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

Full Text 
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SEPARATION WITH ELECTRICAL FIELDFLOW FRACTIONATION By ZHI CHEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006
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Copyright 2006 by Zhi Chen
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This document is dedicated to the graduate students of the University of Florida.
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iv ACKNOWLEDGMENTS This work was performed under the elaborat e instruction of Dr. Anuj Chauhan. He gave me invaluable help and direction dur ing 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 sup port of NASA (NAG 10316) and the National Science Foundation (NSF Grant EEC9402989).
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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv TABLE............................................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................x ii CHAPTER 1 INTRODUCTION TO ELECTRICAL FIELDFLOW 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 e yU 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 FI ELDS AND POISEUILLE FLOW IN A 2D CHANNEL...........................................................................................................30 Theory......................................................................................................................... 32 Model...................................................................................................................32 The diffusive step: No el ectric field and no flow.........................................32 The convective step: Poiseuille flow with no electric field.........................34 Electric field step (El ectric field, no Flow)..................................................36 Long time Analytical Solution............................................................................40 Results and Discussion...............................................................................................42
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vi Mean Velocity.....................................................................................................43 Dispersion Coefficient.........................................................................................44 Separation Efficiency..........................................................................................46 Effect of G....................................................................................................47 Effect of tf/td.................................................................................................48 Effect of 2 f 2t u h ........................................................................................49 Comparison with Constant EFFF........................................................................52 Conclusions.................................................................................................................56 4 TAYLOR DISPERSION IN CYCLIC ELECTRICAL FIELDFLOW 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 comp arison with analytical results................72 Comparison of Sinusoidal and Square fields.......................................................83 Conclusions.................................................................................................................84 5 ELECTROCHEMICAL RESPONSE AN D SEPARATION IN CYCLIC ELECTRIC FIELDFLOW 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 cyc lic 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 asymptotic results.........................................................................113 The effect of changes in .........................................................................118 Conclusions...............................................................................................................123
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vii 6 CONCLUSION AND FUTURE WORK.................................................................126 APPENDIX A DERIVATION OF VELOCITY AND DISPERSION UNDER UNIDIRECTIONAL EFFF......................................................................................134 B DERIVATION OF NUMERICAL CALC ULATION FOR SINUSOIDAL EFFF.138 Analytical Solution to O( ) Problem........................................................................138 Analytical Solution to O( ) 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
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viii TABLE Table page 51 Comparison of the model predictions with experiments of Lao et al....................123
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ix LIST OF FIGURES Figure page 21 Schematic of the 2D channel......................................................................................7 22 Dependency of (D*R)/Pe2 on the product of Pe and e yU.......................................12 23 Dependency of mean velocity U on the product of Pe ande yU..............................12 24 Dependency of L/h on e yU and Pe for separation of DNA strands of different sizes. D2/D1 = 10.....................................................................................................18 25 Dependency of L/h on e yU and Pe for separation of DNA strands of different sizes. D2/D1 = 2.......................................................................................................19 26 Comparison of our predictions with experiments on DNA separation with FlFFF.27 27 Comparison of our predictions with e xperiments on separation of latex particles with EFFF.................................................................................................................28 31 Schematic showing the threestep cycle...................................................................31 32 Dependency of *U on G..........................................................................................44 33 Dependency of *D on G..........................................................................................45 34 Effect of G1 ( 2 0 t td f, 2 0 t u h2 f 2 G2/G1=2) on L/h, /tf and T......................47 35 Effect of d ft t (G1=100, 2 0 t u h2 f 2 G2/G1=2) on L/h, /tf and T.......................49 36 Effect of 2 f 2t u h (G1=100, 2 0 t td f, G2/G1=2) on L/h, /tf and T.......................50 37 Dependency of L/h on G1(pulsating electric field) and e yu(constant electric field). D1/D2=2.........................................................................................................54
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x 38 Dependency of the operating time t on G1(pulsating electric field) and e yu(constant electric field). D1/D2=2........................................................................55 39 Dependency of L/h on G1(pulsating electric field) and e yu(constant electric field). D1/D2=1.2......................................................................................................55 310 Dependency of the operating time t on G1(pulsating electric field) and e yu(constant electric field). D1/D2=1.2.....................................................................56 41 Periodic steady concentration profiles during a period for a square shaped electric field..............................................................................................................67 42 Comparison of the numerically computed (a) mean velocity and (b) dispersion coefficient for a square shaped electric field with the large Pe asymptotes obtained by S&B (Thick line)..................................................................................69 43 gi vs. position for PeR=1, and =100.....................................................................71 44 Time dependent concentration profiles within a period for sinusoidal electric fields......................................................................................................................... 73 45 Time average concentration prof iles for sinusoidal electric field............................74 46 Dependence of *U on PeR......................................................................................76 47 Dependence of (D*1)/Pe2 on PeR...........................................................................79 48 Comparison of the mean velocities for the square (dashed) and the sinusoidal (solid) fields in the large frequency limit.................................................................82 49 Comparison of the mean velocities and the effective diffusivity for the square (dashed) and the sinus oidal (solid) fields.................................................................83 51 Equivalent electric circui t model for an EFFF device..............................................88 52 Transient current profiles after application of step change in voltage in a 500 m thick channel............................................................................................................95 53 Dependence of the electrochemical para meters on salt concentration and applied voltage in a 500 m thick channel...........................................................................97 54 Dependence of the electrochemical pa rameters on channel thickness for V = 0.5 V and DI water.........................................................................................................98 55 Comparison between the experiments (thi n lines) and Eq. (5 24) (thick lines)....101 56 Comparison between the experiments (s tars) and Eq. (5 26) (solid lines)...........102
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xi 57 Dependency of the mean velocity on PeR and ..................................................105 58 Dependence of 210(D*1)/Pe2 on PeR and ........................................................108 59 Dependence of separa tion efficiency on PeR1 and 1 for the case of D1/D2=3 and E2/ E1=3.........................................................................................................109 510 Origin of the singularity in sepa ration efficiency at critical PeR1 and 1 values for 1 = 40 ..........................................................................................................109 511 Dependence of the mean velocity on in the large regime..............................120
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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SEPARATION WITH ELECTRICAL FIELDFLOW FRACTIONATION By Zhi Chen August 2006 Chair: Anuj Chauhan Major Department: Chemical Engineering Separation of colloids such as viruses, cells, DNA, RNA, proteins, etc., is becoming increasingly important due to rapi d advances in the areas of genomics, proteomics and forensics. It is also desirable to separate these colloi ds in free solution in simple microfluidic devices that can be fabricated cheaply by using the microelectromechanical systems (MEMS) tech nology. Electrical fi eldflow fractionation (EFFF) is a technique that can separate char ged particles by combin ing a lateral electric field with an axial pressuredriven flow. EFFF can easily be integrated with other operations such as reaction, preconcen tration, detection, etc., on a chip. The main barrier to implementation of E FFF is the presence of double layers near the electrodes. These double layers cons ume about 99% of th e potential drop, and necessitate application of large fields, wh ich can cause bubble formation and destroy the separation. In this dissertation we have i nvestigated the process of double layer charging and proposed several approaches to mini mize the effect of double layer charging on separations. The essential idea is that if th e applied electric field is either pulsed or
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xiii oscillates with a period shorter than th e 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 investig ated 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 ch annel thickness, appl ied voltage and salt concentration. While investig ating unidirectionalEFFF, pul sed EFFF and cyclicEFFF, we solved the continuum convection diffusion equation for the charged particles to obtain the mean velocity and the dispersion coeffici ents for the particles. Furthermore, we estimated the separation efficiency based on the velocity and dispersion coefficient. Results show that EFFF can separate colloid s 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.
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1 CHAPTER 1 INTRODUCTION TO ELECTRICAL FIELDFLOW FRACTIONATION A number of industrial processes particul arly 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 t hose by bioterrorism have increas ed 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 opera ted by experts. There is a strong demand for simpler processes and devices that can be incorporated on a ch ip and that can accomplish separation in free solution. One approach that has a significant poten tial is electric field flow fractionation (EFFF), which is a variant of a general class of fi eldflow fractionation (FFF) techniques. Fieldflow fractionation re lies on application of a field in the direction perpendicular to the flow to cr eate concentration gradients in the lateral direction. When particles flow through channels in the pres ence 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 st reamline 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 a ccess streamlines closer to the wall, resulting in a reduction of the mean axial velocity. Since the concentration profile in the lateral direction depends on the fielddriven mobility and the diffusion coefficient, molecules
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2 that either have different mobilities or different diffusivities can be separated by this method. Field flow fractionation (FFF) was forma lly defined by J.Calvin Giddings in 1966. However a variant of this approach was used as far back as the Middle Ages to recover gold by sluicing, in which the gravity is combined with a flowing stream to generate separation. Fieldflow fracti onation has many variants depend ing on the types of lateral fields used in separation, such as sedime ntation FFF, electrical FFF, flow FFF, magnetic FFF, etc. There is an extensive lit erature on the use of EFFF [13] and other variants of FFF such as those based on gravity, centrifugal ac celeration [46], lateral fluid flow[7], or thermal fieldflow fractionati on (TFFF) [810]. These techniques have been used in separations of a number of di fferent types of molecules in cluding biomolecules [11,12]. The flowFFF, which is the fractionation techni que that utilizes a combination of lateral fluid flow along with the axial flow, has b een successfully utilized to separate DNA strands [13]. The separation of charged part icles is frequently accomp lished 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 late ral 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 us ed for separation of ch arged particles, such as cells [16,17], proteins [18], DNA molecules and la tex particles [19]. The EFFF technique has received consid erable attention due to its potential application in separation of colloidal part icles [2,20] such as DNA strands, proteins,
PAGE 16
3 viruses, etc. EFFF devices ar e 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 electr odes, causing generation of oxygen and hydrogen. Since bubble formati on could significantly impede separation, the incoming fluid is typically degassed so th at the evolving gases can simply dissolve in the carrier fluid. But even then the amount of latera l electric field that can be applied is limited by the restriction that it should not result in generatio n of gases that can exceed the solubility limit. The time required for the cu rrent 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 th e 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. Acco rdingly 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 addr essed. In order to eliminate these problems
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4 we propose a new technique based on pulsatile fields in chapter 3, and show that this technique is more effective than the conve ntional 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 ch apter 4 for sinusoidal fi elds and in chapter 5 for square fields. Finally chapter 6 summar izes the main conclusions and proposes some future work.
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5 CHAPTER 2 DNA SEPARATION BY E FFF IN A MICROCHANEL The main aims of the research in this ch apter are (i) investigate the feasibility of using EFFF for DNA separation by determining th e field strength required for separation, (ii) study the effect of vari ous system parameters on DNA se paration, (iii) determine the scaling relationships for sepa ration length and time as a function of the DNA length in various parameter regimes, and (iv) determ ine the optimum operating conditions and the minimum channel length and the time require d 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 in dispensable 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 inde pendent 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 counterions [23]. This difficulty is traditionally overcome by performing the electrophoresis in columns or capillaries filled with gels. The field applied in the ge lbased electrophoretic separations can be continuous or pulsed.
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6 Recent advances in microfab rication techniques have led to the production of microfluidic devices frequently referred to as a labonachip that can perform a number of unitoperations su ch as reactions, separations, detection, etc., at a high throughput. Gelbased DNA separa tions are not convenient in such devices because of the difficulty in loading the ge l [24]. Thus, gels have been replaced with polymeric solutions as the sieving mediums. Electrophor esis in a free medium can also separate DNA fragments but it requires precise modifications to the DNA molecules [25]. Microfabricated obstacles such as posts [26], selfassembling co lloids [27], entropic barriers [28], and Brownian ra tchets [29,30] have also been shown to be effective at separating DNA strands. The optimal DNA separation technique s hould accomplish separation without any sieving medium. Electrical fieldflow fractionation (EFFF) [14,20,31], which is a type of fieldflow fractionation (FFF), a technique first proposed in 1966 [32], can separate DNA strands by a combination of a lateral electric field and a Poiseuille flow in the axial direction. The application of the electric fiel d in the lateral direc tion, i.e., the direction perpendicular to the flow, creates a concentrat ion gradient in the lateral direction [33]. The DNA molecules are typically negatively ch arged and thus as they flow through the channels in presence of the lateral fields, th ey are attracted towards the positively charged wall. Thus, the molecules on an average ac cess 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
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7 fields, the differences in the mean velocities lead to separation of the slug into bands, and the band of the smaller mo lecules travels faster. Theory Figure 21 shows the geometry of a 2D ch annel that contains the electrodes for applying the lateral electric fi eld; L and h are the channel le ngth and height respectively, and the channel is infinitely wide in the thir d 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 21. Schematic of the 2D channel The transport of a solute in the channel is governed by the convectiondiffusion equation, 2 2 2 2  e yy c D x c D y c u x c u t c (2 1) where c is the solute concentration, u is th e fluid velocity in th e axial (x) direction, D and D are the diffusion coefficients in the dir ections parallel and perpendicular to the flow, respectively, and e yu 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
PAGE 21
8 velocity e yu can be determined by the Smoluchowski equation, E ur 0 e y, where r and are the fluids dielectric consta nt and viscosity, respectively, is the permittivity of vacuum, and is the zeta potential. Alternatively, E uE e y, where Eis the electrical mobility of DNA, which is independent of length and has a value of about 3.8x108 m2/(Vs) [22]. Outside the thin double layer ne ar the electrodes, the fluid is electroneutral, and the velocity of the charged molecules due to the elec tric fields in the y di rection is constant. Thus Eq. (21) becomes ) y c x c R ( D y c u x c u t c2 2 2 2 e y (2 2) where D / D R, 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 randomcoils and it is equal to 2 if they are fully stretched as cylinders in the flowdirection. The boundary conditions for the a bove differential equation are 0 c u y c De 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 ne glect the total flux of the DNA molecules from the bulk to the doubl e layer. The above boundary condition also assumes that the DNA molecules do not adsorb on the walls.
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9 Due to electroneutrality in the bulk, the velocity profile remains unaffected by the lateral electric field. Thus the fluid velocity pr ofile in the axial direction is parabolic, i.e., ) ) h / y ( h / y ( u 6 u2 (2 4) where is the mean velocity in the ch annel. The convection diffusion equation is solved in Appendix A to determine the dimensionless mean velocity Uand the dimensionless dispersion coefficient *Dfor a pulse of solute introduced into the channel. The results are 1 ) exp( ) ( ) exp( 12 12 ) exp( 6 6 U2 (2 5) ) ) 1 e /(( ) 72 720 2016 e 2016 e 6048 e 720 e 72 e 144 e 24 e 720 e 504 e 6048 e 144 e 24 e 504 e 720 ( Pe R D6 3 2 3 3 2 3 3 2 4 2 2 2 2 2 3 4 2 2 (2 6) In the above expressions Pe = h/D and e yPeU 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 that is about 3h/. 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 adopt ed by other researchers.
