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11, 7402

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Nonlinear electro-osmosis of dilute non-adsorbing
polymer solutions with low ionic strength
Yuki Uematsu
Nonlinear electro-osmotic behaviour of dilute non-adsorbing polymer solutions with low salinity isinvestigated using Brownian dynamics simulations and a kinetic theory. In the Brownian simulations, thehydrodynamic interaction between the polymers and a no-slip wall is considered using the Rotne–Prager
approximation of the Blake tensor. In a plug flow under a sufficiently strong applied electric field, the polymer

Received 18th June 2015,
Accepted 4th August 2015

migrates toward the bulk, forming a depletion layer thicker than the equilibrium one. Consequently, the

DOI: 10.1039/c5sm01507c

nonlinear mobility does not depend qualitatively on the details of the rheological properties of the polymer

electro-osmotic mobility increases nonlinearly with increasing electric field and becomes saturated. Thissolution. Analytical calculations using the kinetic theory for the same system quantitatively reproduce thewww.rsc.org/softmatter

results of the Brownian dynamics simulation well.

1 Introduction
Electro-osmosis is observed widely in many systems such as
colloids, porous materials, and biomembranes. It characterises
the properties of interfaces between solids and electrolyte
solutions.1,2 Interest in the applications of electro-osmosis
has been growing recently. For instance, it is used to pump
fluids in microfluidic devices, as it is more easily implemented
than pressure-driven flow.3 Its application to electrical power
conversion in chemical engineering is also very fascinating.4,5When the electrokinetic properties of a surface are characterisedby a zeta potential, the Smoluchowski equation is often employedin conjunction with measurements of the electro-osmotic orelectrophoretic mobilities. However, the validity of this equation
should be considered more seriously. It is derived from the
Poisson–Boltzmann equation and Newton’s constitutive equationfor viscous fluids. The zeta potential is defined as the electrostatic
potential at the plane where a no-slip boundary conditionis assumed. When these equations cannot be validated, theSmoluchowski equation is also questionable. The Poisson–
Boltzmann equation, the simple hydrodynamic equations, orboth sometimes do not work well for a strong coupling double
layer,6 inhomogeneity of viscosity and dielectric constant near
the interface,7,8 and non-Newtonian fluids,9–12 for example.
To control the electrokinetic properties of charged capillaries,
the structures of liquid interfaces in contact with charged
surfaces are modified by grafting or adding polymers.13 In
capillary electrophoresis, for example, the electro-osmotic flowis reduced by grafted polymers on the interfaces. Several studies
Department of Physics, Kyoto University, Kyoto 606-8502, Japan.
E-mail: y_uematsu@scphys.kyoto-u.ac.jp7402 | Soft Matter, 2015, 11, 7402--7411

of surfaces with end-grafted charged and uncharged polymers
have also been reported.14–19 Under a weak applied electric field,
the grafted polymer remains in the equilibrium configuration,and the resultant electro-osmotic velocity behaves linearly withrespect to the electric field. To measure the mobility of such a
surface, the hydrodynamic screening and anomalous charge
distributions due to the grafted polymers are important.14–19
When a sufficiently strong electric field is applied, the polymers
are deformed by the flow and the electric field; thus, the electroosmotic velocity becomes nonlinear.16 Note that the end-graftedpolymers cannot migrate toward the bulk, as one of the ends isfixed on the surfaces.
When we add polymers to solutions, a depletion or adsorption
layer is often formed near a solid wall, as well as diﬀusive layers ofions in equilibrium states. The interaction between the polymers
and the wall determines whether the polymers are depleted from oradhere to the surfaces. The thickness of the depletion or adsorption
layer is of the same order as the gyration length of the polymers.
When polymers adhere to the wall, the viscosity near the wall
becomes large, so the electro-osmotic mobility is strongly
suppressed.20 Moreover, it is known that an adsorption layerof charged polymers can change the sign of mobility.7,21–24 The
curvature of the surface also modulates the surface charge
density and even increases the mobility beyond the suppression
caused by viscosity enhancement.20
Electro-osmosis of a non-adsorbing polymer solution has
been analysed in terms of two length scales: the equilibrium
depletion length d0 and the Debye length l.7,25 In the depletion
layer, the viscosity is estimated to be approximately the same asthat of the pure solvent, and it is smaller than the solution
viscosity in the bulk. When the Debye length is smaller than thedepletion length, the electro-osmotic mobility is larger thanThis journal is © The Royal Society of Chemistry 2015

