SEPARATION WITH ELECTRICAL FIELD-FLOW...

SEPARATION WITH ELECTRICAL FIELD-FLOW 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.

iv

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 10-316) and the National

Science Foundation (NSF Grant EEC-94-02989).

v

TABLE OF CONTENTS page

ACKNOWLEDGMENTS ................................................................................................. iv

TABLE ............................................................................................................................ viii

LIST OF FIGURES ........................................................................................................... ix

ABSTRACT...................................................................................................................... xii

CHAPTER

1 INTRODUCTION TO ELECTRICAL FIELD-FLOW FRACTIONATION..............1

2 DNA SEPARATION BY EFFF IN A MICROCHANEL............................................5

Application of EFFF in DNA Separation .....................................................................5 Theory...........................................................................................................................7 Results and Discussion ...............................................................................................10

Limiting Cases.....................................................................................................10 Dependence of the Mean Velocity on e

yU and Pe ..............................................13

Dependence of D* on eyU and Pe ........................................................................13

Separation Efficiency ..........................................................................................14 Effect of Pe and e

yU on the Separation Efficiency..............................................16 DNA Separation ..................................................................................................20 Comparison with Experiments ............................................................................25

Summary.....................................................................................................................28

3 SEPARATION OF CHARGED COLLOIDS BY A COMBINATION OF PULSATING LATERAL ELECTRIC FIELDS AND POISEUILLE FLOW IN A 2D CHANNEL ...........................................................................................................30

Theory.........................................................................................................................32 Model...................................................................................................................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 time Analytical Solution ............................................................................40 Results and Discussion ...............................................................................................42

vi

Mean Velocity .....................................................................................................43 Dispersion Coefficient.........................................................................................44 Separation Efficiency ..........................................................................................46

Effect of G....................................................................................................47 Effect of tf/td .................................................................................................48

Effect of ( )2

f

2

tuh ........................................................................................49

Comparison with Constant EFFF ........................................................................52 Conclusions.................................................................................................................56

4 TAYLOR DISPERSION IN CYCLIC ELECTRICAL FIELD-FLOW FRACTIONATION....................................................................................................58

Theory.........................................................................................................................59 Results and Discussion ...............................................................................................65

Square Wave Electric Field .................................................................................65 Transient concentration profiles...................................................................66 Mean 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 FIELD-FLOW FRACTIONATION.......................................................86

Theory.........................................................................................................................87 Equivalent Electric Circuit ..................................................................................87 Model 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

Separation ..........................................................................................................104 Modeling of separation of particles by CEFFF..........................................104 Mean velocity of particles ..........................................................................104 Effective diffusivity of particles.................................................................107 Separation efficiency..................................................................................110

Comparison with Experiments ..........................................................................111 Large Ω asymptotic results.........................................................................113 The effect of changes in Ω .........................................................................118

Conclusions...............................................................................................................123

vii

6 CONCLUSION AND FUTURE WORK .................................................................126

APPENDIX

A DERIVATION OF VELOCITY AND DISPERSION UNDER UNIDIRECTIONAL EFFF ......................................................................................134

B DERIVATION OF NUMERICAL CALCULATION FOR SINUSOIDAL EFFF .138

Analytical Solution to O(ε) Problem ........................................................................138 Analytical Solution to O(ε2) Problem.......................................................................141 Solving for f, g, p and q ............................................................................................144

Solving for p and q ............................................................................................148 Solving for Particular Solution..........................................................................148

REFERENCE LIST .........................................................................................................157

BIOGRAPHICAL SKETCH ...........................................................................................160

viii

TABLE

Table page 5-1 Comparison of the model predictions with experiments of Lao et al. ...................123

ix

LIST OF FIGURES

Figure page 2-1 Schematic of the 2D channel......................................................................................7

2-2 Dependency of (D*-R)/Pe2 on the product of Pe and eyU .......................................12

2-3 Dependency of mean velocity U on the product of Pe and eyU ..............................12

2-4 Dependency of L/h on eyU and Pe for separation of DNA strands of different

sizes. D2/D1 = 10 .....................................................................................................18

2-5 Dependency of L/h on eyU and Pe for separation of DNA strands of different

sizes. D2/D1 = 2 .......................................................................................................19

2-6 Comparison of our predictions with experiments on DNA separation with FlFFF .27

2-7 Comparison of our predictions with experiments on separation of latex particles with EFFF.................................................................................................................28

3-1 Schematic showing the three-step cycle...................................................................31

3-2 Dependency of *U on G..........................................................................................44

3-3 Dependency of *D on G..........................................................................................45

3-4 Effect of G1 ( 2.0tt

d

f = , ( )

2.0tu

h2

f

2

= , G2/G1=2) on L/h, θ/tf and T......................47

3-5 Effect of d

f

tt

(G1=100, ( )

2.0tu

h2

f

2

= , G2/G1=2) on L/h, θ/tf and T.......................49

3-6 Effect of ( )2

f

2

tuh (G1=100, 2.0

tt

d

f = , G2/G1=2) on L/h, θ/tf and T.......................50

3-7 Dependency of L/h on G1(pulsating electric field) and eyu (constant electric

field). D1/D2=2 .........................................................................................................54

x

3-8 Dependency of the operating time t on G1(pulsating electric field) and eyu (constant electric field). D1/D2=2 ........................................................................55

3-9 Dependency of L/h on G1(pulsating electric field) and eyu (constant electric

field). D1/D2=1.2 ......................................................................................................55

3-10 Dependency of the operating time t on G1(pulsating electric field) and eyu (constant electric field). D1/D2=1.2 .....................................................................56

4-1 Periodic steady concentration profiles during a period for a square shaped electric field..............................................................................................................67

4-2 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

4-3 gi vs. position for PeR=1, and Ω =100.....................................................................71

4-4 Time dependent concentration profiles within a period for sinusoidal electric fields. ........................................................................................................................73

4-5 Time average concentration profiles for sinusoidal electric field ............................74

4-6 Dependence of *U on PeR ......................................................................................76

4-7 Dependence of (D*-1)/Pe2 on PeR...........................................................................79

4-8 Comparison of the mean velocities for the square (dashed) and the sinusoidal (solid) fields in the large frequency limit .................................................................82

4-9 Comparison of the mean velocities and the effective diffusivity for the square (dashed) and the sinusoidal (solid) fields .................................................................83

5-1 Equivalent electric circuit model for an EFFF device..............................................88

5-2 Transient current profiles after application of step change in voltage in a 500 µm thick channel ............................................................................................................95

5-3 Dependence of the electrochemical parameters on salt concentration and applied voltage in a 500 µm thick channel ...........................................................................97

5-4 Dependence of the electrochemical parameters on channel thickness for V = 0.5 V and DI water .........................................................................................................98

5-5 Comparison between the experiments (thin lines) and Eq. (5 -24) (thick lines)....101

5-6 Comparison between the experiments (stars) and Eq. (5 -26) (solid lines) ...........102

xi

5-7 Dependency of the mean velocity on PeR and Ω ..................................................105

5-8 Dependence of 210(D*-1)/Pe2 on PeR and Ω ........................................................108

5-9 Dependence of separation efficiency on PeR1 and Ω1 for the case of D1/D2=3 and µE2/µE1=3 .........................................................................................................109

5-10 Origin of the singularity in separation efficiency at critical PeR1 and Ω1 values for Ω1 = 40 π ..........................................................................................................109

5-11 Dependence of the mean velocity on Ω in the large Ω regime ..............................120

xii

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 FIELD-FLOW 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 field-flow fractionation

(EFFF) is a technique that can separate charged particles by combining a lateral electric

field with an axial pressure-driven 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

xiii

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 unidirectional-EFFF, pulsed -EFFF and cyclic-EFFF,

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 effectives of EFFF can be substantially

improved by using either pulsed or cyclic fields.

1

CHAPTER 1 INTRODUCTION TO ELECTRICAL FIELD-FLOW 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 field-flow fractionation

(FFF) techniques.

Field-flow 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 h2/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 field-driven mobility and the diffusion coefficient, molecules

2

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. Field-flow 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 [1-3] and other variants of

FFF such as those based on gravity, centrifugal acceleration [4-6], lateral fluid flow[7], or

thermal field-flow fractionation (TFFF) [8-10]. These techniques have been used in

separations of a number of different types of molecules including biomolecules [11,12].

The flow-FFF, 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,

3

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.

5

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 gel-based electrophoretic

separations can be continuous or pulsed.

6

Recent advances in microfabrication techniques have led to the production of

microfluidic devices frequently referred to as a “lab-on-a-chip” that can perform a

number of unit-operations such as reactions, separations, detection, etc., at a high

throughput. Gel-based 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], self-assembling 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 field-flow fractionation (EFFF) [14,20,31], which is a type of

field-flow 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

7

fields, the differences in the mean velocities lead to separation of the slug into bands, and

the band of the smaller molecules travels faster.

Theory

Figure 2-1 shows the geometry of a 2D channel that contains the electrodes for

applying the lateral electric field; L and h are the channel length and height respectively,

and the channel is infinitely wide in the third direction. 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.

Figure 2-1. Schematic of the 2D channel

The transport of a solute in the channel is governed by the convection-diffusion

equation,

2

2

2

2

||ey y

cDx

cDycu

xcu

tc

∂∂

+∂∂

=∂∂

+∂∂

+∂∂

⊥ (2 - 1)

where c is the solute concentration, u is the fluid velocity in the axial (x) direction, ||D

and ⊥D are the diffusion coefficients in the directions parallel and perpendicular to the

flow, respectively, and eyu 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

8

velocity eyu can be determined by the Smoluchowski equation, E

µζεε

u r0ey = , where εr and

µ are the fluid’s dielectric constant and viscosity, respectively, ε0 is the permittivity of

vacuum, and ζ is the zeta potential. Alternatively, Eµu Eey = , where Eµ is the electrical

mobility of DNA, which is independent of length and has a value of about 3.8x10-8

m2/(V·s) [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. (2-1) becomes

)y

cx

cR(Dycu

xcu

tc

2

2

2

2ey ∂

∂+

∂∂

=∂∂

+∂∂

+∂∂ (2 - 2)

where ⊥= D/DR || , and we denote ⊥D as D. The value of R varies between 1 and 2; it is

equal to 1 if the DNA molecules are random-coils, and it is equal to 2 if they are fully

stretched as cylinders in the flow-direction.

The boundary conditions for the above differential equation are

0cuycD e

y =+∂∂

− at y = 0, h. (2 - 3)

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.

9

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.,

))h/y(h/y(u6u 2−><= (2 - 4)

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

1)αexp()α(

)αexp(1212α

)αexp(66

U2

−

−+

+

= (2 - 5)

)α)1e/(()α72α7202016e2016e6048αe720αe72αe144αe24αe720αe504e6048αe144αe24αe504αe720(PeRD

63α2α3αα32α33α2

4α2α22α2α23α4α2αα2*

−−−−++−+−

++−−−−+−=

(2 - 6)

In the above expressions Pe = h/D and eyPeUα ≡ . 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 δ that is about

3h/α. 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

eyU . If e

yU 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

2* Pe2101RD ; 1U +== (2 - 7)

Also, as eyU becomes large most of the molecules accumulate in a region of thickness δ

and these molecules are subjected to a linear velocity profile, i.e., yhu~u >< . The

dimensional mean velocity of the molecules therefore scales as δ><

hu . Thus

αδ 1~h

~U . The time needed by the molecules to equilibrate in the lateral direction ∆t is

about D

2δ , and the axial distance l traveled by the molecules during this time scales is of

the order of δ><δ

∆hu

D~tu

2

. Since the dispersion arises due to the difference in the

axial motion of the molecules at various lateral positions during the times shorter than the

lateral equilibration time, t∆

l~D2

* . Accordingly, in the large α regime D* is expected to

scale as 4

2223

αPeD~)

Dδ/(

Dhδu

⎟⎟⎠

⎞⎜⎜⎝

⎛ >< .

11

These scalings can also be obtained by expanding the exact solution from Eqs. (2-

5) and (2-6) in the limit of both small and large α. The expansion for D* in the limit of

0α → is

))α(Oα1800

12101(PeRD 422* +++= (2 - 8)

To the leading order, the above expression reduces to 2Pe2101R + , which is the same as

Eq. (2-7). Expanding Eq. (2-6) as α goes to infinity gives

))α1(O

α720

α72(PeRD 654

2* +−+= (2 - 9)

As expected, the leading order term scales as 4

2Peα

. However, the contribution from the

next term, i.e., the O(α5) term, is about 10% of the leading order term for α as large as

100. Figure 2-2 compares the asymptotic solutions obtained above with the exact

solution for D*. The small α and the large α approximations match the analytical

solution for α< 2 and α >8, respectively.

Similarly the asymptotic behavior of U in the limits of small and large α is

0α)α(Oα6011U 4 →+= 2- (2 - 10)

∞→+= α)α1(O

α6U 2 (2 - 11)

The above result for U approaches 1 as α approaches zero, and thus matches the mean

velocity for Poiseuille flow in a channel without any lateral field. Also in the large α

limit, the leading order term is of the order of 1/α, that matches the expected scaling.

Figure 2-3 shows the comparison of these asymptotic results and the exact results from

12

Eq. (2-5). The small α and the large α results match the full solution in the limit of α<2

and α>40, respectively. These asymptotic results help us in understanding the physics of

the dispersion and the DNA separation, as discussed below.

Figure 2-2. Dependency of (D*-R)/Pe2 on the product of Pe and eyU . The dashed line is

the largeα approximation Eq. (2-9), and the dotted line is the small α approximation Eq. (2-8)

Figure 2-3. Dependency of mean velocity U on the product of Pe and eyU . The dashed

line is the large α approximation Eq. (2-11), and the dotted line is the small α approximation Eq. (2-10)

13

Dependence of the Mean Velocity on eyU and Pe

Figure 2-3 shows the dependence of the mean velocity on eyU and Pe. The mean

velocity depends only on α, i.e., the product of eyU and Pe. As discussed above the

product eyPeU 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 α, an

increase in α 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 α at small values of

α is not so clear because with an increase in α, 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 α regime, the mean velocity is reduced with an increase in α. The mean

velocity is thus a monotonically decreasing function of α.

Dependence of D* on eyU and Pe

The effective dispersion coefficient D* depends separately on eyU and Pe.

However, ( ) 2* Pe/RD − depends only on α (Figure 2-2). As discussed above for small α,

with an increase of α, 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 α results in an average deceleration of the particles as reflected in the

reduction of the mean velocity (Figure 2-3), but a significant number of particles still

14

travel at the maximum fluid velocity. This results in a larger spread of a pulse, which

implies an increase in the D*. At larger α, 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 α 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 α

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 α is different in the small and the large α regime, it must

have a maximum. The maximum is expected to occur at the value of α beyond which

there are almost no particles in the region y<h/2, which occurs when ⇒α

−0~e 2

α ~ 5.

Figure 2-2 shows that the maximum value of ( ) 2* Pe/RD − occurs at α ~ 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 α~1, 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 tu and the width

of the Gaussian is tDD4 * . We consider the DNA strands to be separated when the

15

distance between the two pulse centers becomes larger than 3 times of the sum of their

half widths, i.e.,

)tDD4tDD4(3t)uu( *22

*1112 +≥− (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., )u,umax(/L 21 . Substituting for t, and expressing all

the variables in dimensionless form gives

2

12

1

2*2

*1

211

]UU

DD

DD)[U,Umax(

Pe112h/L

−

+≥ (2 - 13)

Eq. (2-13) can also be expressed as

φ=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

+

≥11

*1

2

1

2

*11

*22

11

*1

UPeD

12

UU

1

DDDD

1

UPeD

12h/L (2 - 14)

where φ 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.,

)U,Umin(uLT

21><= (2 - 15)

16

Effect of Pe and eyU on the Separation Efficiency

In Figures 2-4 and 2-5, we show the dependence of L/h on Pe and eyU in the case

of e1yU = e

2yU , which corresponds to DNA fragments of two different lengths. Figure 2-5

is similar to Figure 2-4; the only difference is the value of the ratio D2/D1. Figures 2-4

and 2-5 show that at a small eyU , increasing e

yU , which is physically equivalent to

increasing the electric field, leads to a reduction in L/h required for separation. As eyU Pe

increases, the mean velocities of both kinds of molecules decrease (Figure 2-3). 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

ratio2

121

2

UU1

PeU

⎟⎟⎠

⎞⎜⎜⎝

⎛−

. As shown earlier, in the small α regime 2α6011~U − , thus,

( ) ( )221

22

4ey1

2

121

2

PePeUPe

1~UU

1PeU

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−. Since the ratio Pe2/Pe1 is fixed,

( ) ( ) ( ) 44ey

5122

12

24e

y1Pe1UPe~

PePeUPe

1 −−− α=−

. Thus, an increase in either Pe or eyU

leads to a reduction in L/h in the regime of small α. The constant Pe plots in Figure 2-4

and 2-5 show the ( ) 4eyU − dependency when e

yU is small. Also, the constant Pe curves

shift down with increasing Pe, due to the Pe-5 dependency shown in the above scaling.

The above expression also shows that at a fixed α, an increase in Pe leads to a reduction

in L/h. In the limit of large α, α/6~U , thus,

PeαU~

PePePePe

PePeU

~UU

1PeU e

y

2

12

21

21

ey

2

121

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

−. This implies that even in the large α

17

regime for a fixed α, an increase in Pe leads to a reduction in L/h. It also shows that in

the large α regime and at O(1) Pe, L/h becomes independent of Pe and begins to increase

with an increase in eyU , as shown in Figure 2-4. Since L/h scales as ( ) 4e

yU − in small α

regime, and as eyU in the large α 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 eyU fixed is more difficult to understand

physically. Due to the dedimensionalization of eyU , in order to change Pe while keeping

eyU 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 eyU by the same factor.

Thus, in Figures 2-4 and 2-5, we need to first move to the smaller eyU value and then

follow the constant eyU curve to the larger Pe. This keeps Pe e

yU constant and at O(1) Pe,

18

D* and U remain unchanged, and thus, L/h ~ 1/Pe. Physically, this inverse dependency

of L/h on the mean fluid velocity arises because the dimensional mean velocity of the

molecules depends linearly on . Thus, an increase in results in a linear increase

in the difference between the mean velocities of the two types of molecules, i.e., 21 uu − .

The distance between the peaks at the channel exit is independent of because

although 21 uu − increases linearly with , the time spent by the molecules in the

channel is inversely proportional to . At O(1) Pe, the dispersion coefficients do not

change appreciably with changes in only , and thus the spread of each of the

Gaussians decreases with an increase in due to the reduction of time spent in the

channel. Consequently, the spread of the peaks becomes smaller, making it easier to

separate the two types of DNA.

Figure 2-4. Dependency of L/h on eyU and Pe for separation of DNA strands of different

sizes. e1yU = e

2yU = eyU , Pe1 = Pe, and Pe1/Pe2 = D2/D1 = 10

19

Figure 2-5. Dependency of L/h on eyU and Pe for separation of DNA strands of different

sizes. e1yU = e

2yU = eyU , Pe1 = Pe, and Pe1/Pe2 = D2/D1 = 2

Another interesting regime occurs when Pe>>1 but eyU <

Pe1 . In this regime,

which is relevant for DNA separation, the convective contribution to the dispersion

overwhelms the diffusion. In this regime α can be large or small. By substituting the

asymptotic expressions for *D and U we get the following expressions for L/h.

( ) 33ey

2 αPe144

UPe

144~h/L = for α>>1 (2 - 16)

4

Pe~h/Lα

for α<<1 (2 - 17)

The above expressions show that in this limit for a fixed α, an increase in Pe results

in an increase in L/h, which is contrary to the behavior for Pe<10. This implies the

20

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 10-50 µM.

However, experiments involving DNA are typically done in the range of 10 mM

concentration of electrolytes such as EDTA, tris-HCl and NaCl [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 couple-electrode 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

21

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 3.8x10-8 m2/(V·s), a field of 100 V/m will drive a lateral velocity of

about 3.8 µm/s.

(2) The second restriction on eyu arises from the fact that the thickness of the layer in

which the molecules accumulate, δ, is given by eyu/D3 . 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 kkg Nl6

1R = where lk is the Kuhn length (=2 × 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

N2~300N

6100R g = nm. The diffusivity of the DNA in a 0.1 M PBS buffer is

N102D

10−×= m2/s [40]. Let us choose gR10δ = . This gives

N105.1

N1020103~D3u

2

9

10ey

−

−

− ×=

××

δ= m/s. Thus the condition gRδ >> imposes a

22

smaller value for eyu 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,

rthu ><6 , which is the Weissenberg number Wi, should be less than 1. The

relaxation time kTL~t

5.1

rµ and based on this scaling and the experimental values

reported in literature, tr in water is about 1.6x10-8 N1.5 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 hu6 >< should be less than 100 s-1 to prevent any significant

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 α=Pe eyU

regime. As derived above for the case when α is large but eyU ~

Pe1 , the convective

contribution to the dispersion dominates over molecular diffusion and the length required

for separation is given by

( ) ey

2

ey

3ey

2 uu

huD144

UPe144~h/L ⎟

⎟⎠

⎞⎜⎜⎝

⎛= (2 - 18)

23

By using Eq. (2-15), the time for separation is

( ) ( ) ( )2ey

2ey

ey

3ey

2 u

D24uh

UPe

24u6

hPeU

UPe

144~T == (2 - 19)

In the subsequent discussion we restrict gR10=δ . The above scalings for L and T

can equivalently be expressed in the following forms:hu

Dδ16~L

3

and D3δ8~T

2

.