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10 Results and Discussion Limiting Cases The mean velocity and the dispersion coefficient depend on the Peclet number and e yU. If e yU approaches zero, we expect U and D* to approach the respective values for a 2D pressure driven flow in a channe l without electric field, which are 2 *Pe 210 1 R D ; 1 U (2 7) Also, as e yU becomes large most of the molecule s accumulate in a region of thickness and these molecules are subjected to a linear velocity profile, i.e., y h u ~ u The dimensional mean velocity of th e molecules therefore scales as h u Thus 1 ~ h ~ U. The time needed by the molecules to equilibrate in the lateral direction t is about D2 and the axial distance l traveled by the molecules during this time scales is of the order of h u D ~ t u2. 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 regime D* is expected to scale as 4 2 2 2 3 Pe D ~ ) D /( Dh u
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11 These scalings can also be obtained by e xpanding the exact solution from Eqs. (25) and (26) in the limit of both small and large The expansion for D* in the limit of 0 is )) ( O 1800 1 210 1 ( Pe R D4 2 2 (2 8) To the leading order, the above expression reduces to 2Pe 210 1 R which is the same as Eq. (27). Expanding Eq. (26) as goes to infinity gives )) 1 ( O 720 72 ( Pe R D6 5 4 2 (2 9) As expected, the leading order term scales as 4 2Pe However, the contribution from the next term, i.e., the O( 5) term, is about 10% of th e leading order term for as large as 100. Figure 22 compares the asymptotic solutions obtained a bove with the exact solution for D*. The small and the large approximations match the analytical solution for < 2 and >8, respectively. Similarly the asymptotic behavior of U in the limits of small and large is 0 ) ( O 60 1 1 U4 2(2 10) ) 1 ( O 6 U2 (2 11) The above result for U approaches 1 as approaches zero, and thus matches the mean velocity for Poiseuille flow in a channel w ithout any lateral field. Also in the large limit, the leading order term is of the order of 1/ that matches the expected scaling. Figure 23 shows the comparison of these asym ptotic results and the exact results from
PAGE 25
12 Eq. (25). The small and the large results match the full solution in the limit of <2 and >40, respectively. These asymptotic resu lts help us in understanding the physics of the dispersion and the DNA separation, as discussed below. Figure 22. Dependency of (D*R)/Pe2 on the product of Pe and e yU. The dashed line is the large approximation Eq. (29), and the dotted line is the small approximation Eq. (28) Figure 23. Dependency of mean velocity U on the product of Pe ande yU. The dashed line is the large approximation Eq. (211), and the dotted line is the small approximation Eq. (210)
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13 Dependence of the Mean Velocity on e yU and Pe Figure 23 shows the dependence of the mean velocity on e yU and Pe. The mean velocity depends only on i.e., the product of e yU and Pe. As discussed above the product e yPeUis essentially the inverse of the dimensionless thickness of the thin layer near the wall that contains a majority of th e particles. Thus, it is clear that at large an increase in leads to a reduction in the velocity of most of the particles and thus causes a reduction in the mean velocity. Howe ver, the effect of an increase in at small values of is not so clear because with an increase in the molecules that are attracted to the positive electrode travel with a smaller velo city, 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 regime, the mean velocity is reduced with an increase in The mean velocity is thus a monotoni cally decreasing function of Dependence of D* on e yU and Pe The effective dispersion coefficient D* depends separately on e yU and Pe. However, 2 *Pe / R D depends only on (Figure 22). As discussed above for small with an increase of the particle concentration near th e 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 results in an average deceleration of the particles as reflected in the reduction of the mean velocity (Figure 23), but a significant number of particles still
PAGE 27
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 only a very few particle s exist near the center as most of the particles are concentrated in a thin layer near the wall, and any further increase in leads to a further thinning of this layer. Thus, the velocity of the majority of the particles goes down, resul ting in a smaller spread of the pulse. Finally, as approaches infinity, the mean particle ve locity approaches zero, and the dispersion coefficient approaches the molecular diffusiv ity. Since the behavior of the dispersion coefficient with an increase in is different in the small and the large regime, it must have a maximum. The maximum is e xpected to occur at the value of beyond which there are almost no particles in th e region y50, which is typical for large DNA strands and ~1, the convectiv e 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 se parate into two Gaussian distributions. The axial location of the peak of the DNA molecules at time t is simply t u and the width of the Gaussian is t DD 4*. We consider the DNA strands to be separated when the
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15 distance between the two pulse centers becomes larger than 3 times of the sum of their half widths, i.e., ) t D D 4 t D D 4 ( 3 t ) u u (* 2 2 1 1 1 2 (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 u max( / L2 1. Substituting for t, and expressing all the variables in dimensionless form gives 2 1 2 1 2 2 1 2 1 1] U U D D D D )[ U U max( Pe 1 12 h / L (2 13) Eq. (213) can also be expressed as 1 1 1 2 1 2 1 1 2 2 1 1 1U Pe D 12 U U 1 D D D D 1 U Pe D 12 h / L (2 14) where is a measure of the resolving power of the separation method and we have assumed that species 1 travel faster than 2. In the discussion below, we use L/h to indicate the efficiency of sepa ration, 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 U min( u L T2 1 (2 15)
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16 Effect of Pe and e yU on the Separation Efficiency In Figures 24 and 25, we show the dependence of L/h on Pe and e yU in the case of e 1 yU = e 2 yU, which corresponds to DNA fragments of two different lengths. Figure 25 is similar to Figure 24; the only di fference is the value of the ratio D2/D1. Figures 24 and 25 show that at a small e yU, increasing e yU, which is physically equivalent to increasing the electric field, leads to a reduction in L/h required for separation. As e yU Pe increases, the mean velocities of both kinds of molecules decrease (Figure 23). But the dispersion coefficients do not change signifi cantly because they ar e very close to the diffusive value R for Pe < 10. Thus, L/h is primarily determined by the ratio 2 1 2 1 2U U 1 Pe U As shown earlier, in the small regime 2 60 1 1 ~ U thus, 2 2 1 2 2 4 e y 1 2 1 2 1 2Pe Pe U Pe 1 ~ U U 1 Pe U Since the ratio Pe2/Pe1 is fixed, 4 4 e y 5 1 2 2 1 2 2 4 e y 1Pe 1 U Pe ~ Pe Pe U Pe 1 Thus, an increase in either Pe or e yU leads to a reduction in L/h in the regime of small The constant Pe plots in Figure 24 and 25 show the 4 e yUdependency when e yU is small. Also, the constant Pe curves shift down with increasing Pe, due to the Pe5 dependency shown in the above scaling. The above expression also shows that at a fixed an increase in Pe leads to a reduction in L/h. In the limit of large / 6 ~ U, thus, Pe U ~ Pe Pe Pe Pe Pe Pe U ~ U U 1 Pe Ue y 2 1 2 2 1 2 1 e y 2 1 2 1 2 This implies that even in the large
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17 regime for a fixed an increase in Pe leads to a reducti on in L/h. It also shows that in the large regime and at O(1) Pe, L/h becomes inde pendent of Pe and begins to increase with an increase in e yU, as shown in Figure 24. Since L/h scales as 4 e yUin small regime, and as e yU in the large 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 an d both the mean velocities approach zero. Consequently, the difference of the velocitie s also approaches zero. Therefore, the difference in the mean velocities is zero fo r zero field because both the mean velocities are equal to the fluid velocity, and is also zero at very larg e 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 e yU fixed is more difficult to understand physically. Due to the dedimensionalization of e yU, in order to change Pe while keeping e yU fixed, both the fluid velocity and the el ectric field must be changed by the same factor. As a result, if we want to determin e the effect of only an increase in the mean velocity , we need to incr ease Pe and concurrently reduce e yUby the same factor. Thus, in Figures 24 and 25, we need to first move to the smallere yU value and then follow the constant e yU curve to the larger Pe. This keeps Pee yU constant and at O(1) Pe,
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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 b ecause 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 velocitie s of the two types of molecules, i.e., 2 1u u The distance between the peaks at the cha nnel exit is independent of because although 2 1u u increases linearly with , the ti me 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 24. Dependency of L/h on e yU and Pe for separation of DNA strands of different sizes. e 1 yU = e 2 yU = e yU, Pe1 = Pe, and Pe1/Pe2 = D2/D1 = 10
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19 Figure 25. Dependency of L/h on e yU and Pe for separation of DNA strands of different sizes. e 1 yU = e 2 yU = e yU, Pe1 = Pe, and Pe1/Pe2 = D2/D1 = 2 Another interesting regime occurs when Pe>>1 but e yU< Pe 1 In this regime, which is relevant for DNA separation, the convective contribution to the dispersion overwhelms the diffusion. In this regime can be large or small. By substituting the asymptotic expressions for *D and U we get the following expressions for L/h. 3 3 e y 2 Pe 144 U Pe 144 ~ h / L for >>1 (2 16) 4Pe ~ h / L for <<1 (2 17) The above expressions show th at in this limit for a fixed an increase in Pe results in an increase in L/h, which is contrary to the behavior for Pe<10. This implies the
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20 optimal Pe is the one at which the convective contribution to dispersi on 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 supersaturat e the carrier fluid and causes bubbles to form. The critical field at which bubbl es form depends on a number of factors such as the ionic strength, the electrode reactions, pres ence of redox couple in the solution, fluid velocity, etc. In EFFF, rese archers 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 experi ments reported above [2,15], the carrying fluid was DI water or water with a lo w ionic strength in the range of 1050 M. However, experiments involving DNA are t ypically done in the range of 10 mM concentration of electrolytes such as EDTA, trisHCl and NaCl [37]. EFFF cannot operate at such high ionic st rengths 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 n ecessary 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 c oupleelectrode 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
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21 method can increase the streng th 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. Si nce the DNA mobility for strands longer than 400 bp is 3.8x108 m2/(Vs), a field of 100 V/m will drive a lateral velocity of about 3.8 m/s. (2) The second restriction on e yu arises from the fact that the thickness of the layer in which the molecules accumulate, is given by e yu / D 3. For continuum to be valid the thickness of this layer must be mu ch larger than the radius of gyration of the DNA molecules. On neglecting the excluded volume effects, which is a reasonable assumption for strands shorte r than about 100 kbp, the radius of gyration k k gN l 6 1 R 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 N 2 ~ 300 N 6 100 Rg nm. The diffusivity of the DNA in a 0.1 M PBS buffer is N 10 2 D10 m2/s [40]. Let us choose gR 10 This gives N 10 5 1 N 10 20 10 3 ~ D 3 u2 9 10 e y m/s. Thus the condition gR imposes a
PAGE 35
22 smaller value for e yuthan the condition for preven tion of bubble formation for N>5000. (3) The shear in the microchannels is e xpected to stretch the DNA strands. For Wiessenberg number Wi over 20, the m ean 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, rt h u 6 which is the Weissenberg number Wi, should be less than 1. The relaxation time kT L ~ t5 1 rand based on this scaling a nd the experimental values reported in literature, tr in water is about 1.6x108 N1.5 s, and accordingly it has a value of about 0.01s for N = 10000. T hus for strands that are about 10000bp long, the shear rate h u 6 should be less than 100 s1 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 =Pee yU regime. As derived above for the case when is large but e yU~ Pe 1 the convective contribution to the dispersion dominates ove r molecular diffusion and the length required for separation is given by e y 2 e y 3 e y 2u u hu D 144 U Pe 144 ~ h / L (2 18)
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23 By using Eq. (215), the time for separation is 2 e y 2 e y e y 3 e y 2u D 24 u h U Pe 24 u 6 hPeU U Pe 144 ~ T (2 19)In the subsequent discussion we restrict gR10 The above scalings for L and T can equivalently be expre ssed in the following forms: h u D 16 ~ L3and D 3 8 ~ T2. Substituting gR 10 and expressing Rg and D in terms of N gives the following expressions for L and T: h u N 10 6 ~ L2 13 m (2 20) 2 / 3 6N 10 5 ~ 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 di rectly proportional to the mean velocity. Thus, a reduction in the mean velocity w ill reduce the channel length required for separation. The reason for this effect is th e 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 re gime the expressions for L and T become u hD 6 PeU Pe h 6 ~ L U 6 ~ h Le y e y (2 22) 2 2 2 e yu h D u 6 PeU u hD 6 ~ U u L ~ T (2 23) and accordingly the length and the time requi red for separation begin to increase with a reduction in the mean velocity. Thus, the opt imum channel length re quired for separation occurs when the convective contribution to dispersion is the same as the diffusive
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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 2 13N 10 6 ~ L m (2 24) 2 / 3 6N 10 5 ~ T s (2 25) For DNA the as defined by Eq. (214) is given by 2 2 2 1 4 / 3 2 1 2 1 2 2 / 3 1 2 2 1 2 1 1 2 2N N 16 ~ N N 1 N N 1 D D 1 D D 1 U U 1 D D D D 1 (2 26)where N=N2N1, and we have utilized the large 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 th e diffusive contributions to D* are about the same. Including in the expressions for L and T gives 2 2 12N N N 10 6 9 ~ L m (2 27) 2 2 / 3 5N N N 10 8 ~ 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, sepa ration of larger fragments in the range of
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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 clea rly superior to EFFF because they can separate fragments in th e range of 50 kbp in about 3040 minutes. However, we note that the time for separation ca n be significantly reduced if we relax the restriction of gR 10 But under this situation the c ontinuum equations cannot be used and one will need to perform noncontinuum si mulations to predict the effectiveness of EFFF at separating DNA strands. We also note that in our m odel we have not taken into account the adsorption of DNA on the walls, whic h will need to be carefully considered before designing the EFFF devices for DNA sepa ration. However, our model shows that EFFF has the potential to separate DNA stra nds 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 note d 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 diffu sion equation can be converted to the dispersi on equation of the form 2 0 2 0 0x C DD x C U u t C (2 29)
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26 where U and D* are the dimensionless mean veloci ty and the dimensionless dispersion coefficients, respectively. Accordingly, for a pulse input the concen tration profile at the channel exit (x = L) is given by ) t DD 4 ) t U u L ( exp( t DD 4 M C* 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 elect ric field used in EFFF, the two methods are equivalent, and can be described by the same equations. Figure 26 shows the comparison of the dispersive model with their experiments. In Figure 26, the experimental data of intensity at the detector located at the channel exit is compared with the concentrations predicted by Eq. (230). The vertical scale has been adjusted to ensure that the maximum height of the predicte d profiles matches the maxima of the experiments. All the other parameters require d 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 di spersive model with the experiments of Gale, Caldwell and Frasier in which they sepa rated latex particles of diameters 44, 130 and 207nm by EFFF[1]. Figure 27 shows the co mparison of the intens ity at the channel exit with the concentration predictions from the dispersion model for EFFF. As in Figure 27, the concentrations are scaled to match the experimental maxima. As seen in the Figure, the comparison between the experime nt and the model is reasonable for the 44
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27 nm size particle but the comparison is not sa tisfactory 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 ev en for the 44 nm size particles, the predicted dispersion is significantly less than the observe d dispersion. This discrepancy could be due to the neglect of the wall effect which is known to enhance dispersion. Figure 26. Comparison of our predictions with experiments on DNA separation with FlFFF. Channel geometry: channel height h = 227 m, total channel length L = 30 cm, channel breath b = 2.1cm. Op erational 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 expe rimental results for intensity at the detector and the dashed lines the predicte d concentrations at the channel exit. The two sets of dashed lines correspo nd to the two strands of DNA that were used in the respective experiments.
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28 Figure 27. Comparison of our predictions with experi ments on separation of latex particles with EFFF. Channel geometry: h = 28 m, L = 6cm. Operational parameters: flow velocity u=0.08cm/s, applied voltage Vapp = 1.9v, current I = 165 A. The solid lines are the experimental results for intensity at the detector and the dashed lines the predicted con centrations at the channel exit. The three sets of dashed lines correspond to the three kinds of la tex particles that were used in the respective experiments. Summary Application of lateral fields affects the m ean velocity and the dispersion coefficient of colloidal particles undergoi ng 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 contributi on to the dispersion coefficient is of the form ) PeU ( f Pee y 2. The mean velocity of the partic les decreases monotonically with an increase in Pe Ue y, but 2 *Pe / R D has a maximum at a value of Pe Ue y~ 4. This maximum arises when the thickness of the re gion near the wall where a majority of the particles accumulate is about h/2.
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29 Since the mean velocity of the particle s 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 thei r lengths by applying lateral electric fields. Axial fields cannot accomplish this separati on unless the channel is packed with gel. However, the separation may have to be performed in low ioni c 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 disper sion 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. Howeve r, separation of fragme nts in the range of 100 kbp may take a prohibitively long time. Ap plying a larger electric field may shorten the separation time for the 100kbp fragments, but noncontinuum simulations need to be performed to determine the efficacy of EFFF at separation of DNA fr agments in this size range. The results of this study can serve as a very useful guide in designing the chips for experimentally studying the se paration of DNA strands in the range of 100 kbp and also for separation of other ki nds of particles by EFFF.
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30 CHAPTER 3 SEPARATION OF CHARGED COLLOIDS BY A COMBINATION OF PULSATING LATERAL ELECTRIC FIELDS AND POIS EUILLE 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 fl ow driven by a pressure gradient. The threestep cycle that repeats continually is shown schematically in Figure 31. Initially, after introducing the charged particles into the channel, a strong late ral 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 remo val 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 fl ow because they are farther away from the wall. In the last step, the st rong electric field is reapplied to attract all th e molecules to the vicinity of the wall. As a result of th is cycle, the molecules with a larger diffusion coefficient exhibit a larger axial velocity. This technique shares some similarities with the cyclical fieldflow fractionation techni que developed by Giddings [43] and extended by Shmidt and Cheh [44] and by Chandhok and Leighton [45], which relies on the
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31 application of an oscillator y electric field across the narro w gap of the electrophoretic cell. The motion of the solute species induced by this field interacts with an oscillatory crossflow to cause a separation based on th e electrophoretic mobility of the species. However, the cyclic combination of field a nd flow proposed in this paper is different from the methods proposed in the above references. Figure 31. Schematic show ing the threestep cycle As mentioned above, the potential advantag e of the proposed method is that if the duration of the step in which the field is appl ied is shorter than the time for charging of the double layer then the gas ge neration can be avoided. Al so 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 convec tion 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 channe l at t = 0, and is then subjected to a series of threestep cycles descri bed above. Next, we discuss the dependence of the mean velocity and the dispersion coefficient on the system parameters. Finally, we investigate
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32 the effectiveness of the proposed method at accomplishing separation and compare the proposed method with unidirectional EFFF. Theory Model The diffusive step: No elec tric 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 i (x), which is the number of mo lecules per unit area after the ith cycle. Next, the electric field is re moved 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 2 2 2 2X C Y C C (3 1) The above equation is in a dimensionless form where x X h y Y 0c c C D / h t 2 1 h (3 2) is the characteristic length in the axial direction, h is the channel height, D is the diffusion coefficient of the colloidal particle s, and x and y are the axial and the lateral directions, respectively. We not e that the channel is assumed to extend infinitely in the z direction. The above differential equation is subj ected to the following boundary conditions: 0 ) 1 Y ( Y C ) 0 Y ( Y C (3 3) Additionally, overall mass conservation requires
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33 dX CdXdYi 1 0 (3 4) Since the diffusive step only lasts for time td and in our model dDt 4 h the boundary condition at Y=1 can be replaced by 0 ) Y ( C We solve Eq. (31) by using a the technique of regular perturbation expansions [46,47] in The concentration is expanded as 2 2 0C C C (3 5) By substituting C into Eq. (31) and Eq. (33), we get the differential equations and the boundary conditions for different orders in The equation for the order of 0is 2 0 2 0Y C C (3 6) The solution to Eq. (36) subject to the boundary conditions Eq. (33) and the overall mass conservation is ) 4 Y exp( ) X ( 1 C2 i 0 (3 7) The differential equation for the order of 2is 2 0 2 2 2 2 2X C Y C C (3 8) The solution to Eq. (38) subject to the boundary conditions Eq. (33) is ) 4 Y exp( X ) X ( C2 2 i 2 2 (3 9) The above solution satisfies the overall mass conservation because 0 x as x.
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34 The combination of C0 and C2 gives the concentration profile at the end of diffusion step (d ). ) 4 Y exp( X ) X ( ) 4 Y exp( ) X ( 1 ) ( C Cd 2 d 2 i 2 2 d 2 i d d diff (3 10) We note that d must be smaller than about 1/20 fo r this equation to be valid because otherwise the presence of the wall at Y = 1 w ill 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 d is small and thus we use the simpler similarity solution obtained above. The convective step: Poiseuille flow with no electric field To determine the concentration profile during the convective step, we need to solve the convectiondiffusion equation, a nd apply the solution at the e nd of the diffusive step Eq. (310) as the initial condition. The dime nsionless convectiondiffusion equation is 2 2 2 2 2 xX C Pe Y C Pe 1 X C U C (3 11) In the above equation time has been dedime nsionalized by the convective scaling, i.e., u /, and D h u Pe All the other dimensionless variables are the same as in the diffusive step. We solve the above equation under the conditions1 Pe Accordingly, we assume a regular perturbation expansion for C in terms of and Pe 1 i.e., ) C C ( ) Pe ( 1 ) C C ( Pe 1 ) C C ( C2 ) 2 ( 2 ) 0 ( 2 2 2 ) 2 ( 1 ) 0 ( 1 2 ) 2 ( 0 ) 0 ( 0 (3 12) The boundary conditions for C in the convective step are the same as in the diffusive step.