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Soft Matterthat estimated from the bulk value of viscosity. Typically, for
10 mM electrolyte solutions, one has l E 3 nm and d0 E 100 nm.
In such a case,26–28 an electro-osmotic flow with a high shear rate islocalised at a distance l from the wall. Thus, the electro-osmotic
flow profile and resultant electro-osmotic mobility are almost
independent of the polymers. Such behaviours are experimentallyobserved in solutions of carboxymethyl cellulose with urea.27 On
the other hand, in the solutions of small polymers with low
salinity (typically for 0.1 mM electrolyte solutions, l E 30 nm
and d0 E 5 nm), the electro-osmotic mobility is suppressed by
the polymeric stress.25
When a suﬃciently strong electric field is applied, the electroosmosis of a polymer solution shows nonlinear behaviours.27,29
These nonlinearities are theoretically analysed by models of
uniform non-Newtonian shear-thinning fluids.9–12 Assumingthat polymers remain in the interfacial layers and viscosity
depends on the local shear rate, as in power-law fluids, theirphenomenological parameters differ from those in the bulk, as
the concentration in the interfacial layers differs from the bulk
concentration.27 Thus, the understanding of nonlinear electroosmosis remains phenomenological. Furthermore, when shear
flow is applied to polymer solutions near a wall, it is experimentallyand theoretically confirmed that cross-stream migration is induced
toward the bulk.30–32 The concentration profiles of the polymer
near the wall have been calculated, and the depletion length
dynamically grows tenfold larger than the gyration radius.31 However, these hydrodynamic effects in the electrokinetics have notbeen studied to date, to the best of our knowledge.
In this context, this paper discusses another origin of
nonlinearity, which is induced by hydrodynamic interaction
between the polymer and the wall, considering mainly situations
where d0 { l. For this purpose, this paper is organised as follows.
Section 2 presents a toy model of nonlinear electro-osmosis of dilutepolymer solutions. Section 3 describes the Brownian dynamics
simulation, and Section 4 presents the results of the simulation.
In Section 5, we discuss an analytical approach to nonlinear electroosmosis by using the kinetic theory of cross-stream migration.31
Section 6 outlines the main conclusions.

2 Toy modelFirst, we propose a toy model for electro-osmosis of polymer
solutions. A dilute solution of non-adsorbing polymers is
considered. The viscosity of the solution is given by
Z = Z0(1 + Zsp),

g_ 

m0 E
;
l

(2)

where m0 is the electro-osmotic mobility of the pure solvent andis typically estimated as m0 E 108 m2 (V s)1. According tostudies of cross-stream migration in uniform shear flow,31 thedepletion layer thickness depends on the shear rate,d E d0(t_g)2,(3)

where t is the characteristic relaxation time of the polymers,t

Z0 d0 3
;
kB T

(4)

where kBT is the thermal energy, and is typically 104 s at room
temperature. Using eqn (2) and (3), the depletion length in thepresence of the applied electric field E can be expressed as
8
for E o E0 ;
d0
>
>
>
>
>
<  2Ed(5)d0 for E0  E  E1 ;
>
E
0
>
>
>
>
:
l
for E1 o E;
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where E0 = l/tm0, and E1 ¼ E0 l=d0 . Here, for simplicity, we
assume that the depletion length does not exceed the Debye
length. The eﬀective viscosity in the double layer is given bydZeff  Z0 1 þ Zsp 1 ;(6)land the nonlinear mobility can be estimated as m E m0(Z0/Zeﬀ).
Therefore, the mobility is obtained as
8m0>
for E o E0 ;>> 1 þ Zsp ð1  ðd0 =lÞÞ
>
>
>
<
m0
m
(7)
for E0  E  E1 ;
>
>
1
þ
Z
ð
1
 ðE=E1 ÞÞ2
>
sp
>
>
>
:
m0
for E1 o E:
Fig. 1(a) schematically shows the thickness of the depletion
layer as a function of the electric field strength. Fig. 1(b) shows
the nonlinear electro-osmotic mobility. The mobility increases(1)

where Z0 is the viscosity of the pure solvent, and Zsp is the
specific viscosity of the solution. The gyration length of thepolymers is defined as d0, which is of the same order as theequilibrium depletion length. It is assumed that the polymers
have d0 E 100 nm. Ions are also dissolved in the solution with
the Debye length l. When well-deionised water is considered,the Debye length is on the order of l E 103 nm, although suchsalt-free water is rarely realised owing to the spontaneous
dissolution of carbon dioxides. The interfacial structure near

