UFDC Home  myUFDC Home  Help 
Title Page  
Dedication  
Acknowledgement  
Table of Contents  
List of Tables  
List of Figures  
Abstract  
Introduction to electrical fieldflow...  
DNA separation by EFFF in...  
Separation of charged colloids...  
Taylor dispersion in cyclic electrical...  
Electrochemical response and separation...  
Conclusion and future work  
Appendix A: Derivation of velocity...  
Appendix B: Derivation of numerical...  
References  
Biographical sketch 



Table of Contents  
Title Page
Page i Page ii Dedication Page iii Acknowledgement Page iv Table of Contents Page v Page vi Page vii List of Tables Page viii List of Figures Page ix Page x Page xi Abstract Page xii Page xiii Introduction to electrical fieldflow fractionation Page 1 Page 2 Page 3 Page 4 DNA separation by EFFF in a microchanel Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Separation of charged colloids by a combination of pulsating lateral electric fields and poiseuille flow in a 2D channel Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Taylor dispersion in cyclic electrical fieldflow fractionation Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Electrochemical response and separation in cyclic electric fieldflow fractionation Page 86 Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Conclusion and future work Page 126 Page 127 Page 128 Page 129 Page 130 Page 131 Page 132 Page 133 Appendix A: Derivation of velocity and dispersion under unidirectional EFFF Page 134 Page 135 Page 136 Page 137 Appendix B: Derivation of numerical calculation for sinusoidal EFFF Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 Page 145 Page 146 Page 147 Page 148 Page 149 Page 150 Page 151 Page 152 Page 153 Page 154 Page 155 Page 156 References Page 157 Page 158 Page 159 Biographical sketch Page 160 Page 161 

Full Text  
SEPARATION WITH ELECTRICAL FIELDFLOW FRACTIONATION By ZHI CHEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Zhi Chen This document is dedicated to the graduate students of the University of Florida. ACKNOWLEDGMENTS This work was performed under the elaborate instruction of Dr. Anuj Chauhan. He gave me invaluable help and direction during the research, which guided me when I struggled with difficulties and questions. Also, I deeply appreciate my laboratory colleagues who gave me great help and many suggestions. Furthermore, I would like to thank my wife Xiaoying Sun. Without her help and encouragements in my daily life, I could not have finished my degree. I also acknowledge the financial support of NASA (NAG 10316) and the National Science Foundation (NSF Grant EEC9402989). TABLE OF CONTENTS page ACKNOW LEDGM ENTS ................................................................................................. iv T A B L E ............................................. .............................................. ........................... v iii LIST OF FIGURES ........................................................................ ............................. ix ABSTRACT...................................................................................................................... xii CHAPTER 1 INTRODUCTION TO ELECTRICAL FIELDFLOW FRACTIONATION..............1 2 DNA SEPARATION BY EFFF IN A MICROCHANEL.........................................5... Application of EFFF in DNA Separation.....................................................................5 T h e o ry ............................................................................ .................................. ............7.. Results and Discussion ............................................................................................. 10 Limiting Cases.................................................................................................. 10 Dependence of the M ean Velocity on Uy and Pe ........................................... 13 Dependence of D* on Ut and Pe...................................................................... 13 y Separation Efficiency .......................................................................................... 14 Effect of Pe and U y on the Separation Efficiency........................................... 16 DNA Separation ............................................................... .............................. 20 Comparison with Experiments ..........................................................................25 Summ ary............................................................................................................... 28 3 SEPARATION OF CHARGED COLLOIDS BY A COMBINATION OF PULSATING LATERAL ELECTRIC FIELDS AND POISEUILLE FLOW IN A 2D CHANNEL ................................................................................ ........................ 30 Theory................................................................................................................ ...32 M odel ................................................. ........................... ........................ 32 The diffusive step: No electric field and no flow ...................................... 32 The convective step: Poiseuille flow with no electric field ......................34 Electric field step (Electric field, no Flow)............................................... 36 Long tim e Analytical Solution ......................................................................... 40 Results and Discussion ...............................................................................................42 M ean Velocity ................................................................. ............................... 43 Dispersion Coefficient....................................................................................... 44 Separation Efficiency ........................................................................................ 46 Effect of G ................................................................... ........................... 47 Effect of tf/td .............................................................................................. 48 h2 Effect of ....................... .............. ................................................... 49 Comparison with Constant EFFF ..................................................................... 52 Conclusions............................................................................................................. 56 4 TAYLOR DISPERSION IN CYCLIC ELECTRICAL FIELDFLOW FRACTIONATION .................................................................... ........................... 58 T h eo ry .........................................................................................................................5 9 Results and Discussion ............................................................................................. 65 Square W ave Electric Field.............................................. .............................. 65 Transient concentration profiles....................................... ....................... 66 M ean velocity and dispersion coefficient................................................... 68 Sinusoidal Electric Field.................................................................................... 70 Analytical computations............................................................................. 70 Numerical computations and comparison with analytical results............. 72 Comparison of Sinusoidal and Square fields.................................................... 83 Conclusions..................... .............. .............. ................................................ 84 5 ELECTROCHEMICAL RESPONSE AND SEPARATION IN CYCLIC ELECTRIC FIELDFLOW FRACTIONATION ..................................................... 86 T h eory ................................................................................................................. 87 Equivalent Electric Circuit................................................................................ 87 M odel for Separation in EFFF........................................................................... 88 Result and Discussion............................................................................................... 92 Electrochemical Response............................................... ............................... 92 Current response for a step change in voltage.............................................93 Dependence on applied voltage (V) and salt concentration...................... 97 Dependence on channel thickness (h)....................................................... 99 Current response for a cyclic change in potential..................................... 99 S ep aratio n ........................................................................... ............................ 10 4 M odeling of separation of particles by CEFFF........................................ 104 M ean velocity of particles....................................................................... 104 Effective diffusivity of particles............................................................... 107 Separation efficiency................................................... ............................ 110 Comparison with Experiments ........................................................................111 Large Q asymptotic results..................................................................... 113 The effect of changes in Q ....................................................................... 118 Conclusions................................ ............ ...................................................... 123 6 CONCLUSION AND FUTURE WORK ...............................................................126 APPENDIX A DERIVATION OF VELOCITY AND DISPERSION UNDER U N ID IRECTION AL EFFF .................................. ............................................... 134 B DERIVATION OF NUMERICAL CALCULATION FOR SINUSOIDAL EFFF 138 Analytical Solution to O(s) Problem ...................................................................... 138 A nalytical Solution to O (s2) Problem ...................................................................... 141 Solving for f, g, p and q ........................................................... ........................... 144 Solving for p and q ....................................................... ............................... 148 Solving for Particular Solution.............................................. ......................... 148 R E FE R EN C E L IST ............................................................................. ......................... 157 BIO G R A PH IC A L SK ETCH .......................................................... ............................ 160 TABLE Table page 51 Comparison of the model predictions with experiments of Lao et al. ................. 123 LIST OF FIGURES Figure page 21 Schem atic of the 2D channel...................................................................................7... 22 Dependency of (D*R)/Pe2 on the product of Pe and U ... .........................12 23 Dependency of mean velocity U on the product of Pe and U .........................12 24 Dependency of L/h on Uy and Pe for separation of DNA strands of different sizes. D 2/D 1 = 10 .............................................................................. .................... 18 25 Dependency of L/h on Uy and Pe for separation of DNA strands of different sizes. D 2/D 1 = 2 ........................................................................................... ........ 19 26 Comparison of our predictions with experiments on DNA separation with F1FFF .27 27 Comparison of our predictions with experiments on separation of latex particles w ith E F F F ...................................... ...... ..................................................................2 8 31 Schem atic show ing the threestep cycle................................................................. 31 32 D ependency of U on G ....................................................................................... 44 33 D ependency of D on G ........................................................................................ 45 34 Effect of Gi(= 0.2, = 0.2, G2/Gi=2) on L/h, 0/tf and T...................47 td ((u)tf)2 t h2 35 Effect of I (Gi=100, h = 0.2, G2/G =2) on L/h, 0/tf and T.................... 49 td ((u)tf h2 t 36 Effect of (Gi=100, h= 0.2, G2/Gi=2) on L/h, 0/tf and T.................... 50 ((u)tf)2 td 37 Dependency of L/h on Gi(pulsating electric field) and uy (constant electric field). D I/D 2= 2 .............................. ... ...... ...... ...... ......... ........................................ 54 38 Dependency of the operating time t on Gi(pulsating electric field) and u' (constant electric field). D1/D 2=2...................................................................... 55 39 Dependency of L/h on Gi(pulsating electric field) and u (constant electric field). D I/D 2= 1.2 ................................................................................................... 55 310 Dependency of the operating time t on Gi(pulsating electric field) and u (constant electric field). D1/D 2=1.2.................................................................. 56 41 Periodic steady concentration profiles during a period for a square shaped electric field ............................................................................ .. ........................... 6 7 42 Comparison of the numerically computed (a) mean velocity and (b) dispersion coefficient for a square shaped electric field with the large Pe asymptotes obtained by S& B (Thick line)................................................................................ 