by applications. The objective is to give the reader a practical understanding of the possibilities offered by nanofluidics. 4.2.1  Nanofluidic Resistors as Surface Charge Sensors

Liquids usually contain positive and negative ions originating from dissociation or dissolution of impurities or additives. The liquid in a channel is in a contact with the solid surface of the walls. Most surfaces submerged in a liquid gain a net charge density originating from chemical reactions (e.g., protonation or deprotonation), adsorption, or defects in a crystalline structure. For instance, a glass filled with a solution of pH larger than 3 presents negative charges on its surface attributable to the deprotonation of silanol sites. While ions in the bulk of the liquid are mobile, charges present on the surfaces are fixed. Due to Coulomb interactions, the charged surface attracts mobile ions with opposite charges (counterions) while co-ions are repelled. These arrangements lead to the local formation of a charged layer shielding the surface charges. The so-called electrical double layer (EDL) (Figure 4.8) is often modeled by a fixed layer of adsorbed ions, the Stern layer, and a mobile layer, the diffuse layer. The zeta potential z is the electric potential at the shear plane between these two layers. The thickness of this screening layer is the Debye length 1/k. As a rule of thumb, for a symmetric, monovalent electrolyte in water at 298K:



1 1 [in nm] = k ze

�

ee0 kb T 10 ≈� 2n n[in mM]

where e is the electron charge, e is the permittivity of free space, e is the dielectric constant of the liquid, kb is the Boltzmann constant, T is the temperature, n is the ionic concentration, and z is the valency [202]. Considering a monovalent electrolyte, 1/k is in the range of 0.1–100 nm for a concentration ranging from 0.01 to 10 mM. As will be seen, the proximity between the dimension of a nanochannel and the Debye length has important consequences. In a microchannel, 1/k is negligible

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Figure 4.8  Representation of the electrical double layer at a solid/liquid interface. y is the electrical potential. Negative surface charges are screened by the Stern layer and the diffuse layer. The Stern layer is formed by adsorbed immobile ions. The mobile diffuse layer is located outside the shear plane. The zeta potential z is at the shear plane. The Stern layer and the diffuse layer form the electrical double layer.

compared to the characteristic transverse dimension (i.e., 1–100 mm). As a result, the electric potential is neutral in the bulk of the liquid in the absence of an electric field. As an electric field is applied along the channel, the diffuse layer is mobilized. In a channel small enough (typically with a diameter inferior to the millimeter), the diffuse layer drag the whole liquid leading to the well-known plug-like velocity profile of the electro-osmotic flow (EOF). As the ionic concentration decreases and the dimensions shrink, the situation become slightly different. For a transverse dimension inferior to 2/k, the opposing EDLs overlap (Figure 4.9). As a consequence, the EOF is reduced and its velocity profile is not flat anymore [203–205]. Without any applied voltage, the electroneutrality principle implies a balance between positive and negative charges:

Nsurface charges + Nco-ions = Ncounterions where N is the number of charges from the specified category. In a microchannel, the quantity of surface charges is usually negligible (Nco-ions = Ncounterions in the bulk of the liquid). However, as the ionic concentration decreases and the dimensions shrink, the number of surface charges starts to play an important role in this balance, as it becomes closer to the number of mobile ions. Ultimately, only counterions stay present inside the nanochannel, their number being defined by the number of surface charges independently of the ionic concentration (Nsurface charges = Ncounterions). As demonstrated by Stein and colleagues, this simple scaling effect results in a con-

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Figure 4.9  Electric potential and ionic concentration in a microchannel and in a nanochannel. In a microchannel, the EDL is much smaller than the characteristic transverse dimension. The potential is neutral in most of the channel. In a nanochannel, the Debye length is not negligible anymore, resulting in an excess of counterions in the whole liquid.

ductance G independent of the ionic concentration but governed by surface charges as illustrated in Figure 4.10 [206]. Considering a monovalent solution and a slit-like channel with a width w, a height h, and a density of surface charges s, the authors gave the following expressions for the two limiting cases:

G= G=

I 2|s | ≈ 2ne m wh for n >> (high ionic concentration regime) E eh

� � 2|s | I 4ee0 k0 T for n << ≈ |s | m w 1 + (low ionic concentration regime) E emh eh

where I is the electric current, E is the electric field, and m is the ion mobility. The second equation corresponds to the electrophoretic transport of counterions, with the second term inside the brackets being the conductance enhancement due to electro-osmosis. It should be stressed that the surface-charge-governed regime is observed not only in nanochannels but also in larger channels (50 mm wide and 1 mm high) for ionic concentrations low enough. The quantity 2|s|/eh is the concentration of counterions inside the nanochannel at low ionic strength. In this regime, this concentration is always larger than the concentration in the background electrolyte (leading to the increase of conductance). Thus measuring the conductance in a nanochannel at low ionic concentration provides direct information on the surface state. In their paper, Stein et al. were

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Figure 4.10  Conductance G of a slit-like channel filled of height h as a function of the ionic concentration n for: (a) a constant height h with the density of surface charges s as a parameter and (b) for a constant density of surface charges s with the height h as a parameter. For low concentrations or small dimensions, the conductance becomes governed by the surface charges and independent of the ionic concentration.

able to detect variations of the surface charges as a function of the pH for hydroxyl or amine groups [206]. Using the same approach, Karnik and coworkers observed a change in conductance following the binding of a protein (streptavidin) on the surface of a nanochannel [207]. The silica channel were first treated with a solution of aminosilane APTMS ((3-aminopropyl)trimethoxysilane) in ethanol, leading to amine groups on the surface. Next the surface was biotinylated using a solution of solution of Sulfo-NHS-SS-Biotin (sulfosuccinimidyl 2-(biotinamido)-ethyl-1,3dithiopropionate) and the residual amine groups were passivated using a solution of NHS (n-hydroxysuccinimide). The resulting surface demonstrated a low density of surface charges, highlighting the possible increase of conductance due to a streptavidin-biotin binding event. While the charges of the streptavidin increase the conductance by keeping the concentration of mobile charges constant inside the nanochannel as the surrounding electrolyte concentration decreases, the volume exclusion induced by the presence of the protein (~5.4 ´ 5.8 ´ 4.8 nm3) limits the volume accessible to mobile charges, so decreases the conductance, as the electro-

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lyte concentration increases. Though this device demonstrated a very high sensitivity to surface charges, the response time is limited by the diffusion of the analytes into the nanochannel. This can be improved by applying a pressure-driven or an electro-osmotic flow, decreasing the response time to 1–2 hours [208]. As explained in the first part of this chapter, electrical, label-free detection methods are highly desirable for an easier integration on microfluidic platforms. These surface charge sensors could be used as a detection method for immunoassays, in which the reaction is usually limited by the diffusion of the analytes to the surface. The extreme confinement provided by a nanochannel would allow faster reaction kinetics. 4.2.2  Semipermeability and Preconcentration 4.2.2.1  Donnan Exclusion and Selective Diffusion