This journal is © The Royal Society of Chemistry 2015a charged surface is characterised by l and d0. When an external
electric field is applied, a shear flow is imposed locally within adistance l from the wall, and the resultant shear rate isFig. 1 (a) Depletion length as a function of the applied electric field. (b)Electro-osmotic mobility as a function of the applied electric field.

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and is saturated with increasing electric field. The thresholdelectric field E0 is typically 103 V m1, which is experimentally
accessible.

we neglect lubrication corrections for nearby particles.37 The
Blake tensor for the velocity at x2 induced by a point force at x1
with the no-slip boundary at z = 0 is given by the Oseen tensor
and the coupling fluid–wall tensor as34

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3 Model for simulation

GBab(x2,x1) = Sab(q) + GW
ab(x2,x1),

In this section, our method of Brownian dynamics simulation
is described. As shown in Fig. 2(a), a dumbbell is simulated inan electrolyte solution with a no-slip boundary at z = 0. The
dumbbell behaves like a dilute solution. The solvent is
described as a continuum fluid with the viscosity Z0 and fills
the upper half of the space (z 4 0). Electrolytes are also treated
implicitly with the Debye length l = k1. The dumbbell has twobeads whose hydrodynamic radii are a, and each bead consistsof many monomeric units of the polymer [see Fig. 2(c)]. The
positions of the beads are represented by x1 and x2 [see Fig. 2(b)].
Then we solve the overdamped Langevin equations33 given by

X
dxna
nm
Gnm
¼ u0 ðzn Þdax þab Fmb þ kB Trmb Gab
dt
m;b
(8)
þ xna ;

for n ¼ 1; 2; a ¼ x; y; z;where Xna is the a component of vector Xn. Furthermore, u0(z)is the external plug flow, rna = q/qxna, G is the mobility tensor,
Fn = rnU is the force exerted on the nth bead, and U is theinteraction energy given as a function of xn. nn is the thermal
noise which satisfies the fluctuation–dissipation relation as0
hnna(t)nmb(t 0 )i = 2kBTGnm
ab d(t  t ).

(9)

To include the eﬀects of the no-slip boundary, the Rotne–Prager
approximation for the Blake tensor34 is used as the mobility
tensor for distinct particles (n a m),35,36 although it is valid
only for particles separated by a large distance. In this study,(10)

where q = x2  x1, R = x2  x% 1, and x% 1 is the mirror image of x1
with respect to the plane z = 0 [see Fig. 2(b)]. The Oseen tensor
is given by
1 dab qa qb
þ 3 ;Sab ðqÞ ¼(11)
q
8pZ0 qwhere q is the magnitude of q. The second term in eqn (10) is22
GWab(x2,x1) = Sab(R) + z1 (1  2dbz)rR Sab(R) 2z1(1  2dbz)Saz,b(R),(12)

Sab,g(q) = rqgSab(q),

(13)

where

and rqg = q/qqg. The Rotne–Prager approximation of the Blake
tensor is given by6,35–38
8
a2 2 a2 2
>
>
1
þ
r
r
GBab ðx2 ; x1 Þ þ O a4
þ
>
1
2
>
>
6
6
>
>
>
>
>
>
for q 4 2a;
>
>
>
>
>



<
1
9q
qa qb
dab  2dab 
GRPB
ab ðx2 ; x1 Þ ¼>
6pZ0 a
32a
3q
>
>
>
>


>
>
2
2
>
> þ 1 þ a r 2 þ a r 2 GW ðx ; x Þ þ O a4>>12
ab 2 1
>
6
6
>
>
>
>
:
for q  2a:(14)
The mobility tensor for the self-part is given by6,35–38
RPB
Gself
ab ðzÞ ¼ lim Gab ðx; x1 Þ
x!x1

0

mk ðzÞ

B
B
¼B 0
@
0

0

0

mk ðzÞ

00m? ðzÞ

1
C
C
C;
A

(15)

where
mk ðzÞ ¼
1
9a 1a3
1
þ O a4 ;þ
6pZ0 a16z 8 z

(16)



1
9a 1a3
þ O a4 :1 þ
6pZ0 a8z 2 z(17)m? ðzÞ ¼

Finally we obtain the mobility tensor as
Fig. 2 (a) Dumbbell in the simulated box with L  L  D. A periodic
boundary condition is imposed at the x–y plane. (b) An x–z projection of
the dumbbell in the electro-osmotic flow. (c) Enlarged illustration of thebead in the dumbbell. It is composed of many monomeric units of thepolymers.