69 43 gi vs. position for PeR=l, and Q =100................................................................... 71 44 Time dependent concentration profiles within a period for sinusoidal electric field s. .......................................................... ....................................................... 7 3 45 Time average concentration profiles for sinusoidal electric field.........................74 46 D ependence of U on PeR .................................................................................... 76 47 Dependence of (D * 1)/Pe2 on PeR ............................................. ......................... 79 48 Comparison of the mean velocities for the square (dashed) and the sinusoidal (solid) fields in the large frequency lim it................................. ........................... 82 49 Comparison of the mean velocities and the effective diffusivity for the square (dashed) and the sinusoidal (solid) fields..............................................................83 51 Equivalent electric circuit model for an EFFF device............................................88 52 Transient current profiles after application of step change in voltage in a 500 pm th ick ch ann el ......................................................................................................... .. 9 5 53 Dependence of the electrochemical parameters on salt concentration and applied voltage in a 500 pm thick channel ......................................................................... 97 54 Dependence of the electrochemical parameters on channel thickness for V = 0.5 V an d D I w ater ..................................................... ............................................... 9 8 55 Comparison between the experiments (thin lines) and Eq. (5 24) (thick lines).... 101 56 Comparison between the experiments (stars) and Eq. (5 26) (solid lines) .........102 57 Dependency of the mean velocity on PeR and Q ................................................105 58 Dependence of 210(D*l)/Pe2 on PeR and Q2......................................................108 59 Dependence of separation efficiency on PeRi and Q2 for the case of DI/D2=3 and E2/tElI=3 ........................... ............................................................. 109 510 Origin of the singularity in separation efficiency at critical PeRi and Qi values for Q = 40 .................................................... ............ ........................................ 109 511 Dependence of the mean velocity on Q in the large 2 regime............................120 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SEPARATION WITH ELECTRICAL FIELDFLOW FRACTIONATION By Zhi Chen August 2006 Chair: Anuj Chauhan Major Department: Chemical Engineering Separation of colloids such as viruses, cells, DNA, RNA, proteins, etc., is becoming increasingly important due to rapid advances in the areas of genomics, proteomics and forensics. It is also desirable to separate these colloids in free solution in simple microfluidic devices that can be fabricated cheaply by using the microelectromechanical systems (MEMS) technology. Electrical fieldflow fractionation (EFFF) is a technique that can separate charged particles by combining a lateral electric field with an axial pressuredriven flow. EFFF can easily be integrated with other operations such as reaction, preconcentration, detection, etc., on a chip. The main barrier to implementation of EFFF is the presence of double layers near the electrodes. These double layers consume about 99% of the potential drop, and necessitate application of large fields, which can cause bubble formation and destroy the separation. In this dissertation we have investigated the process of double layer charging and proposed several approaches to minimize the effect of double layer charging on separations. The essential idea is that if the applied electric field is either pulsed or oscillates with a period shorter than the time required for the double layer charging, a much larger fraction of the applied potential drop will occur in the bulk of the channel. Accordingly, in cyclic EFFF (CEFFF) smaller fields may be applied and this may prevent bubble formation. Based on this idea, we proposed a novel separation approach that utilizes pulsed fields while we also investigated both sinusoidal and square shaped cyclic electric fields. We performed experiments to determine the time scales of the double layer charging and studied its dependence on channel thickness, applied voltage and salt concentration. While investigating unidirectionalEFFF, pulsed EFFF and cyclicEFFF, we solved the continuum convection diffusion equation for the charged particles to obtain the mean velocity and the dispersion coefficients for the particles. Furthermore, we estimated the separation efficiency based on the velocity and dispersion coefficient. Results show that EFFF can separate colloids with efficiencies comparable to other methods such as entropic trapping and the effective of EFFF can be substantially improved by using either pulsed or cyclic fields. CHAPTER 1 INTRODUCTION TO ELECTRICAL FIELDFLOW FRACTIONATION A number of industrial processes particularly those related to mining, cosmetics, powder processing, etc. require unit operations to separate particles. Additionally, rapid advances in the area of genomics, proteomics and the threats posed by natural biohazards such as bird flu and also those by bioterrorism have increased the demand for devices that can accomplish separation in free solution. A number of biomolecules such as DNA strands, proteins, etc are currently separated by gel electrophoresis. This is a tedious process that can only be operated by experts. There is a strong demand for simpler processes and devices that can be incorporated on a chip and that can accomplish separation in free solution. One approach that has a significant potential is electric field flow fractionation (EFFF), which is a variant of a general class of fieldflow fractionation (FFF) techniques. Fieldflow fractionation relies on application of a field in the direction perpendicular to the flow to create concentration gradients in the lateral direction. When particles flow through channels in the presence of lateral fields, they experience an attractive force towards one side of the walls. In the absence of any field, each particle has an equal probability of accessing any streamline in a time scale larger than h /D, where h is the height of the channel, and D is the molecular diffusivity. However, in the presence of the lateral fields, the particles access streamlines closer to the wall, resulting in a reduction of the mean axial velocity. Since the concentration profile in the lateral direction depends on the fielddriven mobility and the diffusion coefficient, molecules that either have different mobilities or different diffusivities can be separated by this method. Field flow fractionation (FFF) was formally defined by J.Calvin Giddings in 1966. However a variant of this approach was used as far back as the Middle Ages to recover gold by sluicing, in which the gravity is combined with a flowing stream to generate separation. Fieldflow fractionation has many variants depending on the types of lateral fields used in separation, such as sedimentation FFF, electrical FFF, flow FFF, magnetic FFF, etc. There is an extensive literature on the use of EFFF [13] and other variants of FFF such as those based on gravity, centrifugal acceleration [46], lateral fluid flow[7], or thermal fieldflow fractionation (TFFF) [810]. These techniques have been used in separations of a number of different types of molecules including biomolecules [11,12]. The flowFFF, which is the fractionation technique that utilizes a combination of lateral fluid flow along with the axial flow, has been successfully utilized to separate DNA strands [13]. The separation of charged particles is frequently accomplished by applying electric fields either in the axial or in the lateral direction. Electrical field flow fractionation (EFFF) is a method based on application of lateral electric field, and this technique has been used by a number of researchers for accomplishing separation in microfluidic devices [14,15]. In the past decades, the efficiency of EFFF has improved due to the advances in miniaturization, and it has been used for separation of charged particles, such as cells [16,17], proteins [18], DNA molecules and latex particles [19]. The EFFF technique has received considerable attention due to its potential application in separation of colloidal particles [2,20] such as DNA strands, proteins, viruses, etc. EFFF devices are easy to fabricate and can be integrated in the "Lab on a Chip". While EFFF is a useful technique, it has not yet been commercialized partly because of the problems associated with the charging of the double layers after the application of the electric field. In some instances as much as 99% of the applied potential drop occurs across the double layers [2]. In addition, the constant lateral field results in a flow of current and electrolysis of water at the electrodes, causing generation of oxygen and hydrogen. Since bubble formation could significantly impede separation, the incoming fluid is typically degassed so that the evolving gases can simply dissolve in the carrier fluid. But even then the amount of lateral electric field that can be applied is limited by the restriction that it should not result in generation of gases that can exceed the solubility limit. The time required for the current and the field in the bulk to decrease to the steady value depends on a number of factors including the flow rate, salt concentration, pH, etc. All these factors can be lumped together into an equivalent circuit for current flow in the lateral direction and the RC time constant of this circuit has been reported to vary between 0.02 and 40 s [3]. If the lateral fields are pulsed or varied in a cyclic manner such that the time scale for pulsation is shorter than the RC time constant, a much larger fraction of the applied potential drop occurs in the bulk and this may also reduce or eliminate the bubble formation due to Faradaic processes at the electrodes. The main motivation behind this dissertation was to explore the feasibility of using EFFF for size based DNA separation. Accordingly we began this dissertation by modeling DNA separation in EFFF, and this work is described in chapter 2. The results of chapter 2 show that EFFF can be used for DNA separation but the problems associated with the double layer charging need to be addressed. In order to eliminate these problems 4 we propose a new technique based on pulsatile fields in chapter 3, and show that this technique is more effective than the conventional EFFF. In addition to using pulsed fields one could also minimize the effect of double layer charging by using cyclic fields. Separation by cyclic fields in explored in chapter 4 for sinusoidal fields and in chapter 5 for square fields. Finally chapter 6 summarizes the main conclusions and proposes some future work. CHAPTER 2 DNA SEPARATION BY EFFF IN A MICROCHANEL The main aims of the research in this chapter are (i) investigate the feasibility of using EFFF for DNA separation by determining the field strength required for separation, (ii) study the effect of various system parameters on DNA separation, (iii) determine the scaling relationships for separation length and time as a function of the DNA length in various parameter regimes, and (iv) determine the optimum operating conditions and the minimum channel length and the time required for the DNA separation as a function of the length of the DNA strands. We hope that the results of this study will aid the chip designers in choosing the optimal design and the operating parameters for the separation of DNA. Application of EFFF in DNA Separation DNA electrophoresis has become a very important separation technique in molecular biology. This technique is also indispensable in forensic applications for identifying a person from a tissue sample [21]. However, separation of DNA fragments of different chain lengths by electrophoresis in pure solution is not possible because the velocity of the charged DNA molecules in the electric field is independent of the chain length beyond a length of about 400 bp [22]. This independency is due to the screening of the hydrodynamic interactions in the presence of an electric field by the flowing counter ions [23]. This difficulty is traditionally overcome by performing the electrophoresis in columns or capillaries filled with gels. The field applied in the gelbased electrophoretic separations can be continuous or pulsed. Recent advances in microfabrication techniques have led to the production of microfluidic devices frequently referred to as a "labonachip" that can perform a number of unitoperations such as reactions, separations, detection, etc., at a high throughput. Gelbased DNA separations are not convenient in such devices because of the difficulty in loading the gel [24]. Thus, gels have been replaced with polymeric solutions as the sieving mediums. Electrophoresis in a free medium can also separate DNA fragments but it requires precise modifications to the DNA molecules [25]. Microfabricated obstacles such as posts [26], selfassembling colloids [27], entropic barriers [28], and Brownian ratchets [29,30] have also been shown to be effective at separating DNA strands. The optimal DNA separation technique should accomplish separation without any sieving medium. Electrical fieldflow