Substituting gR10δ = and expressing Rg and D in terms of N gives the following

expressions for L and T:

huN106~L 213 ><

× − m (2 - 20)

2/36 N105~T −× s (2 - 21)

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

δuhD6PeU

Peh6~LU6~

hL e

yey ><

=⇒ (2 - 22)

2

2

2

ey

uh

δD

u6PeU

δuhD6~

UuL~T

><=

><><>< (2 - 23)

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

24

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

213 N106~L −× m (2 - 24)

2/36 N105~T −× s (2 - 25)

For DNA the φ as defined by Eq. (2-14) is given by

2

2

2

1

4/3

2

1

2

1

2

2/3

1

2

2

1

2

*11

*22

NN16~

NN

1

NN

1

DD

1

DD

1

UU

1

DDDD

1⎟⎠⎞

⎜⎝⎛

∆⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

+

=φ (2 - 26)

where ∆N=N2-N1, and we have utilized the large α 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 φ in the expressions for L and T gives

2212

N∆NN106.9~L ⎟

⎠⎞

⎜⎝⎛× − m (2 - 27)

22/35

N∆NN108~T ⎟

⎠⎞

⎜⎝⎛× − s (2 - 28)

The above expressions show that DNA strands in the range of about 10kbp 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

25

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 30-40 minutes.

However, we note that the time for separation can be significantly reduced if we relax the

restriction of gR10δ = . But under this situation the continuum equations cannot be used

and one will need to perform non-continuum 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, FlFFF (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

20

2*00

xC

DDxC

Uut

C∂

><∂=

∂><∂

><+∂

><∂ (2 - 29)

26

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

)tDD4

)tUuL(exp(tDDπ4

MC *

2

*0><−

−>=< (2 - 30)

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 FlFFF. 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 2-6 shows the

comparison of the dispersive model with their experiments. In Figure 2-6, the

experimental data of intensity at the detector located at the channel exit is compared with

the concentrations predicted by Eq. (2-30). 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 2-7 shows the comparison of the intensity at the channel

exit with the concentration predictions from the dispersion model for EFFF. As in Figure

2-7, 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

27

nm size particle but the comparison is not satisfactory when the particle size changes to

130 and 207nm. Especially, when the size is 207nm, the prediction of the position of the

peak is far away from the experiment. This could partially be attributed to the steric

effects of the wall. It is also noted that even for the 44 nm size particles, the predicted

dispersion is significantly less than the observed dispersion. This discrepancy could be

due to the neglect of the wall effect which is known to enhance dispersion.

Figure 2-6. Comparison of our predictions with experiments on DNA separation with FlFFF. Channel geometry: channel height h = 227 µm, total channel length L = 30 cm, channel breath b = 2.1cm. Operational parameters: a. Axial flow rate V = 3.15ml/min, lateral flow rate Vc = 1.05ml/min; b. V = 6.7ml/min, Vc = 1.05ml/min; c. V = 3.15ml/min, Vc = 0.42ml/min; d. V = 3.15ml/min, Vc = 1.05ml/min. The solid lines are the experimental results for intensity at the detector and the dashed lines the predicted concentrations at the channel exit. The two sets of dashed lines correspond to the two strands of DNA that were used in the respective experiments.

28

Figure 2-7. Comparison of our predictions with experiments on separation of latex particles with EFFF. Channel geometry: h = 28µm, L = 6cm. Operational parameters: flow velocity u=0.08cm/s, applied voltage Vapp = 1.9v, current I = 165µA. The solid lines are the experimental results for intensity at the detector and the dashed lines the predicted concentrations at the channel exit. The three sets of dashed lines correspond to the three kinds of latex particles that were used in the respective experiments.

Summary

Application of lateral fields affects the mean velocity and the dispersion coefficient

of colloidal particles undergoing Poiseuille flow in a 2D channel. The dimensionless

mean velocity *U depends on 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 )PeU(fPe ey

2 . The mean velocity of the particles decreases monotonically with an

increase in PeUey , but ( ) 2* Pe/RD − has a maximum at a value of PeUe

y ~ 4. This

maximum arises when the thickness of the region near the wall where a majority of the

particles accumulate is about h/2.

29

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 100-kbp fragments, but non-continuum 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.

30

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 3-1. 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 field-flow fractionation technique developed by Giddings [43] and extended

by Shmidt and Cheh [44] and by Chandhok and Leighton [45], which relies on the

31

application of an oscillatory electric field across the narrow gap of the electrophoretic

cell. The motion of the solute species induced by this field interacts with an oscillatory

cross-flow to cause a separation based on the electrophoretic mobility of the species.

However, the cyclic combination of field and flow proposed in this paper is different

from the methods proposed in the above references.

Figure 3-1. Schematic showing the three-step cycle

As mentioned above, the potential advantage of the proposed method is that if the

duration of the step in which the field is applied is shorter than the time for charging of

the double layer then the gas generation can be avoided. Also there are a number of

design variables in this method that can be controlled to optimize the separation.

In the next section we solve the convection diffusion equation by using the regular

perturbation methods to determine the mean velocity and the dispersion coefficient of a

pulse of solute that is introduced into the channel at t = 0, and is then subjected to a series

of three-step cycles described above. Next, we discuss the dependence of the mean

velocity and the dispersion coefficient on the system parameters. Finally, we investigate

32

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 Γi (x), which is the number of molecules per unit area after the ith 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

2

22

2

2

XCε

YC

τC

∂∂

+∂∂

=∂∂ (3 - 1)

The above equation is in a dimensionless form where

l

xX = , hyY = ,

0ccC = ,

D/htτ 2= , 1hε <<≡

l (3 - 2)

l 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:

0)1Y,(YC)0Y,(

YC

==τ∂∂

==τ∂∂ (3 - 3)

Additionally, overall mass conservation requires

33

∫∫∫∞

∞−

∞

∞−Γ= dXCdXdY i

1

0 (3 - 4)

Since the diffusive step only lasts for time td and in our model dDt4h >> , the boundary

condition at Y=1 can be replaced by 0)Y,(C =∞→τ

We solve Eq. (3-1) by using a the technique of regular perturbation expansions [46,47] in

ε. The concentration is expanded as

L++= 22

0 CεCC (3 - 5)

By substituting C into Eq. (3-1) and Eq. (3-3), we get the differential equations and the

boundary conditions for different orders in ε. The equation for the order of 0ε is

20

20

YC

τC

∂∂

=∂

∂ (3 - 6)

The solution to Eq. (3-6) subject to the boundary conditions Eq. (3-3) and the overall

mass conservation is

)4Yexp()X(1C

2

i0 τ−Γ

τπ= (3 - 7)

The differential equation for the order of 2ε is

20

2

22

22

XC

YC

τC

∂∂

+∂∂

=∂

∂ (3 - 8)

The solution to Eq. (3-8) subject to the boundary conditions Eq. (3-3) is

)τ4

Yexp(πτ

X)X(Γ

C2

2i

2

2 −∂

∂= (3 - 9)

The above solution satisfies the overall mass conservation because 0x

→∂Γ∂ as ±∞→x .

34

The combination of C0 and C2 gives the concentration profile at the end of diffusion step

( dττ = ).

)τ4

Yexp(πτ

X)X(Γ

ε)τ4

Yexp()X(Γπτ

1)ττ(CCd

2d

2i

22

d

2

id

ddiff −

∂∂

+−=== (3 - 10)

We note that τd 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 τd 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

convection-diffusion equation, and apply the solution at the end of the diffusive step Eq.

(3-10) as the initial condition. The dimensionless convection-diffusion equation is

2

22

2

2

x XC

Peεε

YC

Peε1

XCU

τC

∂∂

+∂∂

=∂∂

+∂∂ (3 - 11)

In the above equation time has been dedimensionalized by the convective scaling, i.e.,

>< u/l , andD

huPe ><≡ . All the other dimensionless variables are the same as in the

diffusive step. We solve the above equation under the conditions 1Pe >>ε . Accordingly,

we assume a regular perturbation expansion for C in terms of ε and Pe1

ε, i.e.,

LLLL )εCC()Peε(

1)εCC(Peε1)εCC(C 2)2(

2)0(

222)2(

1)0(

12)2(

0)0(

0 ++++++++= (3 - 12)

The boundary conditions for C in the convective step are the same as in the diffusive

step.

35

The leading order equation for C in Pe1

ε is

0X

)εCC(U

τ)εCC( 2)2(

0)0(

0x

2)2(0

)0(0 =

∂+∂

+∂+∂ (3 - 13)

In the above equation, we transform the X coordinate to

τUXξ x−= (3 - 14)

As a result, Eq. (3-13) becomes

diff2)2(0

)0(0

2)2(0

)0(0

2)2(0

)0(0 C)0,(C)0,(C),(C),(C0

)CC(=εξ+ξ=ετξ+τξ⇒=

τ∂

ε+∂ (3 - 15)

Using Eq. (3-10) in Eq. (3-15) gives

)τ4

Yexp()τUX(Γπτ

1)τ,X(Cd

2

xid

)0(0 −−=

)τ4

Yexp(πτ

X)τUX(Γ

)τ,X(Cd

2d

2xi

2)2(

0 −∂

−∂= (3 - 16)

Similarly, by solving the equations for various orders of ε and 1/(Peε), we get )0(1C , )2(

1C ,

)0(2C and )2(

2C . Substituting them into Eq. (3-12) gives

36

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−+−∂

−∂

+

−+−∂

−∂

+

−∂

−∂

−

+−+−−

+

−+−−−

−−−

−−

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂−∂

+−+−∂

−∂+

+−−−

+

−∂

−∂+−−=

2d

d

22d

2d

2

d2

xi2

d

d

2

2d

2

d2

xi2

3d

d

22d

2xi

2

22dd

d

22

2d

2

dxi

dd

d

2

2d

2

dxi

3dd

d

22

xi

2dd

d

2

xi

2

2

d

2

2xi

2

dd

2

2d

2

d

d2

xi2

2

2d

2

dd

2

xid

d

2

2xi

2d2

d

2

xid

τ

)τ4

Yexp(Yπτ)

τ4Y

τ21(

X)τUX(Γ

81

πt

)τ4

Yexp()τ4

Yτ21(

X)τUX(Γ

41

τ

)τ4

Yexp(Yπτ

X)τUX(Γ

81

ετπτ

)τ4

Yexp(Y)τ4

Yτ21)(τUX(Γ

81

τπτ

)τ4

Yexp()τ4

Yτ21)(τUX(Γ

41

τπτ

)τ4

Yexp(Y)τUX(Γ

41

τπτ

)τ4

Yexp()τUX(Γ

41

)εPe(τ

)τ4

Yexp(X

)τUX(Γπτ

1)τ4

Yexp()τ4

Yτ21(

πτ

X)τUX(Γε

)τ4

Yτ21)(

τ4Yexp()τUX(Γ

πτ1

εPeτ

)τ4

Yexp(X

)τUX(Γπτε)

τ4Yexp()τUX(Γ

πτ1)τ,Y,X(C

(3 - 17)

The axial flow is driven by pressure gradient and thus the velocity profile is parabolic,

i.e.,

))h/y(h/y(u6u 2x −><= (3 - 18)

where 2hxp

µ31u ⎟

⎠⎞

⎜⎝⎛

∂∂

−>=< is the mean velocity in the channel.

In dimensionless form,

)YY(6U 2x −= (3 - 19)

Substituting Ux in Eq. (3-17), 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

fτ=τ in Eq. (3-17). In the third step, the electric field is applied to attract all the

37

molecules to near the wall. Neglecting axial diffusion during this step, the surface

concentration Γ after the end of the i+1st cycle is

∫=+

1

0f1i dY)τ,Y,X(C)X(Γ (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. (3-17) can be expanded by using Taylor series, i.e.,

( )...

X2U

XU)X()UX( 2

i22

xiXixi +

∂Γ∂τ

+∂Γ∂

τ−Γ=τ−Γ

After using the above expansion in Eq. (3-17), and then substituting the expression for C

in Eq. (3-20), and then performing the integration gives

2i

2'i'

i1i xΓD

xΓUΓΓ

∂∂

=∂∂

+−+ (3 - 21)

where X and ε have been replaced by x/l and h/l respectively, and

πhτ2PeDt3

)h

)τ2

1(PeerfDt12

πhτPeDt6

()h

)τ2

1(DPeerftτ12

πhDPetτ12

('U52/3

d

33f

3d

22f

32/1d

22fd

fd

f2/1

d −−+−=

(3 - 22)

)h

)τ2

1(erfDPet216

hπDPetτ108

())τ2

1(erfDt(

)h

)τ2

1(erfDPetτ432

hπDPetτ432

h

)τ2

1(erfDtPe36

()h)t2

1(erfτ(

)h

)t2

1(erfDPetτ216

hπDPetτ288

h

)τ2

1(erfDtPeτ36('D

6d

424f

6

424f

2/1d

df

4d

323fd

4

323f

2/1d

4d

33f

2

2

dd

2d

222f

2d

2

222f

2/3d

2d

22f

2d

+−++

+−++

+−=

−

(3 - 23)

38

If iΓ is known, then Eq. (3-21) can be used to determine iiΓ + , i.e., the surface

concentration at the end of i+1st cycle. Since 0Γ 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 τd<1/20, thus the error-function

( ⎟⎟⎠

⎞⎜⎜⎝

⎛

τd21erf ) 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 convection-diffusion

equation where the convective term arises due to the lateral electric field, i.e.,

2

2ey y

cDycu

tc

∂∂

=∂∂

+∂∂ (3 - 24)

where eyu 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, Eµζεε

u r0ey = , where εr and µ are the fluid’s dielectric constant and viscosity,

respectively, ε0 is the permittivity of vacuum, and ζ is the zeta potential of the colloidal

particle. Alternatively, the electrophoretic velocity can be expressed as Eµu eey = , where

39

µe 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 yΦ

kTZDu

t

ey ∂

∂−≡ where Tt is the absolute

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 eyu only for an approximate estimation of the field required to attract all the molecules

near the wall, we use the simpler expressionyΦ

kTZDu

t

ey ∂

∂−≡ .

The steady state solution to Eq. (3-24) is

)D

yuexp()0y(cc

ey== (3 - 25)

To attract most of molecules into h/100 of the plate, a field satisfying 3~D100hu e

y − must

be used. This gives

ZekT

300∆Φh

D300yΦ

kTZeDu t

t

ey =⇒−=

∂∂

−≡ (3 - 26)

Assuming Z ~ 10, which is a very conservative assumption, gives V77.0=∆Φ . Later we

use a value of about 33 µm for h, and a potential drop of .77V across a 33 µm channel is

about the same voltage as is applied in EFFF [15]. Additionally, under this electric field

the steady state will be attained in a time of about D300

h~uh 2

ey

which is much less than the

diffusive time and thus the assumption of neglecting diffusion during the third step is

40

reasonable. Furthermore, for h = 33 µm and D = 10-10 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 of t

and x.

First, we expand Γ into Taylor series in terms of time

2i

22

dfi

dfidf1i t)tt(

21

t)tt()tt(

∂Γ∂

++∂Γ∂

++Γ=+Γ=Γ + (3 - 27)

Using )tt( df + as time scale gives the following dimensionless equation

2i

2i

i1i TΓ

21

TΓΓΓ

∂∂

+∂∂

+=+ (3 - 28)

Also as shown above

2i

2'i'

i1i xΓD

xΓUΓΓ

∂∂

=∂∂

+−+ (3 - 29)

Substituting Eq. (3-28) into Eq. (3-29) gives

2i

2'i'

2i

2i

xΓD

xΓU

TΓ

21

TΓ

∂∂

=∂∂

+∂∂

+∂∂

(3 - 30)

We again define a new coordinate system,

TUx '−=ξ (3 - 31)

In this moving reference frame Eq. (3-30) becomes

41

2

22''

2'

2

2

ξΓ)

2UD(

TξΓU

TΓ

21

TΓ

∂∂

−=∂∂

∂−

∂∂

+∂∂ (3 - 32)

Where, the subscript has been removed. We shall show later that the long time solution to

the above equation is Gaussian, i.e.,

)DT4ξexp(

TAΓ

2

−= (3 - 33)

Thus,

)T(O~ξTΓ)T(O~

TΓ)T(O~

ξΓ)T(O~

TΓ 2

225

ξ2

223

2

223

−−−−

∂∂∂

∂∂

∂∂

∂∂

(3 - 34)

Keeping the leading order terms in Eq. (3-32) gives

2

22''

ξΓ)

2UD(

TΓ

∂∂

−=∂∂ (3 - 35)

Transferring it into the original coordinates gives

2

22'''

xΓ)

2UD(

xΓU

TΓ

∂∂

−=∂∂

+∂∂ (3 - 36)

Thus, the long time surface concentration is a Gaussian with the dimensional mean

velocity *U and effective diffusion coefficient *D given by

df

2''

*

df

'*

tt

2UD

D;tt

UU+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=+

= (3 - 37)

We dedimensionalize U* and D* with tf/(tf+td) and (tf)2/(tf+td), respectively,

and denote them as *U and *D . The dimensionless mean velocity and dispersion

coefficient are

42

( )2f

2''

*

f

*

tu2

UDD;

tu'UU

><

−=

><= (3 - 38)


Since dτ must be smaller than 1/20, the value of the error functions in Eq. (3-22) and Eq.

(3-23) are very close to 1, thus the expressions for 'U and 'D can be rearranged in the

following form:

)(3U)tt

()(2Utt

)(1U

23

)tt

()126

(tt

)1212

(tu

'U'U

d2

d

fd

d

fd

2/1d2

d

fd

2/1d

d

fd

2/1d

f

τ+τ+τ=

⎥⎦

⎤⎢⎣

⎡

π

τ−⎥

⎦

⎤⎢⎣

⎡τ−

π

τ+⎥

⎦

⎤⎢⎣

⎡τ−

π

τ=

><=

(3 - 39)

5D)tt

()(4D)tu(

htt

)(3Dtt

)(2D)tu(

h)(1D

)216108

()tt

())tu(

h(tt

)432432

36(tt

))tu(

h(

)216288

36()tu(

'D'D

2

d

fd2

f

2

d

fd

d

fd2

f

2

d

2d

2/3d2

d

f2

f

2

dd

f

2d

2/3d

dd

f2

f

2

d

2d

2/3d

d2f

+τ><

+τ+τ><

+τ=

⎥⎦

⎤⎢⎣

⎡τ+

π

τ−+⎥

⎦

⎤⎢⎣

⎡><

τ+

⎥⎦

⎤⎢⎣

⎡τ+

π

τ−τ+⎥

⎦

⎤⎢⎣

⎡><

τ+

⎥⎦

⎤⎢⎣

⎡τ+

π

τ−τ=

><=

(3 - 40)

On tracing the origin of U1, U2, U3, D1, etc, we find that U1 and D1 are contributions

from )0(0C ; D2 arises from )2(

0C ; U2 and D3 originate from )0(1C ; D4 is contributed by )2(

1C ;

and U3 and D5 originate from )0(2C . The )2(

2C does not contribute to either 'U or 'D .

Each of these terms depends only on τd, and accordingly 'U depends strongly on τd and

weakly on d

f

tt . Also 'D depends strongly of τd and weakly on

d

f

tt and

( )2f

2

tuh .

The truncations errors in 'U and 'D are

43

))tt

((O'Ufor error Truncation 3

d

f=

))tt

())ut(

h((O))tt

((O'Dfor error Truncation 3

d

f2

f

3

d

f

><+= (3 - 41)

We note that for the proposed regular expansion solutions to be valid

1)ut(

h and 1tt

fd

f <><

< (3 - 42)

Mean Velocity

Figure 3-2 plots the dependency of the dimensionless mean velocity on G (d

1τ

≡ )

for different values of tf/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. (3-10). But

this trend can be observed as G approaches 20. Figure 3-2 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.

44

In Figure 3-2, the difference between the curves corresponding to different values

of tf/td indicates the contribution from the higher order terms in the expression for *U

(Eq. (3-38), (3-39)). The comparison of these curves shows that the relative importance

of the higher order terms in the expression for *U becomes more important at large

values of G. However at the values of G that are used in the separation scheme described

below (G = 150), the difference between the curve for tf/td = 0, which represents the

leading order contribution and the curve for tf/td =0.3 differ by about 10%.

Figure 3-2. Dependency of *U on G

Dispersion Coefficient

Figure 3-3 plots the dependency of the dimensionless dispersion coefficient *D on

G for different values of tf/td and ( )2

f

2

tuh . The difference between the curves

corresponding to different values of tf/td indicates the effect of tf/td on dispersion.

45

Similarly the difference between the curves corresponding to different values of

( )2f

2

tuh indicates its effect on dispersion. Figure 3-3 shows that

( )2f

2

tuh has a

negligible effect on dispersion and that the effect of tf/td on dispersion is comparable to its

effect on the mean velocity.

Figure 3-3. Dependency of *D on G

From Figure 3-3, we see that the effective diffusion coefficient increases with

decreasing G. Physically, a decrease in G can be interpreted as an increase in td, which

causes the molecules to move close to the center of channel. Since the velocity is larger at

the center, more particles move with large velocity. However, the highest concentration is

still at the wall; thus a large number of particles still move with velocity near zero. As a

result, after a cyclic operation, the particles spread further.

46

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.,

))tt(TD4)tt(TD4(3)tt(T)UU( df*2df

*1df

*2

*1 +++≥+−

⇒ 2*2

*1

*2

*1 )

UU)DD(

(12T−

+≥ (3 - 43)

We use Eq (3-43) to calculate T, i.e., the dimensionless time or, equivalently, the

number of cycles needed for separation. The dimensional time required for separation θ

is equal to T(tf + td), i.e.,

2*2

*1

*2

*1

f

d

f

)UU

)DD((

tt

112t −

+⎟⎟⎠

⎞⎜⎜⎝

⎛+=

θ (3 - 44)

The length of the channel required for separation is equal to the distance traveled by the

faster moving molecule in this time, i.e.,

2*2

*1

*2

*1f

1*

1 )UU

)DD((

htuU12

hLUTL

−

+><=⇒′= (3 - 45)

47

The dimensionless separation time (θ/tf) and channel length (L/h) depend on G1, d

f

tt and

( )2f

2

tuh and G2/G1. Below we discuss this dependence for a G2/G1 = 2.