7404 | Soft Matter, 2015, 11, 7402--7411

self
RPB
Gnm
ab = dnmGab (zn) + (1  dnm)Gab (xn,xm).

(18)

The non-uniform mobility term in eqn (8) can be simplified
within the Rotne–Prager approximation of the Blake tensor

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Soft Matterbecause the following relation holds:X
rmb GRPBab ðxn ; xm Þ ¼ 0:

(19)

Table 1 Simulation parameters. Nt is the number of total steps, Ni is thenumber of interval steps for observation, Nm is the number of sampling foreach parameter set, and Dt is the increase in timeb¼x;y;zHookianThus, the non-uniform mobility term is rewritten asX
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m;b

The interaction energy includes the spring and bead–wallinteraction given byX
U W ðzn Þ;(21)
U ¼ U s ðqÞ þn¼1;2s

where U is the spring energy,
8
H 2
>
>
q ;
>
>
<2
s
"
U ðqÞ ¼
 2 #
>
H
q
>
2
>
;
>
:  2 q0 ln 1  q
0Hookian dumbbells;

FENE dumbbells;(22)

where a FENE dumbbell is a finitely extensible nonlinear elastic
dumbbell, and the parameter b = Hq02/kBT is defined forconvenience. U w is the bead–wall interaction,39 which is purely
repulsive:8   10  4 
2 a
a
3
>
<w
for z  a;

þ
5 z
z
5
U w ðzÞ ¼
(23)
>
:
0
for z 4 a:
Eqn (8) is solved numerically. A reflection boundary conditionis set at z = D. When the centre of the dumbbell crosses theboundary, the z coordinates of each bead are transformed from z
to 2D  z. In the lateral directions, periodic boundary conditions
are imposed. The size of the lateral directions is L  L.

4 Results of simulation
The concentration and velocity profiles are calculated as1D z1 þ z2 E;(24)cðzÞ ¼ 2 d z 
2
L
and
1
duðzÞ ¼
Z0 L2

*

X

+
minðz; zn ÞFnx ;

(25)

n¼1;2

where d(z) is the delta function, du(z) = u(z)  u0(z) is theincrease in velocity due to polymeric stress, and h  i is thestatistical average in the steady state. Eqn (25) is derived inAppendix A.
For a surface with a small zeta potential compared to kBT/e,
where e is the elementary charge, the imposed electro-osmotic
flow u0(z) is given by
u0(z) = m0E(1  ekz),(26)

where m0 is the electro-osmotic mobility in the pure solvent,and E is the applied electric field.2 Eqn (8) is rewritten in aThis journal is © The Royal Society of Chemistry 2015

Nt
Ni
Nm
Dt

10

5  105  1033
0.01

FENE b = 600
10

5  105  1033
0.0025

FENE b = 50
25  1010
25  10350.0001

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dimensionless form with the length scale d0 ¼ kB T=H andtime scale t = 6pZ0a/4H. Diﬀerent types of dumbbells are
simulated using the parameters listed in Table 1.
Note that the simulated systems are treated as dilute systems,
and the linearity with respect to the bulk polymer concentration
is preserved. After sample averaging, we obtain the concentration at the upper boundary c(D), which deviates slightly from(L2D)1 because of the inhomogeneity near the surface. Hereafter, we define the normalised concentration asCðzÞ ¼

cðzÞ
:
cðDÞ

(27)

The velocity increment du(z), as well as the concentration profile, islinear with respect to c(D). For convenience, we set a characteristic
concentration cb = 0.1d03, and the nonlinear electro-osmotic
mobility is defined asmðEÞ ¼ m0 þ

cb duðDÞ
:cðDÞ E(28)