fractionation (EFFF) [14,20,31], which is a type of fieldflow fractionation (FFF), a technique first proposed in 1966 [32], can separate DNA strands by a combination of a lateral electric field and a Poiseuille flow in the axial direction. The application of the electric field in the lateral direction, i.e., the direction perpendicular to the flow, creates a concentration gradient in the lateral direction [33]. The DNA molecules are typically negatively charged and thus as they flow through the channels in presence of the lateral fields, they are attracted towards the positively charged wall. Thus, the molecules on an average access streamlines closer to the wall, which causes a reduction in the mean velocity of the molecules. The enhancement in concentration near the wall is more for the slower diffusing molecules, and thus their mean velocity is reduced more than that of the faster diffusing molecules. Thus, if a slug of DNA molecules of different sizes is introduced into a channel with lateral electric II I velocity u' can be determined by the Smoluchowski equation, u = sr E, where 8r and p are the fluid's dielectric constant and viscosity, respectively, so is the permittivity of vacuum, and C is the zeta potential. Alternatively, u = pEE, where 1E is the electrical mobility of DNA, which is independent of length and has a value of about 3.8x108 m2/(Vs) [22]. Outside the thin double layer near the electrodes, the fluid is electroneutral, and the velocity of the charged molecules due to the electric fields in the y direction is constant. Thus Eq. (21) becomes 9c 8c o c 02c 82c +u+u =D(R+ ) (22) at 8x ay Ox oy where R = DII/D, and we denote D1 as D. The value of R varies between 1 and 2; it is equal to 1 if the DNA molecules are randomcoils, and it is equal to 2 if they are fully stretched as cylinders in the flowdirection. The boundary conditions for the above differential equation are D c +ucc=0 at y = 0,h. (23) ay The above boundary conditions are strictly valid only at the wall and not at the outer edge of the double layer, which is the boundary of the domain in which the differential equation is valid. Still, since the double layer is very thin, and the time scale for attaining steady state inside the double layer is very short, we neglect the total flux of the DNA molecules from the bulk to the double layer. The above boundary condition also assumes that the DNA molecules do not adsorb on the walls. Due to electroneutrality in the bulk, the velocity profile remains unaffected by the lateral electric field. Thus the fluid velocity profile in the axial direction is parabolic, i.e., u= 6 (y/h(y/h)2) (24) where is the mean velocity in the channel. The convection diffusion equation is solved in Appendix A to determine the dimensionless mean velocity U and the dimensionless dispersion coefficient D*for a pulse of solute introduced into the channel. The results are 6+6exp(a) 1212exp(a) U = (25) exp(a) 1 D* = R Pe2 (720e'a + 504eua2 24e a4 144eaa3 6048e2a 504e2aa2 + 720e2Ga + 24e2a 4 144e2aa3 +72e3aa2 720e3a + 6048e( + 2016e3 2016 720a72a2 )/((e 1)3 a6) (2 6) In the above expressions Pe = h/D and a PeU'. As shown in Appendix A the concentration profile of the DNA molecules decays exponentially away from the positive electrode, and all the molecules accumulate in a layer of thickness 8 that is about 3h/cx. The dispersion of molecules in the FFF has also been investigated by Giddings [34], Giddings and Schure [35], and Brenner and Edwards [36], and our results agree with these studies. However, we have used the method of regular expansion in the aspect ratio to determine the mean velocity and the dispersion coefficient, and this approach is different from that adopted by other researchers. 10 Results and Discussion Limiting Cases The mean velocity and the dispersion coefficient depend on the Peclet number and U'. If U' approaches zero, we expect U and D* to approach the respective values for a 2D pressure driven flow in a channel without electric field, which are IU=1 ; D*=R+ 1 Pe2 (27) 210 Also, as Ut becomes large most of the molecules accumulate in a region of thickness 6 y and these molecules are subjected to a linear velocity profile, i.e., u y y The h dimensional mean velocity of the molecules therefore scales as 6 Thus h U The time needed by the molecules to equilibrate in the lateral direction At is ha 62 about , and the axial distance 1 traveled by the molecules during this time scales is of D 82 < > the order of UAt 6 6. Since the dispersion arises due to the difference in the D h axial motion of the molecules at various lateral positions during the times shorter than the 12 lateral equilibration time, D ~. Accordingly, in the large c regime D* is expected to At ( 3 2 Pe2 scale as /() ~ D Dh D a These scalings can also be obtained by expanding the exact solution from Eqs. (2 5) and (26) in the limit of both small and large a. The expansion for D* in the limit of a > 0is D*= R+Pe2( ++ a2+O(a4)) (28) 210 1800 To the leading order, the above expression reduces to R + Pe2, which is the same as 210 Eq. (27). Expanding Eq. (26) as a goes to infinity gives D = R + Pe72 +O( )) (29) Pe2 As expected, the leading order term scales as . However, the contribution from the next term, i.e., the O(a5) term, is about 10% of the leading order term for a as large as 100. Figure 22 compares the asymptotic solutions obtained above with the exact solution for D*. The small a and the large a approximations match the analytical solution for a< 2 and a >8, respectively. Similarly the asymptotic behavior of U in the limits of small and large a is 1 T=l 1 1 +0O(a4) a 0 (210) 60 6 1 U= +0( ) a 0oo (2 11) a a The above result for U approaches 1 as a approaches zero, and thus matches the mean velocity for Poiseuille flow in a channel without any lateral field. Also in the large a limit, the leading order term is of the order of 1/a, that matches the expected scaling. Figure 23 shows the comparison of these asymptotic results and the exact results from Eq. (25). The small a and the large a results match the full solution in the limit of a<2 and 0>40, respectively. These asymptotic results help us in understanding the physics of the dispersion and the DNA separation, as discussed below. o 0 5 10 15 Pe U, 20 25 30 Figure 22. Dependency of (D*R)/Pe2 on the product of Pe and U'. The dashed line is the large approximation Eq. (29), and the dotted line is the small a approximation Eq. (28) 1.5 1 0.5 10 20 30 40 50 PeUl 60 70 80 Figure 23. Dependency of mean velocity U on the product of Pe and U. The dashed line is the large a approximation Eq. (211), and the dotted line is the small a approximation Eq. (210) Dependence of the Mean Velocity on U and Pe Figure 23 shows the dependence of the mean velocity on Uy and Pe. The mean velocity depends only on a, i.e., the product of Uy and Pe. As discussed above the product PeU is essentially the inverse of the dimensionless thickness of the thin layer near the wall that contains a majority of the particles. Thus, it is clear that at large a, an increase in a leads to a reduction in the velocity of most of the particles and thus causes a reduction in the mean velocity. However, the effect of an increase in a at small values of a is not so clear because with an increase in a, the molecules that are attracted to the positive electrode travel with a smaller velocity, but the molecules that move farther away from the negative wall travel at a larger velocity. Due to the exponentially decaying concentration profile away from the positive electrode, the effect of the reduction of the velocity near the positive electrode dominates, and accordingly even in the small a regime, the mean velocity is reduced with an increase in a. The mean velocity is thus a monotonically decreasing function of a. Dependence of D on U and Pe The effective dispersion coefficient D depends separately on Uy and Pe. However, (D* R)/Pe2 depends only on a (Figure 22). As discussed above for small a, with an increase of a, the particle concentration near the positive wall (Y = 1 in our case) begins to increase, and at the same time the particle concentration near Y = 0 begins to decrease. However, a significant number of particles still exist near the center. The increase in a results in an average deceleration of the particles as reflected in the reduction of the mean velocity (Figure 23), but a significant number of particles still travel at the maximum fluid velocity. This results in a larger spread of a pulse, which implies an increase in the D At larger a, only a very few particles exist near the center as most of the particles are concentrated in a thin layer near the wall, and any further increase in a leads to a further thinning of this layer. Thus, the velocity of the majority of the particles goes down, resulting in a smaller spread of the pulse. Finally, as a approaches infinity, the mean particle velocity approaches zero, and the dispersion coefficient approaches the molecular diffusivity. Since the behavior of the dispersion coefficient with an increase in a is different in the small and the large a regime, it must have a maximum. The maximum is expected to occur at the value of a beyond which there are almost no particles in the region y Figure 22 shows that the maximum value of (D* R)/ Pe2 occurs at a ~ 4 and the value at the maximum is about .007. This implies that the convective contribution to dispersion is at most .007 Pe2. Thus, even at Pe = 10, the maximum convective contribution is only about 35% of the diffusive contribution R, which lies between 1 and 2. However, at Pe>50, which is typical for large DNA strands and a~c, the convective contribution dominates the dispersion. Separation Efficiency Consider separation of DNA molecules of two different sizes in a channel. As the DNA molecules flow through the channel they separate into two Gaussian distributions. The axial location of the peak of the DNA molecules at time t is simply ut and the width of the Gaussian is /4DD*t We consider the DNA strands to be separated when the 15 distance between the two pulse centers becomes larger than r3 times of the sum of their half widths, i.e., (U2 U ) t > (4D 1t+ 4D,Dt) (2 12) where the subscripts indicate the two different DNA fragments. If the channel is of length L, the time available for separation is the time taken by the faster moving species to travel through the channel, i.e., L /max(1u ,u2) Substituting for t, and expressing all the variables in dimensionless form gives L/h>12 l max(U1,U2)[ D 1 ]2 (213) Pel U2 U Eq. (213) can also be expressed as / D*2 1+ L/h l2 xU 1 =12 (214) Pe1U, _U PelU U1 where is a measure of the resolving power of the separation method and we have assumed that species 1 travel faster than 2. In the discussion below, we use L/h to indicate the efficiency of separation, i.e., smaller L/h implies a more efficient separation. The time needed for separation is the time required by the slower moving species to travel through the channel, i.e., T = L (215) < u > min(U1 ,U2) Effect of Pe and Uy on the Separation Efficiency In Figures 24 and 25, we show the dependence of L/h on Pe and Uy in the case of U; = U2 which corresponds to DNA fragments of two different lengths. Figure 25 is similar to Figure 24; the only difference is the value of the ratio D2/Dx. Figures 24 and 25 show that at a small U', increasing U;, which is physically equivalent to increasing the electric field, leads to a reduction in L/h required for separation. As Pe U' y increases, the mean velocities of both kinds of molecules decrease (Figure 23). But the dispersion coefficients do not change significantly because they are very close to the diffusive value R for Pe < 10. Thus, L/h is primarily determined by the ratio 2 As shown earlier, in the small a regime ~ 1 a, thus, Pei (U2 U1 60 2 U 1 )4 2 1 e2)2 Since the ratio Pe2/Pel is fixed, PeUU2 _U Pe Uy Pe2 122P 1 ~ Pe, (U )4 = 4 Thus, an increase in either Pe or U' Pe (UyY (e22 Pe2 12Y Pe leads to a reduction in L/h in the regime of small a. The constant Pe plots in Figure 24 and 25 show the (U)4 dependency when Uy is small. Also, the constant Pe curves shift down with increasing Pe, due to the Pe5 dependency shown in the above scaling. The above expression also shows that at a fixed a, an increase in Pe leads to a reduction in L/h. In the limit of large a, U 6 / a, thus, S2 U 2 U2 1 y PelPe2 ~ U This implies that even in the large a Pe U2 _i PeiPe2 Pe2 Pe y Pe Ie *2 ^ G~G f^r^)r regime for a fixed ac, an increase in Pe leads to a reduction in L/h. It also shows that in the large a regime and at 0(1) Pe, L/h becomes independent of Pe and begins to increase with an increase in U as shown in Figure 24. Since L/h scales as (U i in small a regime, and as Uy in the large a regime, it must have a minimum. Physically, the minimum arises because at small field strength, the molecules accumulate near the wall, but the region of accumulation is of finite thickness. Since the thickness of the region is different for the two types of molecules, the mean velocities of the two types of molecules differ. However, as the field strength becomes very large, the thickness of the region of accumulation becomes almost zero and both the mean velocities approach zero. Consequently, the difference of the velocities also approaches zero. Therefore, the difference in the mean velocities is zero for zero field because both the mean velocities are equal to the fluid velocity, and is also zero at very large fields because both the mean velocities approach zero; this implies that a maximum in the difference between the mean velocities of the two types of molecules must exist at some intermediate