Figure 3-4. Effect of G1 ( 2.0tt

d

f = , ( )

2.0tu

h2

f

2

= , G2/G1=2) on L/h, θ/tf and T

Effect of G

Figure 3-4 shows the dependence of T, θ/tf and L/h on G1 for fixed values of d

f

tt ,

( )2f

2

tuh and G2/G1, which are noted in the caption. With increasing G1, the number of

loops, time and the length needed for separation first decrease and then level off. To

48

understand the reasons for this behavior, we calculated the difference between the mean

velocities of the two type of molecules and sqrt( *1D ) as a function of G1. At small G1, an

increase in G1 leads to an increase in the difference in the mean velocities and a decrease

in sqrt( *1D ). Thus, both the factors lead to a better separation, resulting in a reduction in

the number of cycles. Beyond a critical value of G1, the difference in the mean velocities

begins to decrease with a further increase in G1. 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 3-5 shows the dependence of T, θ/tf and L/h on d

f

tt for fixed values of G1,

( )2f

2

tuh and G2/G1, which are noted in the caption. Figure 3-5 shows that the number of

loops required for separation is relatively independent of d

f

tt . With an increase in

d

f

tt ,

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 3-5 shows

that the time required for separation depends strongly on the ratio d

f

tt . This happens

because although the number of steps is unchanged, td decreases as d

f

tt increases and this

leads to a reduction in the time for each step, and consequently a reduction in time for

49

separation. The length required for separation increases with d

f

tt because of the increase

in the mean velocities of both the species.

Figure 3-5. Effect of d

f

tt

(G1=100, ( )

2.0tu

h2

f

2

= , G2/G1=2) on L/h, θ/tf and T

Effect of ( )2

f

2

tuh

Figure 3-6 shows the dependence of T, θ/tf and L/h on ( )2

f

2

tuh for fixed values of

G1, d

f

tt and G2/G1, which are noted in the caption. As noted earlier the mean velocity is

independent of ( )2

f

2

tuh and the dispersion coefficient depends weakly on this ratio.

50

Accordingly, the number of loops required for separation is not expected to depend on

( )2f

2

tuh , as shown in Figure 3-6. In Figure 3-6, θ/tf is plotted on the y-axis, thus tf has to

be kept constant so that the value of the y coordinate can be interpreted as the time

required for separation. Also td is fixed because the ratio d

f

tt is kept constant.

Accordingly, the time required for separation shows the same behavior as the number of

loops. However, L/h depend strongly on ( )2

f

2

tuh because an increase in the x-axis is

equivalent to a reduction in ftu , which leads to a linear reduction in the mean velocity.

Figure 3-6. Effect of ( )2

f

2

tuh (G1=100, 2.0

tt

d

f = , G2/G1=2) on L/h, θ/tf and T

51

The dependency of T, θ and L on the three dimensionless numbers remains the

same for G2/G1=1.2. However, the actual values increase significantly. The description

above shows that the optimum values of G, d

f

tt and

( )2f

2

tuh are about 150, 0.3 and 0.3,

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 D1=10-10 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

d

f

tt =0.3, td is about 0.067 s. By using G = 150 and

( )2f

2

tuh =0.3, the values of h and 

are 32 µm 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 θ and L are reduced by an increase in G. However, if the diffusion

coefficient is about 10-12 m2/s, based on Stokes-Einstein equation the particle size is about

0.2 µm and in this case the proposed continuum model is not valid. An alternate model

52

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 < 10-12 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 D1=10-9 m2/s for G2/G1=2 takes about 17.5 s

in a channel 1 cm in length and 58 µm in height. The time and length become 269 s and

16 cm for G2/G1=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 µm. 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 eyu

due to the electric field is estimated by the equation Eu eey µ= , where eµ is the electrical

mobility of the particles. The value of eµ has been measured for various types of

colloidal particles. It can also be determined by the Smoluchowski equation,

µζεε

=µ r0e , (εr and µ are the fluid’s dielectric constant and viscosity, respectively, ε0 is

the permittivity of vacuum, and ζ is the zeta potential of the colloidal particle). The

mobility of polystyrene latex particles is relatively independent of size and varies in the

53

range of 1.9x10-4 – 3.23 x10-4 cm2/(Vs) for particle diameters in the range of 90 nm-944

nm [2]. For smaller particles the mobility can be estimated by treating them as point

charges and thus eµ can be expressed as kTZD where D and Z are the diffusivity and the

charge of the particle. For D = 10-10 m2/s and Z =10e (e = electronic charge), the

mobility is about 4 x10-4 cm2/(Vs). At these mobilities a field of 1000 V/m will drive a

lateral velocity of the order of 20 µm/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 µm/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, D1, D1/D2 are 30 µm, 10-10 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 eyu is varied from 0 – 100 µm/s which is much larger than the expected

values of the lateral electric velocity. In Figure 3-7 the value of L/h is plotted as a

function of G and eyu 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 3-8) 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

54

is less than the diffusive contribution, and thus the overall dispersion does not decrease

significantly, and the reduction of the mean velocity with a reduction in leads to an

increase in L/h.. As shown in Figure 3-7 the length of the channel required for separation

reduces with increasing eyu and the length required by EFFF at the optimal mean velocity

becomes less than that required by the pulsatile technique for eyu > 30 µm/s, which is

larger than the expected value of the lateral velocity. Also, the time required for

separation is less for the pulsatile technique. Figure 3-9 and 3-10 are very similar to

Figures 3-7 and 3-8; only the value of D1/D2 has been reduced from 2 to 1.2. As shown

in the Figures, the trends discussed above do not change on reducing the value of D1/D2;

only the actual values of L/h and time for separation increase if D1/D2 is smaller.

Figure 3-7. Dependency of L/h on G1(pulsating electric field) and eyu (constant electric

field). Solid lines(Constant EFFF): h=30µm, D=10-10m2/s, the value of are noted on the curves, D1/D2=2; Dashed line(Pulsating EFFF): h=30 µm, D=10-10 m2/s, =0.002 m/s, D1/D2=2, tf=0.02 s

55

Figure 3-8. Dependency of the operating time t on G1(pulsating electric field) and eyu (constant electric field). Constant EFFF: h=30µm, D=10-10m2/s, the value

of are noted on the curves, D1/D2=2; Pulsating EFFF: h=30 µm, D=10-10 m2/s, =0.002 m/s, D1/D2=2, tf=0.02 s

Figure 3-9. Dependency of L/h on G1(pulsating electric field) and eyu (constant electric

field). Solid lines(Constant EFFF): h=30µm, D=10-10m2/s, =0.002, 0.001, 0.0005, 0.00005 m/s, D1/D2=1.2; Dashed line(Pulsating EFFF): h=30 µm, D=10-10 m2/s, =0.002 m/s, D1/D2=1.2, tf=0.02 s

56

Figure 3-10. Dependency of the operating time t on G1(pulsating electric field) and eyu (constant electric field). Constant EFFF: h=30µm, D=10-10m2/s,

=0.002, 0.001, 0.0005, 0.00005 m/s ( does not change the time for separation for the first three velocities), D1/D2=1.2; Pulsating EFFF: h=30 µm, D=10-10 m2/s, =0.002 m/s, D1/D2=1.2, tf=0.02 s

Conclusions

We propose and model a new technique for separating charged colloids of different

sizes. The method relies on a combination of pulsatile lateral fields and an axial flow that

varies in the lateral direction. The method is similar to the EFFF, which also relies on

lateral electric fields for separation. In EFFF only a very small fraction of the applied

fields acts on the particles and the double layers consume the remaining field. In the

pulsatile technique because the time for which the field is applied is smaller than the time

needed for charging of the double layers, the majority of the applied field is expected to

act on the particles, and thus at fields comparable to those applied in EFFF, the particles

will accumulate near the wall if the field is pulsed. The separation efficiency of the

proposed method depends strongly on the rate at which the fluid flow can be switched on

57

and off; the separation improves with a reduction in tf and td, which are the durations of

the flow and the no-flow steps. For reasonable value of design constants, the proposes

technique can separate molecules of diffusivities 10-10 m2/s and 0.5x10-10 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 10-9 m2/s, the length and the time for separation are

1 cm and 17.5 s for D1/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.

58

CHAPTER 4 TAYLOR DISPERSION IN CYCLIC ELECTRICAL FIELD-FLOW

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

eyu/D cannot be separated, where D is the molecular diffusivity and e

yu is the electric

field driven velocity on the lateral direction. Giddings suggested that cyclical electrical

field-flow 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 h/utT 00= , where t0 is the time period of oscillation,

59

u0 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., 1L/hε <<≡ .

The transport of a solute in the channel is governed by the convection-diffusion

equation,

2

2

2

2

||ey y

cDx

cDycu

xcu

tc

∂∂

+∂∂

=∂∂

+∂∂

+∂∂

⊥ (4 - 1)

60

where c is the solute concentration, u is the fluid velocity in the axial (x) direction, ||D


flow, respectively. We assume that the diffusivity tensor is isotropic and thus ||D = ⊥D =

D. In Eq.(4 - 1), eyu 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 eyu can be determined by the Smoluchowski equation, E)µ/ζεε(u r0

ey = , where εr

and µ are the fluid’s dielectric constant and viscosity, respectively, ε0 is the permittivity

of vacuum, and ζ is the zeta potential. Or, it can be simplified as Eµu Eey = where Eµ 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 3.8x10-8

m2/(V·s).[22] In EFFF, researchers have applied an effective electric field of 100V/cm

without gas generation. Thus, typical values of eyu could be as large as 3.8x10-4 m/s.

Eq. (4 - 1) is subjected to the boundary condition of no flux at the walls (y = 0,1),

i.e.,

0cuycD e

y =+∂∂

− (4 - 2)

In a reference moving in the axial direction with velocity *u , Eq. (4 - 1) becomes

)yc

xc(D

ycu

xc)uu(

tc

2

2

2

2ey

*

∂∂

+∂∂

=∂∂

+∂∂

−+∂∂ (4 - 3)

where x is now the axial coordinate in the moving frame. For a sinusoidal electric field

))tsin(EE( max ω= the lateral velocity is

tsinRuVu Eey ω>=<µ= (4 - 4)

61

where is the mean velocity and R is the dimensionless amplitude of the lateral

velocity, which is given by ><µ u/EmaxE . 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

)hy,xDt,tω(CC

ll 2= (4 - 5)

where )π2/(ω is the frequency of the applied field, 1/ω is the short time scale, and D/2l

is the long time scale over which we wish to observe the dispersion. Substituting Eq. (4 -

5) into (4 - 3) gives

2

2

2

22e

y*

sl

2

YC

XC

YCPeU

XC)UU(Pe

TC

TC

∂∂

+∂∂

ε=∂∂

+∂∂

−ε+∂∂

Ω+∂∂

ε (4 - 6)

where tωTs = , l/xX = , h/yY = , 2l /DtT l= ,

DhuPe ><

= , DhωΩ

2

= ,

><=

><=

uuU,

uuU

** ,

><=

uu

Ueye

y and 1lh

<<≡ε .

Since ε<<1, the concentration profile can be expanded in the following regular

expansion.

∑∞

=

ε=0m

lsmm )T,Y,X,T(CC (4 - 7)

62

Substituting Eq. (4 - 7) into Eq. (4 - 6) gives

)ε(θYC

εYC

εYC

XC

ε

YC

PeUεYC

PeUεYC

PeUXC

)UU(Peε

XC

)UU(PeεTC

εΩTC

εΩTC

ΩTC

ε

322

22

21

2

20

2

20

22

2ey

21ey

0ey

1*2

0*

s

22

s

1

s

0

l

02

+∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

−+

∂∂

−+∂∂

+∂∂

+∂∂

+∂∂

(4 - 8)

Eq. (4 - 8) can be separated into a series of equations for different order of ε.

( 0ε ):

20

20e

ys

0

YC

YC

PeUTC

Ω∂∂

=∂∂

+∂∂

(4 - 9)

The solution for C0 can be decomposed into a product of two functions, one of which

depends on Ts and Y and the other depends on X and Tl, i.e., )T,X(A)T,Y(GC ls00 = ,

where G0 satisfies

20

20e

ys

0

YG

YG

PeUTG

∂∂

=∂

∂+

∂∂

Ω (4 - 10)

The above equation is subjected to the following boundary condition at Y = 0, 1:

0ey

0 GPeUY

G=

∂∂

(4 - 11)

Next, we solve the equations at the order of ε. To order ε, the governing equation (4

- 8) becomes

( 1ε ):

21

21e

y0*

s

1

YC

YCPeU

XC

)UU(PeTCΩ

∂∂

=∂∂

+∂∂

−+∂∂ (4 - 12)

63

Integrating the above equation from 0 to 1 in Y and 0 to 2 π in Ts and noting that

0dT)T/C(Ωπ2

0ss1 =∂∂∫ due to periodicity and ∫∫ ∂∂=∂∂

1

0

21

21

01

ey dY)Y/C(dY)Y/C(PeU due

to the boundary conditions gives

∫ ∫

∫ ∫π

π

= 2

0

1

0ss0

2

0

1

0ss0

*

dYdT)T,Y(G

dYdT)Y(U)T,Y(GU (4 - 13)

The solution to C1 is of the form )X/)T,X(A)(T,Y(B ls ∂∂ where B satisfies

2

2eys0

*

s YB

YBPeU)T,Y(G)UU(Pe

TB

∂∂

=∂∂

+−+∂∂

Ω (4 - 14)

and the following boundary conditions:

BPeUYB e

y=∂∂ (4 - 15)

Since we are only interested in the periodic-steady 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 G0 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 Ω/15.0 in all simulations.

The spatial grid size near the wall was set to be smaller than PeR/3.0 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(ε2) problem. To order ε2, the governing equation (4 - 8)

becomes

( 2ε )

22

2

20

22e

ys

21*

l

0

YC

XC

YCPeU

TCΩ

XC)UU(Pe

TC

∂∂

+∂∂

=∂∂

+∂∂

+∂∂

−+∂∂

(4 - 16)

Averaging both sides in Ts and Y gives,

20

21

0

π2

0s2

2*

l

0

XC

dYdTX

AB)UU(PeTC

∂><∂

=∂∂

−+∂

><∂∫ ∫ (4 - 17)

where

∫ ∫π

>=<1

0

2

0s00 dYdTCC (4 - 18)

Rewriting Eq. (4 - 17) gives

20

21

0

2

0s2

02

l

0

XC

BdYdT)UU(Pe1XC

TC

∂><∂

=−φ∂

><∂+

∂><∂

∫ ∫π

(4 - 19)

where

∫ ∫π

≡φ1

0

2

0s0 dYdTG

Now we combine the results for )T/C( s0 ∂∂ and )T/C( l0 ∂∂ .

l

02

s

22

s

1

s

00

TC

lD

TC

TC

TC

tC

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

ε+∂∂

ε+∂∂

ω=∂

∂ (4 - 20)

Averaging the above equation in Y and Ts and using periodicity gives,

l

0

2

0

TC

lD

tC

∂

∂=

∂

∂ (4 - 21)

Using Eq. (4 - 21) in Eq. (4 - 19) gives

65

2

02

*0

x

CDD

tC

∂

∂=

∂

∂ (4 - 22)

where the dimensionless dispersion coefficient is given by

∫ ∫

∫ ∫π

π

−−= 2

0

1

0ss0

2

0

1

0ss

*

dYdT)T,Y(G

dYdT)T,Y(B)U)Y(U(Pe1D (4 - 23)

The numerical solutions for G0 and B that are obtained by solving Eq. (4 - 10) and

(4 - 14), and G0 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.


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 h/utT 00= , where t0 is the time period of oscillation, u0 is the amplitude of

the lateral velocity and h is the channel height. This dimensionless parameter is identical

to 2 πPeR/Ω in terms of the parameters defined in this paper. For the case of square

66

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 fR × where f is simply a square wave function

that oscillates from -1 to 1 with a dimensionless angular frequency of Ω .

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 2T < ( h/utT 00= ), the asymptotic concentration profiles that were

predicted by S&B (Figure 4. of Ref 50) corresponds to a uniform probability outer

solution of width 2/T1− 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 2T < are

shown in Figures 4-1a and these show that the numerical solutions for the concentration

transients agree with the asymptotic solutions. In Figure 4-1a, 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.

67

Figure 4-1. Periodic steady concentration profiles during a period for a square shaped electric field for (a) PeR = 80, Ω = 1500, 2 πPeR/Ω = 0.335 and (b) PeR = 400, Ω = 200 π , 2 πPeR/Ω = 4.

As T becomes larger than 2 (Figure 4-1b), the duration of the time in which the

lateral field is constant is long enough for the lateral concentration profile to attain a

steady state, which is an exponentially decaying concentration from the wall. As the

direction of the field switches, the exponential profile begins to move towards the

opposite wall and spreads into a Gaussian. Eventually, the Gaussian profile touches the

other wall and then achieves the steady state of a decaying exponential. The asymptotic

analysis of S&B predicted the same behavior.

68

Mean velocity and dispersion coefficient

Figures 4-2a and 4-2b compare the numerical results for mean velocity and

dispersion coefficient with those obtained by S&B. In Figure 4-2a and 4-2b 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 4-2a and 4-2b 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 π PeR/Ω > 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 π PeR/Ω < 2 the numerical results approach the asymptotic results but do not reach the

asymptotic limit for Ω as large as 2000. However, based on the trends it can be

concluded that for higher values of Ω, the numerical results will match the asymptotes

obtained by S&B. It is also noted that the kink in Figure 4-2b at Ω

π=PeR2T = 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 2T > , the entire solute band travels as a

delta function and thus there is no spread, and hence no dispersion.

69

Figure 4-2. 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). The markers on each curve represent the results calculated by Brownian dynamics.

70

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

)1Y,T(C)Y,T(C ss α−=θ+π==α=θ= (4 - 24)

Accordingly, both C0 and B satisfy the same symmetry. As shown in Appendix B,

C0 and B be expanded as

))]nTcos()Y(g)nTsin()Y(f()Y(g[const

))nTcos()Y(q)nTsin()Y(p()Y(q)Y,T(B

)T,X(A~))nTcos()Y(g~)nTsin()Y(f~()Y(g~C

sn1n

sn0

sn1n

sn0s

lsn1n

sn00

+++

++=

⎟⎠

⎞⎜⎝

⎛++=

∑

∑

∑

∞

=

∞

=

∞

=

(4 - 25)

Substituting C0 from the above equation into Eq. (4 - 24) gives

))ncos()1(g)1()nsin()1(f)1(()1(g

)))(ncos()1(g))(nsin()1(f()1(g

))ncos()(g)nsin()(f()(g

nn

1nn

n0

n1n

n0

n1n

n0

θα−−+θα−−+α−

=θ+πα−+θ+πα−+α−

=θα+θα+α

∑

∑

∑

∞

=

∞

=

∞

=

(4 - 26)

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-

71

3, in which the functions fi are plotted as a function of Y for N = M = 5 and Pe = R = 1

and Ω =100. The results are similar for gi, pi and qi (plots not shown).

Figure 4-3. gi vs. position for PeR=1, and Ω =100.

Convergence of the series expansions. As shown in Appendix B, the equations

for obtaining G0 and B analytically are two hierarchies of coupled second order ordinary

differential equations, which are closed by setting the coefficients for Nth (for G0) and Mth

(for B) terms to be zero. For PeR = 1, if the values of M and N are taken to be larger than

5, the coefficients of the fifth terms are about 10-6, which is negligible in comparison to

the coefficients of the first terms that are of order 1. Accordingly, both M and N are

chosen to be 5 for the case of PeR = 1. For N = M =5, f5, g5, p5 and q5 are of the order of

10-6. On increasing M and N from 5 to 6, the maximum change in f0 and p0 is less than

0.01%.

The values of M and N required to ensure that the truncation errors are minimal

depends on PeR. On increasing PeR to 30, the values of M and N have to be increased to

7. Thus, determining fi, gi, pi and qi become computationally expensive for PeR larger

than about 40. Additionally, some of the positive eigen values (λ) in the expansions for

fi, gi, pi and qi also become larger on increasing PeR and thus some of these functions

72

grow exponentially in Y, and become very large near Y=1. Accordingly, the matrix that

is inverted to determine the fi, gi, pi and qi 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 Ω on the temporal concentration profiles. Figures 4-4a-d

show the concentration profiles at various time instances during half of a period. In

Figure 4-4a, 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 π, the field is zero, but the concentration

profile is far from uniform, and that the wall concentration keeps increasing beyond ts =

3π/2, even though the field begin to decrease. The profiles in Figure 4-4b correspond to

the same value of PeR as in Figure 4-4a but a much small value of 1 for Ω . 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 Ω.

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 = 3π/2. Figure 4-4c and 4-4d correspond to PeR = 1, and Ω of 1 and 10,

respectively. Since PeR is small, a boundary layer does not develop in both of the cases.

In Figure 4-4c, the concentration profiles are not exactly in phase as evident from the fact

73

that the profiles for ts = 0, π do not overlap, but the profiles are closer to being in phase

with the driving force than those in Figure 4-4 d that correspond to PeR = 1 and Ω = 10.

Figure 4-4. Time dependent concentration profiles within a period for sinusoidal electric fields for (a)Ω =100, Pe =100, R=1; (b)Ω =1, Pe=100, R=1; (c)Ω =10, Pe=1, R=1; and (d)Ω =1, Pe=1, R=1.