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35 The leading order equation for C in Pe 1 is 0 X ) C C ( U ) C C (2 ) 2 ( 0 ) 0 ( 0 x 2 ) 2 ( 0 ) 0 ( 0 (3 13) In the above equation, we transform the X coordinate to U X x (3 14) As a result, Eq. (313) becomes diff 2 ) 2 ( 0 ) 0 ( 0 2 ) 2 ( 0 ) 0 ( 0 2 ) 2 ( 0 ) 0 ( 0C ) 0 ( C ) 0 ( C ) ( C ) ( C 0 ) C C ( (3 15) Using Eq. (310) in Eq. (315) gives ) 4 Y exp( ) U X ( 1 ) X ( Cd 2 x i d ) 0 ( 0 ) 4 Y exp( X ) U X ( ) X ( Cd 2 d 2 x i 2 ) 2 ( 0 (3 16) Similarly, by solving the equa tions for various orders of and 1/(Pe ), we get ) 0 ( 1C, ) 2 ( 1C, ) 0 ( 2Cand ) 2 ( 2C. Substituting them into Eq. (312) gives
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36 2 d d 2 2 d 2 d 2 d 2 x i 2 d d 2 2 d 2 d 2 x i 2 3 d d 2 2 d 2 x i 2 2 2 d d d 2 2 2 d 2 d x i d d d 2 2 d 2 d x i 3 d d d 2 2 x i 2 d d d 2 x i 2 2 d 2 2 x i 2 d d 2 2 d 2 d d 2 x i 2 2 2 d 2 d d 2 xi d d 2 2 x i 2 d 2 d 2 x i d ) 4 Y exp( Y ) 4 Y 2 1 ( X ) U X ( 8 1 t ) 4 Y exp( ) 4 Y 2 1 ( X ) U X ( 4 1 ) 4 Y exp( Y X ) U X ( 8 1 ) 4 Y exp( Y ) 4 Y 2 1 )( U X ( 8 1 ) 4 Y exp() 4 Y 2 1 )( U X ( 4 1 ) 4 Y exp( Y ) U X ( 4 1 ) 4 Y exp( ) U X ( 4 1 ) Pe ( ) 4 Y exp( X ) U X ( 1 ) 4 Y exp( ) 4 Y 2 1 ( X ) U X ( ) 4 Y 2 1 )( 4Y exp( ) U X ( 1 Pe ) 4 Y exp( X ) U X ( ) 4 Y exp( ) U X ( 1 ) 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 ( u 6 u2 x (3 18) where 2h x p 3 1 u is the mean velocity in the channel. In dimensionless form, ) Y Y ( 6 U2 x (3 19) Substituting Ux in Eq. (317), one can determine the concentration profile during the convective step. Electric field step (Ele ctric field, no Flow) The concentration profile at the end of the second step can be calculated by substituting f in Eq. (317). In the third step, the el ectric field is applie d to attract all the
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37 molecules to near the wall. Neglecting axial diffusion during this step, the surface concentration after the end of the i+1st cycle is 1 0 f 1 idY ) Y X ( C ) X ( (3 20) If the convective distance traveled in each cycle is much smaller than the axial length scale, then the expression for C in Eq. (317) can be expanded by using Taylor series, i.e., ... X 2 U X U ) X ( ) U X (2 i 2 2 x i X i x i After using the above expansion in Eq. (317) and then substituting the expression for C in Eq. (320), and then performing the integration gives 2 i 2 i i 1 ix D x U (3 21) where X and have been replaced by x/l and h/l respectively, and h 2 Pe D t 3 ) h ) 2 1 ( Peerf D t 12 h Pe D t 6 ( ) h ) 2 1 ( DPeerf t 12 h DPe t 12 ( U5 2 / 3 d 3 3 f 3 d 2 2 f 3 2 / 1 d 2 2 f d f d f 2 / 1 d (3 22) ) h ) 2 1 ( erf D Pe t 216 h D Pe t 108 ( )) 2 1 ( erf D t ( ) h ) 2 1 ( erf D Pe t 432 h D Pe t 432 h ) 2 1 ( erf D t Pe 36 ( ) h ) t 2 1 ( erf ( ) h ) t 2 1 ( erf D Pe t 216 h D Pe t 288 h ) 2 1 ( erf D t Pe 36 ( D6 d 4 2 4 f 6 4 2 4 f 2 / 1 d d f 4 d 3 2 3 f d 4 3 2 3 f 2 / 1 d 4 d 3 3 f 2 2 d d 2 d 2 2 2 f 2 d 2 2 2 2 f 2 / 3 d 2 d 2 2 f 2 d (3 23)
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38 If i is known, then Eq. (321) can be used to determine i i, i.e., the surface concentration at the end of i+1st cycle. Since 0 is known, by repeating this process one can numerically obtain the surface concentra tion as a function of x and the number of cycles. The above equations are only valid for d<1/20, thus the errorfunction ( d2 1 erf) can be simply replaced by 1.0. In the above derivation, it was assumed th at the colloidal part icles 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 intens ity of the electric fi eld, 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 st ep is governed by the convectiondiffusion equation where the convective term arises due to the lateral electric field, i.e., 2 2 e yy c D y c u t c (3 24) where e yu 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, E ur 0 e y, where r and are the fluids dielectric constant and viscosity, respectively, is the permittivity of vacuum, and is the zeta potential of the colloidal particle. Alternatively, the electroph oretic velocity can be expressed as E ue e y, where
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39 e 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 char ge, the electrophoretic velocity can equivalently be expressed as y kT Z D ut e y where Tt is the absolute temperature, e and Ze are the charge on an electron and on the partic le, respectively. The effective particle charge is in general less than the actual charge due to the electric double layer surrounding the ion. Ho wever, 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 e yu only for an approximate estimation of the field required to attract all the molecules near the wall, we use the simpler expression y kT Z D ut e y The steady state solution to Eq. (324) is ) D y u exp( ) 0 y ( c ce y (3 25) To attract most of molecules into h/ 100 of the plate, a field satisfying 3 ~ D 100 h ue y must be used. This gives Ze kT 300 h D 300 y kT Ze D ut t e y (3 26) Assuming Z ~ 10, which is a very conservative assumption, givesV 77 0 Later we use a value of about 33 m for h, and a potential drop of .77V across a 33 m channel is about the same voltage as is applied in EFFF [15]. Additionally, under this electric field the steady state will be atta ined in a time of about D 300 h ~ u h2 e ywhich is much less than the diffusive time and thus the assumption of ne glecting diffusion duri ng the third step is
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40 reasonable. Furthermore, for h = 33 m and D = 1010 m2/s, the time for attaining steady state is about 3 ms, which is less than the time scale for charging a double layer [3,38]. Thus, gas generations may not be a problem in the third step and a majority of the applied potential difference occurs in the bulk of the channel. Long time Analytical Solution To better understand the physic s of the separation and to avoid repetitive numerical simulations, we also obtain an analytical solution for the surf ace 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 into Taylor series in terms of time 2 i 2 2 d f i d f i d f 1 it ) t t ( 2 1 t ) t t ( ) t t ( (3 27) Using ) t t (d f as time scale gives the following dimensionless equation 2 i 2 i i 1 iT 2 1 T (3 28) Also as shown above 2 i 2 i i 1 ix D x U (3 29) Substituting Eq. (328) into Eq. (329) gives 2 i 2 i 2 i 2 ix D x U T 2 1 T (3 30) We again define a new coordinate system, T U x' (3 31) In this moving reference frame Eq. (330) becomes
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41 2 2 2 ' 2 2 2 ) 2 U D ( T U T 2 1 T (3 32) Where, the subscript has been removed. We sh all show later that th e long time solution to the above equation is Gaussian, i.e., ) DT 4 exp( T A 2 (3 33) Thus, ) T ( O ~ T ) T ( O ~ T ) T ( O ~ ) T ( O ~ T 2 2 2 5 2 2 2 3 2 2 2 3 (3 34) Keeping the leading order terms in Eq. (332) gives 2 2 2 ' ) 2 U D ( T (3 35) Transferring it into the original coordinates gives 2 2 2 ' 'x ) 2 U D ( x U T (3 36) Thus, the long time surface concentration is a Gaussian with the dimensional mean velocity *U and effective diffusion coefficient *D given by d f 2 ' d f *t t 2 U D D ; t t U U (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
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42 2 f 2 ' f *t u 2 U D D ; t u U U (3 38) Results and Discussion Since d must be smaller than 1/20, the value of the error functions in Eq. (322) and Eq. (323) are very close to 1, thus the expressions for' U and D can be rearranged in the following form: ) ( 3 U ) t t ( ) ( 2 U t t ) ( 1 U 2 3 ) t t ( ) 12 6 ( t t ) 12 12 ( t u U Ud 2 d f d d f d 2 / 1 d 2 d f d 2 / 1 d d f d 2 / 1 d f (3 39) 5 D ) t t ( ) ( 4 D ) t u ( h t t ) ( 3 D t t ) ( 2 D ) t u ( h ) ( 1 D ) 216 108 ( ) t t ( ) ) t u ( h ( t t ) 432 432 36 ( t t ) ) t u ( h ( ) 216 288 36 ( ) t u ( D D 2 d f d 2 f 2 d f d d f d 2 f 2 d 2 d 2 / 3 d 2 d f 2 f 2 d d f 2 d 2 / 3 d d d f 2 f 2 d 2 d 2 / 3 d d 2 f (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 ( 2Cdoes not contribute to either U or D. Each of these terms depends only on d and accordingly U depends strongly on d and weakly on d ft t. Also D depends strongly of d and weakly on d ft t and 2 f 2t u h The truncations errors in U and D are
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43 ) ) t t (( O U for error Truncation3 d f ) ) t t ( ) ) u t ( h (( O ) ) t t (( O D for error Truncation3 d f 2 f 3 d f (3 41) We note that for the proposed regular expansion solutions to be valid 1 ) u t ( h and 1 t tf d f (3 42) Mean Velocity Figure 32 plots the dependency of the dimensionless mean velocity on G ( d1 ) for different values of tf/td. When G approaches zero, i.e ., as the diffusion time becomes very large, the concentration profile along th e lateral direction b ecomes 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 beca use of the requirement of Eq. (310). But this trend can be observed as G approaches 20. Figure 32 shows that a decrease in G results in an increase in the mean velocity of the pulse. This happens because smaller G implies larger molecular diffusivity for a fixed td and h. Since molecules with larger D diffuse a longer distance away from the wall, th ey are convected with a larger velocity. However, beyond a certain D, some molecule s 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 fo r this effect, and thus the mean velocity curve exhibits no stationary extremum.
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44 In Figure 32, the difference between the curves corresponding to different values of tf/td indicates the contribution from the hi gher order terms in the expression for *U (Eq. (338), (339)). The comparison of th ese 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 th at 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 32. Dependency of *U on G Dispersion Coefficient Figure 33 plots the dependency of the dimensionless dispersion coefficient *Don G for different values of tf/td and 2 f 2t u h The difference between the curves corresponding to different values of tf/td indicates the effect of tf/td on dispersion.
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45 Similarly the difference between the curv es corresponding to different values of 2 f 2t u h indicates its effect on dispersi on. Figure 33 shows that 2 f 2t u h has a negligible effect on dispersi on and that the effect of tf/td on dispersion is comparable to its effect on the mean velocity. Figure 33. Dependency of *D on G From Figure 33, we see that the effec tive 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 partic les still move with velocity near zero. As a result, after a cyclic operation, the particles spread further.
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46 Separation Efficiency Since the mean velocity of molecules de pends 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 partic les 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., ) ) t t ( T D 4 ) t t ( T D 4 ( 3 ) t t ( T ) U U (d f 2 d f 1 d f 2 1 2 2 1 2 1) U U ) D D ( ( 12 T (3 43) We use Eq (343) to calculate T, i.e., th e dimensionless time or, equivalently, the number of cycles needed for separation. The dimensional time required for separation is equal to T(tf + td), i.e., 2 2 1 2 1 f d f) U U ) D D ( ( t t 1 12 t (3 44) The length of the channel required for separa tion is equal to the distance traveled by the faster moving molecule in this time, i.e., 2 2 1 2 1 f 1 1) U U ) D D ( ( h t u U 12 h L U T L (3 45)
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47 The dimensionless separation time (tf) and channel length (L/h) depend on G1, d ft tand 2 f 2t u h and G2/G1. Below we discuss this dependence for a G2/G1 = 2. Figure 34. Effect of G1 ( 2 0 t td f, 2 0 t u h2 f 2, G2/G1=2) on L/h, /tf and T Effect of G Figure 34 shows the dependence of T, /tf and L/h on G1 for fixed values of d ft t, 2 f 2t u h and G2/G1, which are noted in the caption. With increasing G1, the number of loops, time and the length needed for separati on first decrease and then level off. To
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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 bett er separation, resulti ng 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 35 shows the dependence of T, /tf and L/h on d ft t for fixed values of G1, 2 f 2t u h and G2/G1, which are noted in the caption. Fi gure 35 shows that the number of loops required for separation is relatively independent of d ft t. With an increase in d ft t, the mean velocities and also the difference in the mean velocities increase but this effect is compensated for by an increase in the disp ersion coefficients, and thus the number of loops required for separation does not cha nge appreciably. However Figure 35 shows that the time required for separa tion depends strongly on the ratio d ft t. This happens because although the number of steps is unchanged, td decreases as d ft tincreases and this leads to a reduction in the time for each step and consequently a reduction in time for
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49 separation. The length required for separation increases with d ft tbecause of the increase in the mean velocities of both the species. Figure 35. Effect of d ft t (G1=100, 2 0 t u h2 f 2, G2/G1=2) on L/h, /tf and T Effect of 2 f 2t u h Figure 36 shows the dependence of T, /tf and L/h on 2 f 2t u h for fixed values of G1, d ft tand G2/G1, which are noted in the caption. As noted earlier the mean velocity is independent of 2 f 2t u h and the dispersion coefficient depends weakly on this ratio.
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50 Accordingly, the number of loops required for separation is not expected to depend on 2 f 2t u h as shown in Figure 36. In Figure 36, /tf is plotted on the yaxis, 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 ft tis 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 2t u h because an increase in the xaxis is equivalent to a reduction in ft u, which leads to a linear reduction in the mean velocity. Figure 36. Effect of 2 f 2t u h (G1=100, 2 0 t td f, G2/G1=2) on L/h, /tf and T
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51 The dependency of T, and L on the three dimensionless numbers remains the same for G2/G1=1.2. However, the actual values incr ease significantly. The description above shows that the optimum values of G, d ft tand 2 f 2t u h 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 dimens ional parameters, as shown below. Let us consider separation of two types of molecules with D1=1010 m2/s and D1/D2=2. It is clear that a smaller tf will lead to a reduction in separation time. However, the minimum value of tf is limited by the time in which the flow can be turned on and off in the channel. Rather than turning the pum p 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 ft t=0.3, td is about 0.067 s. By using G = 150 and 2 f 2t u h =0.3, the values of h and are 32 m and 0.003 m/s, respectively. The valu es 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 and L are reduced by an increase in G. However, if the diffusion coefficient is about 1012 m2/s, based on StokesEinstein equa tion the particle size is about 0.2 m and in this case the proposed continuum model is not valid. An alternate model
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52 that takes into account the fin ite particle size may need to be developed to determine the effectiveness of the proposed techniqu e at separating particles with D < 1012 m2/s. On the other extent, as D becomes larger, the chan nel height h must be increased to ensure that G does not becomes smaller than about 50. An increase in h lead s to an increase in L. For instance, separation of molecules for D1=109 m2/s for G2/G1=2 takes about 17.5 s in a channel 1 cm in length and 58 m in height. The time and length become 269 s and 16 cm for G2/G1=1.2. The separation can be signifi cantly 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 sim ilar 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 depe nd on the size and charge of th e molecules. As mentioned above, the electric fields that are applie d in EFFF are limited to about 1 V/ 10 m. Also only about 1% of the applied el ectric 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 e yu due to the electric field is estimated by the equation E ue e y where eis the electrical mobility of the particles. The value of e has been measured for various types of colloidal particles. It can also be determined by the Smoluchowski equation, r 0 e, (r and are the fluids dielectric cons tant and viscosity, respectively, is the permittivity of vacuum, and is the zeta potential of the colloidal particle). The mobility of polystyrene latex particles is rela tively independent of size and varies in the
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53 range of 1.9x104 3.23 x104 cm2/(Vs) for particle diameters in the range of 90 nm944 nm [2]. For smaller particles the mobility can be estimated by treating them as point charges and thus ecan be expressed as kT Z D where D and Z are the diffusivity and the charge of the particle. For D = 1010 m2/s and Z =10e (e = electronic charge), the mobility is about 4 x104 cm2/(Vs). At these mobilities a field of 1000 V/m will drive a lateral velocity of the order of 20 m/s. We note that in our proposed technique most of the applied field is active because the double la yers are not charged and thus the electrical velocity can be as large as 2000 m/s, which as shown earli er can attract all the molecules in a very thin layer in a shor t 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 m, 1010 m2/s and 2, respectively. The value of tf and are 0.02s and 2mm/s, respectively, and the value of G is varied from about 30 to 400, which is equivalent to varying td from 0.3 to 0.0225 s. The value of e yu is varied from 0 100 m/s which is much larger than the expected values of the lateral electric velocity. In Figure 37 the value of L/h is plotted as a function of G and e yu for both the techniques. The multiple curves for the EFFF correspond to different values of the mean ve locity. In EFFF the reduction in the mean velocity reduces the length required for separation because of the reduction in the convective contribution to the di spersion. The time (Figure 38) required for separation does not change appreciably because both the length required for separation decreases almost linearly with the velocity. The tre nd of reduction in L/h w ith reverses at Pe<15 because although the convective contribu tion to dispersion still decreases, its value
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54 is less than the diffusive contribution, a nd thus the overall disp ersion does not decrease significantly, and the reduction of the mean velo city with a reduction in leads to an increase in L/h.. As shown in Figure 37 th e length of the channel required for separation reduces with increasing e yuand the length required by EFFF at the optimal mean velocity becomes less than that require d by the pulsatile technique for e yu > 30 m/s, which is larger than the expected va lue of the lateral velocity. Also, the time required for separation is less for the pulsa tile technique. Figure 39 and 310 are very similar to Figures 37 and 38; 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 a nd time for separation increase if D1/D2 is smaller. Figure 37. Depe ndency of L/h on G1(pulsating electric field) and e yu(constant electric field). Solid lines(Constant EFFF): h=30m, D=1010m2/s, the value of are noted on the curves, D1/D2=2; Dashed line(Pulsating EFFF): h=30 m, D=1010 m2/s, =0.002 m/s, D1/D2=2, tf=0.02 s
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55 Figure 38. Dependency of the operating time t on G1(pulsating electric field) and e yu(constant electric fiel d). Constant EFFF: h=30m, D=1010m2/s, the value of are noted on the curves, D1/D2=2; Pulsating EFFF: h=30 m, D=1010 m2/s, =0.002 m/s, D1/D2=2, tf=0.02 s Figure 39. Depe ndency of L/h on G1(pulsating electric field) and e yu(constant electric field). Solid lines(Constant EFFF): h=30m, D=1010m2/s, =0.002, 0.001, 0.0005, 0.00005 m/s, D1/D2=1.2; Dashed line(Pulsating EFFF): h=30 m, D=1010 m2/s, =0.002 m/s, D1/D2=1.2, tf=0.02 s
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56 Figure 310. Dependency of the operating time t on G1(pulsating electric field) and e yu(constant electric fiel d). Constant EFFF: h=30m, D=1010m2/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 m, D=1010 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 pulsa tile lateral fields a nd an axial flow that varies in the lateral direction. The method is similar to the EFFF, which also relies on lateral electric fields for se paration. In EFFF only a very small fraction of the applied fields acts on the particles and the double la yers consume the remaining field. In the pulsatile technique because the time for which the field is app lied is smaller than the time needed for charging of the double layers, the majo rity 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
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57 and off; the separation impr oves with a reduction in tf and td, which are the durations of the flow and the noflow steps. For reasona ble value of design constants, the proposes technique can separate molecules of diffusivities 1010 m2/s and 0.5x1010 m2/s in 15.7 s in a 3.7 mm long channel. The length and the time increase to 5.45 cm and 231 s if the ratio of the diffusivities is reduced from 2 to 1.2. The separation is easier for larger molecules; however, the model predictions may not be realisti c due to the finite size of the particles. If the diffusivities ar e in the range of 109 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.