field. This maximum results in a minimum in L/h required for separation. The effect of changing Pe while keeping U' fixed is more difficult to understand physically. Due to the dedimensionalization of U;, in order to change Pe while keeping Ui fixed, both the fluid velocity and the electric field must be changed by the same factor. As a result, if we want to determine the effect of only an increase in the mean velocity , we need to increase Pe and concurrently reduce Uy by the same factor. Thus, in Figures 24 and 25, we need to first move to the smallerU' value and then follow the constant U' curve to the larger Pe. This keeps Pe U1 constant and at 0(1) Pe, y I 5, optimal Pe is the one at which the convective contribution to dispersion is about the same as the diffusive component, i.e., Pe~10. DNA Separation To accomplish the separation of DNA by EFFF the applied field and the mean velocity have to satisfy the following constraints: (1) The applied electric field should be less than the value at which the gases that are generated at the electrodes supersaturate the carrier fluid and causes bubbles to form. The critical field at which bubbles form depends on a number of factors such as the ionic strength, the electrode reactions, presence of redox couple in the solution, fluid velocity, etc. In EFFF, researchers have applied an electric field of 100V/cm without gas generation [2]. However, the double layers consume a majority of this field and the active field is only about 1% of the applied field [2], i.e., about 100 V/m. In the EFFF experiments reported above [2,15], the carrying fluid was DI water or water with a low ionic strength in the range of 1050 piM. However, experiments involving DNA are typically done in the range of 10 mM concentration of electrolytes such as EDTA, trisHCI and NaCI [37]. EFFF cannot operate at such high ionic strengths unless a redox couple such as quinone/hydroquinone is added to the carrier fluid [2,38]. Thus, in order to separate duplex DNA by EFFF it may be necessary to study the stability of the DNA in reduced ionic strength fluids or in the presence of various redox couples and then identify a redox coupleelectrode system that does not interfere with the stability of the DNA. Alternatively, the separation could be accomplished under pulsed conditions, which prevent the double layers from getting charged. This method can increase the strength of the active field. In this scheme the field is unidirectional for a majority of the time but the polarity of field is reversed for a short duration (10% of cycle time) in each cycle to discharge the double layer [15]. For the calculations shown below we assume that the active field is about 1% of the applied field of 100V/cm. Since the DNA mobility for strands longer than 400 bp is 3.8x108 m2/(Vs), a field of 100 V/m will drive a lateral velocity of about 3.8 ptm/s. (2) The second restriction on uy arises from the fact that the thickness of the layer in which the molecules accumulate, 6, is given by 3D/u For continuum to be valid the thickness of this layer must be much larger than the radius of gyration of the DNA molecules. On neglecting the excluded volume effects, which is a reasonable assumption for strands shorter than about 100 kbp, the radius of gyration R, = 1k where k is the Kuhn length (=2 x persistence length) and Nk are the number of Kuhn segments in the DNA chain [23]. The persistence length of a double strand DNA is about 50 nm, or about 150 bp [39]. Thus a Kuhn segment is about 100 nm long and contains about 300 bp, and =100 /N R 2 100 2N nm. The diffusivity of the DNA in a 0.1 M PBS buffer is 300 2x1010 2 D= m /s [40]. Let us choose 8 = 10R This gives 3D 3 x 101 1.5 x 102 . ue = x 10 1.5 02 m/s. Thus the condition 6 >> R. imposes a y 8 20 x 109N N smaller value for u than the condition for prevention of bubble formation for N>5000. (3) The shear in the microchannels is expected to stretch the DNA strands. For Wiessenberg number Wi over 20, the mean fractional extension of a long DNA molecule (50kb) is over 40%, and instantaneously can reach 80% of its length [41]. Thus, in order for the molecules to stay coiled, the shear rate in the channel must be much less than the inverse of the relaxation time tr of the DNA, i.e, 6 tr, which is the Weissenberg number Wi, should be less than 1. The h relaxation time t, and based on this scaling and the experimental values kT reported in literature, tr in water is about 1.6x108 N'5 s, and accordingly it has a value of about 0.01s for N = 10000. Thus for strands that are about 10000bp long, the shear rate 6 < u > should be less than 100 s1 to prevent any significant h stretching of the DNA strands. Due to the very small diffusion coefficients of the large (>1 kbp) DNA strands, the Pe number is expected to be large. Thus we focus our attention on the large a=Pe U' y 1 regime. As derived above for the case when a is large but U' the convective contribution to the dispersion dominates over molecular diffusion and the length required for separation is given by L/h~ 144 = 144 K L (2 18) Pe2( hut u By using Eq. (215), the time for separation is 144 hPeUy 24 h 24D Pe2(U 6(u) pe )2(u)(u)2 In the subsequent discussion we restrict 8 = 10R The above scalings for L and T can equivalently be expressed in the following forms: L ~ 163 (u) and T  D h 3D Substituting 8 = 10R, and expressing Rg and D in terms of N gives the following expressions for L and T: L 6 x 1013N L~6xo13N2um (220) h T 5x106N312 s (221) Interestingly the above expressions show that the time for separation is independent of the mean velocity and the channel length is directly proportional to the mean velocity. Thus, a reduction in the mean velocity will reduce the channel length required for separation. The reason for this effect is the reduction in dispersion due to a reduction in the . However, if the mean velocity becomes very small the diffusive contribution dominates the dispersion, and in this regime the expressions for L and T become L 6h 6hD S6Ue = L PeUe = (222) h Pe 8 L 6hD PeUe D h2 T (223) U 66 62 2 and accordingly the length and the time required for separation begin to increase with a reduction in the mean velocity. Thus, the optimum channel length required for separation occurs when the convective contribution to dispersion is the same as the diffusive contribution. But this optimization does not effect the time required for separation, which as shown below in the limiting factor in the separation. So we simply choose the shear rate to be about 1 so that it is less than the inverse relaxation time for DNA strands. Thus the above expressions for L and T become L 6x 1013 Nm (224) T 5 x 106 N312 s (225) For DNA the + as defined by Eq. (214) is given by K 12 3/2 22 3/4 ,2 D3D D2 N 3 1+ 1+ + SDD = i(D= [ N2 ] 16 I (226) 1_ U2 1 D2 Ni AN U1 D1 1 N where AN=N2N1, and we have utilized the large a approximations to relate the mean velocity and the dispersion coefficient to N, and we have assumed that the convective contribution to the dispersion is dominant over the molecular diffusion. Thus, these values do not represent the optimal length because as discussed above the optimal length occurs when the convective and the diffusive contributions to D* are about the same. Including in the expressions for L and T gives L 9.6 x 1012 N2 m (2 27) AN * T 8 x 10N312 s (228) IAN) The above expressions show that DNA strands in the range of about lOkbp that differ in size by about 25% can be separated by EFFF in a channel that is a few mm in size and in a time of about half an hour. However, separation of larger fragments in the range of about 100 kbp will take a prohibitively large time of about 11 hours. Other techniques such as entropic trapping [42] and magnetic beads [27] are clearly superior to EFFF because they can separate fragments in the range of 50 kbp in about 3040 minutes. However, we note that the time for separation can be significantly reduced if we relax the restriction of 8 = 10R But under this situation the continuum equations cannot be used and one will need to perform noncontinuum simulations to predict the effectiveness of EFFF at separating DNA strands. We also note that in our model we have not taken into account the adsorption of DNA on the walls, which will need to be carefully considered before designing the EFFF devices for DNA separation. However, our model shows that EFFF has the potential to separate DNA strands in the range of 10 kpb and the model can serve as a very useful guide in designing the best separation strategy. Furthermore, this model can also be helpful in designing the channels for separation of other types of particles. Comparison with Experiments As mentioned earlier, F1FFF (Flow field flow fractionation) has been used to separate DNA strands and below we compare the predictions of the dispersive model with the experimental results. It is noted that Giddings et al. also compared their experimental results with the model [13], but they only compared the experimental and the predicted resolutions, while we compare the entire temporal concentration profiles at the channel exit. As shown in Appendix A, the convection diffusion equation can be converted to the dispersion equation of the form a S+ < u > U = DD* (229) at 8x 8x2 where U and D* are the dimensionless mean velocity and the dimensionless dispersion coefficients, respectively. Accordingly, for a pulse input the concentration profile at the channel exit (x = L) is given by M (L < u > Ut)2 < Co >= 4 exp( ) DD't 4DD't (230) where M is the mass of the solute present in the pulse. Liu and Giddings separated double stranded DNA molecules of 1107bp and 3254bp, and 692bp and 1975bp successfully with F1FFF. Although the lateral field in their experiments was generated by flow, which is different from the lateral electric field used in EFFF, the two methods are equivalent, and can be described by the same equations. Figure 26 shows the comparison of the dispersive model with their experiments. In Figure 26, the experimental data of intensity at the detector located at the channel exit is compared with the concentrations predicted by Eq. (230). The vertical scale has been adjusted to ensure that the maximum height of the predicted profiles matches the maxima of the experiments. All the other parameters required for the comparison were directly obtained from the experiments. The comparison between the model and the experiments is reasonable. Next, we compare the predictions of the dispersive model with the experiments of Gale, Caldwell and Frasier in which they separated latex particles of diameters 44, 130 and 207nm by EFFF[1]. Figure 27 shows the comparison of the intensity at the channel exit with the concentration predictions from the dispersion model for EFFF. As in Figure 27, the concentrations are scaled to match the experimental maxima. As seen in the Figure, the comparison between the experiment and the model is reasonable for the 44 ..... ... . Since the mean velocity of the particles under a lateral field depends on the Pe, colloidal particles such as DNA molecules that have the same electrical mobility can be separated on the basis of their lengths by applying lateral electric fields. Axial fields cannot accomplish this separation unless the channel is packed with gel. However, the separation may have to be performed in low ionic strength solutions or in the presence of redox couples or with pulsating electric fields. The optimal Pe for separation is the one at which the diffusive contribution to dispersion is about the same as the convective contribution. The model predicts that DNA strands in the range of 10 kpb can be separated in about an hour by EFFF. However, separation of fragments in the range of 100 kbp may take a prohibitively long time. Applying a larger electric field may shorten the separation time for the 100kbp fragments, but noncontinuum simulations need to be performed to determine the efficacy of EFFF at separation of DNA fragments in this size range. The results of this study can serve as a very useful guide in designing the chips for experimentally studying the separation of DNA strands in the range of 100 kbp and also for separation of other kinds of particles by EFFF. CHAPTER 3 SEPARATION OF CHARGED COLLOIDS BY A COMBINATION OF PULSATING LATERAL ELECTRIC FIELDS AND POISEUILLE FLOW IN A 2D CHANNEL The proposed method in this chapter is a cyclic process that combines pulses of lateral electric fields and a pulsating axial flow driven by a pressure gradient. The three step cycle that repeats continually is shown schematically in Figure 31. Initially, after introducing the charged particles into the channel, a strong lateral electric field is applied for a time sufficient to attract all the molecules to the vicinity of the wall. The first step of the cyclic operation requires removal of the electric field for time td that is much less than the diffusive time for the smallest molecules, i.e., h2/D, where h is the height of channel and D is the molecular diffusivity. During this time the molecules diffuse away from the wall, and shorter chains on average diffuse farther due to their larger diffusion coefficients. In the second step, we propose to drive flow through the channel for time tf, which is much shorter than td. Since tf << td, there is only a small diffusion during the flow and the molecules essentially convect in the axial direction with the local fluid velocity. Due to the parabolic velocity profile, the molecules that have a larger diffusivity move a longer distance during the flow because they are farther away from the wall. In the last step, the strong electric field is reapplied to attract all the