At low ionic strength, the flux of co-ions through a nanochannel is prohibited by the fixed charges on its inner surface. Semipermeable ion-exchange membranes take advantage of this filtration mechanism known in membrane science as the Donnan exclusion [209]. In biology, the Gibbs-Donnan equilibrium regulates the flow of charges species between the cell interior and exterior through ion channels [210]. Plecis and colleagues observed the selective diffusion of ions through a nanoslit [211]. The phenomenon is illustrated in Figure 4.11. The authors provided a theoretical model describing the passive transport of ions for varying ionic strength, valid for a low concentration of the diffusing ion low compared to the background electrolyte concentration. The charge of proteins is highly dependent on the pH, the molecule being neutral for a pH equal to its isoelectric point (pI). As a direct consequence, the diffusion of a protein through a charged nanochannel depends on the pH of the buffer. As the pH reaches the pI, the protein becomes uncharged and the interaction with the membrane is purely steric. By adjusting the pH around this value, the molecule is set positively or negatively charged, providing a way to authorize or not its passage through the channel. Using this “on-off” property of molecular transport, Ku and colleagues demonstrated the separation of bovine serum albumin (BSA) and bovine hemoglobin (BHb) according to their respective pI by diffusion through a PC

Figure 4.11  Selective diffusion of counterions though a nanochannel. At low ionic strength, the diffusion of co-ions through the nanochannel is prohibited by the presence of fixed charges on the wall of the nanochannel resulting in a selective permeation.

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membrane [212]. The membrane was coated with gold and modified using a SAM of alkylthiols according to the procedure published by Martin et al. [213, 214]. Schoch and colleagues demonstrated a similar approach using a pyrex nanofluidic chip [215]. The highest permeation was observed when the proteins where neutral at their pI. A shift in the pI was measured and attributed to reversible adsorption on the walls of the channels. 4.2.2.2  Preconcentration

As stated in the first part of this chapter, effective preconcentration schemes are highly desirable for integrated microfluidic devices to enhance detection limits, especially for the analysis of trace levels of proteins. Several alternative nanofluidic methods have been reported. We already introduced a few preconcentration methods in the first part of this chapter, including solid-phase extraction (SPE), isoelectric focusing (IEF), field-amplified stacking (FAS), and isotachophoresis (ITP). More details are available in Section 4.1.1.6. Pu and coworkers observed the electrokinetic-driven flow of fluorescent dyes, either fluorescein (negatively charged) or rhodamine 6G (positively charged) through a 60-nm-deep negatively-charged glass nanochannel [216]. Using this configuration, they observed a 100-fold enrichment of fluorescein and a 500-fold depletion of rhodamine at the cathode by applying 1,000V for 30 seconds. The complete picture of the electrokinetic phenomena involved is relatively complex [217]. Figure 4.12 illustrate the distribution of ionic species around the anodic side of a nanochannel as an electric field is applied. An ideal case is considered where only counterions are flowing through. Co-ions and counterions are moving in opposite directions. As there is no supply of co-ions on the anodic side of the nanochannel, this results locally in a decrease of their concentration. At low electric fields, diffusion balances electromigration resulting in the formation of a diffusion layer. The counterions’ concentration decrease is equal to the co-ions’ concentration decrease. Within the diffusion layer, the concentration is globally reduced and electroneutrality is still maintained. When the electric field is increased, the concentration of the electrolyte is reduced. As the concentration locally tends toward zero, the diffusion reaches a maximum value. The electrical current at this value is known as the limiting current. For larger electric fields, electromigration overcomes diffusion leading to a diffusion-limited regime called “concentration polarization” (CP). The ionic concentration in the diffusion layer linearly decreases, and a space charge layer with an excess of counterions appears between the diffusion layer and the anodic side of the channel. Similar to the more usual EDL, this layer could be mobilized by the application of a tangential electric field, leading to electro-osmosis of the second kind [217]. To complete the picture, it is also necessary to take into account the global flow induced by a difference of pressure or by electro-osmosis of the first kind (going the same direction as counterions). Tallarek’s group studied CP using a simple PDMS microchannel integrating a neutral or negatively-charged hydrogel plug photopolymerized in situ [218–220]. The average size of the pores was around 2 nm. Using BSA as a target molecule, these experiments demonstrated a purely steric sieving in the case of the neutral hydrogel, while CP was obtained for a negatively charged matrix. Numerical simu-

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Figure 4.12  Concentration polarization under an applied electrical field at the anodic entrance of a nanochannel: (a) for a small electric field, diffusion balances electromigration. The diffusion layer presents a reduced concentration, but electroneutrality is still maintained; and (b) for a large electric field, diffusion does not balance electromigration anymore. A space charge layer appears between the diffusion layer and the anodic side of the channel. The mobilization of this space charge layer upon application of a tangential electric field induces a flow called “electro-osmosis of the second kind.”

lations and theoretical models were also given. Ramsey’s group reported the integration of porous silica microfilters able to preconcentrate proteins 580-fold within 10 minutes [221, 222]. A similar concentration approach has been demonstrated using the fast-prototyping fabrication method introduced by Bohn and colleagues from the University of Illinois at Urbana-Champaign [223–226]. The technique consists of inserting a polycarbonate (PC) membrane between two molded PDMS slabs. The same group reported another method to integrate membranes between

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PMMA microchannels [227]. Beyond filtration and concentration, these integrated membranes were also used to precisely control the injection of an analyte in a channel. For instance, applications included the delivery to the tip of an electrospray ionization-mass spectrometry (ESI-MS) device or a miniaturized lead sensor [228, 229]. Han and coworkers from MIT proposed an improved preconcentration device, also taking advantage of the CP effect [230]. Figure 4.13 illustrates the layout and the concentration protocol. Briefly, the silicon device consisted of nanofluidic slits integrated between two microchannels. A space charge layer was induced by application of an electric field normal to the entrance of the nanochannel. Upon application of a tangential electric field, co-ions started to accumulate in front of the space charge layer. A remarkable concentration ratio of 107-fold was achieved within only one hour. The flux of co-ions required to reach such a concentration speed are not achievable with normal electro-osmotic flow. While the details of the strong trapping mechanism are not clear, electro-osmosis of the second kind (introduced above) may be involved. Indeed electro-osmosis of the second kind has been reported to result in velocities greater than those usually predicted by Smoluchowsky’s theory by one or two orders of magnitude [217]. This has been confirmed by the observation of strong recirculation patterns close to the nanochannel’s entrance [231]. This preconcentration mechanism has proven to be useful to increase kinetics of enzymatic reaction normally limited by the diffusion [232, 233].

Figure 4.13  Nanofluidic device for protein preconcentration: (a) layout and (b) concentration mechanism. A space charge layer (ion depletion region) is induced by an electric field En normal to the entrance of the nanochannel. Upon application of a tangential electric field Et, co-ions accumulate in front of the space charge layer. (After: [230].)

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4.2.3  Routing Ions by Nanofluidic Electronics 4.2.3.1  Nanofluidic Diode

Similar to an electronic diode, a nanofluidic diode acts as an ionic current rectifier. For a given absolute value of transmembrane potential, the current-tension I-V curve demonstrates a larger value of current for a forward bias and a reduced value for a reverse bias. Molecular ratchets have received a large interest from the scientific community. This concept has been used to explain the rectification of brownian noise in molecular motors or to design temporally or spatially asymmetric systems where “force-free” motion is induced by rectification of a fluctuating force, even if the temporal average is zero [234–238]. Such behavior was observed in certain conditions by studying electric current through a conical nanopore [239]. The conical shape was formed by etching a single ion track through a polyethylene terephthalate (PET) foil. Only one side of the membrane is in contact with the etchant, while the other side of the chamber is filled with a chemical that stops the etching process monitored by the electrical current. The rectification mechanism is illustrated in Figure 4.14 [240]. No rectification was observed for nanopores without charges, highlighting the importance of electrostatic interactions in this mechanism. Ion pumping was obtained by applying an ac voltage across the membrane.