Effect of PeR and Ω on the mean concentration profiles. The effects of PeR and

Ω on the time averaged concentration are illustrated in Figures 4-5a-e. The short-time

averaged concentration is equal to )T,X(A)Y(g l0 , where ∫π

≡2

0ss00 dT)T,Y(Gg and g0 is

plotted as a function of Y in the plots below. In Figure 4-5a and 4-5b the function g0 is

plotted for various values of PeR for Ω = 20 and 100, respectively.

74

Figure 4-5. Time average concentration profiles for sinusoidal electric field for (a)Ω =20; (b) Ω =100; and Pe=R=1 for Ω ranging from (c) 1-20, (d) 40-100, and (e) 100-1000.

Figures 4-5a-e show that as expected the mean concentration profiles are

symmetric in Y. In Figures 4-5a and 4-5b, due to the presence of the electric field, the

75

particles accumulate near the wall, leading to a higher concentration at the boundaries. In

Figure 4-5a, the concentration profile in the center is relatively flat and the value of g0 in

this central region increases on reducing PeR. However the profiles in Figure 4-5b show

that for Ω = 100, a maxima develop in the central region, when PeR is less than about 40.

The effect of Ω on g0 is further illustrated in Figures 4-5c-e for PeR = 1. The values of

Ω span from 1 to 20 in Figure 4-5c, from 40 to 100 in Figure 4-5d and from 100-1000 in

Figure 4-5e. For Ω 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 Ω beyond 40,

a secondary maximum develops in the center but the maximum concentration is still at

the wall. On increasing Ω further, the value of g0 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 Ω . 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 Ω can be changed by varying

the frequency of the periodic electric field. The mean velocity is only a function of PeR

and Ω and the dispersion coefficient is of the form Pe2 f(PeR, Ω ). Typical microfluidic

channels are about 20-40 µm thick and as stated earlier the lateral electric velocity uye

76

could be as large as 3.8x10-4 m/s and this implies a value of about 1200 for PeR for a D

value of 10-11 m2/s. It is also noted that typical channel lengths are about a 1-2 cm and

thus the value of ε is about 10-3. Accordingly, for our analysis to be valid the values of

PeR and Ω should be much less than about 1000. We now discuss the effect of these

parameters on the mean velocity and the dispersion coefficient.

Figure 4-6. Dependence of *U on PeR for (a) PeR ranging from 0 to 10 and Ω ranging from 1-20; and (b) PeR ranging from 0 to 200 and Ω ranging from 1-100, and comparison with the small Ω asymptote (thick line). The markers on each curve represent the results calculated by Brownian dynamics.

77

Figure 4-6a-b plots the dependency of the mean velocity on PeR for different

values of Ω . Figure 4-6a 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 1-200 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 4-6a-b that are the results of the

Brownian dynamics simulations also match the results computed by finite difference. As

shown in Figures 4-6a-b, 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 4-6a-b also shows the dependence of the mean velocity on Ω ; as Ω

increases, the curve of the mean velocity shifts up. This is due to the fact that as Ω

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 Ω . As a result, on increasing Ω 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

),PeR(fPe1 2 Ω+ . Figures 4-7a-b plots 2* Pe/)1D( − as a function of PeR for different

values of Ω . As for the case of mean velocity, Figures 4-7a and 4-7b were computed by

78

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 4-7a shows that as PeR approaches zero, the curves of

2* Pe/)1D( − 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 4-7a also shows that for small Ω , the curves exhibit a maximum at PeR = 4.

This phenomenon also occurs in constant electric field-flow 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

79

PeR is different in the small and the large PeR regime, it must have a maximum. For a

given PeR, an increase in frequency reduces the concentration differences along various

lateral positions and thus leads to a reduction in the dispersion. Accordingly, the curves

in Figure 4-7a shift down with an increasing Ω and the maximum in the dispersion

coefficient disappears as for Ω larger than about 10.

Figure 4-7. Dependence of (D*-1)/Pe2 on PeR for (a) PeR ranging from 0 to 10 and Ω ranging from 1-20; and (b) PeR ranging from 0 to 200 and Ω ranging from 1-100, and comparison with the small Ω asymptote (thick line). The markers on each curve represent the results calculated by Brownian dynamics.

Small Ω limit. To better understand the effect of Ω on the dispersion, we obtain

expressions for small Ω . In the small Ω limit, it is useful to let the velocity of the

80

reference frame in which we solve the convection diffusion equation to vary during a

period, i.e., )T(UU s** = . In the limit of small Ω , to leading order, Eq. (4 - 9), (4 - 12)

and (4 - 16) become

20

20e

y YC

YC

PeU∂∂

=∂∂

(4 - 27)

21

21e

y0*

YC

YCPeU

XC

)UU(Pe∂∂

=∂∂

+∂∂

− (4 - 28)

22

2

20

22e

y1*

l

0

YC

XC

YCPeU

XC)UU(Pe

TC

∂∂

+∂∂

=∂∂

+∂∂

−+∂∂

(4 - 29)

These equations along with the no-flux boundary conditions are identical to those for

EFFF and thus the short time dependent mean velocity and the dispersion coefficient are

given by[53]

1)αexp()α(

)αexp(1212α

)αexp(66

U2

−

−+

+

= (4 - 30)




−−−−++−+−

++−−−−+−=

(4 - 31)

where )Tsin(PeRPeU sey =≡α . 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 4-6a and 4-7a. Figure 4-7a shows the

comparison of the small Ω expression with the full result from Eq. (4 - 13) for the mean

velocity. The small Ω solution matches the exact solution for Ω <1. Similarly the

81

dispersion coefficient in the small Ω limit is the time average of the dispersion

coefficient for constant electric field-flow fractionation and it matches the full solution

for Ω <1 (Figure 4-7a). The matching of the mean velocity and the dispersion coefficient

with the time averaged EFFF results is expected because as shown earlier for Ω = 1, the

concentration profiles are close to being in phase with the driving force.

Large Ω limit. In the large Ω limit, the mean velocity can be computed by

following the same approach as used by S&B. In this limit, to leading order, the

periodically-steady concentration profiles are given by the following expressions:

For π<T

⎪⎩

⎪⎨

⎧

<<+∆+∆<<∆

∆<<=

1YWp)T(Yfor0Wp)T(YY)T(YforA

)T(YY0for0)T,Y(G

s

ss

s

s0 (4 - 32)

where π−= /T1Wp and ))Tcos(1)(2/T()T(Y ss −π=∆ . For π>T

⎪⎪⎩

⎪⎪⎨

⎧

π<<+πδ+π<<ππ−∆−−δ

π<<−δ<<∆−δ

=

2TTfor ),Y(ATTfor )))T(Y1(Y(A

TT for )1Y(ATT0for ))T(YY(A

)T,Y(G

st

tss

st

tss

s0 (4 - 33)

where A is a constant whose value can be determined by using the normalization

condition, and Tt is the time at which 1)T(Y s =∆ , i.e., the pulse touches the wall. The

mean velocities can then be computed by using Eq. (4 - 13).

In the high frequency limit, the mean velocity depends only on Ω

π=

PeR2T , and

this dependence is shown in Figure 4-8 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

82

duration. It is more reasonable to compare the two types of fields under the stipulation

that the integral of the field during a half period is the same. This stipulation is satisfied

if the amplitude for the periodic fields is set to be 2/π times the amplitude of the square

fields. In Figure 4-8 and also in Figure 4-9 the x scale is chosen to be Ω/PeRπ2 sq ,

where the subscript sq denoted the value of R for the square wave and the value of R for

the sinusoidal fields is 2/Rπ sq . Figure 4-8 shows that the mean velocity for the

sinusoidal fields is smaller than the square fields for 92.6/PeR2 sq <Ωπ and at larger

values of Ωπ /PeR2 sq the mean velocity is higher for the sinusoidal fields. This can be

attributed to the fact that for small values of Ωπ /PeR2 sq , the slope of the )T(Y s∆ at the

time at which Y∆ =1/2 is larger for the sinusoidal fields, and thus the time spent by the

pulse near the center is smaller, and accordingly the mean velocity is smaller. The

situation is reversed for large values of Ωπ /PeR2 sq leading to higher values of mean

velocity for the sinusoidal fields.

Figure 4-8. Comparison of the mean velocities for the square (dashed) and the sinusoidal (solid) fields in the large frequency limit. The R value on the x axis corresponds to that for the square shaped field (Rsq) and the value of R for the sinusoidal field is π/2 times Rsq.

83

Figure 4-9. Comparison of the mean velocities and the effective diffusivity for the square (dashed) and the sinusoidal (solid) fields. The R value on the x axis corresponds to that for the square shaped field (Rsq) and the value of R for the sinusoidal field is π/2 times Rsq.

Comparison of Sinusoidal and Square fields

In this section we compare the results of the mean velocity and the dispersion

coefficient for the sinusoidal and the square fields. The comparisons for the mean

velocity and the dispersion coefficient are shown in Figures 4-9a and 4-9b, respectively.

In the figures the mean velocities and the dispersion coefficients are plotted for a range of

Ω values as a function of Ω/PeRπ2 sq , where as stated above, the subscript sq denoted

84

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 2/Rπ sq . The figures show that for large values of Ω , the

curves for both the mean velocities and the dispersion coefficients are similar and almost

overlap for Ω/PeRπ2 sq <10. To avoid or minimize the decay in the electric field due to

double layer charging, separation will need to be performed at large Ω and for optimal

separation it is best to operate in the region where the mean velocity is most sensitive to

the field strength. Figure 4-9 shows that these requirements suggest that the most

suitable operating parameters are Ω/PeRπ2 sq ~10 and Ω ~ 100. Figures 4-9a and 4-9b

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 eyu/D 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 *U depends on Ω , 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 )Ω,PeR(fPe2 . The mean velocity of the particles decreases monotonically with an

85

increase in PeR , and increases with an increase in Ω ; but ( ) 2* Pe/1D − has a maximum

at a value of PeR ~ 4 for small Ω , and the maximum disappears at large Ω . For Ω <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 Ω , the mean velocities and the dispersion coefficients are similar and almost

overlap for Ω/PeRπ2 sq <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.

86

CHAPTER 5 ELECTROCHEMICAL RESPONSE AND SEPARATION IN CYCLIC ELECTRIC

FIELD-FLOW 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

87

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., 1L/hε <<≡ .

Equivalent Electric Circuit

Figure 5-1 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

88

)/texp(RVi

s

τ−= (5 - 1)

where

dsCR=τ (5 - 2)

If a periodic square shaped voltage is applied, the current is given by

))/texp(1/()/texp()R/V(2i cs τ−+τ−±= (5 - 3)

where tc is half of the time for a period, and the ± sign corresponds to the periods in

which the voltage is positive and negative, respectively.

Figure 5-1. Equivalent electric circuit model for an EFFF device

Model for Separation in EFFF

The transport of charged particles in the channel is governed by the convection-

diffusion equation,

2

2

2

2

||ey y

cDx

cDycu

xcu

tc

∂∂

+∂∂

=∂∂

+∂∂

+∂∂

⊥ (5 - 4)

where c is the particle concentration, u is the fluid velocity in the axial (x) direction, ||D


89

flow, respectively. We assume that the diffusivity tensor is isotropic and thus ||D = ⊥D =

D. In Eq. (5 -4), eyu is the velocity of the particles in the lateral direction due to the

electric field. If the Debye thickness is smaller than the particle size, the lateral velocity

eyu can be determined by the Smoluchowski equation, E)µ/ζεε(u r0

ey = , where εr and µ

are the fluid’s dielectric constant and viscosity, respectively, ε0 is the permittivity of

vacuum, and ζ is the zeta potential. Or, it can be simplified as Eµu Eey = where Eµ is the

electric mobility.

Eq. (5 -4) is subjected to the boundary condition of no flux at the walls (y = 0,1),

i.e.,

0cuycD e

y =+∂∂

− (5 - 5)

In a reference moving in the axial direction with velocity *u , Eq. (5 -4) becomes

)yc

xc(D

ycu

xc)uu(

tc

2

2

2

2ey

*

∂∂

+∂∂

=∂∂

+∂∂

−+∂∂ (5 - 6)

where x is now the axial coordinate in the moving frame.

Below, we use the well established multiple time scale analysis [51] to study the

effect of cyclic lateral fields on Taylor dispersion. In the analysis, we postulate that the

concentration profile is of the form

)hy,

lx

lDt,t(C~C~ 2ω= (5 - 7)

where C~ is the dimensionless concentration, )π2/(ω is the frequency of the applied

field, 1/ω is the short time scale, and D/2l is the long time scale over which we wish to

observe the dispersion. Substituting Eq. (5 -7) into (5 -6) gives

90

2

2

2

22e

y*

sl

2

YC~

XC~

YC~PeU

XC~)UU(Pe

TC~

TC~

∂∂

+∂∂

ε=∂∂

+∂∂

−ε+∂∂

Ω+∂∂

ε (5 - 8)

where tωTs = , l/xX = , h/yY = , 2l /DtT l= ,

DhuPe ><

= , DhωΩ

2

= ,

><=

><=

><=

uuU,

uu

U,uuU

**

eye

y and 1lh

<<≡ε .

Since the aspect ratio ε<<1, the concentration profile can be expanded in the

following regular expansion.

∑∞

=

ε=0m

lsmm )T,Y,X,T(C~C~ (5 - 9)

Substituting Eq. (5 -9) into Eq. (5 -8) gives

)(YC~

YC~

YC~

XC~

YC~PeU

YC~PeU

YC~

PeUXC~)UU(Pe

XC~

)UU(PeTC~

TC~

TC~

TC~

322

22

21

2

20

2

20

22

2ey

21ey

0ey

1*2

0*

s

22

s

1

s

0

l

02

εθ+∂∂

ε+∂∂

ε+∂∂

+∂∂

ε=

∂∂

ε+∂∂

ε+∂∂

+∂∂

−ε+

∂∂

−ε+∂∂

εΩ+∂∂

εΩ+∂∂

Ω+∂∂

ε

(5 - 10)

Eq. (5 -10) can be separated into a series of equations for different order of ε.

( 0ε ):

20

20e

ys

0

YC~

YC~PeU

TC~

∂∂

=∂∂

+∂∂

Ω (5 - 11)

The solution for C~ 0 can be decomposed into a product of two functions, one of which

depends on Ts and Y and the other depends on X and Tl, i.e., )T,X(A)T,Y(GC~ ls00 = ,

where G0 satisfies

20

20e

ys

0

YG

YG

PeUTG

∂∂

=∂

∂+

∂∂

Ω (5 - 12)

91

( 1ε ):

21

21e

y0*

s

1

YC~

YC~PeU

XC~

)UU(PeTC~

∂∂

=∂∂

+∂∂

−+∂∂

Ω (5 - 13)

Integrating the above equation from 0 to 1 in Y and 0 to 2 π in Ts and noting that

0dT)T/C~(2

0ss1 =∂∂Ω∫

π

due to periodicity and ∫∫ ∂∂=∂∂1

0

21

21

01

ey dY)Y/C~(dY)Y/C~(PeU due

to the boundary conditions gives

∫ ∫

∫ ∫π

π

= 2

0

1

0ss0

2

0

1

0ss0

*

dYdT)T,Y(G

dYdT)Y(U)T,Y(GU (5 - 14)

The solution to C~ 1 is of the form )X/)T,X(A)(T,Y(B ls ∂∂ where B satisfies

2

2eys0

*

s YB

YBPeU)T,Y(G)UU(Pe

TB

∂∂

=∂∂

+−+∂∂

Ω (5 - 15)

( 2ε )

22

2

20

22e

ys

21*

l

0

YC~

XC~

YC~PeU

TC~

XC~)UU(Pe

TC~

∂∂

+∂∂

=∂∂

+∂∂

Ω+∂∂

−+∂∂

(5 - 16)

Averaging both sides in Ts and Y gives,

20

21

0

2

0s2

2*

l

0

XC~

dYdTX

AB)UU(PeTC~

∂><∂

=∂∂

−+∂

><∂∫ ∫

π

(5 - 17)

where

φ==>=< ∫ ∫∫ ∫ππ

AdYdTAgdYdTC~C~1

0

2

0s0

1

0

2

0s00 where ∫ ∫≡

1

0

π2

0s0 dYdTgφ (5 - 18)

Substituting A from Eq. (5 -18) into Eq. (5 -17) yields,

92

20

21

0

2

0s2

02

l

0

XC~

BdYdT)UU(Pe1XC~

TC~

∂><∂

=−φ∂

><∂+

∂><∂

∫ ∫π

(5 - 19)

Now we combine the results for )T/C~( s0 ∂∂ and )T/C~( l0 ∂∂ .

l

02

s

22

s

1

s

00

TC~

lD

TC~

TC~

TC~

tC~

∂∂

+∂∂

ωε+∂∂

ωε+∂∂

ω=∂

∂ (5 - 20)

Averaging the above equation in Y and using periodicity gives,

l

0

2

0

T

C~

lD

t

C~

∂

∂=

∂

∂ (5 - 21)

Now using Eq. (5 -19) in Eq. (5 -21) gives

2

02

*0

x

C~DD

t

C~

∂

∂=

∂

∂ (5 - 22)

where the dimensionless dispersion coefficient is given by

∫ ∫ −−=1

0

π2

0s

* BdYdT)UU(φPe1D (5 - 23)

Result and Discussion

Electrochemical Response

Lao et al. measured the current as a function of time in the EFFF device after a step

change in voltage and showed that the current-time relationship in their experiments did

not satisfy the single exponential predicted by the equivalent circuit representation. They

attributed the deviation from the single exponential to flow through the channel. Similar

deviations from the single exponential have also been observed by other researchers.

Such deviations are not unexpected because the equivalent circuit representation shown

in Figure 5-1 is only qualitatively correct. It assumes that the capacitance of the double

layer is constant, which is only accurate if all the ions adsorb on the inner Hehlmoltz

93

plane (IHP). For almost all cases, only a fraction of the ions adsorb on the IHP and the

remaining are present at the outer Hehlmoltz plane (OHP) or in the diffused double layer.

The complex structure of the double layer leads to a potential dependence

capacitance[55,56]. Additionally, the equivalent circuit in Figure 5-1 also neglects the

reactions at the electrode. Accordingly it is expected that the current-time behavior in

experiments will not exactly match the equations given above. However, the single and

double exponential equations serve as a useful guide to fit the experimental data to an

empirical form.

In order to test whether the deviations from the single exponential occur due to

flow, as reported by Lao [15] and Biernacki [54], or due to other processes mentioned

above, we measured the current-time relationship in a channel in the absence of flow. In

our study, the channel was comprised of two gold coated glass plates separated by a layer

of insulating spacer. The glass plates were soaked in acetone for one day and washed

with DI water before they were coated with a 500 nm thick gold layer by sputtering in a

Kurt J. Lesker CMS-18 system. Then, the glass plates were separated by a 500 – 1000

µm thick spacer and clamped. A potentiostat (PGSTAT30, Eco Chemie) was used to

apply either a step or a squarewave cyclic potential of fixed magnitude and measure the

time dependent current.

Current response for a step change in voltage

The current-time behavior after applying a step potential of 0.5 V in a 500 µm thick

channel containing DI water is shown as in Figure 5-2. The current dropped to about

4µA in 100 sec, suggesting that electrode reactions can be neglected. The current

response shown in Figure 5-2 cannot be described by a single exponential, and in fact a

94

double exponential of the form )/texp(c)/texp(cI 2211 τ−+τ−= fits the data very well

(dark line). In this experiment, the fluid in between the plates was stationary, and this

proves that the deviation from the single exponential occurs due to processes other then

flow. We speculate that the multiple time scales occur due to the dependence of the

double layer capacitance on the time dependent voltage drop across the double layer.

The decay time scale is the product of the bulk resistance and the double layer

capacitance, and thus the time dependence of the capacitance will lead to changing decay

time scales. Additionally, there may be an electrode reaction at short times which slows

down with time, and also contributes to the decay of the current.

For the data in Figure 5-2a, the best fit values of the parameters for the double

exponential form are c1 = 6.55x10-4 A, c2 = 1.908x10-4 A, τ1 = 0.197 s, τ2 = 1.575 s. It is

instructive to compare the time constants obtained by us with those reported by other

researchers. Since the time constants depend on the conductivity of the solution, here we

only compare values with those that were also measured in DI water. Additionally the

time constant is expected to scale linearly with channel thickness and this has to be

accounted in the comparison. Palkar et al. obtained a RC time constant of 40 seconds for

DI water in a 178 µm thick channel [38]. This value of time constant is about 25 times

the larger of the two time constants obtained above, and this is unexpected because the

channel in their study is thinner than the channel used in our study. The difference can

partially be attributed to the fact that Palkar et al. used the long time current-time data to

obtain the time constant. Lao et al. also fitted their data to a single exponential. The

channel in their study was 40µm thick and they obtained a value of 0.02 s for τ1. Since

our channel thickness is about 12.5 times of Lao et al., the RC time constant for our

95

device is expected to be about 0.25 s, which is reasonably close to our experimental result

of 0.197s for the shorter time constant. This comparison shows that if a single

exponential is used to fit the data, the value of time constant may differ significantly

depending on whether the short time or the long time data is used. A double exponential,

although an empirical expression, is thus more useful for fitting the current-time data. It

should also be pointed out that the time constants are also expected to depend on the type

of the electrodes and this may also explain the large differences between our results and

those of Palkar et al.

Figure 5-2. Transient current profiles after application of step change in voltage in a 500 µm thick channel for (a) DI water (V = 0.5 volt) and (b) 50 mM NaCl (V = 1 volt)

96

As shown above, the time response of the EFFF device can be represented in terms

of the four parameters: c1, c2, τ1, τ2. Below we investigate the dependence of these

parameters on the applied voltage, channel thickness and the salt concentration. In the

results shown below the error bars represent the standard deviation of 15 experiments (3

different sets of channels, and 5 experiments for each channel).