The top boundary is placed at D = 100d0, the lateral size is set to
L = 1000d0, and the Debye length is set to l = k1 = 10d0. We also
set w = 3kBT and define a hydrodynamic parameter h as31ah ¼ pﬃﬃﬃ ¼ 0:25:
pd0

(29)

Fig. 3 shows the steady-state profiles of the Hookian dumbbell
concentration as functions of the distance from the wall. In theequilibrium state of E = 0, the profile shows a depletion layer
whose width is of the same order as the gyration length d0.
When the applied electric field is increased further, the depletion layer becomes larger than the equilibrium one, and a peakis formed. The inset in Fig. 3 shows the depletion length as afunction of the applied field. The depletion length is defined by
the position of the concentration peak. It shows power-law
behaviour with an exponent of 0.22, which is much smaller
than that of 2.0 for a uniform shear flow.31 The value ofconcentration at the peak also increases as the electric field
increases.
The results given above are for the Hookian dumbbell whichis infinitely extensible with shear deformation. To consider more
realistic polymers, the finitely extensible nonlinear elastic(FENE) dumbbell is simulated. Fig. 4(a) shows the concentration
profiles at E = 1000E0. Interestingly, one-peak behaviour is also
observed in the FENE dumbbells. For the Hookian dumbbell, theconcentration near the surface remains finite. On the other
hand, for the FENE dumbbells, the concentrations near thesurface are negligibly small. Fig. 4(b) plots the electro-osmotic

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Fig. 5 (a) Profiles of hqz2/q2i at E = 1000E0 in diﬀerent types of dumbbells.
(b) Profiles of h(q/d0)2i at E = 1000E0 in diﬀerent types of dumbbells.
Curved lines are calculated using eqn (62) and (71).

Fig. 3 Concentration profiles of Hookian dumbbells under various
applied electric fields. Inset shows the depletion length as a function of
the applied field. The points are obtained by the Brownian dynamics
simulation, and the line is fitted by d/d0 = A(E/E0)B, where A = 7.08, andB = 0.22.

FENE dumbbells, saturation of the dumbbell length is observed.
These behaviours diﬀer greatly from the minor diﬀerences in theconcentration profiles.

5 Kinetic theory
In this section, a kinetic theory for a dumbbell is developed on
the basis of the Ma–Graham theory.31 The probability function
C(x1,x2,t) obeys the continuity equation@C¼ r1  ðx_ 1 CÞ  r2  ðx_ 2 CÞ;@t:where xn is the flux velocity given by43XGnm
x_ na ¼ u0 ðzn Þdxa ab rmb ðU þ kB T ln CÞ:

(30)

(31)

m;b

Fig. 4 (a) Concentration profiles of the polymers at E = 1000E0 in
diﬀerent types of dumbbells. (b) Nonlinear electro-osmotic mobility withrespect to E.

mobilities with respect to the applied electric field. The resultant
electro-osmosis clearly increases nonlinearly with respect to theapplied electric field. When the applied field becomes stronger,
the mobility increases and is saturated, similar to that in the toymodel. The two types of dumbbells have diﬀerent rheologicalproperties in the bulk,40–42 so this nonlinearity is not due to therheological properties of the dumbbells. On the other hand, themobility is almost constant for E t 10E0, and this threshold of
linearity is larger than E0, which is predicted by the toy model.
Likewise, saturation is observed when E E 104E0, which is larger
than E1.
To clarify the diﬀerence in the profiles near the surface, hq2i
and hqz2/q2i are plotted with respect to the distance from thesurface. Fig. 5(a) shows the profiles of hqz2/q2i. In the bulk, they
approach 1/3, which means that the dumbbells are distributed
isotropically. Near the surface, the polymers are inclined by theshear flow. The Hookian dumbbell has the largest angle
between the z axis and the dumbbell direction. Fig. 5(b) plots
the profiles of hq2i. In the bulk, they approach 3d0, which istheir equilibrium value. Near the surface, they become larger
because the polymers are elongated by the shear flow. For the7406 | Soft Matter, 2015, 11, 7402--7411

In the kinetic model, the beads are treated as point-like particles.
Thus, the mobility tensor is obtained by using GB instead of GRPB
for both the self and distinct parts. The continuity equation canbe rewritten using q and r as@C_
_  rq  ðqCÞ;¼ rr  ðrCÞ
@t

(32)

where r = (x1 + x2)/2 is the centre of the mass of the dumbbell.
We also define r1 and r2 as1
r1 ¼ rr  rq ;
2