Figure 4.14  Ionic rectification through a conical nanopore. Overlapping of the electrical double layers occurs at the narrow end of the pore. As a reverse bias is applied, a potential barrier appears trapping counterions.

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Recently, Cheng and Guo observed ionic rectification through 4- and 20-nmdeep silica nanochannels placed between two electrolytes of different ionic strengths [241]. Similar to the conical nanopore, EDLs overlap on one side (low ionic strength), while on the other end they do not (high ionic strength) as long as no voltage is applied. However, opposite to the nanopore, the EDL’s configuration heavily relies on the voltage applied. Upon application of a reverse bias (anode on the high concentration side), the EDL’s overlap extends all along the nanochannel. The electrical current originates solely from the flux of counterions from the low concentration side (equivalent to the conductance in the low ionic concentration regime). As a forward bias is applied, the EDL’s overlap vanishes and the conduction enters the high concentration regime. Pushing further the analogy with electronics, it is also possible to create nanofluidic junctions where main charge carriers are anions on one side while main charge carriers are cations on the other side. As stated above, the concentration of counterions inside a nanochannel (~2|s |/eh) is controlled by the density of surface charges. Two techniques have recently been reported to pattern surface charges inside a nanochannel. Vlassiouk and Siwy from the University of California at Irvine used ion track-etched PET membranes [242]. Half of the conical nanopore was coated with positive amino group, whereas the other half was covered with negatively charged carboxyl groups. Majumdar and coworkers applied a technique named diffusion-limited patterning (DLP) to nanochannels fabricated on silicon using sacrificial layer techniques. In DLP, two different solutions are sequentially injected in a reservoir. During their diffusion, the reactants bind to the channel surface. The pattern is then controlled by the respective time of diffusion. Using both configurations, the authors observed a rectification of the ionic current. The principle of a nanofluidic diode is illustrated in Figure 4.15. It should be stressed out that both counterions and co-ions are flowing through a nanofluidic junction, while only the flux of counterions is rectified in a homogeneously charged conical nanopore. 4.2.3.1  Nanofluidic Transistor

Modification of the surface chemistry is a first way to gain control over the density of surface charges, and ultimately the density and the nature of main charge carriers. This process may be compared to doping in microelectronics. A second way of controlling surface charges is to apply a transverse electrical field through a buried electrode, equivalent to the gate of a field-effect transistor (FET). At the microscale, the direct control of the zeta potential using an external electric field was introduced in the early 1990s, in an attempt to govern electro-osmotic mobility during capillary electrophoresis (CE) [243–245]. Schasfoort and colleagues from MESA+ applied this approach to an integrated microfluidic format [246]. Horiuchi and Dutta demonstrated an electrokinetic control using the liquid flowing into a neighboring channel as a gate, simplifying the fabrication process [247]. At the nanoscale, the potential applied to this gate has an effect on the nature and concentration of ionic species in the whole channel. Gajar and Geis from MIT reported the first nanofluidic transistor as early as 1992 [248]. The microfabricated structure consisted of a silicon nitride channel sandwiched between two polysilicon

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Figure 4.15  Principle of a nanofluidic junction. The diode is composed of a zone with positive surface charges (right) and a zone with negative surface charges (left). Main carriers are anions on the positively charged side while main carriers are anions on the negatively charged side. Upon application of a forward bias, ions overcome electrostatic exclusion and a current is established. On the other hand, the application of a reverse bias induces the formation of a depletion zone and no current passes through the junction.

gates. Solutions of glycerol were used to avoid evaporation but led to relatively long relaxation times. Variation of the conductance was demonstrated as a function of the voltage. More recently, two research groups from EPFL and from Berkeley reported on a similar device [249–251]. The variation of concentration was imaged by fluorescence microscopy. It was shown that surface modification for a nanofluidic FET is equivalent to the concept of doping in microelectronics [252]. Based on this principle, Stern, Geis, and Curtin proposed an interesting device for chemical analyses [253]. The separation mechanism is depicted in Figure 4.16. Briefly, a series of electrodes is integrated over a nanochannel. A packet of counterions is formed by biasing one of these electrodes. This packet can be moved along the nanochannel by switching the potential of the adjacent electrode. However, if the voltage is switched too quickly, larger ions are left behind. Only ions with a diffusion coefficient below a given value (defined by the frequency rate) travel until the end of the channel where a sensor evaluates the ionic concentration by a conductance measurement. By operating the device repeatedly at three different clock rates, it is possible to select ions within a narrow band of diffusion coefficients. While the mechanism illustrates well the level of control given by nanofluidics, no experiments were performed due to fabrication issues. Recently, Karnik and coworkers demonstrated a simple nanofluidic circuit where the flow of proteins to a reservoir was controlled by two nanofluidic FETs. This is a first step towards ionic circuits capable of performing complex operations on biological molecules in an automated fashion.

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Figure 4.16  Separation using the chemical charge coupled device (CCD). A nanochannel is sandwiched between a bottom counter electrode defining the average potential in the channel and a series of top electrodes. (a) As a negative voltage (–V ) is applied on the input gate, a packet of negative ions is formed at the entrance of the channel. (b) This packet of counterions is moved to the right by switching the adjacent electrode to a negative potential (–V2) and adjusting the potential above the packet to a more positive value (–V1 > –V2). (c) If the voltage change happens too fast, larger ions are left behind. By repeating this process with different clock rates, ions can be separated within a narrow range of diffusion coefficients. (After: [253].)

4.3  Separations in Nanofluidic Devices 4.3.1  Batch Separations 4.3.1.1  Free-Solution Electrophoresis

Separation by CE has been the workhorse of sequencing technologies, pushing forward the human genome project. Because of its primary importance, there has been large interest from the scientific community in studying electrophoresis mechanisms and improving separation efficiency. As stated in the first part of this chapter, the free-solution electrophoretic mobility of DNA is independent of the molecular weight in a range from 48 kbp down to 400 bp, and decreases monotonically with decreasing molecular weight for smaller fragments [254]. Electrophoresis experiments in nanochannels demonstrated that the free-solution mobility of oligonucleotides (10–100 bp) depends both on the Debye length 1/k and the characteristic size of the channel h, highlighting the importance of surface interactions [255]. Recent experiments revealed even more intriguing phenomena. Cross and coworkers from the Craighead’s group at Cornell reported a mobility scaling as L–1/2 (where L is the contour length) in a range 2–10 kbp [256]. This behavior was

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attributed to interactions with the surface. Taking these interactions into account, the authors present a model able to predict the mobility within this range of molecular weight. It was explained explained in a previous section (see Figure 4.9) that the velocity profile of an electro-osmotic flow in a nanochannel is not flat as it is at the microscale. Furthermore, ions are more or less repelled from the surface depending on their charges and valency. Combining these two characteristics leads to a separation mechanism close to field-flow fractionation (FFF), a method where a force field is applied perpendicular to a pressure-driven flow. The parabolic velocity profile and the different transverse positions of the analytes lead to the separation. By comparing electrophoretic mobilities of fluorescein (z = –2) and Bodipy (z = –2) in micro- and nanochannels, Pennathur and Santiago demonstrated a method to access both free solution electrophoretic mobility and valencies [257, 258]. 4.3.1.2  Shear-Driven Chromatography