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58 CHAPTER 4 TAYLOR DISPERSION IN CYCL IC ELECTRICAL FIELDFLOW FRACTIONATION This chapter aims to determine the mean velocity and the dispersion coefficient of charged molecules undergoing Poiseuille flow in a channel in the presence of oscillating lateral electric fields. Application of time periodic fields in EFFF techniques was first proposed by Giddings[43] and later explor ed by Shmidt and Cheh[44], Chandhok and Leighton[45] and Shapiro and Brenner[49,50]. In EFFF, particles with same values of e yu / D cannot be separated, where D is the molecular diffusivity and e yu is the electric field driven velocity on the late ral direction. Giddings sugge sted that cyclical electrical fieldflow fractionation (CEFFF) can accomplish separation even in this case. Based on this idea, Giddings developed a model for CEFFF under the assumption that the molecular diffusivity can be neglected while calculating the concentration profile in the lateral direction. Shmidt and Cheh[44] and Chandhok and Leighton[45] extended the idea proposed by Giddings to develop novel techniques for continuous separation of particles by introducing an oscillating flow that is perpendi cular 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. Shap iro and Brenner analyzed the cyclic EFFF for the case of square shaped electric fields. They include d the effects of molecular diffusion in their model and obtained expressions for axial velo city 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 / u t T0 0, where t0 is the time period of oscillation,
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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 a nd 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 so lve the convection diffusion e quation for cyclic EFFF by a multiple time scale analysis to determine the expressions for the mean velocity and the dispersion coefficient. Next, we examine th e effect of the system parameters on the concentration profiles and the mean velocity an d the dispersion coefficient for the case of sinusoidal electric fields. Finally, we com pute the mean velocity and the dispersion coefficient for the square wave and compare th e results with the asymptotic analysis of Shapiro and Brenner. Theory Consider a channel of length L, height h a nd infinite width that contains electrodes for applying the lateral periodic lateral electri c 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.,1 L / h The transport of a solute in the channel is governed by the convectiondiffusion equation, 2 2 2 2  e yy c D x c D y c u x c u t c (4 1)
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60 where c is the solute concentration, u is th e fluid velocity in the axial (x) direction, D and D are the diffusion coefficients in the dir ections parallel and perpendicular to the flow, respectively. We assume that the di ffusivity tensor is isotropic and thus D =D = D. In Eq.(4 1), e yu 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 e yu can be determined by the Smoluchowski equation,E ) / ( ur 0 e y, where r and are the fluids dielectric constant and viscosity, respectively, is the permittivity of vacuum, and is the zeta potential. Or, it can be simplified as E uE e y where E is the electric mobility which has been measured for a number of different types of colloidal particles, e.g., the mobility of DNA beyond a size of about 400 bp is 3.8x108 m2/(Vs).[22] In EFFF, researchers have applied an effective electric field of 100V/cm without gas generation. Thus, typical values of e yu could be as large as 3.8x104 m/s. Eq. (4 1) is subjected to the boundary c ondition of no flux at the walls (y = 0,1), i.e., 0 c u y c De y (4 2) In a reference moving in the axial direction with velocity *u, Eq. (4 1) becomes ) y c x c ( D y c u x c ) u u ( t c2 2 2 2 e y (4 3) where x is now the axial coordinate in the m oving frame. For a sinusoidal electric field )) t sin( E E (max the lateral velocity is t sin R u V uE e y (4 4)
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61 where is the mean velocity and R is the dimensionless amplitude of the lateral velocity, which is given by u / Emax E. The above equation assumes that the solution is dilute in electrolyte and the collo idal 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 ch annel and thus neglect s the presence of the electrical double layer. Inclusion of th e double layers signific antly 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 ) h y x Dt t ( C C l l2 (4 5) where ) 2 /( is the frequency of the applied field, 1/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 2 2 e y s l 2Y C X C Y C PeU X C ) U U ( Pe T C T C (4 6) where t Ts, l/ x X, h / y Y 2 l/ Dt Tl D h u Pe D h 2, u u U u u U* u u Ue y e yand 1 l h Since <<1, the concentration profile can be expanded in the following regular expansion. 0 m l s m m) T Y X T ( C C (4 7)
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62 Substituting Eq. (4 7) into Eq. (4 6) gives ) ( Y C Y C Y C X C Y C PeU Y C PeU Y C PeU X C ) U U ( Pe X C ) U U ( Pe T C T C T C T C 3 2 2 2 2 2 1 2 2 0 2 2 0 2 2 2 e y 2 1 e y 0 e y 1 2 0 s 2 2 s 1 s 0 l 0 2 (4 8) Eq. (4 8) can be separated into a se ries of equations for different order of (0 ): 2 0 2 0 e y s 0Y C Y C PeU T C (4 9) The solution for C0 can be decomposed into a produc t 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 ( G Cl s 0 0 where G0 satisfies 2 0 2 0 e y s 0Y G Y G PeU T G (4 10) The above equation is subjected to the following boundary condition at Y = 0, 1: 0 e y 0G PeU Y G (4 11) Next, we solve the equations at the order of To order the governing equation (4 8) becomes (1 ): 2 1 2 1 e y 0 s 1Y C Y C PeU X C ) U U ( Pe T C (4 12)
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63 Integrating the above equation from 0 to 1 in Y and 0 to Ts and noting that 0 dT ) T / C ( 2 0 s s 1 due to periodicity and 1 0 2 1 2 1 0 1 e ydY ) Y / C ( dY ) Y / C ( PeUdue to the boundary conditions gives 2 0 1 0 s s 0 2 0 1 0 s s 0 *dYdT ) T Y ( G dYdT ) Y ( U ) T Y ( G U (4 13) The solution to C1 is of the form ) X / ) T X ( A )( T Y ( Bl s where B satisfies 2 2 e y s 0 sY B Y B PeU ) T Y ( G ) U U ( Pe T B (4 14) and the following boundary conditions: B PeU Y Be y (4 15) Since we are only interested in the periodi csteady solution to Eq. (4 10) and (4 14), these differential equations can be solv ed 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 / 15 0 in all simulations. The spatial grid size near the wa ll 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 a nd were also compared with the results of the analytical approach presented in Appe ndix B. The simulations are run for times larger than the time required to obtain periodic steady behavior.
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64 We now solve the O( 2) problem. To order the governing equation (4 8) becomes (2 ) 2 2 2 2 0 2 2 e y s 2 1 l 0Y C X C Y C PeU T C X C ) U U ( Pe T C (4 16) Averaging both sides in Ts and Y gives, 2 0 2 1 0 2 0 s 2 2 l 0X C dYdT X A B ) U U ( Pe T C (4 17) where 1 0 2 0 s 0 0dY dT C C (4 18) Rewriting Eq. (4 17) gives 2 0 2 1 0 2 0 s 2 0 2 l 0X C BdYdT ) U U ( Pe 1 X C T C (4 19) where 1 0 2 0 s 0dY dT G Now we combine the results for ) T / C (s 0 and ) T / C (l 0 l 0 2 s 2 2 s 1 s 0 0T C l D T C T C T C t C (4 20) Averaging the above equation in Y and Ts and using periodicity gives, l 0 2 0T C l D t C (4 21) Using Eq. (4 21) in Eq. (4 19) gives
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65 2 0 2 0x C DD t C (4 22) where the dimensionless disper sion coefficient is given by 2 0 1 0 s s 0 2 0 1 0 s s *dYdT ) T Y ( G dYdT ) T Y ( B ) U ) Y ( U ( Pe 1 D (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) a nd (4 23) to obtain the mean velocity and the effective dispersivity, respectively. Additionally, to validate the numerical results we solve Eq. (4 10) and (4 14) analytically. The analytic computations are straightforward but tedious and are outlined in Appendix B. Results and Discussion Below we first describe the results for s quare wave electric field and compare the results with the asymptotic resu lts obtained by S&B, and then we describe the results for the sinsusoidal fields. Finally, the results fo r both shapes of electric fields are compared. Square Wave Electric Field As mentioned in the introduction, S&B de termined 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 larg e PeR and showed that results for both the mean velocity and the dispersion coefficient depend on only a single parameter h / u t T0 0 where t0 is the time period of oscillation, u0 is the amplitude of the lateral velocity and h is the channel height. This dime nsionless parameter is identical to 2PeR/ in terms of the parameters defined in this paper. For the case of square
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66 wave, we can solve (4 10) and (4 14) numeri cally 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 f R where f is simply a square wave function that oscillates from 1 to 1 with a dimensionless angular frequency of Below we compare the results of our simu lations for the case of a square shaped lateral electric field with the asymptotic result s of S&B. First the transient concentration profiles are compared with the asymptotic solu tions and then the mean velocities and the dispersion coefficients are compared. Transient concentration profiles In the case of 2 T( h / u t T0 0), the asymptotic concentration profiles that were predicted by S&B (Figure 4. of Ref 50) co rresponds to a uniform probability outer solution of width 2 / T 1 that executes a periodic moti on 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 beginni ng and the end of each cycle and touch the upper wall at midway in the cycle. The inner solution is zero everywhe re except in a thin region near the edge of the outer so lution. The numerical calculations for 2 Tare shown in Figures 41a and these show that th e numerical solutions for the concentration transients agree with the asymptotic soluti ons. In Figure 41a, 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.
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67 Figure 41. Periodic steady c oncentration profiles during a period for a square shaped electric field for (a) PeR = 80, = 1500, 2 PeR/ = 0.335 and (b) PeR = 400, = 200, 2PeR/ = 4. As T becomes larger than 2 (Figure 41b), the duration of the time in which the lateral field is constant is long enough for th e lateral concentration profile to attain a steady state, which is an exponentially decay ing concentration from the wall. As the direction of the field switch es, the exponential pr ofile 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 stat e of a decaying exponential. The asymptotic analysis of S&B predicted the same behavior.
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68 Mean velocity and dispersion coefficient Figures 42a and 42b compare the numerical results for mean velocity and dispersion coefficient with those obtained by S&B. In Figure 42a and 42b the thick solid lines correspond to the asymptotic result s that were obtained by S&B and the thin solid lines correspond to the r esults of the numerical simu lations. The markers on the curves in Figures 42a and 42b and all the subsequent figures co rrespond 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 bot h the mean velocity and the dispersion coefficient match the results from the Br ownian dynamics simulations. The numerical results for both the mean velocity and the dispersion coefficient agree with the asymptotic expansions for 2PeR/ > 2. The agreement is better for larger Pe, which is expected because the asymptotic expansions are valid for large Pe. For the case of 2PeR/ 2 the numerical results approach the as ymptotic results but do not reach the asymptotic limit for as large as 2000. However, based on the trends it can be concluded that for higher values of the numerical results will match the asymptotes obtained by S&B. It is also noted that the kink in Figure 42b at PeR 2 T = 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 th e band being focused by the nearest wall. In the high frequency (negligible diffusion) limit, at 2 T, the entire solute band travels as a delta function and thus there is no spread, and hence no dispersion.
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69 Figure 42. Comparison of the numerically computed (a) mean velocity and (b) dispersion coefficient for a square shap ed electric field with the large Pe asymptotes obtained by S&B (Thick line). The markers on each curve represent the results calcul ated by Brownian dynamics.
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70 Sinusoidal Electric Field Below, some of the results from the anal ytic calculations ar e described, followed by results from the numerical calculations, a nd comparison of the results from these two approaches. Analytical computations Symmetry in the concentration profile. Since the lateral velo city is sinusoidal (=Rsin(Ts)) and the axial flow and the boundary conditions are symmetric in Y, the concentration profile is expect ed to satisfy the following sy mmetry in the long time limit ) 1 Y T ( C ) Y T ( Cs s (4 24) Accordingly, both C0 and B satisfy the same symmetry. As shown in Appendix B, C0 and B be expanded as ))] nT cos( ) Y ( g ) nT sin( ) Y ( f ( ) Y ( g [ const )) nT cos( ) Y ( q ) nT sin( ) Y ( p ( ) Y ( q ) Y T ( B ) T X ( A ~ )) nT cos( ) Y ( g ~ ) nT sin( ) Y ( f ~ ( ) Y ( g ~ Cs n 1 n s n 0 s n 1 n s n 0 s l s n 1 n s n 0 0 (4 25) Substituting C0 from the above equation into Eq. (4 24) gives )) n cos( ) 1 ( g ) 1 ( ) n sin( ) 1 ( f ) 1 (( ) 1 ( g ))) ( n cos( ) 1 ( g )) ( n sin( ) 1 ( f ( ) 1 ( g )) n cos( ) ( g ) n sin( ) ( f ( ) ( gn n 1 n n n 0 n 1 n n 0 n 1 n n 0 (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
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71 3, in which the functions fi are plotted as a function of Y for N = M = 5 and Pe = R = 1 and =100. The results are similar for gi, pi and qi (plots not shown). Figure 43. gi vs. position for PeR=1, and =100. Convergence of the series expansions. As shown in Appendix B, the equations for obtaining G0 and B analytically are two hierarch ies 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 ta ken to be larger than 5, the coefficients of the fifth terms are about 106, 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 106. 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 ( ) in the expansions for fi, gi, pi and qi also become larger on increasing PeR and thus some of these functions
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72 grow exponentially in Y, and become very larg e 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. Howeve r 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 compar ison with analytical results Effect of PeR and on the temporal concentration profiles. Figures 44ad show the concentration profiles at various time instances during half of a period. In Figure 44a, the value of PeR is 100, and thus most of the molecules aggregate in a thin boundary layer near the wall. The thickness of the boundary layer changes as the field changes during the period. The concentration profiles are not in phase with the driving force as evident by the fact that at ts = 0, 2 the field is zero, but the concentration profile is far from uniform, and that th e wall concentration ke eps increasing beyond ts = 3 /2, even though the field begin to decrease. The profiles in Figure 44b correspond to the same value of PeR as in Figure 44a but a much small value of 1 for Since PeR is still large, the boundary layer w ith time varying thickness still forms but in this case the profiles are almost in phase with the drivi ng electric field due to the small value of Accordingly, at ts = 0, the concentration profile is relatively inde pendent of position, and the wall concentrati on is the maximum in time and the boundary layer thickness is a minimum at ts = 3 /2. Figure 44c and 44d correspond to PeR = 1, and of 1 and 10, respectively. Since PeR is small, a boundary la yer does not develop in both of the cases. In Figure 44c, the concentrati on profiles are not exactly in pha se as evident from the fact
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73 that the profiles for ts = 0, do not overlap, but the profiles ar e closer to being in phase with the driving force than those in Figure 44 d that correspond to PeR = 1 and = 10. Figure 44. Time dependent concentration prof iles within a period for sinusoidal electric fields for (a)=100, Pe =100, R=1; (b)=1, Pe=100, R=1; (c)=10, Pe=1, R=1; and (d)=1, Pe=1, R=1. Effect of PeR and on the mean concentration profiles. The effects of PeR and on the time averaged concen tration are illustrated in Figures 45ae. The shorttime averaged concentration is equal to) T X ( A ) Y ( gl 0, where 2 0 s s 0 0dT ) T Y ( G gand g0 is plotted as a function of Y in the plots belo w. In Figure 45a and 45b the function g0 is plotted for various values of PeR for = 20 and 100, respectively.
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74 Figure 45. Time average concentration profiles for sinusoida l electric field for (a)=20; (b) =100; and Pe=R=1 for ranging from (c) 120, (d) 40100, and (e) 1001000. Figures 45ae show that as expected the mean concentration profiles are symmetric in Y. In Figures 45a and 45b, due to the presence of the electric field, the
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75 particles accumulate near the wall, leading to a higher concentration at the boundaries. In Figure 45a, the concentration profile in the center is relatively flat and the value of g0 in this central region increases on reducing PeR. However the profile s in Figure 45b show that for = 100, a maxima develop in the central re gion, when PeR is less than about 40. The effect of on g0 is further illustrated in Figures 45ce for PeR = 1. The values of span from 1 to 20 in Figure 45c, from 40 to 100 in Figure 45d and from 1001000 in Figure 45e. For 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 beyond 40, a secondary maximum develops in the center but the maximum concentration is still at the wall. On increasing further, the value of g0 at the maximum in the center overshoots the value at the walls, which has be en assigned to be e qual to 1 as a boundary condition. Under these conditions, due to th e 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 ar e 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 For fixed channel geometry and for a given sample, Pe can be changed by adjusting the mean veloc ity of the axial flow, R can be changed by adjusting the magnitude of the periodic electric field, and can be changed by varying the frequency of the periodic electric field. The mean velocity is only a function of PeR and and the dispersion coefficient is of the form Pe2 f(PeR, Typical microfluidic channels are about 2040 m thick and as stated earlier th e lateral electric velocity uy e
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76 could be as large as 3.8x104 m/s and this implies a value of about 1200 for PeR for a D value of 1011 m2/s. It is also noted that typical ch annel lengths are a bout a 12 cm and thus the value of is about 103. Accordingly, for our analysis to be valid the values of PeR and should be much less than about 1000. We now discuss the effect of these parameters on the mean velocity and the dispersion coefficient. Figure 46. Dependence of *U on PeR for (a) PeR ranging from 0 to 10 and ranging from 120; and (b) PeR ranging from 0 to 200 and ranging from 1100, and comparison with the small asymptote (thick line). The markers on each curve represent the re sults calculated by Brownian dynamics.
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77 Figure 46ab plots the dependency of th e mean velocity on PeR for different values of Figure 46a shows the results for PeR< 10 and the data represented in this plot was calculated from the analytical solu tions described in Appe ndix B. In Figure 46b, the values of PeR range from 1200 and th e data shown in this figure was calculated by the numerical approach described above. It is noted that the results from both the methods match for PeR values of around 10, which validates the accuracy of the numerical scheme. The markers on the curves in Figures 46ab that are the results of the Brownian dynamics simulations also match the results computed by finite difference. As shown in Figures 46ab, the mean veloci ty decreases as the product of Pe and R increases. When PeR increases, the particle s experience a larger force in the lateral direction, which pushes them closer to the walls, and consequently reduces the mean velocity. Figure 46ab also shows th e dependence of the mean velocity on as increases, the curve of the mean velocity shifts up. This is due to the fact that as increases, the electric field changes its di rection 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 mi ddle 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 As a result, on increasing 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 di spersion coefficient is of the form ) PeR ( f Pe 12 Figures 47ab plots 2 *Pe / ) 1 D ( as a function of PeR for different values of As for the case of mean velocity, Fi gures 47a and 47b were computed by
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78 the analytical and the numerical methods, respectively, and the results from both the methods merge smoothly for PeR values of ar ound 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 el ectric field is close to zero, the effective diffusivity is expected to approach the value of the Taylor dispersivity for Poiseuille flow through a channel. Figure 47a shows that as PeR approaches zero, the curves of 2 *Pe / ) 1 D ( for all values of frequency approach the expected limit of 1/210. On the other hand, as PeR goes to infinity, which corr esponds to an infinite magnitude of electric field, particles will spend more time in a very thin layer close to the walls. Thus, the effective diffusivity of the particles approaches the molecular diffusivity. Figure 47a also shows that for small the curves exhibit a maximum at PeR = 4. This phenomenon also occurs in constant electric fieldflow fractionation. In the constant EFFF, at small PeR, the particle concentrati on 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 deceler ation of the particles as reflected in the reduction of the mean velocity, but a significan t number of particles still travel at the maximum fluid velocity, resulti ng in a larger spread of a pul se, which implies an increase in the D*. At larger PeR, only a very few partic les exist near the cen ter as most of the particles are concentrated in a thin layer ne ar 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
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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 lead s to a reduction in the dispersi on. Accordingly, the curves in Figure 47a shift dow n with an increasing and the maximum in the dispersion coefficient disappears as for larger than about 10. Figure 47. Dependence of (D*1)/Pe2 on PeR for (a) PeR ranging from 0 to 10 and ranging from 120; and (b) Pe R ranging from 0 to 200 and ranging from 1100, and comparison with the small asymptote (thick line). The markers on each curve represent the results calculated by Brownian dynamics. Small limit. To better understand the effect of on the dispersion, we obtain expressions for small In the small limit, it is useful to let the velocity of the
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80 reference frame in which we solve the c onvection diffusion equation to vary during a period, i.e., ) T ( U Us * In the limit of small to leading order Eq. (4 9), (4 12) and (4 16) become 2 0 2 0 e yY C Y C PeU (4 27) 2 1 2 1 e y 0 *Y C Y C PeU X C ) U U ( Pe (4 28) 2 2 2 2 0 2 2 e y 1 l 0Y C X C Y C PeU X C ) U U ( Pe T C (4 29) These equations along with the noflux boundary conditions are iden tical to those for EFFF and thus the short time dependent mean velocity and the dispersion coefficient are given by[53] 1 ) exp( ) ( ) exp( 12 12 ) exp( 6 6 U2 (4 30) ) ) 1 e /(( ) 72 720 2016 e 2016 e 6048 e 720 e 72 e 144 e 24 e 720 e 504 e 6048 e 144 e 24 e 504 e 720 ( Pe R D6 3 2 3 3 2 3 3 2 4 2 2 2 2 2 3 4 2 2 (4 31) where ) T sin( PeR PeUs e y These results for short time dependent mean velocity and dispersion coefficient can then be averaged over a period to yield the mean velocity and the dispersion coefficient, and these then can be compared with the exact results. These comparisons are shown in Figure 46a and 47a. Figure 47a shows the comparison of the small expression with the full result from Eq. (4 13) for the mean velocity. The small solution matches the exact solution for Similarly the
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81 dispersion coefficient in the small limit is the time average of the dispersion coefficient for constant electric fieldflow fractionation and it matches the full solution for <1 (Figure 47a). The matching of the m ean velocity and the dispersion coefficient with the time averaged EFFF results is expected because as shown earlier for = 1, the concentration profiles are close to bei ng in phase with the driving force. Large limit. In the large limit, the mean velocity can be computed by following the same approach as used by S&B. In this limit, to leading order, the periodicallysteady concentr ation profiles are given by the following expressions: For T 1 Y Wp ) T ( Y for 0 Wp ) T ( Y Y ) T ( Y for A ) T ( Y Y 0 for 0 ) T Y ( Gs s s s s 0 (4 32) where / T 1 Wp and )) T cos( 1 )( 2 / T ( ) T ( Ys s For T 2 T T for ), Y ( A T T for ))) T ( Y 1 ( Y ( A T T for ) 1 Y ( A T T 0 for )) T ( Y Y ( A ) T Y ( Gs t t s s s t t s s s 0 (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 ( Ys 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 PeR 2 T, and this dependence is shown in Figure 48 along wi th 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 subj ected to the same amplitude for the entire
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82 duration. It is more reasona ble to compare the two types of fields under the stipulation that the integral of the field dur ing a half period is the same. This stipulation is satisfied if the amplitude for the periodic fields is set to be 2 / times the amplitude of the square fields. In Figure 48 and also in Figu re 49 the x scale is chosen to be / PeR 2sq, where the subscript sq denoted the value of R for the square wave and the value of R for the sinusoidal fields is 2 / R sq. Figure 48 shows that the mean velocity for the sinusoidal fields is smaller than the square fields for 92 6 / PeR 2sq and at larger values of / PeR 2sq the mean velocity is higher for the sinusoidal fields. This can be attributed to the fact that for small values of / PeR 2sq, the slope of the ) T ( Ys at the time at which Y =1/2 is larger for the sinusoidal fiel ds, and thus the time spent by the pulse near the center is smaller, and accordi ngly the mean velocity is smaller. The situation is reversed for large values of / PeR 2sq leading to higher values of mean velocity for the sinusoidal fields. Figure 48. Comparison of the mean velocitie s for the square (dashed) and the sinusoidal (solid) fields in the large frequenc y limit. The R value on the x axis corresponds to that for th e square shaped field (Rsq) and the value of R for the sinusoidal field is /2 times Rsq.