In many separation systems, ions are added to stabilize the particles and/or to

control the pH of the solutions. Addition of ions alters the electrochemical properties of

the EFFF device by changing the conductivity, electrode kinetics and the capacitance of

the double layer. Below we report the effect of salt addition on the parameters c1, c2, τ1

and τ2. In the experiments described below, we measured the current response in DI

water, 10mM NaCl and 50 mM NaCl solutions for a range of applied voltage V. In these

experiments, we did not observe bubble formation even when the voltage was applied for

very long time, which suggests that there are no electrode reactions involved in these

experiments, except perhaps at short times.

Figure 5-2b shows the current response after a step change in voltage for a 500 µm

thick channel containing 50 mM salt solution. The data shows that the magnitude of

current immediately after the step change is significantly larger than that for the case of

DI water in Figure 5-2a. The figure also shows that the current initially decays very

rapidly on the time scale of 0.005 s, which is much faster than the time scales for DI

water. Both of these observations are expected because the resistance of the salt solution

is significantly less than that of DI water, and therefore initial current which scales as 1/R

is larger and the decay constant which scales as RC is smaller. However, after about 0.01

s, the rate of decay slows down significantly, and the time scales for decay in the

97

remaining time are comparable to those in DI water. The initial current decays so rapidly

that it is not expected to play an important role in the separation. Thus, we neglect this

initial decay, and fit the remaining data to a double exponential of the same form as used

for fitting the data for DI water. The best fit double exponential curve is shown by the

dash line. Below we compare the fitting parameters for the salt solutions with that for DI

water.

Dependence on applied voltage (V) and salt concentration

Based on the equivalent electric circuit model, we can anticipate that for a fixed

channel thickness, the applied voltage will linearly change the magnitude of the current,

and accordingly both c1 and c2 are expected to linearly increase with the voltage. The

results shown in Figure 5-3 a-d demonstrate that as expected both c1 and c2 linearly

increase with V and the value of τ1 and τ2 are relatively constant.

Figure 5-3. Dependence of the electrochemical parameters on salt concentration and applied voltage in a 500 µm thick channel

98

An increase in salt concentration leads to a reduction in the resistance and thus the

slopes of the c1 vs. V and c2 vs. V plots are expected to increase with an increase in salt

concentration. The results shown in Figure 5-3a and 5-3b show that the effect of salt

addition on c1 and c2 is as expected. The values of τ1 and τ2 are relatively unaffected by

addition of salt (Figure 5-3c and 5-3d), which is a surprising result, and can perhaps be

attributed to the fact that increasing salt concentration leads to a smaller resistance and a

higher capacitance, and thus the time constants are relatively unchanged.

Figure 5-4. Dependence of the electrochemical parameters on channel thickness for V = 0.5 V and DI water

99

Dependence on channel thickness (h)

The resistance of the circuit increases with an increase in h and thus both c1 and c2

are expected to decrease. Since the capacitances are not expected to change with increase

in h, both τ1 and τ2 are expected to increase with increasing h. The results in Figure 5-4

a-d show that c1 decreases with increasing h while c2 is relatively constant, and that τ1

and τ2 both increase with h. However, the increase in τ2 is leveling off at large h values.

To understand the exact dependency of c1, τ1, c2 and τ2 on various parameters, one needs

to solve the detailed electrochemical problem that includes solving the Poisson-

Boltzmann equation along with the species conservation, and then coupling it to the

electrode kinetics at the surface. However, for the current paper it suffices to know the

dependence of the four parameters on various system variables so that the current

response can be determined for any set of parameters.

Current response for a cyclic change in potential

The data shown in Figure 5-2 was obtained by applying a step change in voltage.

However, the applied voltage in CEFFF is a periodic square or sinusoidal waveform.

Based on the equivalent circuit shown in Figure 5-1, application of square shaped

periodic waveform leads to a current flow given by Eq. (5 -3). Since the experimental

results for a single step change in voltage fit a double exponential rather than a single

exponential, it may be expected that on applying the periodic square shaped voltage, the

current expression will be given by

( ) ( )

( ))/texp(C)/texp(C))/texp(C)/texp(C1(

I2

)/texp(c)/texp(c))/texp(c)/texp(ccc(

cc2i

22112c21c1

max

22112c21c121

21

τ−+τ−τ−+τ−+

±≡

τ−+τ−τ−+τ−++

+±=

(5 - 24)

100

where s

21max RVccI =+≡ ,

21

11 cc

cC+

≡ , 21

22 cc

cC+

≡ , tc is the half period of the cycle,

i.e., the time in between two successive step changes in the voltage, and t is the cycle

time since the last step change in voltage. Figure 5-5 shows the comparison between the

response predicted above for DI water (Figure 5-5a), 10 mM NaCl (Figure 5-5b) and 50

mM NaCl (Figure 5-5c) and the experimental results in for tc = 0.3 s, channel thickness of

500 µm. There is a reasonable agreement between Eq. (5 -24) and the current transients

in DI water (Fig 5-5a). The agreement is also reasonable for the salt solutions except at

very short times after the change in electric field. This occurs because as mentioned

above the very rapid decay that occurs at time scales of 0.05 s is neglected while fitting

the experimental data. This rapid decay is neglected because it is not expected to make

any contribution to the separation in CEFFF because it is much faster than the diffusive

time scales.

The expression for i given by Eq. (5 -24) can also be expressed as

( ))/texp(C)/texp(Cii 22110 τ−+τ−= (5 - 25)

where

))/texp(C)/texp(C1(RV2i

2c21c1s0 τ−+τ−+

±= (5 - 26)

The above equation predicts that when the frequency is high, i.e., tc is small, the

maximum current is V/Rs and when tc becomes large, the magnitude of the current is

2V/Rs. The experimental data for dependency of i0/(V/Rs) on tc is plotted as the stars in

Figure 5-6a-c for DI water, 10 mM NaCl, and 50 mM NaCl, respectively in a 500 µm

thick channel. The amplitude of the voltage is 1 V in these experiments. The solid lines

101

in these figures correspond to the prediction given by Eq. (5 -26). The figures show that

there is a reasonable agreement between the prediction and the experimental data.

Figure 5-5. Comparison between the experiments (thin lines) and Eq. (5 -24) (thick lines) for current transients on application of a square wave potential of 1V magnitude and time period (tc) 0.3 s. The results in 5a, 5b and 5c are for DI water, 10 mM NaCl, and 50 mM NaCl in a 500 µm thick channel.

102

Figure 5-6. Comparison between the experiments (stars) and Eq. (5 -26) (solid lines) for current transients on application of a square wave potential of 1V magnitude. The results in 6a, 6b and 6c are for DI water, 10 mM NaCl, and 50 mM NaCl in a 500 µm thick channel.

103

Based on the above expression for the current, the field in the bulk of the channel is

given by

( )))/texp(C)/texp(C1(

)/texp(C)/texp(ChV2iRE

2c21c1

2211s τ−+τ−+

τ−+τ−±== (5 - 27)

This electric field drives a lateral velocity of charged particles )u( ey , and the

dimensionless lateral velocity is given by

( ))t(Rf

))/texp(C)/texp(C1()/texp(C)/texp(C

huV2

uE

uu

U2c21c1

2211EEeye

y ≡τ−+τ−+

τ−+τ−><

µ±=

><µ

=><

= (5 - 28)

where Eµ is the electrophoretic mobility of the particles, hu

V2R E

><µ

≡ is the maximum

value of eyU and f(t) characterizes the time dependence of the electric field. By using the

above equation, we can determine the lateral velocity of any type of particles in our EFFF

device. We can also fit the current-time data obtained by other researchers, and then

determine the lateral velocity of particles in EFFF devices.

CEFFF is a useful device for separation partly because there are a number of design

variables such as channel geometry and electrode design and operational variables such

as carrier fluid composition, , V, tc that can be tuned to optimum separation.

However, presence of so many variables also makes it difficult for an experimentalist to

choose the optimal variables. This task can be considerably simplified by using a model,

and below we develop such a model that can help in identifying the key parameters and

the effect of these parameters on separation.

104

Separation

Modeling of separation of particles by CEFFF

The response of the EFFF device can be characterized by seven dimensionless

parameters: C1, C2, 1ωτ , 2ωτ , Pe, R, and Ω . The results reported below were computed

for fixed values of C1, C2, τ1 and τ2 that were obtained in a 500 µm wide channel using

DI water as the carrier fluid and using a voltage of 0.5 V. We have also implicitly

assumed that C1, C2, τ1 and τ2 are independent of V, which is a reasonable assumption

based on the data.

In the previous section we have developed the equations to determine the mean

velocity and the dispersion coefficient of particles in a CEFFF device. To determine the

mean velocity and the dispersion coefficient of particles in CEFFF, we substitute eyU

from Eq. (5 -28) into Eq. (5 -12) and (5 -15) to get G0 and B numerically and we can then

determine the mean velocity and effective diffusivity by Eq. (5 -14) and (5 -23). Below

we first discuss the results for the mean velocity, followed by results for the dispersion

coefficient, and then we combine these to evaluate the separation efficiency of CEFFF.

Mean velocity of particles

Before discussing the results for the mean velocity, we note that many authors

describe separation in terms of retention ratio. The retention ratio RR is defined as the

ratio of the time for uncharged particles to pass the channel to the time for charged

particles to pass, i.e.,

**

r

0R U

uu

tt

R =><

== (5 - 29)

105

Therefore, the retention rate is equivalent to the dimensionless mean velocity of charged

particles.

Figure 5-7. Dependency of the mean velocity on PeR and Ω . The electrochemical parameters are fixed at values that correspond to DI water in a 500 µm

channel. (C1=0.7744, C2=0.2256, 54.2Dh ,3.20

Dh

2

2

1

2

=τ

=τ

)

In Figure 5-7, the mean velocity is plotted as a function of PeR which is the product

of the amplitude of the dimensionless lateral velocity eyU and the Peclet number Pe, for a

range of Dh 2ω

≡Ω , where ω is the frequency of the oscillations. The x axis for this plot

can simply be interpreted as a measure of the applied voltage. It is noted that the x axis

PeR is similar to the parameter 1/λ used by Biernacki et al. while plotting the results for

the retention ratio, which as explained above is identical to the mean velocity. The trends

shown in Figure 5-7 are also similar to those shown by Biernacki et al., except that we

have explored the entire frequency range while Biernacki et al. only focused on the

106

frequencies small enough so that the current essentially decays to zero at the end of each

half cycle [54]. The results in Figure 5-7 show that for a fixed Ω, the mean velocity first

increases with PeR, reaches a maximum and then begins to decrease. The mean velocity

is 1 for PeR = 0 because R = 0 implies absence of any field, and thus the dimensional

mean velocity is simply equal to the flow velocity. As PeR begins to increase, the mean

velocity becomes larger because at these PeR values, the charged particles are oscillating

between the two walls, and the time in each half period is not enough for particles to

travel from one wall to the other. Thus, a large number of particles spend a majority of

the time near the center of the channel resulting in a mean velocity larger than 1. As PeR

increases further, every particle is able to reach the wall during each half period, and thus

most particles accumulate near the wall leading to a reduction in the mean velocity. For

a smaller frequency, the particles can reach the walls at a smaller PeR values because a

longer time is available during the period, and thus the PeR at which the mean velocity is

a maximum moves to smaller values as Ω becomes small.

The dependence of mean velocity on Ω for a fixed PeR is also interesting. At very

small frequencies, the time period of the cycle is much longer than both τ1 and τ2, and

thus the field is zero for a majority of the period, and accordingly the mean velocity is

about 1. As the frequency increases, the field is non-zero during the period leading to a

reduction in the mean velocity. However at very high frequencies, the distance traveled

by the particles during a period becomes small, and thus the concentration profile

becomes uniform and the mean velocity approaches 1.

107

Effective diffusivity of particles

The dispersion coefficient D* is of the form 1 + Pe2f(PeR, Ω, C1, C2, ωτ1, ωτ2).

The first term is the contribution from axial diffusion and the second term represents the

convective contribution. In Figure 5-8, we plot ( ) 2* Pe/1D210 − as a function of PeR and

Ω. The results in Figure 5-8 show that for a fixed Ω, as PeR increases, the dispersion

coefficient initially increases and then decreases. 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 5-8 shows that as PeR

approaches zero, the curves of 2* Pe/)1D(210 − for all values of frequency approach the

expected limit of 1. 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 5-8 also shows that for small Ω , the curves exhibit a maximum at a

value of PeR that becomes small with increasing frequency. This occurs because 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 a pulse. Finally, as PeR approaches infinity, the mean velocity

108

approaches zero, and the dispersion coefficient approaches the molecular diffusivity.

Since the behavior of the dispersion coefficient with an increase in PeR is different in the

small and the large PeR regime, it must have a maximum.

Figure 5-8. Dependence of 210(D*-1)/Pe2 on PeR and Ω. The electrochemical parameters are fixed at values that correspond to DI water in a 500 µm

channel. (C1=0.7744, C2=0.2256, 54.2Dh ,3.20

Dh

2

2

1

2

=τ

=τ

).

For a given PeR, as frequency goes to zero, the field is zero for a majority of the

time and so dispersion coefficient approaches the limit for Poiseuille flow. As frequency

increases, the presence of field causes some particles to accumulate near the walls, while

a number of particles still travel with the mean velocity, and this leads to an increase in

dispersion. As frequency continues to increase, a majority of the particles are present

near the center of the channel and thus the dispersion coefficient begins to decrease.

Finally, at very high frequencies the particles move a small distance in the period, and

109

thus the concentration profile is uniform and the dispersion is coefficient again

approaches the Poiseuille limit.

Figure 5-9. Dependence of separation efficiency on PeR1 and Ω1 for the case of D1/D2=3 and µE2/µE1=3, and thus (PeR)2/(PeR)1 = 9 and Ω2/Ω1=3. The electrochemical parameters are fixed at values that correspond to DI water in a 500 µm

channel. (C1=0.7744, C2=0.2256, 54.2Dh ,3.20

Dh

2

2

1

2

=τ

=τ

).

Figure 5-10. Origin of the singularity in separation efficiency at critical PeR1 and Ω1 values for Ω1 = 40 π. (PeR)2/(PeR)1 = 9, Ω2/Ω1=3, and the electrochemical parameters are fixed at values that correspond to DI water in a 500 µm

channel. (C1=0.7744, C2=0.2256, 54.2Dh ,3.20

Dh

2

2

1

2

=τ

=τ

).

110

Separation efficiency

Consider separation of colloidal particles of two different sizes in a channel. As the

particles flow through the channel, they separate into two Gaussian distributions. The

axial location of the peak of the particles at time t is tu * and the width of the Gaussian is

tDD4 * . We consider the particles to be separated when the distance between the two

pulse centers becomes larger than 3 times of the sum of their half widths, i.e.,

)tDD4tDD4(3t)uu( *22

*11

*1

*2 +≥− (5 - 30)

where the subscripts indicate the two kinds of particles. 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., )u,umax(/L *2

*1 . Substituting for t, and expressing all the

variables in dimensionless form gives

2*1

*2

1

2*2

*1

*2

*1

1

]UU

DD

DD)[U,Umax(

Pe112h/L

−

+≥ (5 - 31)

Below we investigate the separation of two particles that have different mobilities

(µE1/ µE2 = 1/3), and also different diffusivities (D1/D2 = 3), and thus different PeR values

((PeR)2/(PeR)1 = 9).

The ratio L/h required for separation depends on (PeR)1 and also the Ω1 values.

The values of L/h are plotted as a function of (PeR)1 in Figure 5-9 for Ω1 ranging from

0.16 π to 40 π. It is noted that (PeR)2/(PeR)1 = 9 and Ω2/Ω1=3. From Figure 5-9, it can

be seen that choosing the appropriate parameters is extremely important to obtain good

separation. At small frequencies such as at Ω1 = 0.16 π, the separation is inefficient at all

PeR because the mean velocities of both particles are close to 1. The separation improves

111

on increasing Ω1 to 0.4 π because the mean velocities of both the particles decrease, and

the difference between the mean velocities increases. On further increasing the

frequency, the separation improves for certain PeR values, but near a critical PeR1 such

as PeR1=330 for Ω = 40 π, the separation becomes highly inefficient. This occurs

because at these PeR1 values the mean velocities of both the particles are similar, as

shown in Figure 5-10. The optimal (PeR)1 for separation at this frequency is about 120.

Interestingly, for (PeR)1 less than about 330, the smaller particles will elude out of the

channel first but the order of elusion will be reversed for (PeR)1 > 330.

Comparison with Experiments

There is only a small amount of experimental data in literature for separation by

cyclic electric field flow fractionation. As mentioned earlier, Lao et al. and Gale et al.

have investigated separation of charged nanoparticles by CEFFF. The time constant for

decay of electric field was large for the EFFF device used by Gale et al. and thus the

electric field can be treated as constant in their experiments. Such systems in which the

field can be a non-decaying square wave have been investigated by a number of

researchers and asymptotic expressions have been developed for large frequency cases

[50,57]. Gale et al. successfully compared the results of their study with the theoretical

models. The current in the EFFF device fabricated by Lao et al. decayed on a time scale

of about 0.02 s, and thus one needs to include the decaying electric field in the analysis as

is done in this paper. Accordingly we focus this section on comparing the results of our

model with the experiments of Lao et al.

In their experiments, Lao et al. used two kinds of latex carboxylated surface-

modified polystyrene particles with 0.45 and 0.105 µm diameters. They performed the

112

studies with a flow rate of 17µL/min in a 40 µm thick, 1 cm wide and 9 cm long channel

bounded with indium tin oxide(ITO) electrodes that were connected to a potentiostat to

apply the lateral electric field. Lao et al. measured the current response for a step change

in voltage (Figure 6a in Ref 17), and we fitted that data to a double exponential form to

obtain the following constants: C1 = 0.991, C2 = 0.009, τ1 = 0.02 s, and τ2 = 2 s,

respectively. Lao et al. applied square shaped symmetric and asymmetric electric fields.

In the asymmetric fields, in each cycle they applied a +V voltage for dimensionless time

Tpos followed by –V voltage for dimensionless time Tneg. They defined duty cycle Dt as

the ratio Tpos/(Tpos+Tneg)) and it was varied from 0.5 to 0.9. In all the experiments they

pre-equilibrated the sample by first applying the electric field for 20 minutes without

flow and then they started the flow while continuing to apply the electric field. Lao et al.

investigated the effect of frequency, size and mobility on the residence time and

separation. Below each of their experiments are compared with the predictions of the

model developed in this paper.

In order to compare the model predictions with the experiments, one needs the

mobility and the diffusivity of the particles in addition to the parameters listed above.

Since this data were not provided by Lao et al., we measured the electric mobilities with

Brookhaven Zetaplus and obtained values of (1.84 ± 0.11)x10-8 m2s-1V-1 for the 0.105

µm size particles and (2.7 ±0.17)x10-8 m2s-1V-1 for the 0.45 µm size particles. Next, we

used the Stokes-Einstein equation to obtain the molecular diffusivity, i.e.,

aπµ6kTD = (5 - 32)

113

where µ is the viscosity of the fluid and a is the radius of the particle. By this equation,

we get the molecular diffusivity to be 4x10-12 m2/s for the 0.105µm particle and 9x10-13

m2/s for the 0.45µm particle in water.

Based on the parameters listed above, as the applied voltage is 1.75 V and

frequency is 2.2Hz, the Pe numbers are 7087 and 31500, the values of R are 1.173 and

1.994, and the Ω values are 5230 and 24600, for the smaller and the larger particles,

respectively. The values of PeR and Ω are larger than a few thousands in each of the

experiments, and thus rather than using the exact model proposed earlier in this paper, it

is preferable to use the large PeR and Ω asymptotic solutions that are easier to obtain.

Below we first obtain the asymptotic results and then compare the results with the

experiments.

Large Ω asymptotic results

Eq. (5 -11) can be written as

20

20

s

0

YC~

PeR1

YC~)t(f

TC~

PeR ∂∂

=∂∂

+∂∂Ω

(5 - 33)

In the limiting case considered here, both PeR and Ω are large and the ratio is O(1), and

thus in this limit, to leading order in 1/Ω, the above equation becomes

0YC~)t(f

TC~

PeR0

s

0 =∂∂

+∂∂Ω

(5 - 34)

Essentially the above equation implies that as the frequency becomes large, the particles

simply convect in the lateral direction with the time dependent lateral velocity, and lateral

diffusion can be neglected. In this limiting case of large Ω, the behavior of the system

depends strongly on the total distance traveled by the particles in the positive direction

114

during the positive phase of the cycle (V>0) and in the negative direction in the negative

phase. For the case of duty cycle = 0.5, both of these distances are equal. Below we

consider four different cases in the large Ω regime.

(1) First we consider the case when the distance traveled by the particles is larger

than the channel thickness for both the positive and the negative phases of the cycle. In

this case, since the distance traveled by each of the particle in the positive phase of the

cycle is larger than the channel height, all the particles are at the Y = 1 wall at the end of

the positive phase. As the field reverses the particles begin to travel in the negative

direction as a pulse with negligible diffusion, and since the distance traveled in the

negative phase of the cycle is also larger than the channel height, the particles end up

accumulating at the Y = 0 wall at the end of the negative phase. Thus the concentration

profile during a period can be expressed as

⎪⎪⎩

⎪⎪⎨

⎧

<<+×δ+×<<××−∆−−δ

×<<−δ<<∆−δ

=

cttc

ttctctc

tct

t

0

TTTDT ),Y(TDTTDT ))),DTT(Y1(Y(

DTTT ),1Y(TT0 )),T(YY(

)T(G (5 - 35)

where δ(Y) denotes the dirac delta function, T denotes the dimensionless time since the

beginning of the cycle, Tc is the dimensionless cycle time, which due to the choice of

dedimensionalization is equal to 2π, Dt is the duty cycle, and therefore tc DT × is equal to

Tpos, and ∆Y(T) is the distance traveled by the particles in the positive direction in

dimensionless time T, and similarly ∆Y(T-Tpos) is the distance traveled by the particle in

the negative direction in time T-Tpos, and finally Tt is the time at which the particles reach

the Y = 1 wall. Based on the expression for the lateral velocity, ∆Y(T) is given by

115

)]exp(-C)exp(-C[1

)]Texp(1(1CPeR)Texp(1(1CPeR[2)T(Y

22

11

222

111

ωτπ

+ωτπ

+

ωτ−−

ωτΩ+

ωτ−−

ωτΩ=∆ (5 - 36)

We note that the above results are also valid for the case when the duty cycle is equal to

0.5.