The speed of chromatographic separations scales inversely with the diffusion distance. Thus extremely high speeds are expected in nanochannels. However, shrinking the characteristic sizes induces a dramatic increase of the pressure drop. As stated above, electrokinetic-driven flows in nanofluidics exhibit specificities that compromise chromatographic separations. Shear-driven chromatography was introduced in 1999 by Desmet and Baron [259]. The setup is illustrated in Figure 4.17. As a proof of concept, the authors first demonstrated the ability to drive a

Figure 4.17  Shear-driven chromatography in nanochannels. While shrinking dimensionsare expected to result in faster chromatographic separations, specificities of both pressure- and electrokineticdriven flows at the nanoscale compromises chromatographic separation efficiencies. In shear driven chromatography, nanochannels etched in a silicon substrate are moved along the separation axis while the cover slip is stationary. The flow is driven by the resulting shear.

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tracer plug at a speed as high as 2 cm.s–1 through a 125-nm-deep and 4-mm-wide channel (so an incredibly low aspect ratio AR of ~3.10–5!) without inducing any pressure drop or buildup [260]. Using the C18-coated walls as a stationary phase, ultrafast separations of four coumarin dyes were demonstrated, yielding theoretical plate numbers of in a range of 2,200–6,000. The time needed for this separation was only 0.14 second using a flow velocity of 2.5 cm.s−1 [261]. Using the same technique at a velocity of 1.75 cm.s−1, separation of angiotensin I and II was reported within only 0.17 second. The plate numbers were 1,263 and 3,626 for angiotensin I and II, respectively [262]. Despite these remarkable performances, the very small sample volumes (on the order of the picoliter) that can be injected and the detection problems associated with the nanoscale are challenging issues that have to be solved for this promising separation method to reach its full potential [263]. 4.3.1.3  Electrophoresis in Microfabricated Matrices

Within a porous matrix, two regimes of electrophoresis are usually considered depending on the ratio S = RG/h between the gyration radius RG and the characteristic size of the pore h [12, 31, 264, 265]. In three dimensions, considering N = L/LP >> 1 where LP is the persistence length, RG = LP(N/3)1/2. Just to give an idea of the typical dimensions, a nucleotide is 0.34 nm long: thus the persistence length (~60 nm) corresponds to approximately 175 nucleotides under normal physiological conditions [266]. When S < 1, the molecule is smaller than the pore and fractionation occurs by Ogston sieving [267]. The assumption of this model is that the ratio between the mobility in the matrix m and the free solution mobility m0 is equal to the fraction f of the volume available to the particle. The fractional volume available to a spherical particle was calculated by Ogston in the case of a random array of infinitely long (but not cross-linked) fibers and is given by f = exp(-KC) where K is the retardation factor dependent on the size of the analyte, and C is the concentration of the gel. While this assumption has never been properly tested experimentally, this model has been widely used in the analysis of gel electrophoresis results. For S > 1, DNA molecules move through the matrix like a snake in thick grass by a “reptation in a tube” process similar to the mechanism proposed by de Gennes for entangled polymers [12, 268–270]. At low electric fields and considering N >> 1, the electrophoretic mobility of DNA scales inversely with L. However, this dependence vanishes for long DNA molecules (typically 40 kbp) or for sufficiently high electric fields mainly because the molecules tend to be more stretched and oriented in the direction of the electric field [265, 271, 272]. This effect has greatly limited the attainable throughput in DNA analysis. For characteristic pore sizes close to RG (S~1), a mechanism called entropic trapping overlaps, originating from the heterogeneous distribution of the sizes of the pores [12, 273]. Indeed, in this configuration, some of the pores have characteristic sizes slightly larger than RG while others are slightly smaller. Within a pore, the local entropy is given by the number of conformations accessible by the molecule. A DNA molecule in a pore larger or comparable to RG will relax into a spherical shape of radius RG, reaching a maximum of entropy. To thread its way through the gel, the molecule has to move from pores larger than its relaxed volume to pores smaller than its relaxed volume, where fewer conformations are accessible leading

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to reduced entropy. The presence of these entropic barriers results in a slowing down of the electrophoretic mobility. The larger pores surrounded by entropic barriers are entropic traps, which led to the name “entropic trapping.” As highlighted in the first part of this chapter, intensive research has been carried out to improve separation matrices. While standard matrices present a random distribution of the position, sizes, and the shapes of the pores, microfabrication techniques allow one to shape customized porous media with well-defined dimensions, integrated with other elements (e.g., microchannels, electrodes, or valves). In their seminal papers, Volkmuth and Austin described the electrophoresis of DNA in a silicon-microfabricated array [265, 272]. The array consisted of 150-nm-high cylindrical posts with a diameter of 1 mm separated by 2 mm center-to-centers and capped with a pyrex coverslip. Thus, the vertical dimension was more than two times larger than the persistence length LP (~60 nm under normal buffer conditions). By staining the molecule with ethidium bromide, it was possible to monitor its progression using fluorescence microscopy. The authors observed the hooking of DNA around the posts. Surprisingly, the polymer fully stretched to its contour length L despite the relatively low field strength (1 V.cm–1). Fractionation was demonstrated and attributed to the hook-and-slide mechanism (with a velocity scaling as the inverse of the length). Pulsed-field gel electrophoresis (PFGE) overcomes some of the limitations of standard electrophoresis in term of molecular weight [275]. While the method authorizes separation of DNA molecules as long as 10 Mbp, the technique is timeconsuming (runs take 10–200h) and the process is difficult to optimize. Pulsed field capillary gel electrophoresis (PFCGE) overcomes some of these problems but suffers from a lack of reproducibility mainly due to the aggregation of large DNA fragments under strong electric fields and requires relatively complex and expensive instrumentation (as compared to standard methods) [276, 277]. Using a pulsed-field method in a microfabricated hexagonal array, Bakajin et al. reported the separation of 169-kbp DNA molecules within 10 seconds [274]. DNA was first concentrated in a band by entropic trapping. Next, the band was injected and electric fields were switched sequentially along the two main directions of the lattice. The separation mechanism is illustrated in Figure 4.18. Taking advantage of entropic trapping, Han and Craighead demonstrated the separation of DNA as long as 200 kbp within 30 minutes [271, 278]. The device illustrates well the benefits of the geometrical control over the separation matrix. The separation channel (illustrated in Figure 4.19) consists of microchambers (deeper than Rg) connected by shallow nanoslits (shallower than Rg) defining a series of entropic barriers. Fu, Mao, and Han described a similar device working in the Ogston regime (S < 1) for the fast separation of proteins [279, 280]. A major drawback of these sieving matrices is that they are prone to clogging by large molecules, reducing the repeatability and the lifetime of the device. To overcome this issue, Baba and colleagues proposed a size-exclusion chromatography (SEC) device [281]. The array consisted of wide channels oriented along the flow direction, connected by perpendicular narrow channels. Large molecules were excluded from the narrow gap and moved smoothly through the wide channels. Because of Brownian motion, smaller molecules able to enter the narrow gaps traveled a longer path, leading to the size separation. The device was demonstrated to separate 2 kbp, 5 kbp, and 10 kbp DNA within 5 minutes.