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83 Figure 49. Comparison of the mean velocities and the effective diffusivity for the square (dashed) and the sinusoi dal (solid) fields. Th e R value on the x axis corresponds to that for th e square shaped field (Rsq) and the value of R for the sinusoidal field is /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 show n in Figures 49a and 49b, respectively. In the figures the mean velocities and the dispersion coefficients are plotted for a range of values as a function of / PeR 2sq, where as stated above, the subscript sq denoted
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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 / R sq. The figures show that for large values of the curves for both the mean velocities and the di spersion coefficients are similar and almost overlap for / PeR 2sq<10. To avoid or minimize the decay in the electric field due to double layer charging, separation will need to be performed at large and for optimal separation it is best to operat e in the region where the mean velocity is most sensitive to the field strength. Figure 49 shows that these requirements suggest that the most suitable operating parameters are / PeR 2sq~10 and ~ 100. Figures 49a and 49b also show that under these condi tions 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 e yu / 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 the dimensionless frequency, and PeR, the product of the lateral ve locity due to electric field and the Peclet number. The convective contribution to the dispersion coefficient is of the form) PeR ( f Pe2. The mean velocity of the partic les decreases monotonically with an
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85 increase inPeR, and increases with an increase in ; but 2 *Pe / 1 D has a maximum at a value of PeR~ 4 for small and the maximum disappears at large For <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 ex ceeds 1 for the case of small PeR and large frequencies. The results for s quare wave electric fields ma tch the asymptotic expressions obtained by S&B. Also the results of th e finite difference calculations match the Brownian dynamics calculations that were performed with the code provided by Reviewer 2. Comparison of results for sinusoidal and s quare wave fields show that for large values of the mean velocities and the dispersion coefficients are similar and almost overlap for / PeR 2sq<10. These are also the conditi ons most suitable for separation and thus it seems that both types of electri c fields are equally su itable for separation. Since the mean velocity of the particle s 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 a pplying cyclic lateral el ectric fields but only at small or O(1) Pe.
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86 CHAPTER 5 ELECTROCHEMICAL RESPONSE AND SE PARATION IN CYCLIC ELECTRIC FIELDFLOW FRACTIONATION This chapter aims to determine the mean velocity and the dispersion coefficient of charged molecules undergoing Poiseuille flow in a channel in the presence of cyclic lateral electric fields. As introduced in chap ter 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 bul k during half cycle when a constant voltage is applied. In reality, if the double layer char ging time is much shorter than the time for half cycle, the effective electr ic 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 cycl e is comparable to the charging time, the changing of the effective field in the bulk s hould be counted in to give a more rational result. Recently Biernacki et al. included th e 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 calcula te the dispersion of th e molecules, and thus they could not predict the separation effici ency of the devices, which is a balance between the retention and the di spersion. Furthermore, they only focused on determining the mean velocity for frequencies that ar e small enough so that the current decays to
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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 fo llows: In the next section we present the theory for the flow of curre nt during the operation of the CEFFF and the theory for the calculation of the mean velocity and the disper sion coefficient. The theory for the flow of current is based on the equi valent 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 effici ency of the CEFFF. Finally, some of the available experimental data on CEFFF is disc ussed and compared with theory. Theory Consider a channel of length L, height h a nd infinite width that contains electrodes for applying the lateral periodic lateral electri c 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 ra tio is much less than 1, i.e.,1 L / h Equivalent Electric Circuit Figure 51 is the commonly used equivalent electric circuit model for EFFF channel for the case when the applied voltage is low enough such that there is no electrode reaction. The capacitor Cd in the circuit can be attr ibuted to the double layers and the resistance Rs represents the resistance of the solution. On application of a potential V, the charging of capacitance lead s to an exponentially decaying current given by
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88 ) / t exp( R V is (5 1) where d sC R (5 2) If a periodic square shaped voltage is applied, the current is given by )) / t exp( 1 /( ) / t exp( ) R / V ( 2 ic s (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 51. Equivalent electric circuit model for an EFFF device Model for Separation in EFFF The transport of charged particles in th e channel is governed by the convectiondiffusion equation, 2 2 2 2  e yy c D x c D y c u x c u t c (5 4) where c is the particle concentration, u is th e fluid velocity in the axial (x) direction, D and D are the diffusion coefficients in the dir ections parallel and perpendicular to the
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89 flow, respectively. We assume that the di ffusivity tensor is isotropic and thus D =D = D. In Eq. (5 4), e yu 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 e yu can be determined by the Smoluchowski equation,E ) / ( ur 0 e y, where r and are the fluids dielectric consta nt and viscosity, respectively, is the permittivity of vacuum, and is the zeta potential. Or it can be simplified as E uE e y where E is the electric mobility. Eq. (5 4) is subjected to the boundary c ondition of no flux at the walls (y = 0,1), i.e., 0 c u y c De y (5 5) In a reference moving in the axial direction with velocity *u, Eq. (5 4) becomes ) y c x c ( D y c u x c ) u u ( t c2 2 2 2 e y (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 ) h y l x l Dt t ( C ~ C ~2 (5 7) where C ~ is the dimensionless concentration, ) 2 /( is the frequency of the applied field, 1/ is the short time scale, and D /2l is the long time scale over which we wish to observe the dispersion. Substituti ng Eq. (5 7) into (5 6) gives
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90 2 2 2 2 2 e y s l 2Y C ~ X C ~ Y C ~ PeU X C ~ ) U U ( Pe T C ~ T C ~ (5 8) where t Ts l/ x X ,h / y Y 2 l/ Dt Tl D h u Pe D h 2 u u U u u U u u U* e y e y and 1 l h Since the aspect ratio <<1, the concentration prof ile can be expanded in the following regular expansion. 0 m l s m m) T Y X T ( C ~ C ~ (5 9) Substituting Eq. (5 9) into Eq. (5 8) gives ) ( Y C ~ Y C ~ Y C ~ X C ~ Y C ~ PeU Y C ~ PeU Y C ~ PeU X C ~ ) U U ( Pe X C ~ ) U U ( Pe T C ~ T C ~ T C ~ T C ~3 2 2 2 2 2 1 2 2 0 2 2 0 2 2 2 e y 2 1 e y 0 e y 1 2 0 s 2 2 s 1 s 0 l 0 2 (5 10) Eq. (5 10) can be separated into a seri es of equations for different order of (0 ): 2 0 2 0 e y s 0Y C ~ Y C ~ PeU T C ~ (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 ( G C ~ l s 0 0, where G0 satisfies 2 0 2 0 e y s 0Y G Y G PeU T G (5 12)
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91 (1 ): 2 1 2 1 e y 0 s 1Y C ~ Y C ~ PeU X C ~ ) U U ( Pe T C ~ (5 13) Integrating the above equation from 0 to 1 in Y and 0 to in Ts and noting that 0 dT ) T / C ~ (2 0 s s 1 due to periodicity and 1 0 2 1 2 1 0 1 e ydY ) Y / C ~ ( dY ) Y / C ~ ( PeUdue to the boundary conditions gives 2 0 1 0 s s 0 2 0 1 0 s s 0 *dYdT ) T Y ( G dYdT ) Y ( U ) T Y ( G U (5 14) The solution to C ~ 1 is of the form ) X / ) T X ( A )( T Y ( Bl s where B satisfies 2 2 e y s 0 sY B Y B PeU ) T Y ( G ) U U ( Pe T B (5 15) (2 ) 2 2 2 2 0 2 2 e y s 2 1 l 0Y C ~ X C ~ Y C ~ PeU T C ~ X C ~ ) U U ( Pe T C ~ (5 16) Averaging both sides in Ts and Y gives, 2 0 2 1 0 2 0 s 2 2 l 0X C ~ dYdT X A B ) U U ( Pe T C ~ (5 17) where A dY dT Ag dY dT C ~ C ~1 0 2 0 s 0 1 0 2 0 s 0 0 where 1 0 2 0 s 0dY dT g (5 18) Substituting A from Eq. (5 18) into Eq. (5 17) yields,
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92 2 0 2 1 0 2 0 s 2 0 2 l 0X C ~ BdYdT ) U U ( Pe 1 X C ~ T C ~ (5 19) Now we combine the results for ) T / C ~ (s 0 and ) T / C ~ (l 0 l 0 2 s 2 2 s 1 s 0 0T C ~ l D T C ~ T C ~ T C ~ t C ~ (5 20) Averaging the above equation in Y and using periodicity gives, l 0 2 0T C ~ l D t C ~ (5 21) Now using Eq. (5 19) in Eq. (5 21) gives 2 0 2 0x C ~ DD t C ~ (5 22) where the dimensionless disper sion coefficient is given by 1 0 2 0 s *BdYdT ) U U ( Pe 1 D (5 23) Result and Discussion Electrochemical Response Lao et al. measured the current as a func tion of time in the EFFF device after a step change in voltage and showed that the curren ttime relationship in their experiments did not satisfy the single exponential predicted by th e equivalent circuit re presentation. They attributed the deviation from the single expone ntial to flow through the channel. Similar deviations from the single exponential have al so been observed by other researchers. Such deviations are not unexpected because the equivalent circuit representation shown in Figure 51 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
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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 la yer leads to a potential dependence capacitance[55,56]. Additionally, the equivalent circuit in Figure 51 also neglects the reactions at the electrode. Accordingly it is expected that the currenttime behavior in experiments will not exactly match the equati ons given above. However, the single and double exponential equations serve as a useful guide to fit th e 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 currenttime relationshi p in a channel in the absence of flow. In our study, the channel was comprised of two gol d 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 CMS18 system. Then, the glass plates were separated by a 500 1000 m thick spacer and clamped. A potentiost at (PGSTAT30, Eco Chemie) was used to apply either a step or a squa rewave cyclic potential of fixed magnitude and measure the time dependent current. Current response for a step change in voltage The currenttime behavior after applyi ng a step potential of 0.5 V in a 500 m thick channel containing DI water is shown as in Figure 52. The current dropped to about 4 A in 100 sec, suggesting that electrode re actions can be neglected. The current response shown in Figure 52 cannot be descri bed by a single exponential, and in fact a
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94 double exponential of the form ) / t exp( c ) / t exp( c I2 2 1 1 fits the data very well (dark line). In this experime nt, the fluid in between the plates was stationary, and this proves that the deviation from the single e xponential 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 contribut es to the decay of the current. For the data in Figure 52a, the best fit values of the parameters for the double exponential form are c1 = 6.55x104 A, c2 = 1.908x104 A, 1 = 0.197 s, 2 = 1.575 s. It is instructive to compare the time constants ob tained by us with t hose 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. ob tained a RC time constant of 40 seconds for DI water in a 178 m thick channel [38]. This value of time constant is about 25 times the larger of the two time constants obtaine d above, and this is unexpected because the channel in their study is thinne r than the channel used in our study. The difference can partially be attributed to the f act that Palkar et al. used th e long time currenttime data to obtain the time constant. Lao et al. also fitted their data to a single exponential. The channel in their study was 40 m thick and they obtained a value of 0.02 s for 1. Since our channel thickness is about 12.5 times of Lao et al., the RC time constant for our
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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 valu e of time constant may differ significantly depending on whether the short time or the l ong time data is used. A double exponential, although an empirical expression, is thus more useful for fitting the currenttime data. It should also be pointed out that the time cons tants 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 52. Transient current profiles after application of st ep change in voltage in a 500 m thick channel for (a) DI water (V = 0.5 volt) and (b) 50 mM NaCl (V = 1 volt)
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96 As shown above, the time response of the E FFF device can be represented in terms of the four parameters: c1, c2, 1, 2. Below we investigate the dependence of these parameters on the applied voltage, channel thic kness 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 adde d to stabilize the particles and/or to control the pH of the solutions. Addition of ions alters the electr ochemical properties of the EFFF device by changing the conductivity, electrode kinetics a nd the capacitance of the double layer. Below we report the eff ect of salt addition on the parameters c1, c2, 1 and 2. In the experiments described below, we measured the current response in DI water, 10mM NaCl and 50 mM NaCl solutions fo r a range of applied voltage V. In these experiments, we did not observe bubble forma tion even when the voltage was applied for very long time, which suggests that there are no electrode re actions involved in these experiments, except perhaps at short times. Figure 52b shows the current response af ter a step change in voltage for a 500 m thick channel containing 50 mM salt solution. The data shows that the magnitude of current immediately after the step change is si gnificantly larger than that for the case of DI water in Figure 52a. The figure also s hows 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 expect ed because the resistan ce of the salt solution is significantly less than that of DI water, a nd therefore initial current which scales as 1/R is larger and the decay constant which scales as RC is smalle r. However, after about 0.01 s, the rate of decay slows down significantl y, and the time scales for decay in the
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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 ro le 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 electr ic circuit model, we can anticipate that for a fixed channel thickness, the applied voltage will linea rly change the magnitude of the current, and accordingly both c1 and c2 are expected to li nearly increase with the voltage. The results shown in Figure 53 ad dem onstrate that as expected both c1 and c2 linearly increase with V and the value of 1 and 2 are relatively constant. Figure 53. Dependence of the electrochemi cal parameters on salt concentration and applied voltage in a 500 m thick channel
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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 53a and 53b show that the effect of salt addition on c1 and c2 is as expected. The values of 1 and 2 are relatively unaffected by addition of salt (Figure 53c and 53d), whic h is a surprising result, and can perhaps be attributed to the fact that increasing salt co ncentration leads to a smaller resistance and a higher capacitance, and thus the time constants are relatively unchanged. Figure 54. Dependence of the electrochemi cal parameters on channel thickness for V = 0.5 V and DI water
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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 capacitan ces are not expected to change with increase in h, both 1 and 2 are expected to increase with increasing h. The results in Figure 54 ad show that c1 decreases with increasing h while c2 is relatively constant, and that 1 and 2 both increase with h. However, the increase in 2 is leveling off at large h values. To understand the exact dependency of c1, 1, c2 and 2 on various parameters, one needs to solve the detailed electrochemical pr oblem that includes solving the PoissonBoltzmann equation along with the species c onservation, 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 vari ous 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 52 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 Fi gure 51, application of square shaped periodic waveform leads to a current flow gi ven by Eq. (5 3). Since the experimental results for a single step change in voltage fit a double exponential ra ther than a single exponential, it may be expected that on appl ying the periodic square shaped voltage, the current expression will be given by ) / t exp( C ) / t exp( C )) / t exp( C ) / t exp( C 1 ( I 2 ) / t exp( c ) / t exp( c )) / t exp( c ) / t exp( c c c ( c c 2 i2 2 1 1 2 c 2 1 c 1 max 2 2 1 1 2 c 2 1 c 1 2 1 2 1 (5 24)
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100 where s 2 1 maxR V c c I 2 1 1 1c c c C 2 1 2 2c c c C 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 55 shows the comparison between the response predicted above for DI water (Figur e 55a), 10 mM NaCl (Figure 55b) and 50 mM NaCl (Figure 55c) and the experimental results in for tc = 0.3 s, channel thickness of 500 m. There is a reasonable agreement between Eq. (5 24) and the current transients in DI water (Fig 55a). 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 tim e 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 separa tion 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 ) / t exp( C ) / t exp( C i i2 2 1 1 0 (5 25) where )) / t exp( C ) / t exp( C 1 ( R V 2 i2 c 2 1 c 1 s 0 (5 26) The above equation predicts that wh en 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 56ac for DI water, 10 mM NaCl and 50 mM NaCl, respectively in a 500 m thick channel. The amplitude of the voltage is 1 V in these experiments. The solid lines
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101 in these figures correspond to the prediction gi ven by Eq. (5 26). The figures show that there is a reasonable agreement between th e prediction and the experimental data. Figure 55. Comparison between the experiments (thin lines) a nd 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 m thick channel.
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102 Figure 56. Comparison between th e experiments (stars) and Eq. (5 26) (solid lines) for current transients on application of a s quare 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 m thick channel.
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103 Based on the above expression for the current, the field in the bulk of the channel is given by )) / t exp( C ) / t exp( C 1 ( ) / t exp( C ) / t exp( C h V 2 iR E2 c 2 1 c 1 2 2 1 1 s (5 27) This electric field drives a late ral velocity of charged particles) u (e y, and the dimensionless lateral velocity is given by ) t ( Rf )) / t exp( C ) / t exp( C 1 ( ) / t exp( C ) / t exp( C h u V 2 u E u u U2 c 2 1 c 1 2 2 1 1 E E e y e y (5 28) where E is the electrophoretic mob ility of the particles, h u V 2 RE is the maximum value of e yU and f(t) characterizes the time dependen ce of the electric field. By using the above equation, we can determine the lateral velo city of any type of particles in our EFFF device. We can also fit the currenttime data obtained by other researchers, and then determine the lateral velocity of particles in EFFF devices. CEFFF is a useful device for separation pa rtly because there ar e a number of design variables such as channel geometry and el ectrode design and opera tional 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 th e key parameters and the effect of these parameters on separation.