(2) Now we consider the case when the distance traveled in one of the phases

(positive or negative) is larger than the thickness but the distance traveled in the other

phase is smaller than the height. We note that the EFFF device is symmetric in Dt around

0.5, and in our computations we assume that Dt < 0.5. Thus in the negative phase the

particles travel a distance larger than the thickness, and accordingly the concentration

profile at the end of the negative phase is a pulse at Y = 0 wall. During the positive

phase, the pulse moves towards the Y = 1 wall but it does not reach the wall in the

positive phase. As the field direction changes, the pulse begins to travel back towards the

Y = 0 wall. Thus, the concentration profile during a period can be expressed as

⎪⎩

⎪⎨

⎧

<<δ

<<−∆−∆−δ

<<∆−δ

=

cpos

pospospospos

pos

0

TTT2 ),Y(

T2TT ))),TT(Y)T(Y(Y(

TT0 )),T(YY(

)T,Y(G (5 - 37)

(3) We now consider the case when the distance traveled in both the positive and

the negative phases is less than the channel height and the duty cycle is not equal to 0.5.

To understand the physical situation that corresponds to this case, we assume that the

concentration is uniform in the lateral direction at T = 0, and then the electric field is

applied. Let us define the distances that the particles can travel in the positive and

negative phases as ∆Ypos and ∆Yneg, respectively. As the duty cycle is not 0.5, ∆Ypos is

different from ∆Yneg. Without a loss of generality, we assume that Dt < 0.5, i.e., ∆Yneg >

116

∆Ypos. As the field is applied in the negative direction, the particles begin to move

towards the Y = 0 plate, and at the end of the negative phase, the particles that were

located in the region Y < ∆Yneg will accumulate at the Y = 0 wall and there will be no

particles in the region between Y = 1 and Y = 1 - ∆Yneg. In the remaining region,

particles will be uniformly distributed. In the positive phase of the cycle, all the particles

will travel a distance ∆Ypos towards the Y = 1 wall, and at the end of this phase, there will

be no particles in the region between Y = 0 and ∆Ypos, and also between Y = 1 and 1-

∆Yneg+∆Ypos. The pulse of particles that accumulated at the Y = 0 wall in the positive

phase will be located at Y = ∆Ypos, and particles will be uniformly distributed in the

remaining region. Since in each cycle there is a net motion towards the Y = 0 wall, after

1/ (∆Yneg- ∆Ypos) cycles, all the particles will be located in a pulse at the Y = 0 wall.

Subsequently, the concentration profiles will represent the periodic steady state solutions,

in which a traveling pulse moves a distance ∆Ypos towards the Y = 1 wall during the

positive phase, and then moves back towards the Y = 0 wall during the negative phase.

The concentration profiles after attainment of the periodic steady state are

⎪⎪⎩

⎪⎪⎨

⎧

<<δ

<<−∆−∆−δ

<<∆−δ

=

cpos

pospospospos

pos

0

TTT2),Y(T2TT)),TT(Y)T(Y(Y(

TT0)),T(YY(

)T,Y(G (5 - 38)

It is noted that the periodic steady concentration profiles are the same for both case (1)

and case (2).

(4) Now we consider the final case in which both ∆Ypos and ∆Yneg are less than 1

but the duty cycle equals 0.5, and thus ∆Ypos = ∆Yneg = ∆Y. In this case, let us assume

that the particles are uniformly distributed at T = 0. As the positive field is applied, the

117

particles begin to move in the positive direction, and at the end of the positive phase, the

particles that were located in the region Y > 1-∆Y will accumulate at the Y = 1 wall, and

there will be no particles in the region between the Y = 0 and ∆Y. In the negative phase

of the cycle, all the particles will travel a distance ∆Y towards Y = 0 wall and at the end

of this phase, there will be no particles in the region between Y = 1 and 1-∆Y, and the

pulse of particles that accumulated at the Y = 1 wall in the positive phase will be located

at Y = 1 - ∆Y, and particles will be uniformly distributed in the remaining region. This is

the periodic steady solution that will move equal distances in both the positive and the

negative phases but in opposite directions. Thus the concentration profile for this case is

given by

⎪⎪⎩

⎪⎪⎨

⎧

<<+∆+∆=+∆−δ

+∆<<∆∆<<

=

1YWp)T(Y,0Wp)T(YY)),Wp)T(Y(Y(

Wp)T(YY)T(Y,1)T(YY0,0

)T,Y(G 0 (5 - 39)

where Wp is the width of the block area and it is given by

Y1Wp ∆−= (5 - 40)

In this case, the concentration profile is a sum of a moving pulse and a moving uniform

concentration block, and these two distributions travel with individual mean velocities,

and can be detected as two separate peaks at the channel outlet. This phenomenon was

observed by Bruce Gale in their experiments [58].

If the diffusion is negligible the concentration profile predicted above will be

maintained for a long time. However, due to diffusion some of the particles from the

pulse distribution diffuse into the block. Given sufficient time, the pulse disappears and

118

the periodic steady concentration profile is simply a uniform distribution that moves

equal distances back and forth. The concentration profile for this case is given by

⎪⎩

⎪⎨

⎧

<<+∆+∆<<∆

∆<<=

1YWp)T(Y,0Wp)T(YY)T(Y,1

)T(YY0,0)T(C (5 - 41)

It is noted that attaining this periodic steady state requires a dimensional time of

D/h 2 , which implies a dimensionless time of Ω. In the experiments conducted by Lao

et al. sufficient time was provided for equilibration before starting the flow, so the dual

peaks observed by Gale et al. were not observed in their experiments even for the case of

duty cycle = 0.5 and ∆Y < 1.

Now that the periodic steady concentration profiles are determined, the mean

velocity can be computed by using Eq. (5 -14).

The effect of changes in Ω

Lao et al. focused on measuring the effects of changes in the frequency on the mean

velocity. To further understand the physics of the various cases considered above and for

comparison with experiments of Lao et al., it is instructive to analyze the effect of the

frequency on the mean velocity of particles. In Figure 5-11a and 5-11b, we show results

of the dependency of the mean velocity on frequency in the high frequency regime for

cases of Duty cycle different from 0.5 (Figure 5-11 a) and Duty cycle = 0.5 (Figure 5-11

b).

In Figure 5-11a, the two dashed vertical lines divide the Ω domain into three

regions. The first and the second dashed lines indicate the Ω values at which ∆Ypos = 1

and ∆Yneg = 1, respectively. As above we consider the case of Dt < 0.5, while noting that

the response is symmetric in Dt around 0.5. According to Eq. (5 -36), when Ω is less

119

than the value where ∆Ypos = 1, both ∆Yneg and ∆Ypos are larger than 1 and this domain

corresponds to the case (1) that we discussed above. Similarly, the region in between the

two dashed lines corresponds to case (2), where ∆Yneg>1 and ∆Ypos<1. And the remaining

area corresponds to case (3) in which both ∆Yneg and ∆Ypos are less than 1. In Figure 5-

11b, since the duty cycle equals 0.5, ∆Yneg = ∆Ypos, and thus the two lines shown in

Figure 5-11a overlap with each other, and the region to the left of the vertical line

corresponds to case (1), and the remaining area corresponds to case (4). In both Figure 5-

11a and 5-11b, the dash-dot line near Ω = 0 indicates the values of Ω below which the

large Ω expressions derived above cannot be used to calculate the mean velocity, and the

general analysis shown earlier in the paper needs to be used to calculate the mean

velocity. It is important to note that the positions of the lines indicating ∆Yneg = 1 and

∆Ypos = 1 are dependent on the PeR value. As PeR decreases, the time needed for ∆Yneg =

1 or ∆Ypos = 1 increases, and thus all the lines shift to the left, i.e., to small Ω values.

Furthermore, at a critical value of PeR, the line for ∆Ypos = 1 will disappear from the

figure because the distance traveled by the particles in the negative half period can never

reach the channel thickness. Similarly, the line for ∆Yneg = 1 could also disappear from

the figure at very small PeR.

In region 1, the frequency is small and thus the time period is larger than the time

constants of the equivalent RC circuit. Accordingly the electric field and the lateral

velocity decay to zero during the period. Also since both ∆Yneg and ∆Ypos are larger than

1, the particles are positioned at the wall during the period when the field is zero, which

we refer as the ‘resting’ period. The velocity of the particles in the resting period is close

120

to zero, and thus an increase in the duration of the ‘resting’ period, which occurs with a

reduction in Ω, leads to a decrease in the mean velocity.

Figure 5-11. Dependence of the mean velocity on Ω in the large Ω regime. The values of other parameters are fixed at Pe=31500, R=1.7319, C1=0.991, C2=0.009,

889Dh ,88889

Dh

2

2

1

2

=τ

=τ

, and (a) Dt = 0.2 (b) Dt = 0.5.

It is noted that as Ω becomes very small, the resting period is comparable to the

time required for lateral equilibration of concentration profile due to diffusion. Diffusion

is neglected while analyzing the large Ω asymptotes, and thus one has to use the more

121

general analysis valid for all Pe and Ω that was developed earlier in the paper to predict

the mean velocity in this regime. The mean velocity goes to 1 as the frequency

approaches zero because the fractional time during which the field is operating

approaches zero as Ω approaches zero. However, this regime cannot be analyzed by the

large Ω asymptotes, and thus the mean velocity does not seem to be approaching 1 for

small frequencies in Figure 5-11.

The Ω regime at frequencies larger than the value at which ∆Ypos = 1 corresponds

to case (2). In this regime, increasing Ω leads to a reduction in the distance traveled by

particles in the positive half period. As particles do not reach the Y = 1 wall in the

positive half period, particles spend more time near the center of the channel, and thus the

mean velocity keep increasing until it reach a maximum near the frequency at which

∆Ypos is close to 0.5. After that, the velocity will decrease because the traveling distance

is less than 0.5, and particles do not spend time near the center where the axial velocity is

the highest. Interestingly, this maximum might not be in case (2) domain because in

some situations, ∆Yneg becomes less than 1 before ∆Ypos reaches about 0.5. In that

condition, the maximum will appear in case (3) area. Case (3) is essentially an extension

of case (2). In case (3), both ∆Yneg and ∆Ypos keep decreasing with increasing Ω.

Although ∆Yneg <1 in case (3), particles still accumulate at the wall of Y = 0 at the end of

the negative half period because ∆Yneg >∆Ypos. And in this regime, if Ω is so large that

∆Ypos is close to 0, the particles spend almost all time near the wall of Y=0 and the

velocity goes to 0.

The mean velocity profiles in region 1 in Figure 5-11b are similar to the region 1 in

Figure 5-11a. However in region 4, the behavior is significantly different. As discussed

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above, in region 4, there may be two different pulses detected at the channel exit, and the

mean velocities of these two pulses are indicated by two separate profiles in region 4

Figure 5-11b. If sufficient equilibration time is provided, the equilibrium profile

resembles a block that moves back and forth as the direction of field reverses. The

concentration within the block is constant and is zero everywhere else. As Ω increases in

case (4), ∆Yneg = ∆Ypos decreases, thus the width of the block area increases as the width

of the block is 1- ∆Yneg. As Ω goes to infinity, the concentration profile becomes

uniform, and the mean velocity approaches 1. If sufficient equilibration time is not

provided, then in addition to the block profile, there is a pulse of solute that is located at

the one of the edges of the block. This pulse moves a distance ∆Yneg = ∆Ypos away from

the wall in each cycle and then moves back as the direction of field changes. As Ω

increases, ∆Yneg = ∆Ypos decreases and thus the distance traveled by the pulse decreases,

and accordingly the particles in this pulse sample streamlines near the wall that have a

small velocity, and thus the velocity of the pulse decreases. Eventually as Ω goes to

infinity, the pulse is always located at the wall, and thus the mean velocity associated

with this pulse becomes zero.

Based on this model, we calculated the mean velocity under the conditions in Lao’s

experiments. Table 5-1 shows the comparison of the experimental data with our

simulation results. The comparison between the experiments and the model predictions

is good except at 10 Hz frequency and 0.5 duty cycle. In view of the fact that the

agreement is good for all other conditions including the case of 10 Hz frequency and 0.9

duty cycle, the significant difference between the predictions and the experiments for the

specific case of 10 Hz and 0.5 duty cycle is surprising.

123

Table 5-1 Comparison of the model predictions with experiments of Lao et al. Frequency

(Hz) Duty cycle

Particle size (µm)

Voltage (V)

Measured Dimensionless Velocity

Model Prediction

Figure 7 b-I 5 0.5 0.45 1.75 0.343 0.289 Figure 7 b-II 10 0.5 0.45 1.75 0.254 0.572 Figure 7 b-III 15 0.5 0.45 1.75 ~1 1.23 Figure 7 b-IV 20 0.5 0.45 1.75 ~1 1.23 Figure 8 (d) 9 0.9 0.45 1.52 0.254 0.248 Figure 9 (c) 2.2 0.8 0.45 1.4 0.212 0.219 Figure 9 (d) 2.2 0.8 0.105 1.4 0.508 0.389

The Figure numbers listed in the first column correspond to the figures in Ref. 17.

The model predictions are based on the large frequency asymptotes, where the

experimental conditions for Figure 7 b-I, 7 b-II and 9(c) are in case (1); those for Figure

8(d) are in case (2); those for Figure 9(d) are in case (3); and finally those for Figure 7 b-

III and 7 b-VI are in case (4).

Conclusions

Techniques based on lateral electric fields can be effective in separating colloidal

particles in microfluidic devices. However, application of such fields results in a very

large potential drops across the double layer and consequently large fields have to applied

for separation. These large fields could result in bubble generation, which can destroy

the separation. It has been proposed that periodic fields can be used effectively in such

cases because if the frequency of the periodic fields is faster than the RC time constant

for the equivalent electrical circuit then the majority of the potential drop will occur

across the bulk and thus smaller fields will have to be applied and this will reduce the

extend of Faradaic reaction at the electrode.

In this paper we investigate the separation of charged particles by cyclic EFFF by

measuring the electrochemical response of the CRFFF device, developing an equivalent

124

circuit and then using the parameters of the equivalent circuit into a continuum model to

determine the mean velocity and the dispersion coefficient for charged particles. The

continuum model is solved by using the method of multiple time scales and regular

expansions. Also analytical expressions are determined for the large frequency limit.

The results for the mean velocity and the dispersion coefficient are utilized to predict the

separation efficiency of the CEFFF device. Also the theoretical predictions are compared

with experimental data available in literature.

The experimental results for current transients show that even in the absence of

flow there are two distinct time scales for current flow through an EFFF channel

containing DI water, and thus a single RC model is not sufficient to model the equivalent

circuit. In the presence of salt there is an additional very rapid decay immediately

following the time at which the field reverses. This time scale can be neglected while

analyzing separations in EFFF because it is much shorter than all relevant time scales for

separation. After neglecting this initial decay, the transient current can be described by a

double exponential with time constants that are similar to the values for DI water. The

multiple time scales can partly be attributed to the dependence of the double layer

capacitance on the instantaneous potential droop across the double layer.

The theoretical analysis for the CEFFF shows that for a given set of

electrochemical parameters, the dimensionless mean velocity *U depends on Ω , 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 )Ω,PeR(fPe2 . In the high frequency limit the mean velocity and the dispersion

coefficient depend only on the ratio Ω/PeR . Since this ratio is independent of

125

diffusivity, colloidal particles such as DNA strands that have the same electrical mobility

cannot be separated on the basis of their lengths at high frequencies. However, at small

frequencies these particles can be separated because *U and D* depend separately on

PeR and Ω.

In the small frequency regime with a fixed value of Ω, *U increases with

increasing PeR, reaches a maximum and then begins to decrease. The location of the

maximum shifts to larger PeR values with increasing Ω. The dispersion coefficient D* is

in form of 1+Pe2*f(PeR, Ω), and thus (D*-1)/Pe2 depends only on PeR and Ω. At large

PeR, particles accumulate near the walls and the dispersion coefficient approaches

molecular diffusivity. At very small PeR, the concentration is uniform and the dispersion

is close to the classic Taylor dispersion coefficient for Poiseuille flow in a channel. In

addition, the effective diffusivity is large both at very large and very small Ω values,

while a minimum exists at intermediate frequencies.

It is advantageous that there is a very large number of geometric and operating

parameter in CEFFF which can be optimized for separation. However choosing these

parameters is a difficult task for an experimentalist, and the model developed in this

paper could be a very valuable tool in understanding the effect of various parameters and

determining the best conditions for separation.

126

CHAPTER 6 CONCLUSION AND FUTURE WORK

Separation of colloidal particles such as DNA strands, viruses, proteins, etc is

becoming increasingly important due to rapid advances in genomics, proteomics, and due

to threats posed by bioterrorism. A number of these colloids are charged making these

amenable to separation by using electric fields. Currently, techniques such as gel

electrophoresis can separate these particles, but this technique is complex and only

suitable for use by experts. There is a growing demand for simple chip based devices

that can accomplish separations in free solution, and this dissertation has focused on

electric field flow fractionation (EFFF), which is a simple approach for separation in free

solution that can be implemented on a chip.

Although this dissertation focuses on EFFF, the results of this study are applicable

to a majority of field flow fractionation (FFF) devices. The difference between other

types of FFF and EFFF is that EFFF uses electric field as the body force that drives

lateral transport, while other variants of FFF use other forms of fields, such as lateral

fluid flow, gravity/centrifugation, thermal gradients, magnetic field, etc. For all these

cases the lateral force is independent of the position, and thus the theory developed in this

dissertation can be applied by replacing the expression for the electric field driven lateral

velocity with the appropriate expression.

Amongst all the FFF techniques, EFFF is the simplest for implementation on a

chip. However, application of EFFF for separations is hindered by issues related to the

electrochemical response of the devices. It is well known that when a constant electric

127

field is applied between two electrodes separated by only a few tens of microns, a double

layer forms at each electrode, and the potential drop across these double layers can be as

much as 99% of the applied voltage. Thus, only a very weak electric field is present in

the bulk of the channel, and this leads to inefficient separation. This problem is further

compounded by the fact that in order to have a steady electric field in the channel, one

need electrodic reactions which may generate gases. The evolving gases may lead to

bubble formation that could destroy separation. It may be feasible to operate EFFF in

presence of a redox couple such that the electrode reactions do not lead to bubble

formation. The problems associated with unidirectional EFFF are discussed in chapter 2

of this dissertation in the context of DNA separation. In this chapter we have also

developed scaling relationships for separation of colloidal particles, particularly DNA

strands. To separate DNA strands by EFFF, it may be best to operate below the shear

rate at which the strands unfold. Our reasoning if the shear rate is sufficiently high so

that the Weissenberg number is larger than 1, the DNA strands will unfold near the wall

but will stay coiled near the center. This would cause very large dispersion, and offset

the effect of the unfolding on the mean velocity. Based on this hypothesis, we obtained

the optimal separation conditions for DNA strands, and show that for DNA strands in the

range of 10 kbp (kilo base pairs) EFFF could potentially achieve separate efficiencies

comparable to entropic trapping devices.

While use of redox couples could eliminate some problems associated with double

layers, it is not the optimal solution because the redox couple will electroplate the

channel. In chapter 3, we propose a novel approach based on a combination of pulsed

electric field and pulsed Poiseuille flow to separate colloidal particles. Since the electric

128

field is pulsed, the problems associated with the unidirectional EFFF are not expected to

occur in our proposed technique. The essential idea of the proposed technique is to

accumulate all the particles near a wall by applying electric field, and then switch off the

field. After switching off the field, the particles diffuse and the ones with a larger

diffusivity travel a larger distance from the wall. If a flow is now turned on for a short

time, the smaller particles travel a larger axial distance because these are further from the

wall. In chapter 3 we developed a model for this technique and solved the model

analytically. The separation efficiency of this method depends strongly on the rate at

which the fluid flow can be switched on and off, and the separation improves with a

reduction in tf and td, which are the durations of the flow and the diffusive steps. We

showed that this method can be tuned to yield separation efficiencies that are better than

those for EFFF. While this method eliminates problems associated with the double layers

and it has improved separation efficiencies, it is more difficult to implement than the

conventional EFFF because of problems associated with switching the flow.

A more convenient method to minimize the problems associated with double layer

charging is to use oscillating electric fields that change direction sufficiently rapidly so

that the double layers do not get charged completely in each cycle, and thus most of the

applied potential appears in the bulk. The applied field could be either sinusoidal or

square shaped. We investigate cyclic electric field flow fractionation (CEFFF), i.e.,

EFFF with cyclic fields in chapters 4 and 5. Chapter 4 deals with sinusoidal fields and

chapter 5 focuses on square shaped fields.

In addition to the fact that the cyclic EFFF can increase the effective electric fields

in the bulk, there are other potential advantages of using cyclic fields in EFFF. CEFFF is

129

potentially a more universal separation technique because it has additional operating

variables such as frequency that could be utilized in separation. In constant EFFF, the

velocity of the particles in the flow direction is only a function of PeR, which is

essentially D

hu ey , where h is the channel height, and D and e

yu are the diffusivity and the

lateral velocity of the particle. Thus, two types of particles with the same ratio of Du e

y

cannot be separated with constant EFFF. In CEFFF, the mean velocity is function of PeR

as well as of D

h 2ω≡Ω , where ω is the frequency of the applied field. Therefore, if two

types of particles have different molecular diffusivity, they can be separated by CEFFF

even if they have same ratio of Du e

y .