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Figure 4.18  Cross-section of an entropic trapping separation device. The design consists of a series of microchambers connected by nanoslits. Molecules larger than the size of the pore have to unfold to thread their way through the channel, reducing their entropy. As the entropy barrier is larger for longer molecules, it was expected that the mobility scales inversely with the length. However, this is not true for a rectangular nanoslit (height < size of the molecule < width), where larger molecules have a higher contact area with the entrance, so a higher probability to form a hernia entering the channel, resulting in a larger mobility. (After: [271].)

4.3.2  Continuous-Flow Separations

In a batch separation, a plug of sample is injected into a separation channel. Some microfluidic methods of injection were introduced earlier (see Figures 4.1 and 4.2). The operation of batch separation devices consists of sequences of injection and separation steps repeated for each run. On the other hand continuous-flow separation devices operate continuously. The generic principle of continuous-flow separation is illustrated in Figure 4.20. In terms of operation, one of the main differences is the relative direction of the flow and the force field. While in batch separations, flow and separation occurs in the same direction, in a continuous-flow separation, flow and force field are at an angle leading to different trajectories for different compounds. This simple difference results in many advantages. Not only

Figure 4.19  Pulsed-field electrophoresis of DNA in a microfabricated hexagonal array. To illustrate the principle, the limit case of a very long polymer is depicted. For this long DNS, half a period is not sufficient to escape the post around which it is hooked. During the second half of the period, the molecule simply retraces the same path. On the other hand, a smaller DNA has enough time to unhook during half a period and is thus able to progress through the array. (After: [274].)

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Figure 4.20  Batch and continuous-flow separations. (a) In a batch separation device, sequences of injection and separation are repeated for each run. Flow and separation occur in the same direction, leading after detection at the end of the channel to a signal as a function of the time. (b) In a continuous-flow separation process, flow and force field are at an angle. Therefore analytes with different responses to this force field have different trajectories. The signal is given as a function of the position. (After: [282].).

is it possible to detect changes in the composition of the samples in real time, but also variations in the parameters of the separation make optimization easier and faster compared to batch processes where changing a parameter means starting a new run. In batch separation, clogging of the single channel drastically reduces the device reproducibility and the lifetime, while this is usually less of a concern in a continuous-flow device. The spatial distribution of the analytes authorizes collection after separation. The spatial localization also allows an enhancement of the detection by integration of the signal over the time. From a more general point of view, system integration with upstream and downstream functions in the context of a lab-on-a-chip is more straightforward. Continuous-flow separation has been widely investigated in microfluidics using many forces, including electric, magnetic, acoustic, optical, and dielectrophoreti. Interested readers should refer to the excellent review article from Pamme about continuous-flow separations in micro- and nanofluidic devices [282]. Examples include free-flow electrophoresis and free-flow isoelectric focusing, which are continuousflow versions of electrophoresis and isoelectric focusing, respectively [59, 283]. Again, the ability to tune the geometry of a nanofluidic matrix defined with microfabrication techniques has allowed a new generation of analytical devices to emerge. More specifically, this high level of control has allowed the introduction of anisotropy in the sieving properties of the matrix, making possible the integration of new separation principles in continuous-flow devices. Austin’s group at Princeton has pioneered the application of smart designs to nanofluidic continuous-flow separation matrices. The device illustrated in Figure 4.21 takes advantage of the arrangement of silicon obstacles to rectify lateral diffusion

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Figure 4.21  Asymmetric diffusion array for continuous flow separation of DNA. Black rectangles represent microfabricated obstacles. The molecule at point A moves under the action of an electric field. Diffusion results in a stochastic displacement perpendicular to the field axis. Because of the arrangement of the obstacles, two trajectories are possible depending on the extent of the lateral diffusion. Most probably the molecule will arrive in B1, but may also arrive in B2 in the case of a large diffusion on the right. At the macroscale, this asymmetry results in different trajectories for analytes with different sizes. (The smaller the molecule, the larger the diffusion coefficient.) (After: [284].)

during the electrophoresis of molecules [284]. As a result, molecules are deflected at an angle depending on their diffusion coefficients. This principle was applied to the continuous flow separation of DNA in a range 15–167 kbp [285–288]. A drawback of the method is that resolution scales with the inverse of the flow rate, since higher velocities mean reduced lateral displacement by diffusion. Obtaining a homogeneous distribution of the force field over a large array has often been a key issue in the design of continuous-flow devices. Huang et al. introduced an improved, continuous flow version (called the “DNA prism”) of the pulsed-field method illustrated in Figure 4.19 [289]. By integrating a fluidic interface able to generate and maintain uniform electric fields all over the array and authorizing a precise sample injection, the authors demonstrated the continuous flow separation of DNA in a range 61–209 kbp. Using a similar interface, Fu and colleagues extended the concept of the entropic trapping separation device introduced in Figure 4.18 to a two-dimensional array and applied it to continuous-flow separations [290]. Briefly, the array (illustrated in Figure 4.22) was composed of thousands of parallel deep channels connected by shallow nanoslits. Upon application of diagonal electric field, molecules travel through the array. From time to time, they pass from a deep channel to another through one of the nanoslits at a rate depending on the molecule properties and the sieving regime. Using this device, the authors demonstrated continuous-flow separation of DNA or proteins according to their size or charge. In fact, by adjusting experimental conditions, it was possible to switch the separation mode between the Ogston regime (analytes smaller than the pore, smaller molecules are faster), entropic trapping (analytes larger than the pore, larger molecules are faster), and electrostatic sieving (low buffer ionic concentration, analytes smaller than the pore, counterions with the large valence are faster). Because of the small dimensions involved, flow in micro- and nanofluidics is always considered laminar. Streamlines around obstacles are well-defined, as illus-

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Figure 4.22  Continuous-flow separation in an anisotropic nanofilter array. Upon application of a diagonal electric field, molecules travel along the array. From time to time, they pass from a deep channel to another through a nanoslit at a rate depending on the molecule properties and the sieving regime. For instance, in the Ogston regime, small molecules (trajectory 2) thread more easily through the slit than large molecules (trajectory 1) and thus are more deflected. (After: [290].)

trated in Figure 4.23. The method named “deterministic lateral displacement” takes advantage of these streamlines to control the deflection of molecules according to their size [291]. Small molecules follow one of the streams and have a trajectory that is globally straight. On the other hand, because of the longer distance between their center-of-mass and the wall of the obstacles, large molecules have to move from

Figure 4.23  Continuous-flow separation through deterministic lateral displacement. In micro- and nanofluidics, streamlines around obstacles are well-defined. In this example, the small molecule follows one of the streamlines. Globally, its trajectory is straight. The second mechanism behind this device is size exclusion. Because of its size, the center of mass of the larger molecule is always located in the central stream. This molecule is forced to move from a stream to another (for instance, from 2 to 3) at each constriction and is globally deflected to the right. (After: [291].)