molecules to the vicinity of the wall. As a result of this cycle, the molecules with a larger diffusion coefficient exhibit a larger axial velocity. This technique shares some similarities with the cyclical fieldflow fractionation technique developed by Giddings [43] and extended by Shmidt and Cheh [44] and by Chandhok and Leighton [45], which relies on the t . the effectiveness of the proposed method at accomplishing separation and compare the proposed method with unidirectional EFFF. Theory Model The diffusive step: No electric field and no flow Let us assume that after the application of the electric field all the molecules have accumulated near the wall. Although the molecules are present in a thin layer near the wall, we treat the thickness of this layer to be zero, and accordingly define a surface concentration Fi (x), which is the number of molecules per unit area after the i cycle. Next, the electric field is removed and the molecules begin to diffuse in both the axial and lateral directions. Since there is no flow in this step, the diffusion of the molecules is governed by the unsteady diffusion equation aC 2C 22C + s (3 1) at 8Y2 TX2 The above equation is in a dimensionless form where x y c t h X=, Y=, C= 9<<1 (32) A h co h /D C t is the characteristic length in the axial direction, h is the channel height, D is the diffusion coefficient of the colloidal particles, and x and y are the axial and the lateral directions, respectively. We note that the channel is assumed to extend infinitely in the z direction. The above differential equation is subjected to the following boundary conditions: (, Y = 0) = (T,Y = 1) = 0 (33) Additionally, overall mass conservation requiresY Additionally, overall mass conservation requires J JCdXdY JFijdX (34) cc 0 00 Since the diffusive step only lasts for time td and in our model h >> D the boundary condition at Y=1 can be replaced by C(r, Y > oo) = 0 We solve Eq. (31) by using a the technique of regular perturbation expansions [46,47] in s. The concentration is expanded as C=Co +2C2 +... (35) By substituting C into Eq. (31) and Eq. (33), we get the differential equations and the boundary conditions for different orders in s. The equation for the order of s is c 0 = 0 (36) Ot aY2 The solution to Eq. (36) subject to the boundary conditions Eq. (33) and the overall mass conservation is Y2 Co = (X) exp( ) (3 7) VrT7t 4T The differential equation for the order of s2 is C2 2C a2C 2 a2 0 (3 8) O aY 2 8X2 The solution to Eq. (38) subject to the boundary conditions Eq. (33) is a2F, (X) Y C2 X2 exp(~ ) (39) The above solution satisfies the overall mass conservation because > 0 as x > +o. Ox The combination of Co and C2 gives the concentration profile at the end of diffusion step (T =d). 1Y[ Y a (X) Y 3 Cff = C(= t)= F (X)exp( )+ exp() (3 10) Fd7 4,du8X x 4,d We note that td must be smaller than about 1/20 for this equation to be valid because otherwise the presence of the wall at Y = 1 will affect the concentration profile. It is possible to obtain analytical solutions that can include the effect of the wall at Y = 1, but as shown later, the separation is more effective for the case when Id is small and thus we use the simpler similarity solution obtained above. The convective step: Poiseuille flow with no electric field To determine the concentration profile during the convective step, we need to solve the convectiondiffusion equation, and apply the solution at the end of the diffusive step Eq. (310) as the initial condition. The dimensionless convectiondiffusion equation is CC aC 1 2 2 P 2 +U a2c (3 11) t x 8X sPe 8Y2 sPe 8X2 In the above equation time has been dedimensionalized by the convective scaling, i.e., h S/ < u >, and Pe All the other dimensionless variables are the same as in the diffusive step. We solve the above equation under the conditions sPe >> 1. Accordingly, 1 we assume a regular perturbation expansion for C in terms of s and  i.e., sPe C=(C)+C2)2 + +1 (C2 +C2 )+( +C2) 2+...)... (3 12) sPe (sPe) The boundary conditions for C in the convective step are the same as in the diffusive step. The leading order equation for C in is sPe O(C)r + C' 2) 2 O(c) +C2)__) + U Cx 0 X (3 13) In the above equation, we transform the X coordinate to (3 14) As a result, Eq. (313) becomes (C) + C2)2) C (2) 2 OT = 0 0 C O) ({, x)+ C2) ({, T)s2 O) (,o)+C(2) (,)62 C0 (~O) + 0 (, cdiff (3 15) Using Eq. (310) in Eq. (315) gives C()(X,T)= T (X UxC) exp( ) C( (X, ) = ) exp( ) (3 16) SX2 4Td Similarly, by solving the equations for various orders of s and 1/(Pes), we get C), C(2) C) and C 2. Substituting them into Eq. (312) gives C(XY,)= (XU )exp( 2)+. 82 (X exp( ) t dl 4X Vt OX2 4 CY 1 Y4) I (XUT)exp( )(+ ) Pes a2 (X )t 1y2 Y 1 2 (XU) exp + F E + )exp( )+ exp( ) L X VaC 2t, 44T 4 c X 4t/J 4 (XUt)exp( ) (X )Yexp( ) F, (XUt)( + exp( 4 471 4 1 72 4 4c4 21 (XU2 ) y2 exp(4 1 XX 4c 1 Y2 2 y2 I (XuT)( + _)Y2 exp( ) 1 2t4 4 4t 2 8a2' (XtUU) 1 0X2 4 02r, (X U,) 1 8X2 ( I 1 Y2 Y2  + _,2)exp() 2u, 44 4id 1 Y2 Y + )Y2 exp( ) 2T4 4 7 d 4T (3 17) The axial flow is driven by pressure gradient and thus the velocity profile is parabolic, i.e., (3 18) where < u >= O h2 is the mean velocity in the channel. 3ul Ox) In dimensionless form, Ux =6(YY2) (3 19) Substituting Ux in Eq. (317), one can determine the concentration profile during the convective step. Electric field step (Electric field, no Flow) The concentration profile at the end of the second step can be calculated by substituting T = Tf in Eq. (317). In the third step, the electric field is applied to attract all the 2 +g ux = 6 < u > (y / h (y / h)2) molecules to near the wall. Neglecting axial diffusion during this step, the surface concentration F after the end of the i+lst cycle is 1 Fr,+ (X) = C(X, Y, ,)dY (3 20) If the convective distance traveled in each cycle is much smaller than the axial length scale, then the expression for C in Eq. (317) can be expanded by using Taylor series, i.e., Fi(XUx~)= Er(X)Uxt a OX (Ut)2 X2, +2 x2 2 8X2 After using the above expansion in Eq. (317), and then substituting the expression for C in Eq. (320), and then performing the integration gives r i+1 Fi + U 8x (321) = D OX2 3x2 where X and s have been replaced by x/1 and h/1 respectively, and 1 1 12z t f DPeerf ( ) 12t D2Peerf( ) S12z' tfDPe 2 ,u) 6tD2Pe 26t7 3tfD Pe U h h /2h h3) 2/72h5 (3 22) 36, dPe2t D2erf( ) 216z2t2Pe2D2erf( ) D'= ( 24,7 288z2tPe D2 24t h2 Jzh2 h2 36Pe2tD 3erf(1 ) + (Tderf( )h 2) 2 24it h 432i /2t3Pe2D3 432TdtfPe Derf( ) 27 , 216tf Perf( 1108 l/2t4pe 2D4 2477 +(tfD erf(  ))+( + ) 2.dF /h h (3 23) If Fi is known, then Eq. (321) can be used to determine Fi, i.e., the surface concentration at the end of i+lst cycle. Since F0 is known, by repeating this process one can numerically obtain the surface concentration as a function of x and the number of cycles. The above equations are only valid for Td<1/20, thus the errorfunction (erf 4 ) can be simply replaced by 1.0. In the above derivation, it was assumed that the colloidal particles accumulate at the wall at the end of the third step. When the electric field is applied in the lateral direction, a concentration gradient will build within a thin layer near the wall. The thickness of this layer depends on the intensity of the electric field, and the model proposed is only valid if the thickness of this layer is much smaller than h. Below, we estimate the intensity of the field required to accumulate most of the molecules in a thin layer of thickness h/100. The motion of molecules in the third step is governed by the convectiondiffusion equation where the convective term arises due to the lateral electric field, i.e., Oc &c 0C +ue = D (3 24) Bt Y y 6 y 2 where Uy is the velocity of the molecules in the lateral direction due to the electric field and in the limit of thin electrical double layer can be estimated by the Smoluchowski equation, u, = r E, where sr and 4 are the fluid's dielectric constant and viscosity, respectively, so is the permittivity of vacuum, and C is the zeta potential of the colloidal particle. Alternatively, the electrophoretic velocity can be expressed as ue = g.E, where p, is the electrophoretic mobility of the particles, which has been measured for a variety of colloidal particles [2]. By treating the colloid as a point charge, the electrophoretic velocity can equivalently be expressed as ue D Z (D where Tt is the absolute y kTt oy temperature, e and Ze are the charge on an electron and on the particle, respectively. The effective particle charge is in general less than the actual charge due to the electric double layer surrounding the ion. However, for a weakly charged polyion in the limit of low ionic strength, Z approaches the actual charge on the polyion. Since we need an equation for ue only for an approximate estimation of the field required to attract all the molecules near the wall, we use the simpler expressionue D Z kTt, y The steady state solution to Eq. (324) is u'y c=c(y = 0)exp( ) (325) D uTh To attract most of molecules into h/100 of the plate, a field satisfying 3 must 100D be used. This gives Ze 8_ 300D kTt ue D Ze 300D =>Ad=300 (326) y kTt Oy h Ze Assuming Z ~ 10, which is a very conservative assumption, gives AD = 0.77V. Later we use a value of about 33 ptm for h, and a potential drop of .77V across a 33 pm channel is about the same voltage as is applied in EFFF [15]. Additionally, under this electric field h h2 the steady state will be attained in a time of about  which is much less than the ue 300D diffusive time and thus the assumption of neglecting diffusion during the third step is reasonable. Furthermore, for h = 33 pm and D = 1010 m2/s, the time for attaining steady state is about 3 ms, which is less than the time scale for charging a double layer [3,38]. Thus, gas generations may not be a problem in the third step and a majority of the applied potential difference occurs in the bulk of the channel. Long time Analytical Solution To better understand the physics of the separation and to avoid repetitive numerical simulations, we also obtain an analytical solution for the surface concentration in the long time limit, in which the surface concentration can be treated as a continuous function oft and x. First, we expand F into Taylor series in terms of time aFi 1 a2i Fi =(t +td) = Fi +(tf +t) +(tf +td)2 (3 27) Using (tf + td) as time scale gives the following dimensionless equation ar i a 1 a2 r = r. l (3 28) ST 2 T2 Also as shown above ri+1 Fi + U = D 2 (329) Ox x2 (329) Substituting Eq. (328) into Eq. (329) gives r 1i 1 &L Dri 2. + i + U= D' (330) aT 2 T 2 ax Ox2 We again define a new coordinate system, =xU'T (331) In this moving reference frame Eq. (330) becomes 41 a 1 a2F a2F U'2 a2 + U (D' (332) aT 2 T2 T 2 (332) Where, the subscript has been removed. We shall show later that the long time solution to the above equation is Gaussian, i.e., F = exp( ) (333) IfT 4DT Thus, Or 2] a32F 5 I2 SO(T 2) O(T 2) O(T 2) O(T2) aT 82 T2] T (3 34) Keeping the leading order terms in Eq. (332) gives aF U'2 a2 (D ) (335) aT 2 8t Transferring it into the original coordinates gives ar ar U a2 (336) +U =(D ) (336) OT ax 2 ax2 Thus, the long time surface concentration is a Gaussian with the dimensional mean velocity U* and effective diffusion coefficient D* given by U* D*= (337) tf +td tf +td We dedimensionalize U* and D* with tf/(tfrtd) and (tf)2/(tf+td), respectively, and denote them as U* and D*. The dimensionless mean velocity and dispersion coefficient are U2 U1U U* ; 2 (338) < u > tf (< u > t)2 Results and Discussion Since d must be smaller than 1/20, the value of the error functions in Eq. (322) and Eq. (323) are very close to 1, thus the expressions for U' and D' can be rearranged in the following form: U ( 12d) ( 12d) ( )2 / tf L td, )+ I td 2( (339) =Ul(Tr)+ tU2(r)+(tf )2U3(r) td td D' 288t3d/2 162 D'(< u > tf2 (36Td + 2167 ) +( h2 + [t (36Td 432 +432C)2 +(a (< u > tf)2 t d 4 tht2 108 C/2] (td) )2( 216 )1 t+ (< u > tf) 1t, d h t tf h2 = D1(d)+ h 2 D2(d) + tfD3(r )+ 2 D4(d)+ (tf)2 D5 (< u > tf)2 td t (< U > tf)2 td (3 40) On tracing the origin ofU1, U2, U3, Dl, etc, we find that Ul and Dl are contributions fromC); D2 arises fromC2) ; U2 and D3 originate from C(0) ; D4 is contributed byC2) ; and U3 and D5 originate from C (. The C() does not contribute to either U' or D'. Each of these terms depends only on 'd, and accordingly U' depends strongly on rd and t t, h2 weakly on . Also D' depends strongly of Td and weakly on L and 2 td td ((u)tf)2 The truncations errors in U' and D' are 43 Truncation error for U'= O((t)3) td  t h t 3 Truncation error for D'= O(() )+ O(( )2 (' ) 3) (3 41) td (tf < u >) td We note that for the proposed regular expansion solutions to be valid t h < 1 and < 1 (342) td (tf < u >) Mean Velocity Figure 32 plots the dependency of the dimensionless mean velocity on G ( ) Id for different values oftf/td. When G approaches zero, i.e., as the diffusion time becomes very large, the concentration profile along the lateral direction becomes uniform. Thus, the mean velocity of the pulse should be close to the mean velocity of the flow, i.e., the dimensionless mean velocity approaches 1. However, we cannot capture this effect because our model is only valid for G > 20 because of the requirement of Eq. (310). But this trend can be observed as G approaches 20. Figure 32 shows that a decrease in G results in an increase in the mean velocity of the pulse. This happens because smaller G implies larger molecular diffusivity for a fixed td and h. Since molecules with larger D diffuse a longer distance away from the wall, they are convected with a larger velocity. However, beyond a certain D, some molecules move beyond the centerline and get closer to the other wall and consequently convect at a smaller velocity. The molecules that get closer to the center, however, compensate for this effect, and thus the mean velocity curve exhibits no stationary extremum. .......... ........ ... Separation Efficiency Since the mean velocity of molecules depends strongly only on G, molecules with different values of G can be separated by this technique. We are interested in determining the time and the length of the channel required to accomplish separation of colloidal particles of different sizes. Consider separation of two types of particles in a channel with diffusion coefficient D1 and D2 respectively. We assume that when the distance between two pulse centers is larger than 3 times of the sum of their half widths, they are separated, i.e., (U7 U;)T(tf +td) ,3( 4D7T(tf +td) + 4D;T(tf +td)) (JBY J*+ ) 2 T> T 12( ) (343) U U* We use Eq (343) to calculate T, i.e., the dimensionless time or, equivalently, the number of cycles needed for separation. The dimensional time required for separation 0 is equal to T(tf+ td), i.e., 12 1+ ( (344) tf tf U U2 The length of the channel required for separation is equal to the distance traveled by the faster moving molecule in this time, i.e., L < tf (D +4 X ) L=TU' => 12Ui h ( ) 2 (345) h h 2 H understand the reasons for this behavior, we calculated the difference between the mean velocities of the two type of molecules and sqrt(D ) as a function of Gi. At small Gi, an increase in Gi leads to an increase in the difference in the mean velocities and a decrease in sqrt( D ). Thus, both the factors lead to a better separation, resulting in a reduction in the number of cycles. Beyond a critical value of Gi, the difference in the mean velocities begins to decrease with a further increase in Gi. Thus, the effect of reduction in the dispersion is compensated for by a reduction in the difference in mean velocities, leading to an almost constant value on the number of cycles needed for separation. Effect of tf/td Figure 35 shows the dependence of T, 0/tf and L/h on for fixed values of G1, td ( )and G2/GI, which are noted in the caption. Figure 35 shows that the number of u)t loops required for separation is relatively independent of t. With an increase in , td td the mean velocities and also the difference in the mean velocities increase but this effect is compensated for by an increase in the dispersion coefficients, and thus the number of loops required for separation does not change appreciably. However Figure 35 shows that the time required for separation depends strongly on the ratio L. This happens td because although the number of steps is unchanged, td decreases as L increases and this td leads to a reduction in the time for each step, and consequently a reduction in time for F ...... The dependency of T, 0 and L on the three dimensionless numbers remains the same for G2/Gi=1.2. However, the actual values increase significantly. The description above shows that the optimum values of G, t and are about 150, 0.3 and 0.3, td ((u)tf)2 respectively. Based on these optimum values of dimensionless parameters we can choose the appropriate values of the dimensional parameters, as shown below. Let us consider separation of two types of molecules with Di=1010 m2/s and D1/D2=2. It is clear that a smaller tf will lead to a reduction in separation time. However, the minimum value of tf is limited by the time in which the flow can be turned on and off in the channel. Rather than turning the pump on and off it is much faster to switch the flow between the channel and a bypass system by using a valve. Since the flow rates in microfluidic devices are small, the valves can switch in time scales of 1 ms [48]. To eliminate the effects of the ramping up and ramping down of flow during the opening and the closing of the valve, we choose tf to be 20 ms in our calculations. Since we fix t h2  =0.3, td is about 0.067 s. By using G = 150 and =0.3, the values of h and td (u)t f) are 32 pm and 0.003 m/s, respectively. The values of length and the time for this separation are 3.7 mm and 15.7 s, respectively. If G2/G1 is reduced to 1.2, the values of length and time increase to 5.45 cm and 231 s, respectively. For the same design, a further reduction in D1 improves separation because changes in D1 only change G and as shown above 0 and L are reduced by an increase in G. However, if the diffusion coefficient is about 10"12 m2/s, based on StokesEinstein equation the particle size is about 0.2 pm and in this case the proposed continuum model is not valid. An alternate model that takes into account the finite particle size may need to be developed to determine the effectiveness of the proposed technique at separating particles with D < 1012 m2/s. On the other extent, as D becomes larger, the channel height h must be increased to ensure that G does not becomes smaller than about 50. An increase in h leads to an increase in L. For instance, separation of molecules for Di=109 m2/s for G2/Gi=2 takes about 17.5 s in a channel 1 cm in length and 58 jim in height. The time and length become 269 s and 16 cm for G2/Gi=1.2. The separation can be significantly improved if faster switches can be designed so that tf can be reduced below 20 ms. Comparison with Constant EFFF The technique proposed above is very similar to the commonly used EFFF. In both the techniques the electric field is used to create concentration gradients in the lateral direction and the axial Poiseuille flow is used to move the molecules in the axial direction with mean velocities that depend on the size and charge of the molecules. As mentioned above, the electric fields that are applied in EFFF are limited to about 1 V/ 10 pm. Also only about 1% of the applied electric field (= 1000 V/m) is active in the channel and the rest is applied across the double layers at the electrodes. The lateral electric velocity Ue due to the electric field is estimated by the equation uy = pgE, where wo is the electrical mobility of the particles. The value of gt has been measured for various types of colloidal particles. It can also be determined by the Smoluchowski equation, p = V (sr and p. are the fluid's dielectric constant and viscosity, respectively, so is the permittivity of vacuum, and C is the zeta potential of the colloidal particle). The mobility of polystyrene latex particles is relatively independent of size and varies in the range of 1.9x10"4 3.23 x10"4 cm2/(Vs) for particle diameters in the range of 90 nm944 nm [2]. For smaller particles the mobility can be estimated by treating them as point charges and thus Kt can be expressed as D Z where D and Z are the diffusivity and the kT charge of the particle. For D = 1010 m2/s and Z =10e (e = electronic charge), the mobility is about 4 x104 cm2/(Vs). At these mobilities a field of 1000 V/m will drive a lateral velocity of the order of 20 pm/s. We note that in our proposed technique most of the applied field is active because the double layers are not charged and thus the electrical velocity can be as large as 2000 pnm/s, which as shown earlier can attract all the molecules in a very thin layer in a short amount of time. Below we compare the separation time and length required by the proposed technique with those required by the EFFF. For these comparisons, the values of h, DI, Di/D2 are 30 pm, 1010 m2/s and 2, respectively. The value of tf and are 0.02s and 2mm/s, respectively, and the value of G is varied from about 30 to 400, which is equivalent to varying td from 0.3 to 0.0225 s. The value of u' is varied from 0 100 gtm/s which is much larger than the expected values of the lateral electric velocity. In Figure 37 the value of L/h is plotted as a function of G and u' for both the techniques. The multiple curves for the EFFF correspond to different values of the mean velocity. In EFFF the reduction in the mean velocity reduces the length required for separation because of the reduction in the convective contribution to the dispersion. The time (Figure 38) required for separation does not change appreciably because both the length required for separation decreases almost linearly with the velocity. The trend of reduction in L/h with reverses at Pe<15 because although the convective contribution to dispersion still decreases, its value . ... ....... ... ........ . and off; the separation improves with a reduction in tf and td, which are the durations of the flow and the noflow steps. For reasonable value of design constants, the proposes technique can separate molecules of diffusivities 10"10 m2/s and 0.5x1010 m2/s in 15.7 s in a 3.7 mm long channel. The length and the time increase to 5.45 cm and 231 s if the ratio of the diffusivities is reduced from 2 to 1.2. The separation is easier for larger molecules; however, the model predictions may not be realistic due to the finite size of the particles. If the diffusivities are in the range of 109 m2i/s, the length and the time for separation are 1 cm and 17.5 s for DI/D2=2, and 16 cm and 269 s for D1/D2 = 1.2. The performance of the proposed technique is expected to be better than the EFFF. CHAPTER 4 TAYLOR DISPERSION IN CYCLIC ELECTRICAL FIELDFLOW FRACTIONATION This chapter aims to determine the mean velocity and the dispersion coefficient of charged molecules undergoing Poiseuille flow in a channel in the presence of oscillating lateral electric fields. Application of time periodic fields in EFFF techniques was first proposed by Giddings[43] and later explored by Shmidt and Cheh[44], Chandhok and Leighton[45] and Shapiro and Brenner[49,50]. In EFFF, particles with same values of D / uy cannot be separated, where D is the molecular diffusivity and uy is the electric field driven velocity on the lateral direction. Giddings suggested that cyclical electrical fieldflow fractionation (CEFFF) can accomplish separation even in this case. Based on this idea, Giddings developed a model for CEFFF under the assumption that the molecular diffusivity can be neglected while calculating the concentration profile in the lateral direction. Shmidt and Cheh[44], and Chandhok and Leighton[45] extended the idea proposed by Giddings to develop novel techniques for continuous separation of particles by introducing an oscillating flow that is perpendicular to both the electric field and the main flow. But the molecular diffusion in the lateral direction was still neglected in both of these papers. Shapiro and Brenner analyzed the cyclic EFFF for the case of square shaped electric fields. They included the effects of molecular diffusion in their model and obtained expressions for axial velocity and effective diffusivity in CEFFF in the limit of large Pe. They concluded that the axial velocity and effective diffusivity depends only on a single parameter T = t0u0 / h, where to is the time period of oscillation, uO is the amplitude of the lateral velocity and h is the channel height. There are two main differences between the work of Shapiro and Brenner and the work described in this paper. Firstly, our results are valid for all Pe whereas the results of Shapiro and Brenner are valid only for large Pe. Secondly, we examine both sinusoidal and square shaped electric fields whereas Shapiro and Brenner obtained the asymptotic results for square shaped electric fields only. In the next section we solve the convection diffusion equation for cyclic EFFF by a multiple time scale analysis to determine the expressions for the mean velocity and the dispersion coefficient. Next, we examine the effect of the system parameters on the concentration profiles and the mean velocity and the dispersion coefficient for the case of sinusoidal electric fields. Finally, we compute the mean velocity and the dispersion coefficient for the square wave and compare the results with the asymptotic analysis of Shapiro and Brenner. Theory Consider a channel of length L, height h and infinite width that contains electrodes for applying the lateral periodic lateral electric field. The approximate values of L and h are about 2 cm and 20 microns, respectively. Thus, continuum is still valid for flow in the channel. Also, the aspect ratio is much larger than 1, i.e., s h / L << 1. The transport of a solute in the channel is governed by the convectiondiffusion equation, 8c 8c e c 02c 0c +u +U =D  +D (41) at Ox ay ax2 Y 2 where c is the solute concentration, u is the fluid velocity in the axial (x) direction, D11 and D1 are the diffusion coefficients in the directions parallel and perpendicular to the flow, respectively. We assume that the diffusivity tensor is isotropic and thus D11 = D1 = D. In Eq.(4 1), u is the velocity of the molecules in the lateral direction due to the electric field. If the Debye thickness is smaller than the particle size, then the lateral velocity ue can be determined by the Smoluchowski equation,uy = (sosr,/ g)E, where sr and p are the fluid's dielectric constant and viscosity, respectively, so is the permittivity of vacuum, and is the zeta potential. Or, it can be simplified as u' = gEE where gE is the electric mobility which has been measured for a number of different types of colloidal particles, e.g., the mobility of DNA beyond a size of about 400 bp is 3.8x108 m2/(V*s).