In chapter 5 we investigated CEFFF with sinusoidal fields. If a sinusoidal field is

applied to a channel, the field experienced by the particles is also sinusoidal with a

reduced amplitude and phase delay. In chapter 5 we calculated the mean velocity and

dispersion coefficient for the case of sinusoidal lateral velocity by using a multiple time

scale analysis. In this chapter, we also considered the case of square wave fields under

the condition that the decay of the field due to the double layer effects can be neglected.

For the square shaped fields, we solved the model by combining numerical and analytical

techniques and validated the result by comparing with Brownian Dynamics simulations.

For the CEFFF with sinusoidal field, in addition to solving the model numerically, we

also developed an analytical approach to obtain the velocity and dispersion coefficient

that is suitable in small PeR and Ω regime. The results from both methods agreed with

130

each other and also with the Brownian dynamics simulations. We also developed

asymptotic results for small frequencies. We compared the separation efficiencies of

square wave and sinusoidal fields, and showed that these have similar separation

efficiencies in the range of the desirable operation conditions.

We also obtained asymptotic results for large frequencies for the case of sinusoidal

fields, and also compared the high frequency results for square fields with the asymptotic

results reported in literature. In the large PeR and Ω regime, the molecular diffusivity is

negligible and the particles convect in the lateral direction without much spreading.

Accordingly, in this regime the velocity of particles on the axial direction is only function

of Ω/PeR , i.e., )h/(u ey ω⋅ , and thus the separation depends only on e

yu . Therefore,

particles with similar mobilities such as DNA strands of various sizes cannot be separated

by CEFFF in the large frequency regime.

While analyzing CEFFF in chapter 4, we did not explicitly consider the decay of

the electric field that could distort the shape of the field experienced by the particles. We

explored this phenomenon in chapter 5 experimentally, and included the temporal decay

in the transport model for separation. For the case of CEFFF with sinusoidal voltages,

the electric field in the bulk still follows sinusoidal form, but there is a frequency

dependent phase lag and a reduction in potential drop in the bulk. Since the bulk field is

sinusoidal, the theory developed in chapter 4 is valid for sinusoidal fields even after

accounting for the effects of the double layers. However, for the case of CEFFF with

square wave voltages, the field experienced in the bulk is not square shaped.

In the first part of chapter 5, we experimentally explored the electrochemistry of a

CEFFF device fabricated in our lab. We applied a fixed step voltage or a cyclic square

131

shaped voltage, and measured current. The current is a measure of the bulk field, and is

thus a critical parameter in the CEFFF. We compare the experimental results with a

commonly used equivalent circuit, and show that the equivalent circuit is correct only if

the dependence of the double layer capacitance on the potential drop across the double

layer is taken into account. We also explored the dependence of the electrochemical

response on channel thickness, magnitude of applied voltage and the salt concentration.

We show that the decay of current in DI water can be fitted to a double exponential and

in the presence of salt there is a very rapid initial decay in current but the remaining

current transients can still be fitted to a double exponential with time constants that are

similar to that for DI water. We attribute this to the fact that the salt is only a supporting

electrolyte, i.e., it does not react at the electrodes and thus it only plays an important role

in the short times at which the salt concentration evolves due to the field. These time

constants are relatively insensitive to the applied voltage. We incorporated the current

transients that were experimentally measured into the transport model for the particles

undergoing a pressure driven Poiseuille flow along with the lateral electric fields. We

solved the model by using a multiple time scale approach, and calculated the velocity and

dispersion coefficient numerically, and utilized these results to determine the separation

efficiency. The results show that the mean velocity and the dispersion coefficients are a

complex functions of the electrochemical response and also of PeR and Ω. Therefore

choosing the optimal parameters for separation is not simple, and can only be

accomplished by using the model developed in this dissertation. To illustrate the

complexity of this technique, we investigated the effect of PeR and Ω for separating two

types of particles and showed that the length of channel required for separation varies

132

over several orders of magnitude for a reasonable range of operating parameters. The

complex interplay of electrochemistry and hydrodynamics makes CEFFF a useful

equipment because there are a large number of operating variables that could be tuned for

separation. However it also makes the process of choosing the operating variables

difficult and we hope that our model will serve as a useful guide for experimentalists in

designing and operating CEFFF devices.

We believe that CEFFF based on both cyclic and pulsatile fields have the potential

to perform a wide variety of separations on a variety of scales. While this dissertation

has focused on microchannels, EFFF may be utilized for a large scale industrial

separation by using larger channels. However successful implementation of CEFFF,

particularly in microchannels requires a more detailed investigation of the

electrochemical response. We hope that this dissertation has shown the usefulness of

CEFFF and will encourage researchers to solve the complete electrochemical problem by

solving the Poisson-Boltzmann equation along with the species conservation, and then

couple it to the electrode kinetics at the surface. This is a complex task mainly due to the

lack of details on the electrode reactions for a majority of electrolytes. It is hoped that

molecular level techniques can help in identifying and characterizing the electrode

reactions, and these can then be coupled with continuum simulations.

In this dissertation we have compared the model predictions with the meager

experimental data available in literature. In addition to developing a better theoretical

understanding of CEFFF devices, it is also important to generate more experimental data

that can validate the models developed in this dissertation. Additionally we hope that

133

researchers will attempt to fabricate the pulsed EFFF described in chapter 3 and compare

the results with our predictions.

134

APPENDIX A DERIVATION OF VELOCITY AND DISPERSION UNDER UNIDIRECTIONAL

EFFF

Our aim is to determine the Taylor dispersion of a pulse of solute introduced into the

channel at t = 0. In a reference frame moving with a velocity u , the mean velocity of the

pulse, Eq. (2-2) becomes,

)y

cx

cR(Dycu

xc)uu(

tc

2

2

2

2ey ∂

∂+

∂∂

=∂∂

+∂∂

−+∂∂ (A-1)

Since we are interested in long-term dispersion, the appropriate time scale is L/

where L is the total channel length, and is the mean fluid velocity. In this time, a

pulse will spread to a width of about ><≡ u/DLl , which is the appropriate length

scale in the x direction. These scales ensure that the convective time scale is comparable

to the diffusive time scale in the axial direction. The scaling gives

2

22

ε1

Dhu

hDhu

hL

uL

D><

≡⎟⎠⎞

⎜⎝⎛><

=⇒><

=ll (A-2)

where 1Lh~hε <<≡

l. We use the following de-dimensionalization:

><=

u/LtT , U = u/, ><= u/uU ,

U ey = u e

y /, C = c/c0, X = x/l, Y = y/h, Pe =h/ D

(A-3)

135

where L is the length of the channel; l is the width of the pulse as it exits the channel; h is

the height; is the average velocity of the flow; and Pe is the Peclet number based on

D = ⊥D . In dimensionless form, Eq. (A-1) and the boundary conditions Eq. (2-3) become

2

2

22

2ey2 Y

Cε1

XCR

YCU

εPe

XC)UU(

εPe

TC

∂∂

+∂∂

=∂∂

+∂∂

−+∂∂ (A-4)

0CPeUYC e

y =+∂∂

− at Y = 0,1. (A-5)

We assume a regular expansion for C in ε,

................εCεCCC 2210 +++= (A-6)

Substituting the regular expansion for C into (A-4) and (A-5) gives the following sets

of equations and boundary conditions to different orders in ε.

(1/ε2):

20

20e

y YC

YC

PeU∂∂

=∂∂

; 0CPeUYC

0ey

0 =−∂∂

at Y = 0, 1

⇒ )YPeUexp()T,X(AC ey0 = (A-7)

(1/ε):

21

21e

y0

YC

YCPeU

XC

)UU(Pe∂∂

=∂∂

+∂∂

− ; 0CPeUYC

1ey

1 =−∂∂ (A-8)

Substituting C0 from Eq. (A-7) gives

21

21e

yey Y

CYCPeU)YPeUexp(

XA)UU(Pe

∂∂

=∂∂

+∂∂

− (A-9)

Integrating Eq. (A-9) in Y from 0 to 1 gives

∫∫ =1

0

ey

1

0

ey dY)YPeUexp(UdY)YPeUexp(U (A-10)

136

Eq. (A-10) gives the average velocity of the pulse.

1)αexp()α(

)αexp(1212α

)αexp(66

U2

−

−+

+

= (A-11)

where

α= eyPeU (A-12)

In Eq. (A-9) we assume

)Y(GXAC1 ∂

∂= (A-13)

This gives

2

2ey

ey Y

GYGPeU)YPeUexp()UU(Pe

∂∂

=∂∂

+− (A-14)

Solving Eq. (A-14) with boundary conditions gives

Yα3

2

22

α3

Yα

e)constαY2

αY6

αY3

)1e(α)Yαe(12(PeG +−++

−+

= −

−

(A-15)

and the constraint 0GdY1

0

=∫ determines the const in the equation. However, this const

does not affect the mean velocity and the dispersion coefficient.

ε0:

22

2

20

22e

y10

YC

XC

RYCPeU

XC)UU(Pe

TC

∂∂

+∂∂

=∂∂

+∂∂

−+∂

∂ ; 0CPeU

YC

2ey

2 =−∂∂ at Y = 0,1

(A-16)

Integrating the above equation, using the boundary conditions and using

137

]1)PeU[exp(PeU

AdYCC eye

y

1

0 00 −=>=< ∫ (A-17)

gives

]dY)Y(G)UU(1e

UPeR[

XC

TC 1

0PeU

ey2

20

20

ey ∫ −

−−

∂><∂

=∂

><∂ (A-18)

Thus the dimensionless dispersion coefficient D* is

]dY)Y(G)UU(1e

UPeR[D

1

0PeU

ey2*

ey

∫ −−

−= (A-19)

Substituting G from Eq. (A-15) into Eq. (A-19) and integrating gives




−−−−++−+−

++−−−−+−=

(A-20)

138

APPENDIX B DERIVATION OF NUMERICAL CALCULATION FOR SINUSOIDAL EFFF

Analytical Solution to O(ε) Problem

The solution to Eq. (4 - 10) can be expressed as

⎟⎠

⎞⎜⎝

⎛++= ∑

∞

=

))nTcos()Y(g)nTsin()Y(f()Y(gG sn1n

sn00 (B - 1)

where fn(Y) and gn(Y) can be determined by substituting the postulated form of G0 into

Eq. (4 - 10) and equating coefficients of sin(nTs) and cos(nTs). The equations for f and g

represent a hierarchy of coupled second order ordinary differential equations, which we

close by assuming that fN and gN are zero, where N is large enough not to cause

significant truncation errors.

Substituting Eq. (B-1) into (4 - 10) yields,

)]nTcos(Yg

)nTsin(Yf

[Yg

)]nTcos(Y

f)nTsin(

Yg

[PeR21

)]nTsin(Y

g)nTcos(Y

f[PeR21

Yg

)Tsin(PeR)]nTsin(ng)nTcos(nf[Ω

s2n

2

s1n

2n

2

20

2

2ns

1ns

1n

0ns

1ns

1n0s

1nsnsn

∂∂

+∂∂

+∂∂

=∂

∂−

∂∂

+

∂∂

−∂

∂+

∂∂

+−

∑∑

∑∑∞

=

∞

=

−−

∞

=

++∞

=

(B - 2)

where R=r/. Comparing both sides of Eq. (B-2) and equating the time independent

terms and the coefficients of sin(nTs) and cos(nTs) gives the following 2N-1 coupled

second order differential equations.

(Time independent terms): 0Yg

Yf

PeR21

20

21 =

∂∂

−∂∂

(B - 3)

139

(sin(Ts)): 0Yf

Yg

PeR21

Yg

PeRgΩ 21

220

1 =∂∂

−∂∂

−∂∂

+− (B - 4)

(cos(Ts)): 0Yg

Yf

PeR21fΩ 2

12

21 =

∂∂

−∂∂

+ (B - 5)

(sin(nTs))(n=2..N-1): 0Yf

Yg

PeR21

Yg

PeR21gΩn 2

n2

1n1nn =

∂∂

−∂

∂+

∂∂

−− −+ (B - 6)

(cos(nTs))(n=2..N-1): 0Yg

Yf

PeR21

Yf

PeR21fΩn 2

n2

1n1n1 =

∂∂

−∂

∂−

∂∂

+ −+ (B - 7)

The boundary conditions at O(ε0) are

0ey

0 GPeUY

G=

∂∂

at Y=0,1 (B - 8)

Substituting Eq. (B-1) into Eq. (B-8) yields

∑∑

∑∞

=−−

∞

=++

∞

=

−+−+=

∂∂

+∂∂

+∂∂

2ns1ns1n

0ns1ns1n0s

1ns

ns

n0

)]nTcos(f)nTsin(g[PeR21)]nTsin(g)nTcos(f[PeR

21g)Tsin(PeR

)]nTcos(Yg

)nTsin(Yf

[Yg

(B - 9)

Comparison of the time independent terms on both sides gives

10 PeRf

21

Yg

=∂∂

(B - 10)

Although Eq. (B-10) is valid at both walls, i.e., at Y= 0 and 1, imposing Eq. (B-10) at

either wall automatically satisfies the same condition at the other wall. To demonstrate

this, we average Eq. (4 - 9) in Ts and Y.

∫∫∫∫∫∫ ∂∂

=∂∂

+∂∂ dYdT

YCdYdT

YCPeUdYdT

TCΩ s2

02

s0e

yss

0 (B - 11)

Using Eq. (B-1) and (B-8) in the above equation gives

140

1

0

01

01 Y

gPeRf

21

∂∂

= (B - 12)

which proves that Eq. (B-10) can only be implemented at either Y = 0 or at Y = 1. To get

around this issue we rewrite Eq. (B-1) as

)T,X(A~))nTcos()Y(g~)nTsin()Y(f~()Y(g~C lsn1n

sn00 ⎟⎠

⎞⎜⎝

⎛++= ∑

∞

=

(B - 13)

where )1Y(g/gg~ 0nn == , )1Y(g/ff~ 0nn == , and )1Y(g)T,X(A)T,X(A~ 0ll == . Thus,

by definition 1)1Y(g~0 == . This is equivalent to stating that we can set the concentration

scale arbitrarily since the problem is homogenous. This is also equivalent to utilizing a

normalizing condition such as ensuring that the integral of the concentration in the lateral

direction is conserved, which was the normalization utilized by Shapiro and Brenner. In

the equations below, we remove the decorator ~ for convenience.

By equating the coefficients of sin(nTs) and cos(nTs) in the boundary condition, we

get the following:

(sin(Ts)): 20

1 PeRg21PeRg

Yf

−=∂∂

(B - 14)

(cos(Ts)): 2

1 PeRf21

Yg

=∂∂

(B - 15)

(sin(nTs))(n=2..N-1): 1n1n

n PeRg21PeRg

21

Yf

+− −=∂∂

(B - 16)

(cos(nTs))(n=2..N-1): 1n1n

n PeRf21PeRf

21

Yg

−+ −=∂∂

(B - 17)

Eq. (B-14)–(B-17) are valid at Y=0 and 1 and thus represent 4(N-1) boundary conditions.

These along with Eq. (B-10) at Y=0 and the only nonhomogenous condition

141

1)1Y(g0 == can be used to solve Eq. (B-3)-(B-7) to determine fi for i = 0: N-1 and gi for

i = 1: N-1.

Since all the differential equations (B-3)-(B-7) are linear with constant coefficients, the

solutions for fi and gi can be expressed as

∑ λ=j

Yj,ii

jeff

∑ λ=j

Yj,ii

jegg (B - 18)

The detailed equations for determining the eigenvalues and the eigenfunctions and

thereby determining fi and gi are provided below.

Analytical Solution to O(ε2) Problem

The solution to C1 is of the form )X/)T,X(A)(T,Y(B ls ∂∂ where B satisfies

2

2eys0

*

s YB

YBPeU)T,Y(G)UU(Pe

TB

∂∂

=∂∂

+−+∂∂

Ω (B - 19)

Substituting the expression for G0 in Eq. (B-19) gives

2

2eysn

1nsn0

*

s YB

YBPeU))nTcos()Y(g)nTsin()Y(f()Y(g)UU(Pe

TB

∂∂

=∂∂

+⎟⎠

⎞⎜⎝

⎛++−+

∂∂

Ω ∑∞

=

(B - 20)

The solution for B can be expressed as

]))nTcos()Y(g)nTsin()Y(f()Y(g[const

))nTcos()Y(q)nTsin()Y(p()Y(q)Y,T(B

1nsnsn0

sn1n

sn0s

∑

∑∞

=

∞

=

++×+

++= (B - 21)

The value of the const in Eq. (B-21) does not affect the value of either the mean velocity

or the effective diffusivity. Thus, in the rest analysis, we set it to be zero. The functions

pi and qi in Eq. (B-21) can be determined by substituting the postulated form for B into

Eq. (B-20) and equating coefficients of sin(nTs) and cos(nTs). Still, the equations for p

142

and q represent a hierarchy, which we close by assuming pM and qM are zero where M is

large enough.

Substituting Eq. (B-21) into Eq. (B-20) gives

∑

∑∑

∑∑

∞

=

∞

=

∞

=

∞

=

∞

=

∂∂

+∂

∂+

∂∂

=

−−+∂∂

++−−∂∂

+∂∂

+

⎟⎠

⎞⎜⎝

⎛++−+−

1ns2

n2

sn

2

2

2

1nss

n

1nss

n0s

sn1n

sn0*

1nsnsn

)]nTcos(Yq

)nTsin(Yp

[Y

q

]T)1nsin(T)1n[sin(Yq

PeR21]T)1ncos(T)1n[cos(

Yp

PeR21

Yq

)Tsin(PeR

))nTcos()Y(g)nTsin()Y(f()Y(g)UU(Pe)]nTsin(nq)nTcos(np[Ω

(B - 22)

Equating both sides gives

(time independent): 0*

20

21 g)UU(Pe

Yq

YpPeR

21

−−=∂∂

−∂∂

(B - 23)

(sin(Ts)): 1*

21

220

1 f)UU(PeYp

YqPeR

21

Yq

PeRqΩ −−=∂∂

−∂∂

−∂∂

+− (B - 24)

(cos(Ts)): 1*

21

22

1 g)UU(PeYq

Yp

PeR21pΩ −−=

∂∂

−∂∂

+ (B - 25)

(sin(nTs)): n*

2n

21n1n

n f)UU(PeYp

Yq

PeR21

Yq

PeR21qΩn −−=

∂∂

−∂

∂−

∂∂

+− +− (B - 26)

(cos(nts)): n*

2n

21n

1 g)UU(PeYq

Yp

PeR21pΩn −−=

∂∂

−∂

∂+ + (B - 27)

Boundary Conditions:

1ey

1 CPeUYC

=∂∂ (B - 28)

Substituting the expression for C1 into Eq. (B-28) and comparing both sides gives the

following boundary conditions at Y = 0 and 1,

(time independent terms): 10 PeRp

21

Yq

=∂∂

(B - 29)

143

(sin(Ts)): 201 PeRq

21PeRq

Yp

−=∂∂ (B - 30)

(cos(Ts)): 21 PeRp

21

Yq

=∂∂ (B - 31)

(sin(nTs)) (n=2..M-1): 1n1nn PeRq

21PeRq

21

Yp

+− −=∂∂ (B - 32)

(cos(nTs)) (n=2..M-1): 1n1nn PeRp

21PeRp

21

Yq

−+ −=∂∂ (B - 33)

These 4N-2 boundary conditions can be used to solve the 2N-1 second order differential

equations (B-23)-(B-27) to determine pi and qi. Again, since all the differential equations

(B-23)-(B-27) are linear with constant coefficients, the solution for pi and qi can be

expressed as

∑ λ=j

Yj,ii

jepp ∑ λ=j

Yj,ii

jeqq (B - 34)

The detailed equations for determining the eigenvalues and the eigenfunctions are

provided below.