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one stream to another, resulting in a deflection. Therefore to obtain a deflection, the molecule size has to be close to the gap size. To extend the separation range, it is then necessary to implement a gradient of the gap dimension along the direction of the flow. Using this principle, the authors demonstrated the continuous-flow separation of DNA (61 kbp and 158 kbp). Though the characteristic dimensions of the array are on the order of a micron (thus not purely “nanofluidic”), the principle illustrates well the opportunities offered by microfabricated matrices. Opposite to the asymmetric diffusion array described above (see Figure 4.21), the deterministic lateral displacement device does not rely on a random mechanism. Increasing the flow rate does not reduce separation performances. In fact, increasing the flow rate improves not only the throughput but also the resolution as diffusion is limited.

4.4  Linear Analysis of Biomolecules 4.4.1  Why Stretching DNA?

The information is stocked all along the linear DNA molecule. However DNA in solution is naturally coiled into a sphere of radius RG. An attractive way to access this information would be to stretch DNA and read the information along the strand as one can read the sequence of letters in an open book. After linearization, the molecule length can be directly measured. The position of specific hybridization probes or transcription factors tagged with fluorescent probes can be read, indicating the presence of target sequences and their relative locations and give insight into transcriptional regulation. Pushed by the need for faster, cheaper sequencing methods, scientists are now investigating devices that may be able to directly read the nucleotides sequence along the stretched DNA. Human chromosomes have a length in a range 1.7–8.5 cm (50–250 Mbp). Despite its remarkable strength, directly handling such a long molecule is still out of reach. Thus DNA is usually cut into small pieces, either by applying a shear or using restriction enzymes that bind onto a specific recognition sequence of the molecule and cleave it by making an incision at the phosphate backbone without damaging the bases. The localization of the restriction sites along the chromosome (restriction mapping) is useful to target the right piece of sequence or get a complete picture after analysis of each strand. Restriction mapping is usually based on the sizing of the DNA fragments after digestion. Jacobson and Ramsey reported a microfluidic device integrating restriction and sizing by CE [292]. Despite the success of the electrophoresis-based approach, numerous direct linear analysis (DLA) techniques have been investigated in an attempt to find a cheaper, faster alternative. Schwartz and colleagues reported the direct optical sizing of restriction fragments as early as 1993 [293]. The fragments were stretched by a fluid flow and fixed in place by the gelation of agarose, without application of electrical fields. Michalet and coworkers from Institut Pasteur introduced a simple and reliable method named “molecular combing” [294]. In this method, a silanized glass slides is dipped 5 minutes into a buffer solution containing genomic DNA. During this incubation period, one end of the molecule binds onto the surface. As the slide is pulled out of the solution, the forces exerted at the meniscus “comb” the

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DNA over the surface. Because of their hydrophobicity, the silanized surfaces dry instantaneously as they are pulled out of the solution. This results in irreversibly fixed, parallel DNA fibers, aligned in a single direction all over the surface. Hydrodynamic approaches have also been demonstrated where DNA are stretched in flow [295]. Recently a method to stretch and embed DNA fragments into a polymer nanofiber was reported [296]. The technique consists of electrospinning a solution of polyethylene oxide containing DNA strands. The company US Genomics has marketed a microfluidic device stretching DNA in flow and reading fluorescent hybridization probes in a continuous fashion, similar to flow cytometry [297]. It is worth noting that tools such as AFM and optical, magnetic, or MNEMS tweezers can also be used to stretch DNA, but have been more dedicated to fundamental studies of the mechanical properties [298–301]. 4.4.2  Sizing Confined DNA



The confinement of DNA in a nanochannel also induces a linearization. The recent need for design rules in nanofluidics has driven many researchers to investigate the conformation of DNA under confinement [302]. Figure 4.24 illustrates two regimes well described by established theories. The extent LEXT of a molecule of contour length L in a planar nanochannel of width w and height h is given by: � �1 wLp 3 LEXT ∼ for w, h >> Lp (de Gennes regime) =L h2

� � �2 � h 3 for w, h << LP (Odijk regime) LEXT = L 1 − A LP where A ~ 0.361 [302]. However, the cross-over behavior between these two regimes is not clearly understood. On the other hand, this cross-over is particularly important in nanofluidics as it occurs for a value around the persistence length, which is ~60 nm in normal conditions. Recently, experiments in nanofluidic devices

Figure 4.24  Linearization of a polymer under confinement in a nanochannel of height h and width w. (a) For h >> LP (persistence length), the molecule can be subdivided into a series of blobs (de Gennes regime). (b) For h << LP, the confined polymer is in the Odijk regime.

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and theoretical progress have given a deeper insight into the physics of polymers under confinement [303–305]. The effect of the ionic environment is also a topic of study, as confined DNA extension was found to increase with decreasing ionic strength [306, 307]. Tegenfeldt and colleagues from the Austin group demonstrated the direct optical sizing of genomic DNA (> 1 Mbp) confined in 100-nm-wide channels [308]. They showed that, at this scale, the measurements of the extent are still in agreement with the de Gennes theory. The statistical analysis allowed the authors to measure the extension of DNA with a standard deviation of 130 nm SD in 1 minute. Using a similar approach, the same group demonstrated restriction mapping in a nanofluidic device [309]. The digestion was triggered in the nanochannel by diffusion of the catalyst ion Mg2+ at one end of the nanochannels. Using single DNA molecules, the positions of restriction sites have been measured with a precision of 1.5 kbp within 1 minute. Taking advantage of the focal volume confinement, scientists from the Craighead’s group have developed advanced optical microscopy techniques and the associated algorithms to measure the conformation, length, and speed of DNA molecules flowing into a nanochannel at a very high rate [305, 310–312]. In their latest report, the authors claimed a molecular length resolution of 114 nm and an analysis time of only 20 ms per molecule. The algorithm was able to recognize seven different conformations depending on the fluorescence profile of the molecule. In a recent paper, Krishnan, Mönch, and Schwille reported a simple method to stretch DNA in a silicon/glass nanochannel [313]. They observed the spontaneous extension of the polymer along the lateral sidewalls of a nanoslit during the capillary filling with a solution of l-DNA labeled with YOYO-1. The effect was observed in a 100-nm-deep nanochannel. While the details of the mechanism are not fully understood, the confinement seems to be induced by the electrostatic interactions between the charged DNA molecule and the charged walls. The high curvature at the edge of the slit is supposed to induce a potential well along the sidewall, leading to the trapping of the molecule. 4.4.3  Nanopore Sequencing

In 2008, the company 454 Life Sciences announced the de novo sequencing of James Watson for a cost of $100,000. Two months have been required to complete the operation using the pyrosequencing method. While there has already been enormous progress since the end of the $3 billion genome project, the need for faster, cheaper sequencing techniques has never been greater than it is today. With the objective of the $1,000 genome in mind, scientists proposed many alternatives to Sanger and pyrosequencing techniques [314]. Among them, nanopore sequencing is a promising candidate [315]. Figure 4.25 illustrates the principle of nanopore sequencing. This concept was first proposed in the 1990s, and the first experimental results were reported by Kasianowicz and colleagues at NIST using biological nanopores [316, 317]. The authors took advantage of transmembrane proteins (a-haemolysin) that spontaneously insert themselves into a lipid membrane. The width of this pore is 1.4 nm at the narrowest point. Typically, an electric field is applied across the membrane. A sharp decrease of the electrical current corresponds to a translocation event (current blockade due to the presence of DNA in the pore). While biological