(33)

1
r2 ¼ rr þ rq :2

(34)Then, the probability function is also regarded as a function ofr and q. Here we neglect the interaction between the wall andthe beads. The flux velocities for r and q are obtained as11
sr_a ¼ ½u0 ðz1 Þ þ u0 ðz2 Þdxa þ G
ab Fb
2
2
þ

kB T 
Gab rqb ln C  DK
ab rrb ln C;
2(35)^ ab F s
q_ a ¼ ½u0 ðz2 Þ  u0 ðz1 Þdxa  GbþkB T 
^ ab rqb ln C;
Gab rrb ln C  kB T G
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where F s = r1U s is the spring force, and DK is the Kirkwood
diffusion tensor, which characterises the diffusivity of macromolecules and is given by
DK ¼

kB T G11 þ G12 þ G21 þ G22
:
4

(37)

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% and Ĝ are variants of the mobility tensors defined as
G% = G11  G12 + G21  G22,
G

(38)

Ĝ = G11  G12  G21 + G22.

(39)

The concentration field c(r,t) can be obtained by integrating theprobability function with respect to the spring coordinate. It is
given byðcðr; tÞ ¼ Cðr; q; tÞdq:(40)
We also define the probability function only for the spring
coordinate as^ t; rÞ ¼ Cðr; q; tÞ:Cðq;cðr; tÞ

(41)

These new fields satisfy the continuity conditions, such that


@c
_q ;
¼ rr  chri
(42)
@t
^
@C
^ q_ ;¼ rq  C@t(43)

where h  iq represents the average with the spring coordinate:Ðð   Cðr; q; tÞdq^ q; tÞdq:¼    Cðr;(44)h  iq ¼
cðr; tÞ
% and DK can be expanded using r.
For the limit of q { r, G
Keeping only the leading term, we obtain
1
0
qz
0
wqx
C
3 B
C
B
 ¼
qz wqy C þ    ;
(45)
G
B 0
2
A
32pZ0 z @wqx wqy 2qz

DK ¼



kB T
3a
I þ SðqÞ þ    ;12pZ0 a
4

(46)

(47)

Note that the approximation is more accurate than that in aprevious study31 as that study considered only w E 1, which is
not satisfied near the surface. With the approximation, eqn (36)
^ and finally we obtain the concentration flux
is averaged by C,
for the z direction as
i
dh
c DK
chr_z iq ¼ cumig ðzÞ (48)
zz q ;dz

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umig ðzÞ ¼
¼

E
1D 
 zb
Gzb Fbs  kB Trqb G
q2
3
64pZ0 z2D E
 w qx Fxs þ qy Fys  2qx Fzs  2kB Tðw  1Þ :q

(49)
Eqn (48) indicates two opposite fluxes of the polymers due tothe external flow field. One is the migration flux from the wall
toward the bulk and originates from the hydrodynamic interaction between the wall and the force dipoles.31 The other is thediﬀusion flux from the bulk to the surface wall and is not foundfor polymers in uniform shear flows.31 Note that the second
flux includes not only the ordinary diﬀusion flux hDKzziqrrzc, butalso the diﬀusion flux due to q inhomogeneity, crrzhDKzziq.
When the external shear flow is uniform, the second flux
vanishes, and the depletion length is proportional to the squareof the shear rate, as the migration velocity is proportional to thenormal stress diﬀerence.31 In a plug flow, the diﬀusion flux
suppresses the growth of the depletion layer, and this behaviour
may explain why the exponent of the depletion length is muchsmaller than 2.0 in a uniform shear flow. In the steady states ofelectro-osmosis, the total flux in eqn (48) becomes zero; thus,
!
d DK
dc
c
zz q
¼
:
(50)
umig 
dz
dz
DK
zz q
This equation shows that the migration flux and the diﬀusion
flux are balanced at the peak of the concentration profiles.
Finally the concentration profile is calculated as
! #
"ð
z
d DK
1
zz q
0
c ¼ cb exp
dz0 :umig ðz Þ 
(51)
K
dz
D Dzz q
The resultant flow profile can be calculated asð
1 z p 0 0s ðz Þdz ;duðzÞ ¼ 
Z0 0 xz(52)

where rp is the polymeric part of the stress tensor:
rp = chqF siq  ckBT I.