144

Solving for f, g, p and q

To solve Eq. (B-3)-(B-7) and (B-10)-(B-17) fn ,gn, pm and qm are expanded as

∑ λ=j

Yj,ii

jeff ∑ λ=j

Yj,ii

jegg ∑ λ=j

Yj,ii

jepp ∑ λ=j

Yj,ii

jeqq (B - 35)

where λs are eigenvalues that can be determined by substituting the above expansions in

Eq. (B-3)-(B-7). It is noted that fN and gN are assumed to be zero to close the hierarchy

of equations for fi and gi. Thus, to determine fi and gi, there are 2N-1 that need to be

determined for each λ. Substituting the above expressions in Eq. (B-3)-(B-7) and

collecting the terms for λj gives

2jj,0jj,1 gPeRf

21

λ=λ (B - 36)

2jj,1jj,2jj,0j,1 fPeRg

21PeRgg λ=λ−λ+Ω− (B - 37)

2jj,1jj,2j,1 gPeRf

21f λ=λ+Ω (B - 38)

1-2..Nnfor fPeRg21PeRg

21gn 2

jj,njj,1njj,1nj,n =λ=λ+λ−Ω− −+ (B - 39)

1N..2n for gPeRf21PeRf

21fn jj,njj,1njj,1nj,n −=λ=λ−λ+Ω −+ (B - 40)

The above set of 2N-1 equations leads to 4N-2 values of λ’s and of these λ1 and λ2 are

zero. Thus, fi and gi must be of the form

∑−

=

λ++=)1N2(2

3j

Yj,i2,i1,ii

jefYfff

∑−

=

λ++=)1N2(2

3j

Yj,i2,i1,ii

jegYggg (B - 41)

145

Substituting Eq. (B-41) into the equations (B-3)-(B-7) and then equating coefficients of

sin(nTs) and cos(nTs) yields the following

(time independent terms): ∑∑−

=

−

=

=+)1k2(2

3j

Yλ2jj,0

)1k2(2

3j

Yλjj,12,1

jj eλgeλfPeR21PeRf

21

(B - 42)

(sinTs):

∑

∑∑∑−

=

λ

−

=

λ−

=

λ−

=

λ

λ=

λ+−λ++++Ω−

)1k2(2

3j

Y2jj,1

)1k2(2

3j

Yjj,22,2

)1k2(2

3j

Yjj,02,0

)1k2(2

3j

Yj,12,11,1

j

jjj

ef

)egg(PeR21)egg(PeR)egYgg(

(B - 43)

(cosTs):

∑∑∑−

=

−

=

−

=

=++++)1k2(2

3j

Yλ2jj,1

)1k2(2

3j

Yλjj,22,2

)1k2(2

3j

Yλj,12,11,1

jjj eλg)eλff(PeR21)efYff(Ω

(B - 44)

(sin nTs): n = 2..N-1

∑

∑∑∑−

=

−

=++

−

=−−

−

=

=

+−++++−

)1k2(2

3j

Yλ2jj,n

)1k2(2

3j

Yλjj,1n2,1n

)1k2(2

3j

Yλjj,1n2,1n

)1k2(2

3j

Yλj,n2,n1,n

j

jjj

eλf

)eλgg(PeR21)eλgg(PeR)egYgg(Ωn

(B - 45)

(cos nTs): n = 2..N-1

∑

∑∑∑−

=

−

=++

−

=−−

−

=

=

+++−++

)1k2(2

3j

Yλ2jj,n

)1k2(2

3j

Yλjj,1n2,1n

)1k2(2

3j

Yλjj,1n2,1n

)1k2(2

3j

Yλj,n2,n1,n

j

jjj

eλg

)eλff(PeR21)eλff(PeR

21)efYff(Ωn

(B - 46)

146

It is noted that for a given λ, the 2N-1 equations given above are not linearly

independent and thus one of them has to be eliminated. For the 4N-2 values of λ’s, the

total number of independent equations given in (B-42)-(B-46) are (4N-2)(2N-2) and the

total number of unknowns are (4N-2)(2N-1). The other 4N-2 equations are provided by

the boundary conditions. In equations (B-42)-(B-46), collecting the terms for a given Y

functionality and then further collecting the terms for different time dependencies results

in the following equations:

For j = 1(corresponding to terms independent of Y)

(time independent terms): 0PeRf21

2,1 =

(sinTs): 0PeRg21PeRggΩ 2,22,01,1 =−+−

(cosTs): 0PeRf21fΩ 2,21,1 =+

(sinnTs): 0PeRg21PeRggΩn 2,1n2,1n1,n =−+− +−

(cosnTs): 0PeRf21PeRffΩn 2,1n2,1n1,n =+− +−

(B - 47)

For j = 2 (terms linear in Y)

(time independent terms): No equation

(sinTs): 0gΩ 2,1 =−

(cosTs): 0fΩ 2,1 =

(sinnTs): 0gΩn 2,n =−

(cosnTs): 0fΩn 2,n =

147

(B - 48)

This shows that all the terms linear in Y are identically zero.

For j = 3..2(2N-1)

These equations are identical to those given in (B-36)-(B-40).

Similarly, substituting Eq. (B-41) into boundary equations (B-10)-(B-17) and then

collecting terms on the basis of the time dependencies, we get the following 2(2N-1)

equations:

(time independent terms):

∑∑−

=

−

=

++=+)1k2(2

3j

Yλj,12,11,1

)1k2(2

3j

Yλjj,02,0

jj efPeR21YPeRf

21PeRf

21eλgg

(sinTs):

)egPeR21YPeRg

21PeRg

21(

)egPeRYPeRgPeRg(eλff

)1k2(2

3j

Yλj,22,21,2

)1k2(2

3j

Yλj,02,01,0

)1k2(2

3j

Yλjj,12,1

j

jj

∑

∑∑−

=

−

=

−

=

++−

++=+

(cosTs):

∑∑−

=

−

=

++=+)1k2(2

3j

Yλj,22,21,2

)1k2(2

3j

Yλjj,12,1

jj efPeR21YPeRf

21PeRf

21eλgg

(sinnTs):

)egPeR21YPeRg

21PeRg

21(

)egPeR21YPeRg

21PeRg

21(eλff

)1k2(2

3j

Yλj,1n2,1n1,1n

)1k2(2

3j

Yλj,1n2,1n1,1n

)1k2(2

3j

Yλjj,n2,n

j

jj

∑

∑∑−

=+++

−

=−−−

−

=

++−

++=+

(cosnTs):

148

)efPeR21YPeRf

21PeRf

21(

)efPeR21YPeRf

21PeRf

21(eλgg

)1k2(2

3j

Yλj,1n2,1n1,1n

)1k2(2

3j

Yλj,1n2,1n1,1n

)1k2(2

3j

Yλjj,n2,n

j

jj

∑

∑∑−

=+++

−

=−−−

−

=

+++

++−=+

(B - 49)

In the above set, the first equation is valid only at Y = 0 and the others are valid at both Y

= 0 and Y = 1. Thus, these represent 2(2N-2)+1 equations. The last equation is the

nonhomogeneity

1gg)0Y(g)1N2(2

3jj,01,00 =+== ∑

−

=

(B - 50)

Combining the governing equations and the boundary equations gives all the fi,j and gi,j.

Solving for p and q

Comparing Eq. (B-3)-(B-7) with Eq. (B-23)-(B-27), we can find that they have

almost the same structure except that the latter ones have a non-homogeneous term,

which is a product of )UU(Pe *− with fi or gi. Thus, the solution for pi and qi can be

separated into the particular and the homogeneous solution.

homi

parii ppp += hom

iparii qqq += (B - 51)

Furthermore, based on the form of the nonhomogeneity we propose the following forms

for the particular solutions

∑∑

∑∑

=

λ−

=

λ−

++++=

++++=

3j

Y34j,i

23j,i

2j,i

1j,i

m

1mm1,i

pari

3j

Y34j,i

23j,i

2j,i

1j,i

m

1mm1,i

pari

j

j

e)YqYqYqq()Yq(q

e)YpYpYpp()Yp(p (B - 52)

Below we use MAPLE to develop the equations for determining the particular solution.

Solving for Particular Solution

149

Substituting Eq. (B-52) into (B-23)-(B-27) gives

(time independent terms ):

0eY)λqλPeRp21(

eY)λq6λqPeRp23λPeRp

21Peg6(

Ye)q6λq4λqPeRpλPeRp21Peg6(

e)q2λq2λqPeRp21λPeRp

21UPeg(

Y)q)1m)(2m(PeRp2

1m(

Y)q20PeRp2Peg6(Y)q12PeRp23Peg6Peg6(

Y)q6PeRpgUPePeg6()q2PeRp21gUPe(

Yλ32j

4j,0j

4j,1

Yλ2j

4j,0

2j

3j,0

4j,1j

3j,1j,0

Yλ4j,0j

3j,0

2j

2j,0

3j,1j

2j,1j,0

Yλ3j,0j

2j,0

2j

1j,0

2j,1j

1j,1

*j,0

4m

m3m1,0

2m1,1

361,0

51,12,0

251,0

41,11,02,0

41,0

31,12,0

*1,0

31,0

21,11,0

*

j

j

j

j

=−+

−−++−+

−−−+++

−−−++−+

++−+

+

−+−+−+−+

−+−+−+−

∑

∑

∑

∑

∑=

++

(B - 53)

150

(sinTs):

0eY)λpλPeRq21λPeRqqΩ(

eY)λp6λp

PeRq23PeRq3λPeRq

21λPeRqPef6qΩ(

Ye)p6λp4λp

PeRqPeRq2λPeRq21λPeRqPef6qΩ(

e)p2λp2λp

PeRq21PeRqλPeRq

21λPeRqfUPeqΩ(

Y)p)2m)(1m(PeRq2

1mPeRq)1m(qΩ(

Y)p20PeRq2PeRq4qΩPef6(

Y)p12PeRq23PeRq3qΩPef6Pef6(

Y)p6PeRqPeRq2qΩPef6fUPe(

)p2PeRq21PeRqqΩfUPe(

Yλ32j

4j,1j

4j,2j

4j,0

4j,1

Yλ2j

4j,1

2j

3j,1

4j,2

4j,0j

3j,2j

3j,0j,1

3j,1

Yλ4j,1j

3j,1

2j

2j,1

3j,2

3j,0j

2j,2j

2j,0j,1

2j,1

Yλ3j,1j

2j,1

2j

1j,1

2j,2

2j,0j

1j,2j

1j,0j,1

*1j,1

4m

m3m1,1

2m1,2

2m1,0

1m1,1

361,1

51,2

51,0

41,12,1

251,1

41,2

41,0

31,11,12,1

41,1

31,2

31,0

21,11,12,1

*

31,1

21,2

21,0

11,11,1

*

j

j

j

j

=−−+−+

−−

−+−+−−+

−−−

−+−++−+

−−−

−+−+−−+

++−+

−++−+

−−+−−+

−−+−−++

−−+−+−+

−−+−−

∑

∑

∑

∑

∑=

++++

(B - 54)

151

(cosTs):

0eY)λqλPeRp21pΩ(

eY)λq6λqPeRp23λPeRp

21Peg6pΩ(

Ye)q6λq4λqPeRpλPeRp21Peg6pΩ(

e)q2λq2λqPeRp21λPeRp

21gUPepΩ(

Y)q)2m)(1m(PeRp2

1mpΩ(

Y)q20PeRp2pΩPeg6(

Y)q12PeRp23pΩPeg6Peg6(

Y)q6PeRppΩPeg6gUPe(

)q2PeRp21pΩgUPe(

Yλ32j

4j,1j

4j,2

4j,1

Yλ2j

4j,1

2j

3j,1

4j,2j

3j,2j,1

3j,1

Yλ4j,1j

3j,1

2j

2j,1

3j,2j

2j,2j,1

2j,1

Yλ3j,1j

2j,1

2j

1j,1

2j,2j

1j,2j,1

*1j,1

4m

m3m1,1

2m1,2

1m1,1

361,1

51,2

41,12,1

251,1

41,2

31,11,12,1

41,1

31,2

21,11,12,1

*

31,1

21,2

11,11,1

*

j

j

j

j

=−++

−−++−+

−−−++++

−−−++−+

++−+

++

−++−+

−++−+

−+++−+

−++−

∑

∑

∑

∑

∑=

+++

(B - 55)

152

(sinnTs):

0eY)λpλPeRq21λPeRq

21qΩn(

eY)λp6λpPeRq23

PeRq23λPeRq

21λPeRq

21Pef6qΩn(

Ye)p6λp4λpPeRq

PeRqλPeRq21λPeRq

21Pef6qΩn(

e)p2λp2λpPeRq21

PeRq21λPeRq

21λPeRq

21fUPeqΩn(

Y)p)2m)(1m(PeRq2

1mPeRq2

1mqΩn(

Y)p20PeRq2PeRq2qΩnPef6(

Y)p12PeRq23PeRq

23qΩnPef6Pef6(

Y)p6PeRqPeRqqΩnPef6fUPe(

)p2PeRq21PeRq

21qΩnfUPe(

Yλ32j

4j,nj

4j,1nj

4j,1n

4j,n

Yλ2j

4j,n

2j

3j,n

4j,1n

4j,1nj

3j,1nj

3j,1nj,n

3j,n

Yλ4j,nj

3j,n

2j

2j,n

3j,1n

3j,1nj

2j,1nj

2j,1nj,n

2j,n

Yλ3j,nj

2j,n

2j

1j,n

2j,1n

2j,1nj

1j,1nj

1j,1nj,n

*1j,n

4m

m3m1,n

2m1,1n

2m1,1n

1m1,n

361,n

51,1n

51,1n

41,n2,n

251,n

41,1n

41,1n

31,n1,n2,n

41,n

31,1n

31,1n

21,n1,n2,n

*

31,n

21,1n

21,1n

11,n1,n

*

j

j

j

j

=−−+−+

−−−

+−+−−+

−−−−

+−++−+

−−−−

+−+−−+

++−+

−+

+−+

−−+−−+

−−+−−++

−−+−+−+

−−+−−

∑

∑

∑

∑

∑

+−

+

−+−

+

−+−

+

−+−

=

+++

+−

+

+−

+−

+−

+−

(B - 56)

153

(cosnTs):

0

eY)λqλPeRp21λPeRp

21pΩn(

eY)λq6λqPeRp23

PeRp23λPeRp

21λPeRp

21Peg6pΩn(

Ye)q6λq4λq

PeRpλPeRp21PeRpλPeRp

21Peg6pΩn(

e)q2λq2λq

PeRp21λPeRp

21PeRp

21λPeRp

21gUPepΩn(

Y)q)2m)(1m(PeRp2

1mPeRp2

1mqΩn(

Y)q20PeRp2PeRp2qΩnPeg6(

Y)q12PeRp23PeRp

23pΩnPeg6Peg6(

Y)q6PeRpPeRppΩnPeg6gUPe(

)q2PeRp21PeRp

21pΩngUPe(

Yλ22j

4j,nj

4j,1nj

4j,1n

4j,n

Yλ2j

4j,n

2j

3j,n

4j,1n

4j,1nj

3j,1nj

3j,1nj,n

3j,n

Yλ4j,nj

3j,n

2j

2j,n

3j,1nj

2j,1n

3j,1nj

2j,1nj,n

2j,n

Yλ3j,nj

2j,n

2j

1j,n

2j,1nj

1j,1n

2j,1nj

1j,1nj,n

*1j,n

4m

m3m1,n

2m1,1n

2m1,1n

1m1,n

361,n

51,1n

51,1n

41,n2,n

251,n

41,1n

41,1n

31,n1,n2,n

41,n

31,1n

31,1n

21,n1,n2,n

*

31,n

21,1n

21,1n

11,n1,n

*

j

j

j

j

=

−+−+

−−+

−+−−+

−−−

++−−++

−−−

++−−−+

++−+

++

−+

−+−+−+

−+−+−+

−+−++−+

−+−+−

∑

∑

∑

∑

∑

+−

+

−+−

++−−

++−−

=

+++

+−

+

+−

+−

+−

+−

(B - 57)

Rearranging these equations and equations various Y dependencies gives

For Y independent terms:

1,0*3

1,02

1,1 gUPeq2PeRp21

=−

1,1*3

1,12

1,22

1,01

1,1 fUPep2PeRq21PeRqqΩ =−−+−

1,1*3

1,12

1,21

1,1 gUPeq2PeRp21pΩ =−+

1,n*3

1,n2

1,1n2

1,1n1

1,n fUPep2PeRq21PeRq

21qΩn =−−+− +−

154

1,n*3

1,n2

1,1n2

1,1n1

1,n gUPeq2PeRp21PeRp

21pΩn =−+− +−

(B - 58)

For Y-Ym terms:

These equations are identical to Eq. (B-58).

For Yλ je terms:

*j,0

3j,0j

2j,0

2j

1j,0

2j,1j

1j,1 UPegq2λq2λqPeRp

21λPeRp

21

=−−−+

j,1*3

j,1j2

j,12j

1j,1

2j,2

2j,0j

1j,2j

1j,0

1j,1 fUPep2λp2λpPeRq

21PeRqλPeRq

21λPeRqqΩ =−−−−+−+−

j,1*3

j,1j2

j,12j

1j,1

2j,2j

1j,2

1j,1 gUPeq2λq2λqPeRp

21λPeRp

21pΩ =−−−++

j,n*

3j,nj

2j,n

2j

1j,n

2j,1n

2j,1nj

1j,1nj

1j,1n

1j,n

fUPe

p2λp2λpPeRq21PeRq

21λPeRq

21λPeRq

21qΩn

=

−−−−+−+− +−+−

j,n*

3j,nj

2j,n

2j

1j,n

2j,1nj

1j,1n

2j,1nj

1j,1n

1j,n

gUPe

q2λq2λqPeRp21λPeRp

21PeRp

21λPeRp

21pΩn

=

−−−++−− ++−−

(B - 59)

For Yλ jYe terms:

j,04

j,0j3

j,02j

2j,0

3j,1j

2j,1 Peg6q6λq4λqPeRpλPeRp

21

−=−−−+

j,14

j,1j3

j,12j

2j,1

3j,2

3j,0j

2j,2j

2j,0

2j,1 Pef6p6λp4λpPeRqPeRq2λPeRq

21λPeRqqΩ −=−−−−+−+−

j,14

j,1j3

j,12j

2j,1

3j,2j

2j,2

2j,1 Peg6q6λq4λqPeRpλPeRp

21pΩ −=−−−++

155

j,n

4j,nj

3j,n

2j

2j,n

3j,1n

3j,1nj

2j,1nj

2j,1n

2j,n

Pef6

p6p4pPeRqPeRqPeRq21PeRq

21qn

−=

−λ−λ−−+λ−λ+Ω− +−+−

j,n

4j,nj

3j,n

2j

2j,n

3j,1nj

2j,1n

3j,1nj

2j,1n

2j,n

Peg6

q6q4qPeRpPeRp21PeRpPeRp

21pn

−=

−λ−λ−+λ+−λ−Ω ++−−

(B - 60)

For Yλ2 jeY terms:

j,0j4

j,02j

3j,0

4j,1j

3j,1 Peg6λq6λqPeRp

23λPeRp

21

=−−+

j,1j4

j,12j

3j,1

4j,2

4j,0j

3j,2j

3j,0

3j,1 Pef6λp6λpPeRq

23PeRq3λPeRq

21λPeRqqΩ =−−−+−+−

j,1j4

j,12j

3j,1

4j,2j

3j,2

3j,1 Peg6λq6λqPeRp

23λPeRp

21pΩ =−−++

j,n

j4

j,n2j

3j,n

4j,1n

4j,1nj

3j,1nj

3j,1n

3j,n

Pef6

p6pPeRq23PeRq

23PeRq

21PeRq

21qn

=

λ−λ−−+λ−λ+Ω− +−+−

j,n

j4

j,n2j

3j,n

4j,1n

4j,1nj

3j,1nj

3j,1n

3j,n

Peg6

q6qPeRp23PeRp

23PeRp

21PeRp

21pn

=

λ−λ−+−λ+λ−Ω +−+−

(B - 61)

For Yλ3 jeY :

0λqλPeRp21 2

j4

j,0j4

j,1 =−

0λpλPeRq21λPeRqqΩ 2

j4

j,1j4

j,2j4

j,04

j,1 =−−+−

0λqλPeRp21pΩ 2

j4

j,1j4

j,24

j,1 =−+

0λpλPeRq21λPeRq

21qΩn 2

j4

j,nj4

j,1nj4

j,1n4

j,n =−−+− +−

156

0λqλPeRp21λPeRp

21pΩn 2

j4

j,nj4

j,1nj4

j,1n4

j,n =−+− +−

(B - 62)

The pi and qi in the particular solution can be obtained by solving all equations

simultaneously along with the boundary conditions.

The form of equations for the homogeneous solution for pi and qi are identical to

those for fi and gi and can be obtained in an analogous manner.

157

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BIOGRAPHICAL SKETCH

I was born on 23 April 1976, in He Zhang, a small town in Gui Zhou province,

China. My father is a physical teacher in high school and my mother is a doctor. I

established a strong interest in science since childhood due to the intellectual surrounding

provided by my family. My wide region of reading earned me honors in various

competitions in high school. With competitive scores in the National Entrance

Examination, I was admitted by the most prestigious university of China—Tsinghua

University. I urged myself in undergraduate study in Tsinghua University, took five-year

courses in four years, graduated one year earlier than my peers, and ranked in the top 5%

in my department of 120 students. After that I entered the graduate program of

biochemical engineering in 1998, waived of the entrance examination. In graduate stage,

I ranked in the top 10% in my class.

During seven years in Tsinghua University, I participated in several projects. In

my undergraduate diploma project, I studied the measurement of solubility of sodium

sulfate in supercritical fluid, which is a part of the research of supercritical water

oxidation (SCWO), a promising method for dealing with wastewater. In 1999, I took part

in a project to undertake middle-scaled amplification of the production of PHB (poly-β-

hydroxybutyrate, a kind of biodegradable plastic) with E.Coli., which was part of a Ninth

Five-year National Key Project of China. Under my active and successful participation,

we found and eliminated the scattering of nitrogen during sterilization and improved the

distribution of air input. The density of bacteria reached 120g/l and the production of

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PHB extended to 80g/l, far beyond the original goal. The amplification succeeded and

won me the honor of the first prize for outstanding performance in the field practice of

my department.

My master’s thesis was under the guidance of Prof. Zhongyao Shen, the Vice-Dean

of the School of Life Sciences and Engineering in Tsinghua University. My work

focused on coupling of fermentation and separation. In the first year, I applied the

coupling of fermentation and ion exchange on the production of 2-keto gulonic acid, the

direct precursor of vitamin C. However, this research was abandoned because of an

unfeasibility resulting from the fermentation system. After that, my main interest was on

the coupling of fermentation and membrane separation in the production of acrylamide

from acrylotrile. During the process, I acquired insights on membrane, fermentation, ion

exchange, and operation of analytical equipment. Finally, I got the high enzyme activity

from the fermentation that is the highest value on documents.

After I graduated from Tsinghua University in 2001, I came to the Department of

Chemical Engineering, University of Florida, to pursue advanced education. My research

focuses on separation process with microchannel and electric fields. Under the guidance

of Dr. Anuj Chauhan, I obtained promising results in modeling of separation with EFFF

and we understood the advantages and problems of this method in applications. I believe

our research can stimulate and propel the commercialization of EFFF.

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