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Figure 4.25  Conceptual illustration of nanopore sequencing. DNA is stretched as it threads its way through a nanopore, while an integrated sensor read the sequence.

nanopores were widely used in early studies, solid-state nanopores fabricated using silicon technologies have brought many benefits including a better stability, better control of the dimensions, and, of course, new possibilities in terms of integration. Recent progress in solid-state nanopores have recently been reviewed by Dekker [317]. We report in this section only a few examples of nanofluidic platforms aiming to integrate nanopore sequencing. Fan and colleagues at Berkeley reported the integration of a carbon nanotube between two microchannels [318]. Translocation of DNA was observed by monitoring the electrical current through the tube under various conditions, demonstrating the potential of this platform. Recently, Liang and Chou at Princeton reported the fabrication of a silicon device with a nanochannel 45-nm-wide, 45-nm-deep and 50-mm-long [319]. Two electrodes separated by only a 9-nm-wide gap were integrated inside this nanochannel. Using this configuration, DNA was stretched through the channel, and the electrical current was measured during the translocation. According to theoretical models, it should be possible to detect differences of conduction depending on the nucleotides crossing the electrical field. However, the resolution of the signal was not good enough to detect these subtle variations. To reach their objective, the authors proposed to decrease the DNA flow speed and shrink further the dimensions of the gap.

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Chapter 5

Conclusion We introduced nanofluidics as a new discipline of engineering. The definition given in the introduction stated that nanofluidics “encompasses science and technologies that involve a fluid flowing in a system with a least one dimension in the nanoscale,” usually defined in the MEMS/NEMS community by the range 1–100 nm. Throughout this book, it was shown that the renewed interest in this field stems from: (1) effects specific to this scale, and (2) the new possibilities offered by microand nanofabrication technologies in terms of design and integration of the channel network. Most of the theoretical background of nanofluidics has been well established and studied during the last century through disciplines such as membrane sciences, analytical chemistry (electrophoresis), physics of fluids, and tribology. The ability to tailor the geometry of nanoporous media and integrate them with microchannels and electrodes in a planar format (i.e., a nanofluidic chip) has allowed the fabrication of devices able to perform operations on molecules at the nanoscale in an automated and well-controlled fashion. There are three modeling levels for nanofluidic applications: the molecular level, the continuum level, and the macro level. In the molecular level, the system is described by interactions of single molecules. Suitable techniques are molecular dynamics and DSMC. Because of the large number of molecules or particles involved in such a simulation, the computational expense including the hardware and the computing time is huge. A modified continuum model is more justified for design problems in nanofluidics that need a quick answer. Although nanofluidics involves channel dimensions less than 100 nm, the continuum assumption still can be applied to model fluid flow in nanochannels. Considering the channel size and the interactions with other physical effects, the model can be modified to be used with conventional continuum-based governing equations. For instance, a slip condition and a higher apparent viscosity can replace the conventional no-slip condition and the bulk viscosity. The higher apparent viscosity is caused by many sources, such as the electroviscous effect or tiny bubbles trapped in the liquid. A macro model further simplifies the fluidic system into fluidic elements such as fluidic resistance, fluidic inertance, and fluidic capacitance. However, the macro model is still based on the continuum assumption and may need further modification for describing nanofluidic devices. Micro- and nanofabrication lies at the heart of nanofluidics. Chapter 3 introduces standard silicon fabrication techniques. Planar nanochannels (or nanoslits) can easily be fabricated using conventional photolithography. However, as the lateral dimension decreases, slow and expensive patterning tools (e.g., EBL or FIB) have to be involved. Alternative nanolithography methods have emerged to overcome this situation: (1) A simple network of nanostructures (e.g., a regular array of lines or dots) can be patterned using low-cost, parallel techniques such as edge 197

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lithography or interference lithography; and (2) more complex structures can be replicated from a negative template using techniques similar to plastic molding (e.g., nanoimprint lithography, step and flash imprint lithography, and reverse lithography). Large-scale production of disposable plastic devices at low cost is a key to the marketing of nanofluidic products. The transition from silicon to polymer as a structural material has been made challenging by their lower Young’s modulus. Despite these technical issues, promising methods have recently been demonstrated, paving the way towards a wider acceptance of nanofluidics. Owing to these outstanding advances in fabrication, nanofluidics has been finding applications wherever a random, isotropic porous media can be advantageously replaced by a well-defined array of nanochannels. Because of the huge impact of sequencing technologies, there has been a large interest in: (1) studying and (2) improving DNA fragmentation by electrophoresis using smart nanofluidic designs instead of random porous gels. Chapter 3 discusses microfluidic and nanofluidic approaches. Using anisotropic arrays, continuous flow separations of DNA and proteins have been demonstrated by a few groups, promising faster and better biomolecular analyses. Stretching DNA into a nanochannel has allowed scientists to directly access information such as the molecule length, its conformation, or the position of fluorescently tagged hybridization probes or transcription factors. However, the “holy grail” of direct linear analysis, the direct reading of a DNA sequence, has yet to be demonstrated. The ability to control the ionic flux through a nanochannel, combined with control over the design, has led to the first nanofluidic diodes and transistors (“nanofluidic electronics”). These preliminary results pave the way to ionic circuits able to perform complex, automated operations on chemical species. Beyond this, nanofluidics may find applications in fields where semipermeable membranes are traditionally employed, such as water purification or fuel cells. The close proximity between the wavelength of visible light and the dimensions of nanochannels may also give rise to potential applications in optofluidics. However, there is still a huge gap between these academic demonstrations and the marketing of nanofluidic devices. Using a micro- or nanofluidic chip is not straightforward. Designing simple, cheap, and reliable packaging (the interface with the “macro-world”) is still challenging. The instrumentation around these miniaturized systems often consists of bulky equipment (e.g., computers, acquisition cards, fluorescence microscopes with detectors, high-voltage power supplies, and syringe pumps). Only simple nanofluidic functions have been integrated. It is clear that integration of all these building blocks into a fully automated lab-on-chip is necessary for the field to reach its full potential.

About the Authors Patrick Abgrall studied microelectronics and microtechnologies at the University of Rennes I in France and at the Chemnitz University of Technology in Germany. After a Ph.D. thesis on polymer fabrication for lab-on-chips at LAAS-CNRS in Toulouse, France, he joined the Singapore-MIT alliance where his research was focused on fabrication and applications of polymer nanofluidic devices. He is now a postdoctoral researcher at the Biomedical Diagnostics Institute in Dublin. His research is directed towards the development of novel micro- and nanofluidic technologies and their applications in analytical sciences. Nam-Trung Nguyen received a Diplom-Ingenieur (M.S.) in 1993, a Doktor-Ingenieur (Ph.D.) in 1997, and a Doktor-Ingenieur Habilitatus (a German postdoctoral degree) in 2004 from Chemnitz University of Technology in Germany, all in electrical engineering. From 1992 to 1993, he worked on microfluidics at Robert Bosch GmbH (Reutlingen) as a research student. Dr. Nguyen was also a postdoctoral research engineer at the University of California, Berkeley, in the Berkeley Sensors and Actuators Center (BSAC). He is currently an associate professor with the School of Mechanical and Aerospace Engineering at Nanyang Technological University, Singapore. Dr. Nguyen’s past and present research interests are the development of microtechnology and devices for microfluidic applications, especially in biomedical instrumentation. Dr. Nguyen is a member of the American Society of Mechanical Engineers (ASME) and the Institute of Electrical and Electronics Engineers (IEEE).