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104 Separation Modeling of separation of particles by CEFFF The response of the EFFF device can be characterized by seven dimensionless parameters: C1, C2, 1, 2, Pe, R, and The results reported below were computed for fixed values of C1, C2, 1 and 2 that were obtained in a 500 m wide channel using DI water as the carrier fluid and using a vo ltage of 0.5 V. We have also implicitly assumed that C1, C2, 1 and 2 are independent of V, whic h is a reasonable assumption based on the data. In the previous section we have devel oped the equations to determine the mean velocity and the dispersion coefficient of pa rticles in a CEFFF device. To determine the mean velocity and the dispersion coeffici ent of particles in CEFFF, we substitute e yU 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 diffu sivity by Eq. (5 14) and (5 23). Below we first discuss the results for the mean ve locity, followed by results for the dispersion coefficient, and then we combine these to ev aluate the separation e fficiency of CEFFF. Mean velocity of particles Before discussing the results for the m ean velocity, we note that many authors describe separation in terms of rete ntion 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 0 RU u u t t R (5 29)
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105 Therefore, the retention rate is equivalent to the dimensionless mean velocity of charged particles. Figure 57. Dependency of th e mean velocity on PeR and The electrochemical parameters are fixed at values th at correspond to DI water in a 500 m channel. (C1=0.7744, C2=0.2256, 54 2 D h 3 20 D h2 2 1 2 ) In Figure 57, the mean velocity is plotte d as a function of PeR which is the product of the amplitude of the dimensionless lateral velocity e yU and the Peclet number Pe, for a range of D h2 where is the frequency of the oscillat ions. 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/ used by Biernacki et al. wh ile plotting the results for the retention ratio, which as explained above is identical to the mean velocity. The trends shown in Figure 57 are also similar to thos e shown by Biernacki et al., except that we have explored the entire frequency range while Biernacki et al. only focused on the
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106 frequencies small enough so that the current es sentially decays to zero at the end of each half cycle [54]. The results in Figure 57 show that for a fixed 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 abse nce of any field, and thus the dimensional mean velocity is simply equal to the flow ve locity. 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 e nough for particles to travel from one wall to the other. Thus, a la rge number of particles spend a majority of the time near the center of the channel resulti ng 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 partic les can reach the walls at a smaller PeR values because a longer time is available during the period, and th us the PeR at which the mean velocity is a maximum moves to smaller values as becomes small. The dependence of mean velocity on for a fixed PeR is also interesting. At very small frequencies, the time period of the cycle is much longer than both 1 and 2, and thus the field is zero for a majority of th e period, and accordingly the mean velocity is about 1. As the frequency increases, the fiel d is nonzero during the period leading to a reduction in the mean velocity. However at very high frequencies, the distance traveled by the particles during a peri od becomes small, and thus the concentration profile becomes uniform and the mean velocity approaches 1.
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107 Effective diffusivity of particles The dispersion coefficient D* is of the form 1 + Pe2f(PeR, C C). The first term is the contribution from axia l diffusion and the second term represents the convective contribution. In Figure 58, we plot 2 *Pe / 1 D 210 as a function of PeR and The results in Figure 58 show that for a fixed as PeR increases, the dispersion coefficient initially increases and then decrea ses. As PeR goes to zero, i.e., the electric field is close to zero, the effective diffusivity is expected to appr oach the value of the Taylor dispersivity for Poiseuille flow thr ough a channel. Figure 58 shows that as PeR approaches zero, the curves of 2 *Pe / ) 1 D ( 210 for all values of frequency approach the expected limit of 1. On the other hand, as Pe R 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 diffusiv ity of the particles approaches the molecular diffusivity. Figure 58 al so shows that for small 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 partic les as reflected in the reduction of the mean velocity, but a significant numbe r of particles still travel at the maximum fluid velocity, resulting in a larger spre ad 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 particle s are concentrated in a thin layer near the wall, and any further in crease 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 in finity, the mean velocity
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108 approaches zero, and the dispersion coefficien t 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 58. Dependence of 210(D*1)/Pe2 on PeR and The electrochemical parameters are fixed at values th at correspond to DI water in a 500 m channel. (C1=0.7744, C2=0.2256, 54 2 D h 3 20 D h2 2 1 2 ). 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 pa rticles 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 conti nues to increase, a majority of the particles are present near the center of the channel and thus the dispersion coefficient be gins to decrease. Finally, at very high frequenc ies the particles move a sma ll distance in the period, and
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109 thus the concentration prof ile is uniform and the dispersion is coefficient again approaches the Poiseuille limit. Figure 59. Dependence of se paration efficiency on PeR1 and 1 for the case of D1/D2=3 and E2/ E1=3, and thus (PeR)2/(PeR)1 = 9 and 2/ 1=3. The electrochemical parameters are fixed at values th at correspond to DI water in a 500 m channel. (C1=0.7744, C2=0.2256, 54 2 D h 3 20 D h2 2 1 2 ). Figure 510. Origin of the singularity in separation efficiency at critical PeR1 and 1 values for 1 = 40 (PeR)2/(PeR)1 = 9, 2/ 1=3, and the electrochemical parameters are fixed at values th at correspond to DI water in a 500 m channel. (C1=0.7744, C2=0.2256, 54 2 D h 3 20 D h2 2 1 2 ).
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110 Separation efficiency Consider separation of colloidal particles of two different si zes in a channel. As the particles flow through the channe l, they separate into two Gaussian distributions. The axial location of the peak of the particles at time t is t u* and the width of the Gaussian is t DD 4*. We consider the particles to be sepa rated when the distance between the two pulse centers becomes larger than 3 times of the sum of th eir half widths, i.e., ) t D D 4 t D D 4 ( 3 t ) u u (* 2 2 1 1 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 u max( / 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] U U D D D D )[ U U max( Pe 1 12 h / L (5 31) Below we investigate the separation of two particles that have different mobilities ( E1/ E2 = 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 values. The values of L/h are plotted as a function of (PeR)1 in Figure 59 for ranging from 0.16 to 40 It is noted that (PeR)2/(PeR)1 = 9 and 2/ 1=3. From Figure 59, it can be seen that choosing the a ppropriate parameters is extr emely important to obtain good separation. At small frequencies such as at = 0.16 the separation is inefficient at all PeR because the mean velocities of both par ticles are close to 1. The separation improves
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111 on increasing to 0.4 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 certa in PeR values, but near a critical PeR1 such as PeR1=330 for = 40 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 510. 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 experiment al 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 na noparticles by CEFFF. The time constant for decay of electric field was large for the E FFF 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 nondecaying 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 th eir study with the theoretical models. The current in the EFFF device fabr icated 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 focu s 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 surfacemodified polystyrene particles with 0.45 and 0.105 m diameters. They performed the
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112 studies with a flow rate of 17 L/min in a 40 m thick, 1 cm wide and 9 cm long channel bounded with indium tin oxide(ITO) electrodes that were connect ed 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 fitte d that data to a double exponential form to obtain the following constants: C1 = 0.991, C2 = 0.009, 1 = 0.02 s, and 2 = 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 preequilibrated the sample by first applyi ng the electric field for 20 minutes without flow and then they started the flow while con tinuing to apply the electr ic field. Lao et al. investigated the effect of frequency, si ze 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 th e parameters listed above. Since this data were not provided by Lao et al ., we measured the electric mobilities with Brookhaven Zetaplus and obt ained values of (1.84 0.11)x108 m2s1V1 for the 0.105 m size particles and (2.70.17)x108 m2s1V1 for the 0.45 m size particles. Next, we used the StokesEinstein equation to obtain the molecular diffusivity, i.e., a 6 kT D (5 32)
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113 where is the viscosity of the fluid and a is the radius of the particle By this equation, we get the molecular diffusivity to be 4x1012 m2/s for the 0.105 m particle and 9x1013 m2/s for the 0.45 m 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 ar e 7087 and 31500, the values of R are 1.173 and 1.994, and the values are 5230 and 24600, for the sma ller and the larger particles, respectively. The values of PeR and are larger than a few thousands in each of the experiments, and thus rather than using the exact model propo sed earlier in this paper, it is preferable to use the large PeR and asymptotic solutions th at are easier to obtain. Below we first obtain the asymptotic result s and then compare the results with the experiments. Large asymptotic results Eq. (5 11) can be written as 2 0 2 0 s 0Y C ~ PeR 1 Y C ~ ) t ( f T C ~ PeR (5 33) In the limiting case considered here, both PeR and are large and the ratio is O(1), and thus in this limit, to leading order in 1/ the above equation becomes 0 Y C ~ ) t ( f T C ~ PeR0 s 0 (5 34) Essentially the above equation implies that as the frequency becomes large, the particles simply convect in the lateral direction with th e time dependent latera l velocity, and lateral diffusion can be neglected. In this limiting case of large the behavior of the system depends strongly on the total di stance traveled by the particles in the positive direction
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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 regime. (1) First we consider the case when the di stance traveled by the particles is larger than the channel thickness for bot h the positive and the negative phases of the cycle. In this case, since the distance traveled by each of the particle in th e positive phase of the cycle is larger than the channel height, all th e particles are at the Y = 1 wall at the end of the positive phase. As the field reverses th e particles begin to travel in the negative direction as a pulse with ne gligible 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 c t t c t t c t c t c t c t t 0T T T D T ), Y ( T D T T D T ))), D T T ( Y 1 ( Y ( D T T T ), 1 Y ( T T 0 )), T ( Y Y ( ) T ( G (5 35) where (Y) denotes the dirac delta function, T de notes the dimensionless time since the beginning of the cycle, Tc is the dimensionless cycle ti me, which due to the choice of dedimensionalization is equal to 2 Dt is the duty cycle, and therefore t cD T is equal to Tpos, and Y(T) is the distance traveled by the pa rticles in the positive direction in dimensionless time T, and similarly Y(TTpos) is the distance traveled by the particle in the negative direction in time TTpos, and finally Tt is the time at which the particles reach the Y = 1 wall. Based on the e xpression for the lateral velocity, Y(T) is given by
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115 )] exp(C ) exp(C [1 )] T exp( 1 ( 1 C PeR ) T exp( 1 ( 1 C PeR [ 2 ) T ( Y2 2 1 1 2 2 2 1 1 1 (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 th ickness but the distance traveled in the other phase is smaller than the height. We not e 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 th e 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 prof ile during a period can be expressed as c pos pos pos pos pos pos 0T T T 2 ), Y ( T 2 T T ))), T T ( Y ) T ( Y ( Y ( T T 0 )), T ( Y Y ( ) T Y ( G (5 37) (3) We now consider the case when the di stance traveled in both the positive and the negative phases is less than the channel he ight and the duty cycle is not equal to 0.5. To understand the physical situa tion that corresponds to this case, we assume that the concentration is uniform in the lateral direct ion 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 genera lity, we assume that Dt < 0.5, i.e., Yneg >
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116 Ypos. As the field is applied in the negativ e 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 th e 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 1Yneg+ 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/ ( YnegYpos) cycles, all the particles will be locat ed 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 attain ment of the periodic steady state are c pos pos pos pos pos pos 0T T T 2 ), Y ( T 2 T T )), T T ( Y ) T ( Y ( Y ( T T 0 )), T ( Y Y ( ) T Y ( G (5 38) It is noted that the periodic steady concen tration 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
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117 particles begin to move in th e positive direction, and at th e end of the positive phase, the particles that were located in the region Y > 1Y 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 partic les in the region between Y = 1 and 1Y, 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 distribu ted 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 di rections. Thus the concentra tion profile for this case is given by 1 Y Wp ) T ( Y 0 Wp ) T ( Y Y )), Wp ) T ( Y ( Y ( Wp ) T ( Y Y ) T ( Y 1 ) T ( Y Y 0 0) T Y ( G0 (5 39) where Wp is the width of the block area and it is given by Y 1 Wp (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 peak s at the channel outl et. This phenomenon was observed by Bruce Gale in their experiments [58]. If the diffusion is negligible the concen tration 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. Gi ven sufficient time, the pulse disappears and
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118 the periodic steady concentration profile is simply a uniform distribution that moves equal distances back and forth. The concen tration profile for this case is given by 1 Y Wp ) T ( Y 0 Wp ) T ( Y Y ) T ( Y 1 ) T ( Y Y 0 0 ) T ( C (5 41) It is noted that attaining this periodic steady state requires a dimensional time of D / h2, which implies a dimensionless time of In the experiments conducted by Lao et al. sufficient time was provided for equilib ration before starting the flow, so the dual peaks observed by Gale et al. were not observe d 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 Lao et al. focused on measuring the effects of changes in the frequency on the mean velocity. To further understa nd the physics of the various ca ses considered above and for comparison with experiments of Lao et al., it is instructive to analy ze the effect of the frequency on the mean velocity of particles. In Figure 511a and 511b, 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 (Fi gure 511 a) and Duty cycle = 0.5 (Figure 511 b). In Figure 511a, the two dashed vertical lines divide the domain into three regions. The first and the sec ond dashed lines indicate the 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 is less
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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 511b, since the duty cycle equals 0.5, Yneg = Ypos, and thus the two lines shown in Figure 511a overlap with each other, and th e region to the left of the vertical line corresponds to case (1), and the remaining area corresponds to case (4 ). In both Figure 511a and 511b, the dashdot line near = 0 indicates the values of below which the large expressions derived above cannot be used to calculate the mean velocity, and the general analysis shown earlie r 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 values. Furthermore, at a critical value of PeR, the line for Ypos = 1 will disappear from the figure because the distance traveled by the pa rticles 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
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120 to zero, and thus an increase in the duration of the resting period, which occurs with a reduction in leads to a decrease in the mean velocity. Figure 511. Dependence of the mean velocity on in the large regime. The values of other parameters are fixed at Pe=31500, R=1.7319, C1=0.991, C2=0.009, 889 D h 88889 D h2 2 1 2 and (a) Dt = 0.2 (b) Dt = 0.5. It is noted that as becomes very small, the resti ng period is comparable to the time required for lateral equilibr ation of concentration profile due to diffusion. Diffusion is neglected while analyzing the large asymptotes, and thus one has to use the more
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121 general analysis valid for all Pe and 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 ti me during which the field is operating approaches zero as approaches zero. However, this regime cannot be analyzed by the large asymptotes, and thus the mean velocity does not seem to be approaching 1 for small frequencies in Figure 511. The regime at frequencies larger than the value at which Ypos = 1 corresponds to case (2). In this regime, increasing 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 tim e near the center where the axial velocity is the highest. Interestingly, this maximum mi ght 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) ar ea. Case (3) is essentially an extension of case (2). In case (3), both Yneg and Ypos keep decreasing with increasing 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 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 511b are similar to the region 1 in Figure 511a. However in region 4, the behavi or is significantly di fferent. As discussed
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122 above, in region 4, there may be two different pulses detected at the channel exit, and the mean velocities of these two pulses are indi cated by two separate profiles in region 4 Figure 511b. If sufficient equilibration time is provide d, the equilibrium profile resembles a block that moves back and forth as the direction of field reverses. The concentration within the block is consta nt and is zero ever ywhere else. As increases in case (4), Yneg = Ypos decreases, thus the width of th e block area increases as the width of the block is 1Yneg. As goes to infinity, the con centration profile becomes uniform, and the mean velocity approaches 1. If sufficient equilibration time is not provided, then in addition to th e 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 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 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 m ean velocity under the conditions in Laos experiments. Table 51 shows the comparis on 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 in cluding 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.
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123 Table 51 Comparison of the model predic tions with experiments of Lao et al. Frequency (Hz) Duty cycle Particle size ( m) Voltage (V) Measured Dimensionless Velocity Model Prediction Figure 7 bI 5 0.5 0.45 1.75 0.343 0.289 Figure 7 bII 10 0.5 0.45 1.75 0.254 0.572 Figure 7 bIII 15 0.5 0.45 1.75 ~1 1.23 Figure 7 bIV 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 Figur e 7 bI, 7 bII and 9(c) are in case (1); those for Figure 8(d) are in case (2); those for Figure 9(d) ar e in case (3); and finally those for Figure 7 bIII and 7 bVI 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 re sult in bubble generati on, which can destroy the separation. It has been proposed that peri odic 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 separa tion of charged particles by cyclic EFFF by measuring the electrochemical response of the CRFFF device, developing an equivalent
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124 circuit and then using the para meters of the equivalent circuit into a continuum model to determine the mean velocity and the dispersi on coefficient for charged particles. The continuum model is solved by using the method of multiple time scales and regular expansions. Also analytical expressions ar e 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. Al so the theoretical predictions are compared with experimental data available in literature. The experimental results for current transi ents 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 suffici ent 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 consta nts that are similar to the values for DI water. The multiple time scales can part ly be attributed to the dependence of the double layer capacitance on the instanta neous potential droop ac ross the double layer. The theoretical analysis for the CE FFF shows that for a given set of electrochemical parameters, the dimensionless mean velocity *U depends on the dimensionless frequency, and PeR, the product of the lateral velocity due to electric field and the Peclet number. The convective contribu tion to the dispersion coefficient is of the form) PeR ( f Pe2. In the high frequency limit the mean velocity and the dispersion coefficient depend only on the ratio / PeR. Since this ratio is independent of
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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 In the small frequency regi me with a fixed value of *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 The dispersion coefficient D* is in form of 1+Pe2*f(PeR, ), and thus (D*1)/Pe2 depends only on PeR and At large PeR, particles accumulate near the walls and the dispersion coefficient approaches molecular diffusivity. At very small PeR, th e concentration is uniform and the dispersion is close to the classic Taylor dispersion coeffi cient for Poiseuille flow in a channel. In addition, the effective diffusivity is la rge both at very large and very small values, while a minimum exists at intermediate frequencies. It is advantageous that there is a ve ry 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 experi mentalist, and the mode l developed in this paper could be a very valuable tool in understanding the eff ect of various parameters and determining the best conditions for separation.