[22] In EFFF, researchers have applied an effective electric field of 100V/cm without gas generation. Thus, typical values of u' could be as large as 3.8x104 m/s. Eq. (4 1) is subjected to the boundary condition of no flux at the walls (y = 0,1), i.e., Dc D+uc = 0 (42) ay In a reference moving in the axial direction with velocity u*, Eq. (4 1) becomes ac _. ac ac 2c 02c + (uu ) +u = D( ) (43) 8t 6x Q'ly 8x2 y where x is now the axial coordinate in the moving frame. For a sinusoidal electric field (E = Emax sin(o)t))the lateral velocity is u = ItEV =< u > R sin cot (44) where is the mean velocity and R is the dimensionless amplitude of the lateral velocity, which is given by pEmax / < u > The above equation assumes that the solution is dilute in electrolyte and the colloidal particles so that the presence of these particles does not alter the electric field. Additionally, the above equation assumes that the electric field is uniform in the entire channel and thus neglects the presence of the electrical double layer. Inclusion of the double layers significantly increases the complexity of the model and will be treated separately in the future. Below, we use the well established multiple time scale analysis [51] to study the effect of time periodic lateral fields on Taylor dispersion. In the multiple time scale analysis, we postulate that the concentration profile is of the form C C(otDt x, y) (45) 12 l h where o /(27t) is the frequency of the applied field, l/@O is the short time scale, and 2 /D is the long time scale over which we wish to observe the dispersion. Substituting Eq. (4  5) into (4 3) gives 2 ac ac , ac C 2 a2C a2C s +Q +sPe(UU )+ PeU =  + (46) aT, DT, ax ) ay ax 2 Y Y2 where Ts=ot,X=x/1, Y=y/h, T, = Dt/2 ,Pe < u > h  D D e u u uy h U= ,U = ,U U ands<<1. 1 Since s<<1, the concentration profile can be expanded in the following regular expansion. C= s'mCm(Ts,X,Y,TI) (47) m=0 62 Substituting Eq. (4 7) into Eq. (4 6) gives 2ac aC 0 C 1C2 1 ac 82 o + 1 L 2+sPe(UU ) 0 aTBC T, HC ax +&2Pe(UU )L1+PeUe 0 +sPeU e 1 2 PeUe 2 (48) S ax Y y Y ay Y Y 2 Co a2Co F2C1 2 a2C2 = + + + +0(6) X 2 ay2 a8Y2 aY2 Eq. (4 8) can be separated into a series of equations for different order of s. (s80): C ac C 02 Q + PeU ay C (4 9) BTS y Y BY2 The solution for Co can be decomposed into a product of two functions, one of which depends on Ts and Y and the other depends on X and TI, i.e., CO = Go (Y, TS)A(X, T,), where Go satisfies DG BGG 8 2G4 Q o +PeUe = (410) aT, Y ay aY2 The above equation is subjected to the following boundary condition at Y = 0, 1: S PeUGo (411) LY Next, we solve the equations at the order ofs. To order s, the governing equation (4  8) becomes (s1): C* CC 8Ce422C1 Q '+Pe(UTU) O+PeUe (4 12) xT X xY a Y2 63 Integrating the above equation from 0 to 1 in Y and 0 to 27 in Ts and noting that 271 1 1 fQ(C, /1 T,)dT, = 0 due to periodicity and PeU(OC, /OY)dY = (02C / OY2)dY due 0 0 0 to the boundary conditions gives J Go (Y, T)U(Y)dYdT, = 0 21 (413) J fGo (Y, T,)dYdTs 00 The solution to Ci is of the form B(Y, T)(OA(X, Ti)/ OX) where B satisfies QOB+ Pe(UU*)Go(Y,T)+PeUe OB 2B (4 14) BT, Y aY2 and the following boundary conditions: aB PeUyB (4 15) aY Since we are only interested in the periodicsteady solution to Eq. (4 10) and (4  14), these differential equations can be solved numerically for any arbitrary initial conditions. In our simulations, we chose uniform distributions for Go and B as the initial conditions. Eq. (4 10) and (4 14) were solved by an implicit finite difference scheme with a dimensionless time step that was kept smaller than 0.15 / Q in all simulations. The spatial grid size near the wall was set to be smaller than 0.3 / PeR near the walls to ensure accurate results in the boundary layers and the grid size was increased by a factor of about 10 near the center. To establish the accuracy of the numerical scheme, the solutions were tested for grid independence and were also compared with the results of the analytical approach presented in Appendix B. The simulations are run for times larger than the time required to obtain periodic steady behavior. 64 We now solve the O(S2) problem. To order s2, the governing equation (4 8) becomes (F2) o+Pe(U*T) 1 2+PeU 2 0 (416) aT1 X Ts BY aX2 aY2 Averaging both sides in Ts and Y gives, a < Eo> a+T 8T, SPe(UU )B dYdT < o Sa00x ax2 (417) where 1 2r < To >=JJ CdTdY 00 (4 18) Rewriting Eq. (4 17) gives a < CO > + OT, 2 < C > 1Pe(U)BdYdT = o > ax2 f o aX2 (419) where 1 27 Sf GodTsdY 0 0 Now we combine the results for (OCo / DTs) and (OCo / 8T,). ac0o aCo0 +Eacl 2 ac2 D aC0 at DT, aT, aT, 12 aT, Averaging the above equation in Y and Ts and using periodicity gives, aC) D1CaTI Bt 12 BT, (4 20) (421) Using Eq. (4 21) in Eq. (4 19) gives = DD* (422) Ot ox2 where the dimensionless dispersion coefficient is given by J J (U(Y) U)B(Y, T,)dYdT. D=lPe00 1 (423) J JGo (Y, T,)dYdT, 00 The numerical solutions for Go and B that are obtained by solving Eq. (4 10) and (4 14), and Go and B can be used in Eq. (4 13) and (4 23) to obtain the mean velocity and the effective dispersivity, respectively. Additionally, to validate the numerical results we solve Eq. (4 10) and (4 14) analytically. The analytic computations are straightforward but tedious and are outlined in Appendix B. Results and Discussion Below we first describe the results for square wave electric field and compare the results with the asymptotic results obtained by S&B, and then we describe the results for the sinsusoidal fields. Finally, the results for both shapes of electric fields are compared. Square Wave Electric Field As mentioned in the introduction, S&B determined the mean velocity and the dispersion coefficient for CEFFF for the case of a square wave [50]. They developed asymptotic expansions that are valid for large PeR and showed that results for both the mean velocity and the dispersion coefficient depend on only a single parameter T = t0u0 / h, where to is the time period of oscillation, uo is the amplitude of the lateral velocity and h is the channel height. This dimensionless parameter is identical to 2 x PeR/ Q in terms of the parameters defined in this paper. For the case of square wave, we can solve (4 10) and (4 14) numerically and then use (4 13) and (4 23) to compute the mean velocity and the dispersion coefficient. The lateral velocity for the case of a square wave field is given by R x f where f is simply a square wave function that oscillates from 1 to 1 with a dimensionless angular frequency of Q. Below we compare the results of our simulations for the case of a square shaped lateral electric field with the asymptotic results of S&B. First the transient concentration profiles are compared with the asymptotic solutions and then the mean velocities and the dispersion coefficients are compared. Transient concentration profiles In the case of T < 2 ( T = t0u0 /h ), the asymptotic concentration profiles that were predicted by S&B (Figure 4. of Ref 50) corresponds to a uniform probability outer solution of width 1 T / 2 that executes a periodic motion between the walls in phase with the driving force. Since the time period of the oscillation is T, the edges of the outer solution touch the lower wall at the beginning and the end of each cycle and touch the upper wall at midway in the cycle. The inner solution is zero everywhere except in a thin region near the edge of the outer solution. The numerical calculations for T < 2 are shown in Figures 4 la and these show that the numerical solutions for the concentration transients agree with the asymptotic solutions. In Figure 4la, the outer solution is constant at a value of about 1 as predicted by S&B and that there is a thin boundary layer near the wall of thickness 1/PeR and then there is a transition region in which the boundary layer solution merges with the outer solution. Mean velocity and dispersion coefficient Figures 42a and 42b compare the numerical results for mean velocity and dispersion coefficient with those obtained by S&B. In Figure 42a and 42b the thick solid lines correspond to the asymptotic results that were obtained by S&B and the thin solid lines correspond to the results of the numerical simulations. The markers on the curves in Figures 42a and 42b and all the subsequent figures correspond to results obtained by using a Brownian dynamics code that was provided by Professor David Leighton. This Brownian dynamics code is similar to the one used by Molloy and Leighton [52]. The numerical results for both the mean velocity and the dispersion coefficient match the results from the Brownian dynamics simulations. The numerical results for both the mean velocity and the dispersion coefficient agree with the asymptotic expansions for 2 7t PeR/ Q > 2. The agreement is better for larger Pe, which is expected because the asymptotic expansions are valid for large Pe. For the case of 2 7 PeR/ < 2 the numerical results approach the asymptotic results but do not reach the asymptotic limit for Q as large as 2000. However, based on the trends it can be concluded that for higher values of Q, the numerical results will match the asymptotes PeR obtained by S&B. It is also noted that the kink in Figure 42b at T = 2n = 2 is real, and corresponds to the frequency at which the entire solute band gets tightly focused at both walls, rather than just the edges of the band being focused by the nearest wall. In the high frequency (negligible diffusion) limit, at T > 2, the entire solute band travels as a delta function and thus there is no spread, and hence no dispersion. Sinusoidal Electric Field Below, some of the results from the analytic calculations are described, followed by results from the numerical calculations, and comparison of the results from these two approaches. Analytical computations Symmetry in the concentration profile. Since the lateral velocity is sinusoidal (=Rsin(Ts)) and the axial flow and the boundary conditions are symmetric in Y, the concentration profile is expected to satisfy the following symmetry in the long time limit C(T, = 0, Y = a) = C(T = 7 + 0, Y = 1 a) (424) Accordingly, both Co and B satisfy the same symmetry. As shown in Appendix B, CO and B be expanded as Co = f0(Y)+ (f (Y)sin(nT,)+gn(Y)cos(nT,)) A(X,T,) B(T, Y) = qo(Y) + (pn (Y)sin(nT5) + qn (Y)cos(nT,)) (4 25) n=l + const[go (Y) + (f, (Y)sin(nT,) + g. (Y) cos(nT,))] n=l Substituting Co from the above equation into Eq. (4 24) gives go (a) + Z (fa (a) sin(nO) + g, (a) cos(nO)) = n=I (4 26) go (1 a) + (f (1 a) sin(n(7 + 0)) + g, (1 a) cos(n(7 + 0))) = n=l go (1 a) + ((1) f (1 a) sin(n0) + (1)" g. (1 a) cos(nO)) Therefore, both fn and gn are symmetric in Y if n is even, and are antisymmetric if n is odd. Similarly, symmetry of B implies that both pm and qm are symmetric in Y for even m, and are antisymmteric for odd values of m. These symmetries are evident in Figure 4 ..... .......... ........ .... ..... grow exponentially in Y, and become very large near Y=I. Accordingly, the matrix that is inverted to determine the fi, gi, Pi and qj becomes close to singular. Thus, the analytical method does not provide reliable result for PeR > 50. However the analytical method is useful because comparison of the analytical predictions with the numerical computations help to establish the accuracy of our computations. Numerical computations and comparison with analytical results Effect of PeR and 0 on the temporal concentration profiles. Figures 44ad show the concentration profiles at various time instances during half of a period. In Figure 44a, the value of PeR is 100, and thus most of the molecules aggregate in a thin boundary layer near the wall. The thickness of the boundary layer changes as the field changes during the period. The concentration profiles are not in phase with the driving force as evident by the fact that at ts = 0, 2 nt, the field is zero, but the concentration profile is far from uniform, and that the wall concentration keeps increasing beyond t, = 37/2, even though the field begin to decrease. The profiles in Figure 44b correspond to the same value of PeR as in Figure 44a but a much small value of 1 for Qi. Since PeR is still large, the boundary layer with time varying thickness still forms but in this case the profiles are almost in phase with the driving electric field due to the small value of Q. Accordingly, at ts = 0, the concentration profile is relatively independent of position, and the wall concentration is the maximum in time and the boundary layer thickness is a minimum at ts = 3t7/2. Figure 44c and 44d correspond to PeR = 1, and Q of 1 and 10, respectively. Since PeR is small, a boundary layer does not develop in both of the cases. In Figure 44c, the concentration profiles are not exactly in phase as evident from the fact .. ... ..... .... ... .. . particles accumulate near the wall, leading to a higher concentration at the boundaries. In Figure 45a, the concentration profile in the center is relatively flat and the value of go in this central region increases on reducing PeR. However the profiles in Figure 45b show that for Q = 100, a maxima develop in the central region, when PeR is less than about 40. The effect of Q on go is further illustrated in Figures 45ce for PeR = 1. The values of 2 span from 1 to 20 in Figure 45c, from 40 to 100 in Figure 45d and from 1001000 in Figure 45e. For 2 values less than 20, the wall concentration is the highest and it levels off in the center. The distance from the wall at which it levels off and also the value in the center decrease with an increase in frequency. However on increasing Q beyond 40, a secondary maximum develops in the center but the maximum concentration is still at the wall. On increasing 2 further, the value of go at the maximum in the center overshoots the value at the walls, which has been assigned to be equal to 1 as a boundary condition. Under these conditions, due to the accumulation of the molecules near the center, the mean velocity exceeds 1. Mean velocity and dispersion coefficient. In the process of separation by cyclic lateral electric fields there are three dimensionless parameters that control the separation. These are the Peclet number Pe, the