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Index A

D

Adhesion, 72 Agarose, 3 Anisotropic wet etching, 89, 120, 126 Anodic bonding, 102 Anodic dissolution, 96 Atomic force microscope (AFM), 82, 125, 126

de Gennes regime, 179 Deal and Grove model, 74 Debye length, 4, 32, 50, 159 Deep reactive ion etching process (DRIE), 99 Detection, 149 Dimensionless channel size, 4 Dimensionless number, 9 Dip pen nanolithography (DPN), 82 Direct Simulation Monte Carlo (DSMC), 46, 48 DNA, 5, 146 Donnan exclusion, 163 Dry etching, 96 Dynamic Reynolds number, 17

B Block copolymer lithography, 92 Boltzmann constant, 23 Bosch process, 99 Bulk micromachining, 109, 118

C Capillary electrochromatography (CEC), 146 Capillary flow, 19 Capillary gel electrophoresis (CGE), 147 Capillary number, 9 Capillary zone electrophoresis (CZE), 145 Chapmann-Enskog theory, 48 Chemical vapor deposition (CVD), 75 Chemiluminescence (CL), 150 Colloid science, 1 Concentration, 153, 163 Concentration polarization (CP), 164 Conformality, 72 Contact angle, 20 Continuity equation, 12 Continuous-flow separation, 174 Corrosion, 96, 121 Cracking, 91 Cryogenic etching, 98 Czochralski, 72

E Electric double layer (EDL), 5, 23, 160 Elasticity number, 11 Electrocapillary effect, 32 Electrochemical etching, 95 Electrochemiluminescence (ECL), 150 Electrokinetics, 2, 22, 159 Electron beam lithography (EBL), 80 Electro-osmosis, 22, 23, 160 Electro-osmotic velocity, 24, 160 Electrophoresis, 5, 31, 142 Electrophoretic velocity, 32 Electroplating, 77 Electrowetting, 32 End-label free-solution electrophoresis (ELFSE), 147 Energy equation, 12 Entrance length, 14 Entropic trapping, 172 201

202

Etching, 92 Evaporation, 76 Exclusion-enrichment effect, 64, 163 Eyring fluid, 47

F Fanning friction factor, 14 Fick’s law, 35 Field-amplified stacking (FAS), 164 Float zone, 72 Fluidic capacitance, 15 Fluidic inertance, 15 Fluidic resistance, 15 Fluorescence-activated cell sorter (FACS), 156 Focused ion beam (FIB), 81 Free solution CE (FSCE), 145 Furnace, 75 Fusion bonding, 101

G Gas chromatography (GC), 146 Glass, 72, 111 Gouy-Chapman layer, 23, 50

H Hagen-Poiseuille model, 21, 32 High-aspect ratio nanochannel, 118, 126 High-pressure liquid chromatography (HPLC), 146 Hindered diffusion, 49 Human genome project, 146, 170 Hydraulic diameter, 14 Hydrogen bond, 52

I Injection, 143 Interference lithography (IL), 91, 126 Interface gas-liquid, 2, 21 solid-gas, 2, 21 solid-liquid, 2, 21

Index

Ion beam lithography, 81 Ionic transport, 5, 162 Isoelectric point (pI), 163 Isotachophoresis (ITP), 154, 164 Isotropic wet etching, 94 K Kinetic theory, 43 Knudsen diffusion, 49 Knudsen number, 45 L Lab-on-chip, 142, 154 Lamination, 78, 115 Laser machining, 79, 108 Laser-induced fluorescence (LIF), 149 Lennard-Jones potential, 46 Lift-off, 100, 120 Linear analysis, 178 Lippmann equation, 33 Low pressure chemical vapor deposition (LPCVD), 176 M Mach number, 44 Macro model, 15 Mass flow source, 15 Mass spectrometry (MS), 150 Maxwell-Boltzmann distribution, 44 Mean free path, 43 Micellar electrokinetic chromatography (MEKC), 145 Microelectromechanical system (MEMS), 1, 71 MicroTAS 142 Miniaturization, 9 Molding, 103, 116 Molecular dynamics (MD), 46 Moore’s law 3, 83 N Nanoelectromechanical system (NEMS), 1 Nanofluidic diode, 5, 167

Index

Nanofluidic electronics, 167 Nanofluidic transistor, 168 Nanopores 5, 126, 167, 180 Navier-Stokes equation, 11, 25, 27 Near-field scanning optical microscope (NSOM), 83 Newton’s second law, 46 Newtonian fluid, 12, 47 Non-Newtonian fluid, 47

O Odijk regime, 179 Off-axis illumination, 85 Ogston-Morris-Rodbard-Charambach (OMRC) model, 66 Ogston regime, 172 Open-tubular CEC (OTCEC), 146 Optical proximity correction (OPC) techniques, 85 Oxidation, 74, 119

P Parylene 76, 113 PDMS casting, 116 Peclet number, 11, 36 Phase-shift lithography, 86, 90 Photolithography, 83 Photomask, 85 Photoresist, 85 Physical vapor deposition (PVD), 76 Planar nanochannel, 117 Plasma, 96 Plasma-enhanced chemical vapor deposition (PECVD), 76 Poincaré section, 41 Poisson coefficient, 125 Poisson-Boltzmann equation, 23, 26, 27, 54 Polymer, 103 Polymerase chain reaction (PCR), 151 Pressure source, 15 Pressure-driven flow, 13 Pretreatment, 151 Pyrosequencing, 146, 180

203

R Radial distribution function, 42 RCA cleaning, 83 Reactive ion etching (RIE), 98 Replication, 104 Reptation, 65, 172 Reynolds number, 9, 44 S Sacrificial layer, 111, 120, 126 Scaling laws, 9 Scanning beam lithography (SBL), 78 Scanning probe lithography (SPL), 82 Sedimentation potential, 23 Self-assembled monolayer (SAM), 82, 83, 91 Self-assembly, 91 Semipermeability, 163 Silicon-on-insulator (SOI), 101 Slip boundary condition, 12 Smoluchowski velocity, 24 Spin-coating, 78 Spray-coating, 78 Sputtering, 77 Square nanochannel, 117 Stepper, 87 Steric effect, 66 Stern layer, 23, 49, 160 Stokes-Einstein model, 34 Stoney equation, 73 Streaming potential, 23 Stress, 73 SU-8, 108, 114 Surface micromachining, 111 T Tangential momentum coefficient, 13 Taylor-Aris dispersion, 35, 37 Temperature accommodation coefficient, 13 Thin films, 72 Tribology, 1 U Ultraviolet (UV) absorbance, 149 Uniformity, 72

204

Index

W

Y

Wafer bonding, 100 Wafer direct bonding, 101 Weber number, 9 Weisenberg number, 9 Wrinkling, 91

Young equation, 20, 33 Young-Laplace equation, 20, 54

Z X XeF2 etching, 100

Zeolite, 3 Zeta potential, 23, 53, 159