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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 bioterrori sm. A number of these coll oids are charged making these amenable to separation by using electric fi elds. Currently, techniques such as gel electrophoresis can separate th ese particles, but this technique is complex and only suitable for use by experts. There is a gr owing demand for simple chip based devices that can accomplish separations in free solu tion, and this dissertation has focused on electric field flow fractionati on (EFFF), which is a simple approach for separation in free solution that can be implemented on a chip. Although this dissertation focuses on EFFF, th e 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 elec tric field as the body force that drives lateral transport, whil e other variants of FFF use other fo rms of fields, such as lateral fluid flow, gravity/centrifugation, thermal grad ients, magnetic field, etc. For all these cases the lateral force is indepe ndent of the position, and thus the theory developed in this dissertation can be applied by replacing the expr ession 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 sepa rations is hindered by i ssues related to the electrochemical response of the devices. It is well known that when a constant electric
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127 field is applied between two electrodes sepa rated by only a few tens of microns, a double layer forms at each electrode, and the potenti al drop across these double layers can be as much as 99% of the applied voltage. Thus, onl y 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 dest roy 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 uni directional EFFF are discussed in chapter 2 of this dissertation in the c ontext of DNA separation. In this chapter we have also developed scaling relationships for separati on of colloidal partic les, particularly DNA strands. To separate DNA strands by EFFF, it may be best to operate below the shear rate at which the strands unfol d. 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 w ould cause very large dispersion, and offset the effect of the unfolding on the mean velo city. 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 eliminat e some problems associated with double layers, it is not the optimal solution b ecause 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 Poiseuil le flow to separate colloidal particles. Since the electric
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128 field is pulsed, the problems associated with the unidirectional EFFF are not expected to occur in our proposed technique. The essent ial idea of the propos ed technique is to accumulate all the particles near a wall by appl ying electric field, and then switch off the field. After switching off the field, the pa rticles diffuse and the ones with a larger diffusivity travel a larger distance from the wa ll. If a flow is now turned on for a short time, the smaller particles travel a larger ax ial 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 a nd off, and the separation improves with a reduction in tf and td, which are the durations of the fl ow and the diffusive steps. We showed that this method can be tuned to yiel d 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 a ssociated with switching the flow. A more convenient method to minimize the problems associated with double layer charging is to use oscillating el ectric fields that change dir ection sufficiently rapidly so that the double layers do not ge t 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 elec tric 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 th e effective electric fields in the bulk, there are other poten tial advantages of using cyclic fields in EFFF. CEFFF is
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129 potentially a more universal separation t echnique because it has additional operating variables such as frequency that could be ut ilized in separation. In constant EFFF, the velocity of the particles in the flow dir ection is only a function of PeR, which is essentially D h ue y, 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 D ue y cannot be separated with constant EFFF. In CEFFF, the mean velocity is function of PeR as well as of D h2 where 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 D ue y. In chapter 5 we investigated CEFFF with si nusoidal 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 c onsidered 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 th e model by combining numerical and analytical techniques and validated the result by compar ing with Brownian Dynamics simulations. For the CEFFF with sinusoidal field, in a ddition 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 regime. The results from both methods agreed with
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130 each other and also with the Brownian dyna mics simulations. We also developed asymptotic results for small frequencies. We compared the separation efficiencies of square wave and sinusoidal fields, and s howed that these have similar separation efficiencies in the range of th e desirable operation conditions. We also obtained asymptotic results for la rge frequencies for the case of sinusoidal fields, and also compared the high frequency resu lts for square fields with the asymptotic results reported in literature. In the large PeR and regime, the molecular diffusivity is negligible and the particles c onvect in the lateral directi on without much spreading. Accordingly, in this regime the velocity of particles on the axial di rection is only function of / PeR, i.e., ) h /( ue y 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 di d not explicitly cons ider 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 experi mentally, 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 reduc tion in potential drop in the bu lk. 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 experien ced in the bulk is not square shaped. In the first part of chapter 5, we experime ntally explored the el ectrochemistry of a CEFFF device fabricated in our lab. We applie d a fixed step voltage or a cyclic square
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131 shaped voltage, and measured current. The curr ent 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 circui t, and show that the equivale nt circuit is correct only if the dependence of the doubl e layer capacitance on the pot ential drop across the double layer is taken into account. We also e xplored the dependence of the electrochemical response on channel thickness, magnitude of a pplied 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 rapi d 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 electr odes and thus it only pl ays an important role in the short times at which the salt concentr ation evolves due to the field. These time constants are relatively insensitive to the a pplied voltage. We incorporated the current transients that were experimentally measured into the transport model for the particles undergoing a pressure driven Pois euille flow along with the lateral electric fields. We solved the model by using a multiple time scal e 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 disper sion coefficients are a complex functions of the electrochemi cal response and also of PeR and Therefore choosing the optimal parameters for sepa ration 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 for separating two types of particles and showed that the length of channel required for separation varies
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132 over several orders of magnitude for a reas onable 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 pr ocess 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. Wh ile 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 th is dissertation has s hown the usefulness of CEFFF and will encourage researchers to solv e the complete electrochemical problem by solving the PoissonBoltzmann equation along with the species conservation, and then couple it to the electrode kineti cs at the surface. This is a complex task mainly due to the lack of details on the electrode reactions for a majority of el ectrolytes. It is hoped that molecular level techniques can help in id entifying and character izing 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 genera te more experimental data that can validate the models developed in th is dissertation. Additionally we hope that
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133 researchers will attempt to fabricate the pul sed EFFF described in chapter 3 and compare the results with our predictions.
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134 APPENDIX A DERIVATION OF VELOCI TY 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 refere nce frame moving with a velocity u, the mean velocity of the pulse, Eq. (22) becomes, ) y c x c R ( D y c u x c ) u u ( t c2 2 2 2 e y (A1) Since we are interested in longterm disper sion, the appropriate time scale is L/ where L is the total channel le ngth, 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 ax ial direction. The scaling gives 2 2 2 1 D h u h D h u h L u L D l l (A2) where 1 L h ~ h l. We use the following dedimensionalization: u / L t T U = u/, u / u U, Ue y = ue y/, C = c/c0, X = x/l, Y = y/h, Pe =h/D (A3)
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135 where L is the lengt h 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. (A1) and the boundary conditions Eq. (23) become 2 2 2 2 2 e y 2Y C 1 X C R Y C U Pe X C ) U U ( Pe T C (A4) 0 C PeU Y Ce y at Y = 0,1. (A5) We assume a regular expansion for C in ...... .......... C C C C2 2 1 0 (A6) Substituting the regular expansion for C into (A4) and (A5) gives the following sets of equations and boundary conditio ns to different orders in (1/ 2): 2 0 2 0 e yY C Y C PeU ; 0 C PeU Y C0 e y 0 at Y = 0, 1 ) Y PeU exp( ) T X ( A Ce y 0 (A7) (1/ ): 2 1 2 1 e y 0Y C Y C PeU X C ) U U ( Pe ; 0 C PeU Y C1 e y 1 (A8) Substituting C0 from Eq. (A7) gives 2 1 2 1 e y e yY C Y C PeU ) Y PeU exp( X A ) U U ( Pe (A9) Integrating Eq. (A9) in Y from 0 to 1 gives 1 0 e y 1 0 e ydY ) Y PeU exp( U dY ) Y PeU exp( U (A10)
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136 Eq. (A10) gives the averag e velocity of the pulse. 1 ) exp( ) ( ) exp( 12 12 ) exp( 6 6 U2 (A11) where = e yPeU (A12) In Eq. (A9) we assume ) Y ( G X A C1 (A13) This gives 2 2 e y e yY G Y G PeU ) Y PeU exp( ) U U ( Pe (A14) Solving Eq. (A14) with boundary conditions gives Y 3 2 2 2 3 Y e ) const Y 2 Y 6 Y 3 ) 1 e ( ) Y e ( 12 ( Pe G (A15) and the constraint 0 GdY1 0determines the const in the equation. However, this const does not affect the mean velocity and the dispersion coefficient. 0: 2 2 2 2 0 2 2 e y 1 0Y C X C R Y C PeU X C ) U U ( Pe T C ; 0 C PeU Y C2 e y 2 at Y = 0,1 (A16) Integrating the above equation, us ing the boundary conditions and using
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137 ] 1 ) PeU [exp( PeU A dY C Ce y e y 1 0 0 0 (A17) gives ] dY ) Y ( G ) U U ( 1 e U Pe R [ X C T C1 0 PeU e y 2 2 0 2 0e y (A18) Thus the dimensionless dispersion coefficient D* is ] dY ) Y ( G ) U U ( 1 e U Pe R [ D1 0 PeU e y 2 *e y (A19) Substituting G from Eq. (A15) into Eq. (A19) and integrating gives ) ) 1 e /(( ) 72 720 2016 e 2016 e 6048 e 720 e 72 e 144 e 24 e 720 e 504 e 6048 e 144 e 24 e 504 e 720 ( Pe R D6 3 2 3 3 2 3 3 2 4 2 2 2 2 2 3 4 2 2 (A20)
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138 APPENDIX B DERIVATION OF NUMERICAL CALC ULATION FOR SINUSOIDAL EFFF Analytical Solution to O() Problem The solution to Eq. (4 10) can be expressed as )) nT cos( ) Y ( g ) nT sin( ) Y ( f ( ) Y ( g Gs n 1 n s n 0 0 (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. (B1) into (4 10) yields, )] nT cos( Y g ) nT sin( Y f [ Y g )] nT cos( Y f ) nT sin( Y g [ PeR 2 1 )] nT sin( Y g ) nT cos( Y f [ PeR 2 1 Y g ) T sin( PeR )] nT sin( ng ) nT cos( nf [ s 2 n 2 s 1 n 2 n 2 2 0 2 2 n s 1 n s 1 n 0 n s 1 n s 1 n 0 s 1 n s n s n (B 2) where R=r/. Comparing both sides of Eq (B2) and equating the time independent terms and the coefficients of sin(nTs) a nd cos(nTs) gives the following 2N1 coupled second order differential equations. (Time independent terms): 0 Y g Y f PeR 2 12 0 2 1 (B 3)
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139 (sin(Ts)): 0 Y f Y g PeR 2 1 Y g PeR g 2 1 2 2 0 1 (B 4) (cos(Ts)): 0 Y g Y f PeR 2 1 f 2 1 2 2 1 (B 5) (sin(nTs))(n=2..N1): 0 Y f Y g PeR 2 1 Y g PeR 2 1 g n2 n 2 1 n 1 n n (B 6) (cos(nTs))(n=2..N1): 0 Y g Y f PeR 2 1 Y f PeR 2 1 f n2 n 2 1 n 1 n 1 (B 7) The boundary conditions at O( ) are 0 e y 0G PeU Y G at Y=0,1 (B 8) Substituting Eq. (B1) into Eq. (B8) yields 2 n s 1 n s 1 n 0 n s 1 n s 1 n 0 s 1 n s n s n 0)] nT cos( f ) nT sin( g [ PeR 2 1 )] nT sin( g ) nT cos( f [ PeR 2 1 g ) T sin( PeR )] nT cos( Y g ) nT sin( Y f [ Y g (B 9) Comparison of the time independe nt terms on both sides gives 1 0PeRf 2 1 Y g (B 10) Although Eq. (B10) is valid at both walls, i.e ., at Y= 0 and 1, imposing Eq. (B10) at either wall automatically satisfies the same condition at the other wall. To demonstrate this, we average Eq. (4 9) in Ts and Y. dY dT Y C dY dT Y C PeU dY dT T C s 2 0 2 s 0 e y s s 0 (B 11) Using Eq. (B1) and (B8) in the above equation gives
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140 1 0 0 1 0 1Y g PeRf 2 1 (B 12) which proves that Eq. (B10) can only be implem ented at either Y = 0 or at Y = 1. To get around this issue we rewrite Eq. (B1) as ) T X ( A ~ )) nT cos( ) Y ( g ~ ) nT sin( ) Y ( f ~ ( ) Y ( g ~ Cl s n 1 n s n 0 0 (B 13) where ) 1 Y ( g / g g ~ 0 n n ) 1 Y ( g / f f ~ 0 n n and ) 1 Y ( g ) T X ( A ) T X ( A ~ 0 l l Thus, by definition1 ) 1 Y ( g ~ 0 This is equivalent to stating that we can set the concentration scale arbitrarily since the probl em is homogenous. This is also equivalent to utilizing a normalizing condition such as ensuring that the in tegral of the concentr ation 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)): 2 0 1PeRg 2 1 PeRg Y f (B 14) (cos(Ts)): 2 1PeRf 2 1 Y g (B 15) (sin(nTs))(n=2..N1): 1 n 1 n nPeRg 2 1 PeRg 2 1 Y f (B 16) (cos(nTs))(n=2..N1): 1 n 1 n nPeRf 2 1 PeRf 2 1 Y g (B 17) Eq. (B14)(B17) are valid at Y=0 and 1 a nd thus represent 4(N1) boundary conditions. These along with Eq. (B10) at Y= 0 and the only nonhomogenous condition
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141 1 ) 1 Y ( g0 can be used to solve Eq. (B3)(B7) to determine fi for i = 0: N1 and gi for i = 1: N1. Since all the differential equations (B3)(B7 ) are linear with constant coefficients, the solutions for fi and gi can be expressed as j Y j i ije f f j Y j i ije g g (B 18) The detailed equations for determining th e eigenvalues and the eigenfunctions and thereby determining fi and gi are provided below. Analytical Solution to O() Problem The solution to C1 is of the form ) X / ) T X ( A )( T Y ( Bl s where B satisfies 2 2 e y s 0 sY B Y B PeU ) T Y ( G ) U U ( Pe T B (B 19) Substituting the expression for G0 in Eq. (B19) gives 2 2 e y s n 1 n s n 0 sY B Y B PeU )) nT cos( ) Y ( g ) nT sin( ) Y ( f ( ) Y ( g ) U U ( Pe T B (B 20) The solution for B can be expressed as ] )) nT cos( ) Y ( g ) nT sin( ) Y ( f ( ) Y ( g [ const )) nT cos( ) Y ( q ) nT sin( ) Y ( p ( ) Y ( q ) Y T ( B1 n s n s n 0 s n 1 n s n 0 s (B 21) The value of the const in Eq. (B21) does not affect the value of eith er the mean velocity or the effective diffusivity. Thus, in the rest an alysis, we set it to be zero. The functions pi and qi in Eq. (B21) can be determined by su bstituting the postulated form for B into Eq. (B20) and equating coefficients of sin(nTs) and cos(nTs). Still, the equations for p
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142 and q represent a hierarchy, which we close by assuming pM and qM are zero where M is large enough. Substituting Eq. (B21) into Eq. (B20) gives 1 n s 2 n 2 s n 2 2 2 1 n s s n 1 n s s n 0 s s n 1 n s n 0 1 n s n s n)] nT cos( Y q ) nT sin( Y p [ Y q ] T ) 1 n sin( T ) 1 n [sin( Y q PeR 2 1 ] T ) 1 n cos( T ) 1 n [cos( Y p PeR 2 1 Y q ) T sin( PeR )) nT cos( ) Y ( g ) nT sin( ) Y ( f ( ) Y ( g ) U U ( Pe )] nT sin( nq ) nT cos( np [ (B 22) Equating both sides gives (time independent): 0 2 0 2 1g ) U U ( Pe Y q Y p PeR 2 1 (B 23) (sin(Ts)): 1 2 1 2 2 0 1f ) U U ( Pe Y p Y q PeR 2 1 Y q PeR q (B 24) (cos(Ts)): 1 2 1 2 2 1g ) U U ( Pe Y q Y p PeR 2 1 p (B 25) (sin(nTs)): n 2 n 2 1 n 1 n nf ) U U ( Pe Y p Y q PeR 2 1 Y q PeR 2 1 q n (B 26) (cos(nts)): n 2 n 2 1 n 1g ) U U ( Pe Y q Y p PeR 2 1 p n (B 27) Boundary Conditions: 1 e y 1C PeU Y C (B 28) Substituting the expression for C1 into Eq. (B28) and comparing both sides gives the following boundary conditions at Y = 0 and 1, (time independent terms): 1 0PeRp 2 1 Y q (B 29)
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143 (sin(Ts)): 2 0 1PeRq 2 1 PeRq Y p (B 30) (cos(Ts)): 2 1PeRp 2 1 Y q (B 31) (sin(nTs)) (n=2..M1): 1 n 1 n nPeRq 2 1 PeRq 2 1 Y p (B 32) (cos(nTs)) (n=2..M1): 1 n 1 n nPeRp 2 1 PeRp 2 1 Y q (B 33) These 4N2 boundary conditions can be used to solve the 2N1 second order differential equations (B23)(B27) to determine pi and qi. Again, since all the differential equations (B23)(B27) are linear with consta nt coefficients, the solution for pi and qi can be expressed as j Y j i ije p p j Y j i ije q q (B 34) The detailed equations for determining th e eigenvalues and the eigenfunctions are provided below.