dimensionless amplitude of the lateral velocity R, and the dimensionless frequency Q. For fixed channel geometry and for a given sample, Pe can be changed by adjusting the mean velocity of the axial flow, R can be changed by adjusting the magnitude of the periodic electric field, and Q can be changed by varying the frequency of the periodic electric field. The mean velocity is only a function of PeR and 2 and the dispersion coefficient is of the form Pe2 f(PeR, K ). Typical microfluidic channels are about 2040 pm thick and as stated earlier the lateral electric velocity uye Figure 46ab plots the dependency of the mean velocity on PeR for different values of Q. Figure 46a shows the results for PeR<10 and the data represented in this plot was calculated from the analytical solutions described in Appendix B. In Figure 4 6b, the values of PeR range from 1200 and the data shown in this figure was calculated by the numerical approach described above. It is noted that the results from both the methods match for PeR values of around 10, which validates the accuracy of the numerical scheme. The markers on the curves in Figures 46ab that are the results of the Brownian dynamics simulations also match the results computed by finite difference. As shown in Figures 46ab, the mean velocity decreases as the product of Pe and R increases. When PeR increases, the particles experience a larger force in the lateral direction, which pushes them closer to the walls, and consequently reduces the mean velocity. Figure 46ab also shows the dependence of the mean velocity on Q ; as Q increases, the curve of the mean velocity shifts up. This is due to the fact that as Q increases, the electric field changes its direction more rapidly, and thus, the solute molecules in the bulk of the channel simply move back and forth. Therefore, the concentration profile is almost uniform in the middle of the channel. In a thin region near the wall, the concentration is different from that in the center but the thickness of this region becomes smaller on increasing K2. As a result, on increasing Q the concentration profile becomes more uniform in the lateral direction and accordingly the dimensionless mean velocity approaches a value of 1. As mentioned above, the dimensionless dispersion coefficient is of the form 1 + Pe'f (PeR, ) ). Figures 47ab plots (D* 1)/Pe 2 as a function of PeR for different values of Q. As for the case of mean velocity, Figures 47a and 47b were computed by the analytical and the numerical methods, respectively, and the results from both the methods merge smoothly for PeR values of around 10. Also the markers that represent the calculations from the Brownian dynamics code match the results computed by finite difference. As PeR goes to zero, i.e., the electric field is close to zero, the effective diffusivity is expected to approach the value of the Taylor dispersivity for Poiseuille flow through a channel. Figure 47a shows that as PeR approaches zero, the curves of (D* 1)/Pe2 for all values of frequency approach the expected limit of 1/210. On the other hand, as PeR goes to infinity, which corresponds to an infinite magnitude of electric field, particles will spend more time in a very thin layer close to the walls. Thus, the effective diffusivity of the particles approaches the molecular diffusivity. Figure 47a also shows that for small Q, the curves exhibit a maximum at PeR = 4. This phenomenon also occurs in constant electric fieldflow fractionation. In the constant EFFF, at small PeR, the particle concentration near the walls begins to increase with an increase in PeR; however, a significant number of particles still exist near the center. The increase in PeR results in an average deceleration of the particles as reflected in the reduction of the mean velocity, but a significant number of particles still travel at the maximum fluid velocity, resulting in a larger spread of a pulse, which implies an increase in the D*. At larger PeR, only a very few particles exist near the center as most of the particles are concentrated in a thin layer near the wall, and any further increase in PeR leads to a further thinning of this layer. Thus, the velocity of the majority of the particles decreases, resulting in a smaller spread of the pulse. Finally, as PeR approaches infinity, the mean velocity approaches zero, and the dispersion coefficient approaches the molecular diffusivity. Since the behavior of the dispersion coefficient with an increase in . ......... . . ....... .. reference frame in which we solve the convection diffusion equation to vary during a period, i.e., T* = T*(T,). In the limit of small Q to leading order, Eq. (4 9), (4 12) and (4 16) become PeU a2 (427) y 3Y BY2 Pe(UU) +PeUe (428) aX Y Y Y2 aO +Pe(U U *)C +PeUe a 0 + 2 (429) aT, 9X a xY ax2 aY2 These equations along with the noflux boundary conditions are identical to those for EFFF and thus the short time dependent mean velocity and the dispersion coefficient are given by[53] 6 + 6exp(a) 12 12exp(a) U a (a)2 (430) exp(a) 1 D* = R Pe2 (720e'a + 504e'a2 24e'a4 144eaa3 6048e2a 504e2aa2 + 720e2"a + 24e2aa4 144e2'a3 + 72e3Ga2 720e3aa + 6048e' + 2016e3a 2016 720a 72a2) /((e 1)3 a6 ) (431) where a PeUy = PeR sin(Ts). These results for short time dependent mean velocity and dispersion coefficient can then be averaged over a period to yield the mean velocity and the dispersion coefficient, and these then can be compared with the exact results. These comparisons are shown in Figure 46a and 47a. Figure 47a shows the comparison of the small Q expression with the full result from Eq. (4 13) for the mean velocity. The small Q solution matches the exact solution for Q <1. Similarly the dispersion coefficient in the small Q limit is the time average of the dispersion coefficient for constant electric fieldflow fractionation and it matches the full solution for Q <1 (Figure 47a). The matching of the mean velocity and the dispersion coefficient with the time averaged EFFF results is expected because as shown earlier for 2 = 1, the concentration profiles are close to being in phase with the driving force. Large 0 limit. In the large Q limit, the mean velocity can be computed by following the same approach as used by S&B. In this limit, to leading order, the periodicallysteady concentration profiles are given by the following expressions: For T < 7 0 for 0 < Y < AY(T,) Go,(Y,T)) =A for AY(T) < Y < AY(T) + Wp (432) 0 for AY(Ts)+Wp < Y < 1 where Wp = 1 T / 7 and AY(T,) = (T / 27)(1 cos(T,)). For T > 7t A6(Y AY(T,)) for 0 < T8 < Tt A5(Y1) for Ti < T < Go (Y, T) = (433) SA6(Y (1 AY(T, ))) for 7 < T < t + Tt A5(Y), for in+Tt where A is a constant whose value can be determined by using the normalization condition, and Tt is the time at which AY(T8) = 1, i.e., the pulse touches the wall. The mean velocities can then be computed by using Eq. (4 13). 2riPeR In the high frequency limit, the mean velocity depends only on T = and this dependence is shown in Figure 48 along with the results for square fields obtained by S&B. For the same amplitude, the mean velocity is expected to be smaller for the square fields because the molecules are subjected to the same amplitude for the entire . .......... ... .. ..... that that the value of R used in the x scale is that for the square wave and the value of R for the sinusoidal fields is 7Rsq / 2. The figures show that for large values of Q, the curves for both the mean velocities and the dispersion coefficients are similar and almost overlap for 2inPeRq / Q <10. To avoid or minimize the decay in the electric field due to double layer charging, separation will need to be performed at large 2 and for optimal separation it is best to operate in the region where the mean velocity is most sensitive to the field strength. Figure 49 shows that these requirements suggest that the most suitable operating parameters are 2inPeRq / Q10 and Q~ 100. Figures 49a and 49b also show that under these conditions the mean velocities and the dispersion coefficients are similar for sinusoidal and square fields. Conclusions Techniques based on lateral electric fields can be effective in separating colloidal particles in microfluidic devices. However, application of such fields can effectively immobilize the colloidal particles at the wall, and furthermore, particles with same values of D/u' cannot be separated by EFFF. It has been proposed that these problems could potentially be alleviated by cyclic electric field flow fractionation. In this paper the mean velocity and the dispersion coefficient for charged molecules in CEFFF are determined by using the method of multiple time scales and regular expansions. The dimensionless mean velocity W* depends on Q the dimensionless frequency, and PeR, the product of the lateral velocity due to electric field and the Peclet number. The convective contribution to the dispersion coefficient is of the form Pe2f(PeR, Q). The mean velocity of the particles decreases monotonically with an increase in PeR, and increases with an increase in Q; but (D* 1)/Pe2 has a maximum at a value of PeR ~ 4 for small Q, and the maximum disappears at large (Q. For Q <1 the lateral concentration profile oscillates in phase with the electrical field and the mean velocity and the dispersion coefficient simply become the time averaged values of the results for the EFFF. The mean velocity exceeds 1 for the case of small PeR and large frequencies. The results for square wave electric fields match the asymptotic expressions obtained by S&B. Also the results of the finite difference calculations match the Brownian dynamics calculations that were performed with the code provided by Reviewer 2. Comparison of results for sinusoidal and square wave fields show that for large values of Q, the mean velocities and the dispersion coefficients are similar and almost overlap for 27xPeR /Q <10. These are also the conditions most suitable for separation and thus it seems that both types of electric fields are equally suitable for separation. Since the mean velocity of the particles under a periodic lateral field depends on Pe, colloidal particles such as DNA molecules that have the same electrical mobility can be separated on the basis of their lengths by applying cyclic lateral electric fields but only at small or O(1) Pe. CHAPTER 5 ELECTROCHEMICAL RESPONSE AND SEPARATION IN CYCLIC ELECTRIC FIELDFLOW FRACTIONATION This chapter aims to determine the mean velocity and the dispersion coefficient of charged molecules undergoing Poiseuille flow in a channel in the presence of cyclic lateral electric fields. As introduced in chapter 4, some researchers have done some work on modeling and experiments on CEFFF. But, many of the researchers assumed that the effective electric field is constant in the bulk during half cycle when a constant voltage is applied. In reality, if the double layer charging time is much shorter than the time for half cycle, the effective electric field will be close to zero for most of time; if the double layer charging time is much longer than the time for half cycle, the effective electric field will be close to the maximum value for most of time. In these two cases, this assumption does not result in great discrepancy between the theoretical estimation and the experiments. But if the time for half cycle is comparable to the charging time, the changing of the effective field in the bulk should be counted in to give a more rational result. Recently Biernacki et al. included the effect of the decaying electric field in the calculations of the retention ration, which is essentially the inverse of the mean velocity [54]. However Biernacki et al. did not calculate the dispersion of the molecules, and thus they could not predict the separation efficiency of the devices, which is a balance between the retention and the dispersion. Furthermore, they only focused on determining the mean velocity for frequencies that are small enough so that the current decays to almost zero during every cycle. The model that we develop in this paper does not require the current to decay to zero and so we also explore the high frequency regime. The arrangement of this chapter is as follows: In the next section we present the theory for the flow of current during the operation of the CEFFF and the theory for the calculation of the mean velocity and the dispersion coefficient. The theory for the flow of current is based on the equivalent circuit model and in the next section we present some experimental data that is used to obtain the parameters for the equivalent circuit. These parameters are subsequently used to predict the mean velocity and the dispersion coefficient. Subsequently, the mean velocity and the dispersion coefficient are utilized to analyze the separation efficiency of the CEFFF. Finally, some of the available experimental data on CEFFF is discussed and compared with theory. Theory Consider a channel of length L, height h and infinite width that contains electrodes for applying the lateral periodic lateral electric field. The approximate values of L and h are about 9 cm and 40 microns, respectively. Thus, continuum is still valid for flow in the channel. Also, the aspect ratio is much less than 1, i.e., 8 h / L << 1. Equivalent Electric Circuit Figure 51 is the commonly used equivalent electric circuit model for EFFF channel for the case when the applied voltage is low enough such that there is no electrode reaction. The capacitor Cd in the circuit can be attributed to the double layers and the resistance Rs represents the resistance of the solution. On application of a potential V, the charging of capacitance leads to an exponentially decaying current given by 