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144 Solving for f, g, p and q To solve Eq. (B3)(B7) and (B10)(B17) fn ,gn, pm and qm are expanded as j Y j i ije f f j Y j i ije g g j Y j i ije p p j Y j i ije q q (B 35) where s are eigenvalues that can be determined by substituting the above expansions in Eq. (B3)(B7). 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 2N1 that need to be determined for each Substituting the above expressions in Eq. (B3)(B7) and collecting the terms for j gives 2 j j 0 j j 1g PeRf 2 1 (B 36) 2 j j 1 j j 2 j j 0 j 1f PeRg 2 1 PeRg g (B 37) 2 j j 1 j j 2 j 1g PeRf 2 1 f (B 38) 1 2..N n for f PeRg 2 1 PeRg 2 1 g n2 j j n j j 1 n j j 1 n j n (B 39) 1 N .. 2 n for g PeRf 2 1 PeRf 2 1 f nj j n j j 1 n j j 1 n j n (B 40) The above set of 2N1 equations leads to 4N2 values of s and of these 1 and 2 are zero. Thus, fi and gi must be of the form ) 1 N 2 ( 2 3 j Y j i 2 i 1 i ije f Y f f f ) 1 N 2 ( 2 3 j Y j i 2 i 1 i ije g Y g g g (B 41)
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145 Substituting Eq. (B41) into the equations (B3 )(B7) and then equating coefficients of sin(nTs) and cos(nTs) yields the following (time independent terms): ) 1 k 2 ( 2 3 j Y 2 j j 0 ) 1 k 2 ( 2 3 j Y j j 1 2 1j je g e f PeR 2 1 PeRf 2 1 (B 42) (sinTs): ) 1 k 2 ( 2 3 j Y 2 j j 1 ) 1 k 2 ( 2 3 j Y j j 2 2 2 ) 1 k 2 ( 2 3 j Y j j 0 2 0 ) 1 k 2 ( 2 3 j Y j 1 2 1 1 1j j j je f ) e g g ( PeR 2 1 ) e g g ( PeR ) e g Y g g ( (B 43) (cosTs): ) 1 k 2 ( 2 3 j Y 2 j j 1 ) 1 k 2 ( 2 3 j Y j j 2 2 2 ) 1 k 2 ( 2 3 j Y j 1 2 1 1 1j j je g ) e f f ( PeR 2 1 ) e f Y f f ( (B 44) (sin nTs): n = 2..N1 ) 1 k 2 ( 2 3 j Y 2 j j n ) 1 k 2 ( 2 3 j Y j j 1 n 2 1 n ) 1 k 2 ( 2 3 j Y j j 1 n 2 1 n ) 1 k 2 ( 2 3 j Y j n 2 n 1 nj j j je f ) e g g ( PeR 2 1 ) e g g ( PeR ) e g Y g g ( n(B 45) (cos nTs): n = 2..N1 ) 1 k 2 ( 2 3 j Y 2 j j n ) 1 k 2 ( 2 3 j Y j j 1 n 2 1 n ) 1 k 2 ( 2 3 j Y j j 1 n 2 1 n ) 1 k 2 ( 2 3 j Y j n 2 n 1 nj j j je g ) e f f ( PeR 2 1 ) e f f ( PeR 2 1 ) e f Y f f ( n(B 46)
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146 It is noted th at for a given the 2N1 equations give n above are not linearly independent and thus one of them has to be eliminated. For the 4N2 values of s, the total number of independent equations given in (B42)(B46) are (4N2)(2N2) and the total number of unknowns are (4N2)(2N1). The other 4N2 equations are provided by the boundary conditions. In equations (B42) (B46), collecting the terms for a given Y functionality and then furthe r collecting the terms for diffe rent time dependencies results in the following equations: For j = 1(corresponding to terms independent of Y) (time independent terms): 0 PeRf 2 12 1 (sinTs): 0 PeRg 2 1 PeRg g 2 2 2 0 1 1 (cosTs): 0 PeRf 2 1 f 2 2 1 1 (sinnTs): 0 PeRg 2 1 PeRg g n2 1 n 2 1 n 1 n (cosnTs): 0 PeRf 2 1 PeRf f n2 1 n 2 1 n 1 n (B 47) For j = 2 (terms linear in Y) (time independent terms): No equation (sinTs): 0 g 2 1 (cosTs): 0 f 2 1 (sinnTs): 0 g n2 n (cosnTs): 0 f n2 n
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147 (B 48) This shows that all the terms lin ear in Y are identically zero. For j = 3..2(2N1) These equations are identical to those given in (B36)(B40). Similarly, substituting Eq. (B41) into boundary equations (B10)(B17) and then collecting terms on the basis of the time dependencies, we get the following 2(2N1) equations: (time independent terms): ) 1 k 2 ( 2 3 j Y j 1 2 1 1 1 ) 1 k 2 ( 2 3 j Y j j 0 2 0j je f PeR 2 1 Y PeRf 2 1 PeRf 2 1 e g g (sinTs): ) e g PeR 2 1 Y PeRg 2 1 PeRg 2 1 ( ) e g PeR Y PeRg PeRg ( e f f) 1 k 2 ( 2 3 j Y j 2 2 2 1 2 ) 1 k 2 ( 2 3 j Y j 0 2 0 1 0 ) 1 k 2 ( 2 3 j Y j j 1 2 1j j j (cosTs): ) 1 k 2 ( 2 3 j Y j 2 2 2 1 2 ) 1 k 2 ( 2 3 j Y j j 1 2 1j je f PeR 2 1 Y PeRf 2 1 PeRf 2 1 e g g (sinnTs): ) e g PeR 2 1 Y PeRg 2 1 PeRg 2 1 ( ) e g PeR 2 1 Y PeRg 2 1 PeRg 2 1 ( e f f) 1 k 2 ( 2 3 j Y j 1 n 2 1 n 1 1 n ) 1 k 2 ( 2 3 j Y j 1 n 2 1 n 1 1 n ) 1 k 2 ( 2 3 j Y j j n 2 nj j j (cosnTs):
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148 ) e f PeR 2 1 Y PeRf 2 1 PeRf 2 1 ( ) e f PeR 2 1 Y PeRf 2 1 PeRf 2 1 ( e g g) 1 k 2 ( 2 3 j Y j 1 n 2 1 n 1 1 n ) 1 k 2 ( 2 3 j Y j 1 n 2 1 n 1 1 n ) 1 k 2 ( 2 3 j Y j j n 2 nj j j (B 49) In the above set, the first equation is valid on ly 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 1 g g ) 0 Y ( g) 1 N 2 ( 2 3 j j 0 1 0 0 (B 50) Combining the governing equations and th e boundary equations gives all the fi,j and gi,j. Solving for p and q Comparing Eq. (B3)(B7) with Eq. (B23) (B27), we can find that they have almost the same structure except that th e latter ones have a nonhomogeneous term, which is a product of ) U U ( Pe* with fi or gi. Thus, the solution for pi and qi can be separated into the particular and the homogeneous solution. hom i par i ip p p hom i par i iq q q (B 51) Furthermore, based on the form of the nonhomogeneity we propose the following forms for the particular solutions 3 j Y 3 4 j i 2 3 j i 2 j i 1 j i m 1 m m 1 i par i 3 j Y 3 4 j i 2 3 j i 2 j i 1 j i m 1 m m 1 i par ij je ) Y q Y q Y q q ( ) Y q ( q e ) Y p Y p Y p p ( ) Y p ( p (B 52) Below we use MAPLE to develop the equations for determining the particular solution. Solving for Particular Solution
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149 Substituting Eq. (B52) into (B23)(B27) gives (time independent terms ): 0 e Y ) q PeRp 2 1 ( e Y ) q 6 q PeRp 2 3 PeRp 2 1 Peg 6 ( Ye ) q 6 q 4 q PeRp PeRp 2 1 Peg 6 ( e ) q 2 q 2 q PeRp 2 1 PeRp 2 1 U Peg ( Y ) q ) 1 m )( 2 m ( PeRp 2 1 m ( Y ) q 20 PeRp 2 Peg 6 ( Y ) q 12 PeRp 2 3 Peg 6 Peg 6 (Y ) q 6 PeRp g U Pe Peg 6 ( ) q 2 PeRp 2 1 g U Pe (Y 3 2 j 4 j 0 j 4 j 1 Y 2 j 4 j 0 2 j 3 j 0 4 j 1 j 3 j 1 j 0 Y 4 j 0 j 3 j 0 2 j 2 j 0 3 j 1 j 2 j 1 j 0 Y 3 j 0 j 2 j 0 2 j 1 j 0 2 j 1 j 1 j 1 j 0 4 mm 3 m 1 0 2 m 1 1 3 6 1 0 5 1 1 2 0 2 5 1 0 4 1 1 1 0 2 0 4 1 0 3 1 1 2 0 1 0 3 1 0 2 1 1 1 0 *j j j j (B 53)
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150 (sinTs): 0 e Y ) p PeRq 2 1 PeRq q ( e Y ) p 6 p PeRq 2 3 PeRq 3 PeRq 2 1 PeRq Pef 6 q ( Ye ) p 6 p 4 p PeRq PeRq 2 PeRq 2 1 PeRq Pef 6 q ( e ) p 2 p 2 p PeRq 2 1 PeRq PeRq 2 1 PeRq f U Pe q ( Y ) p ) 2 m )( 1 m ( PeRq 2 1 mPeRq ) 1 m ( q ( Y ) p 20 PeRq 2 PeRq 4 q Pef 6 ( Y ) p 12 PeRq 2 3 PeRq 3 q Pef 6 Pef 6 ( Y ) p 6 PeRq PeRq 2 q Pef 6 f U Pe ( ) p 2 PeRq 2 1 PeRq q f U Pe (Y 3 2 j 4 j 1 j 4 j 2 j 4 j 0 4 j 1 Y 2 j 4 j 1 2 j 3 j 1 4 j 2 4 j 0 j 3 j 2 j 3 j 0 j 1 3 j 1 Y 4 j 1 j 3 j 1 2 j 2 j 1 3 j 2 3 j 0 j 2 j 2 j 2 j 0 j 1 2j 1 Y 3 j 1 j 2 j 1 2 j 1 j 1 2 j 2 2 j 0 j 1 j 2 j 1 j 0 j 1 1 j 1 4 m m 3 m 1 1 2 m 1 2 2 m 1 0 1 m 1 1 3 6 1 1 5 1 2 5 1 0 4 1 1 2 1 2 5 1 1 4 1 2 4 1 ,0 3 1 1 1 1 2 1 4 1 1 3 1 2 3 1 0 2 1 1 1 1 2 1 3 1 1 2 1 2 2 1 0 1 1 1 1 1 *j j j j (B 54)
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151 (cosTs): 0 e Y ) q PeRp 2 1 p ( e Y ) q 6 q PeRp 2 3 PeRp 2 1 Peg 6 p ( Ye ) q 6 q 4 q PeRp PeRp 2 1 Peg 6 p ( e ) q 2 q 2 q PeRp 2 1 PeRp 2 1 g U Pe p ( Y ) q ) 2 m )( 1 m ( PeRp 2 1 m p ( Y ) q 20 PeRp 2 p Peg 6( Y ) q 12 PeRp 2 3 p Peg 6 Peg 6 ( Y ) q 6 PeRp p Peg 6 g U Pe ( ) q 2 PeRp 2 1 p g U Pe (Y 3 2 j 4 j 1 j 4 j 2 4 j 1 Y 2 j 4 j 1 2 j 3 j 1 4 j 2 j 3 j 2 j 1 3 j 1 Y 4 j 1 j 3 j 1 2 j 2 j 1 3 j 2 j 2 j 2 j 1 2 j 1 Y 3 j 1 j 2 j 1 2 j 1 j 1 2 j ,2 j 1 j 2 j 1 1 j 1 4 m m 3 m 1 1 2 m 1 2 1 m 1 1 3 6 1 1 5 1 2 4 1 1 2 1 2 5 1 1 4 1 2 3 1 1 1 1 2 1 4 1 1 3 1 2 2 1 1 1 1 2 1 3 1 1 2 1 2 1 1 1 1 1*j j j j (B 55)
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152 (sinnTs): 0 e Y ) p PeRq 2 1 PeRq 2 1 q n ( e Y ) p 6 p PeRq 2 3 PeRq 2 3 PeRq 2 1 PeRq 2 1 Pef 6 q n ( Ye ) p 6 p 4 p PeRq PeRq PeRq 2 1 PeRq 2 1 Pef 6 q n ( e ) p 2 p 2 p PeRq 2 1 PeRq 2 1 PeRq 2 1 PeRq 2 1 f U Pe q n (Y ) p ) 2 m )( 1 m ( PeRq 2 1 m PeRq 2 1 m q n ( Y ) p 20 PeRq 2 PeRq 2 q n Pef 6 ( Y ) p 12 PeRq 2 3 PeRq 2 3 q n Pef 6 Pef 6 ( Y ) p 6 PeRq PeRq q n Pef 6 f U Pe ( ) p 2 PeRq 2 1 PeRq 2 1 q n f U Pe (Y 3 2 j 4 j n j 4 j 1 n j 4 j 1 n 4 j n Y 2 j 4 j n 2 j 3 j n 4 j 1 n 4 j 1 n j 3 j 1 n j 3 j 1 n j n 3 j n Y 4 j n j 3 j n 2 j 2 j n 3 j 1 n 3 j 1 n j 2 j 1 nj 2 j 1 n j n 2 j n Y 3 j n j 2 j n 2 j 1 j n 2 j 1 n 2 j 1 n j 1 j 1 n j 1 j 1 n j n 1 j n 4 m m 3 m 1 n 2 m 1 1 n 2 m 1 1 n 1 m 1 n 3 6 1 n 5 1 1 n 5 1 1 n 41 n 2 n 2 5 1 n 4 1 1 n 4 1 1 n 3 1 n 1 n 2 n 4 1 n 3 1 1 n 3 1 1 n 2 1 n 1 n 2 n 3 1 n 2 1 1 n 2 1 1 n 1 1 n 1 n *j j j j (B 56)
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153 (cosnTs): 0 e Y ) q PeRp 2 1 PeRp 2 1 p n ( e Y ) q 6 q PeRp 2 3 PeRp 2 3 PeRp 2 1 PeRp 2 1 Peg 6 p n ( Ye ) q 6 q 4 q PeRp PeRp 2 1 PeRp PeRp 2 1 Peg 6 p n ( e ) q 2 q 2 q PeRp 2 1 PeRp 2 1 PeRp 2 1 PeRp 2 1 g U Pe p n (Y ) q ) 2 m )( 1 m ( PeRp 2 1 m PeRp 2 1 m q n ( Y ) q 20 PeRp 2 PeRp 2 q n Peg 6 ( Y ) q 12 PeRp 2 3 PeRp 2 3 p n Peg 6 Peg 6 ( Y ) q 6 PeRp PeRp p n Peg 6 g U Pe ( ) q 2 PeRp 2 1 PeRp 2 1 p n g U Pe (Y 2 2 j 4 j n j 4 j 1 n j 4 j 1 n 4 j n Y 2 j 4 j n 2 j 3 j n 4 j 1 n 4 j 1 n j 3 j 1 n j 3 j 1 n j n 3 j n Y 4 j n j 3 j n 2 j 2 j n 3 j 1 n j 2 j 1 n 3 j 1 nj 2 j 1 n j n 2 j n Y 3 j n j 2 j n 2 j 1 j n 2 j 1 n j 1 j 1 n 2 j 1 n j 1 j 1 n j n 1 j n 4 m m 3 m 1 n 2 m 1 1 n 2 m 1 1 n 1 m 1 n 3 6 1 n 5 1 1 n 5 1 1 n 41 n 2 n 2 5 1 n 4 1 1 n 4 1 1 n 3 1 n 1 n 2 n 4 1 n 3 1 1 n 3 1 1 n 2 1 n 1 n 2 n 3 1 n 2 1 1 n 2 1 1 n 1 1 n 1 n *j j j j (B 57) Rearranging these equations and equa tions various Y dependencies gives For Y independent terms: 1 0 3 1 0 2 1 1g U Pe q 2 PeRp 2 1 1 1 3 1 1 2 1 2 2 1 0 1 1 1f U Pe p 2 PeRq 2 1 PeRq q 1 1 3 1 1 2 1 2 1 1 1g U Pe q 2 PeRp 2 1 p 1 n 3 1 n 2 1 1 n 2 1 1 n 1 1 nf U Pe p 2 PeRq 2 1 PeRq 2 1 q n
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154 1 n 3 1 n 2 1 1 n 2 1 1 n 1 1 ng U Pe q 2 PeRp 2 1 PeRp 2 1 p n (B 58) For YYm terms: These equations are identical to Eq. (B58). For Y je terms: j 0 3 j 0 j 2 j 0 2 j 1 j 0 2 j 1 j 1 j 1U Peg q 2 q 2 q PeRp 2 1 PeRp 2 1 j 1 3 j 1 j 2 j 1 2 j 1 j 1 2 j 2 2 j 0 j 1 j 2 j 1 j 0 1 j 1f U Pe p 2 p 2 p PeRq 2 1 PeRq PeRq 2 1 PeRq q j 1 3 j 1 j 2 j 1 2 j 1 j 1 2 j 2 j 1 j 2 1 j 1g U Pe q 2 q 2 q PeRp 2 1 PeRp 2 1 p j n 3 j n j 2 j n 2 j 1 j n 2 j 1 n 2 j 1 n j 1 j 1 n j 1 j 1 n 1 j nf U Pe p 2 p 2 p PeRq 2 1 PeRq 2 1 PeRq 2 1 PeRq 2 1 q n j n 3 j n j 2 j n 2 j 1 j n 2 j 1 n j 1 j 1 n 2 j 1 n j 1 j 1 n 1 j ng U Pe q 2 q 2 q PeRp 2 1 PeRp 2 1 PeRp 2 1 PeRp 2 1 p n (B 59) For Y jYe terms: j 0 4 j 0 j 3 j 0 2 j 2 j 0 3 j 1 j 2 j 1Peg 6 q 6 q 4 q PeRp PeRp 2 1 j 1 4 j 1 j 3 j 1 2 j 2 j 1 3 j 2 3 j 0 j 2 j 2 j 2 j 0 2 j 1Pef 6 p 6 p 4 p PeRq PeRq 2 PeRq 2 1 PeRq q j 1 4 j 1 j 3 j 1 2 j 2 j 1 3 j 2 j 2 j 2 2 j 1Peg 6 q 6 q 4 q PeRp PeRp 2 1 p
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155 j n 4 j n j 3 j n 2 j 2 j n 3 j 1 n 3 j 1 n j 2 j 1 n j 2 j 1 n 2 j nPef 6 p 6 p 4 p PeRq PeRq PeRq 2 1 PeRq 2 1 q n j n 4 j n j 3 j n 2 j 2 j n 3 j 1 n j 2 j 1 n 3 j 1 n j 2 j 1 n 2 j nPeg 6 q 6 q 4 q PeRp PeRp 2 1 PeRp PeRp 2 1 p n (B 60) ForY 2je Yterms: j 0 j 4 j 0 2 j 3 j 0 4 j 1 j 3 j 1Peg 6 q 6 q PeRp 2 3 PeRp 2 1 j 1 j 4 j 1 2 j 3 j 1 4 j 2 4 j 0 j 3 j 2 j 3 j 0 3 j 1Pef 6 p 6 p PeRq 2 3 PeRq 3 PeRq 2 1 PeRq q j 1 j 4 j 1 2 j 3 j 1 4 j 2 j 3 j 2 3 j 1Peg 6 q 6 q PeRp 2 3 PeRp 2 1 p j n j 4 j n 2 j 3 j n 4 j 1 n 4 j 1 n j 3 j 1 n j 3 j 1 n 3 j nPef 6 p 6 p PeRq 2 3 PeRq 2 3 PeRq 2 1 PeRq 2 1 q n j n j 4 j n 2 j 3 j n 4 j 1 n 4 j 1 n j 3 j 1 n j 3 j 1 n 3 j nPeg 6 q 6 q PeRp 2 3 PeRp 2 3 PeRp 2 1 PeRp 2 1 p n (B 61) ForY 3je Y: 0 q PeRp 2 12 j 4 j 0 j 4 j 1 0 p PeRq 2 1 PeRq q 2 j 4 j 1 j 4 j 2 j 4 j 0 4 j 1 0 q PeRp 2 1 p 2 j 4 j 1 j 4 j 2 4 j 1 0 p PeRq 2 1 PeRq 2 1 q n2 j 4 j n j 4 j 1 n j 4 j 1 n 4 j n
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156 0 q PeRp 2 1 PeRp 2 1 p n2 j 4 j n j 4 j 1 n j 4 j 1 n 4 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.
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157 REFERENCE LIST 1. B.K. Gale, K.D. Caldwell, A.B. Fr azier, IEEE Trans. Biomed. Eng. 45 (1998) 1459. 2. K.D. Caldwell, Y.S. Gao, Anal. Chem. 65 (1993) 1764. 3. B.K. Gale, K.D. Caldwell and A.B. Frazier. Anal. Chem. 74 (2002) 1024. 4. C. Contado, P. Reschiglian, S. Faccini A. Zattoni, F. Dondi, J. Chromatogr. A 871 (2000) 449. 5. R. Hecker, P.D. Fawell, A. Jefferson, J.B. Farrow, J. Chromatogr. A 837 (1999) 139. 6. P. Reschiglian, G. Torsi, Chromatographia 40 (1995) 467. 7. B Chen, J.P. Selegue, Anal. Chem. 74 (2002) 4474. 8. E.P.C. Mes, W.T. Kok, R. Tijssen, Chromatographia 53 (2001) 697. 9. P. Vastamki, M. Jussila, M.L. Riekkola, Sep. Sci Technol. 36 (2001) 2535. 10. S.N. Semenov, Anal. Commun. 35 (1998) 229. 11. R. Sanz, B. Torsello, P. Reschigl ian, L. Puignou, M.T. Galceran, J. Chromatogr. A 966 (2002) 135. 12. S. Saenton, H.K. Lee, Y. Gao, J. Ranville, S.K.R. Williams, Sep. Sci. Technol. 35 (2000) 1761. 13. M.K. Liu, J.C. Giddings, Macromolecules 26 (1993) 3576. 14. B.K. Gale, K.D. Caldwell, A.B. Frazier, Anal. Chem. 73 (2001) 2345. 15. A.I.K. Lao, D. Trau, I.M. Hsing, Anal. Chem. 74 (2002) 5364. 16. X.B. Wang, J. Yang, Y. Huang, J. Vykouka l, F.F. Becker, P.R.C. Gascoyne, Anal. Chem. 72 (2000) 832. 17. P. Reschiglian, A. Zattoni, B. Roda, S. Casolari, Anal. Chem. 74 (2002) 4895. 18. K.D. Caldwell, L.F. Kesner, M.N. Myer, J.C. Giddings, Science 176 (1972) 296.
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160 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 ChinaTsinghua University. I urged myself in undergraduate study in Tsinghua Univ ersity, took fiveyear courses in four years, graduate d one year earlier than my p eers, and ranked in the top 5% in my department of 120 students. After that I entered the gr aduate 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 seve ral 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 deali ng with wastewater. In 1999, I took part in a project to undertake mi ddlescaled amplification of the production of PHB (polyhydroxybutyrate, a kind of biode gradable plastic) with E.Coli., which was part of a Ninth Fiveyear National Key Project of China. Under my active and successful participation, we found and eliminated the sc attering of nitrogen during sterilization and improved the distribution of air i nput. The density of bacteria reached 120g/l and the production of
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161 PHB extended to 80g/l, far beyond the original goal. The amplification succeeded and won me the honor of the first prize for outst anding performance in the field practice of my department. My masters thesis was under the guid ance of Prof. Zhongyao Shen, the ViceDean of the School of Life Sciences and Engin eering in Tsinghua University. My work focused on coupling of fermentation and separa tion. In the first year, I applied the coupling of fermentation and ion exchange on the production of 2keto gulonic acid, the direct precursor of vitamin C. However, this research was abandoned because of an unfeasibility resulting from the fermentation syst em. 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 ac quired insights on membrane, fermentation, ion exchange, and operation of analytical equipm ent. 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 microchanne l and electric fields. Under the guidance of Dr. Anuj Chauhan, I obtai ned 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 prope l the commercialization of EFFF.