where

5=2
qx 2 þ qy 2
:
w¼ 1þ
4z2

where

(53)

To obtain explicit expressions for c and du, it is necessary toestimate umig, hDKzziq, and rp. For this purpose, eqn (43) should
be analysed. However, eqn (43) is highly complicated. Even
without the wall eﬀects, it cannot be solved exactly, so several
approximation methods have been proposed.44 For simplicity,
all the hydrodynamic interactions are ignored; thus, the continuity equation is given as



^
@C
du0
2Fas ^ 2kB T^ : (54)¼ rqaqz dza 
C
rqa C
@t
6pZ0 adz
6pZ0 aFor the Hookian dumbbell, eqn (54) can be solved for the second
moment of q, and for the FENE dumbbell, a pre-averaged

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Soft Matterapproximation41,42 is employed. The curved lines in Fig. 5 arecalculated using these approximations, and they exhibit goodquantitative agreement with the simulation results. Appendix B
gives approximate expressions for these quantities of the Hookian
and FENE dumbbells.
Fig. 6(a), (c), and (e) show the concentration profiles for theapplied field E = 1000E0. The points are obtained by the Brownian
dynamics simulation and the curved lines are obtained by the
kinetic theory. The theoretical calculations quantitatively cover
the simulations well. Moreover, they reproduce the diﬀerences in
the concentration near the surface between the Hookian and
FENE dumbbells, as the migration velocity can be approximately
proportional to hwiq (see Appendix B), and it is greatly suppressedin the Hookian dumbbells. Fig. 6(b), (d), and (f) show the nonlinear electro-osmotic mobilities with respect to the applied field.
The theoretical curved lines also exhibit acceptable agreementwith the simulation results. However, they are not as consistent
with the simulation results under weak applied electric fields, as
the equilibrium depletion layer is not considered in the kinetic
theory.

Paper

6 Summary and remarks
Brownian dynamics simulations are used to study nonlinear
electro-osmotic behaviour of dilute polymer solutions. The simulation results agree with a toy model and analytical calculations usinga kinetic theory. The main results are summarised below.
(i) Under an external plug flow, the polymer migrates toward
the bulk. The concentration profile of the polymer shows adepletion layer and a single peak. The thickness of the depletion
layer depends on the electric field. At the peak, the migration
flux is balanced by the diﬀusion flux.
(ii) The growth of the depletion layer leads to an increase
and saturation of the electro-osmotic mobility. This behaviour
does not depend qualitatively on the rheological properties ofthe dumbbells.
(iii) The results of analytical calculation of the concentrationand the nonlinear mobility using the kinetic theory agree withthe results of the Brownian dynamics simulation. The threshold of the electric field for nonlinear growth and saturation of
the mobility is much larger than that predicted by the toymodel, as the diﬀusive flux suppresses the migration toward
the bulk due to the inhomogeneous shear flow.
We conclude this study with the following remarks.
(1) Nonlinear electro-osmosis with l { d0 has already been
observed experimentally.26,27 These studies reported that themobility increased with increasing electric field. However, nonlinear electro-osmosis with l c d0 has not been reported
experimentally; therefore, experimental verification of our findings
is highly desired.
(2) It remains a future problem to determine whether thehydrodynamic interaction between the polymers and the surface
plays an important role in the electro-osmosis of polymer solutions
with l { d0. In this case, the elongation of the polymers is strongly
inhomogeneous under plug flow with a short Debye length; thus,
more realistic chain models should be considered.
(3) Addition of charged polymers to solutions can change thedirection of the linear electro-osmotic flow.7,23 When a suﬃciently
strong electric field is applied to this system, the flow might recoverits original direction. This needs to be investigated theoretically andexperimentally.

A Derivation of the velocity equationfor Brownian dynamics simulation
In this appendix, the derivation of eqn (25) is explained. The
velocity field induced by the polymer is given byduðzÞ ¼ 
Fig. 6 (a) Concentration profiles of the Hookian dumbbell as a function of
the distance from the surface. Points show the simulation results; thecurved line is calculated by the kinetic theory. (b) Nonlinear electroosmotic mobilities of the Hookian dumbbell as a function of an applied
electric field. (c) and (d) same as (a) and (b), respectively, but for the FENE
dumbbell with b = 600. (e) and (f) Same as (a) and (b), respectively, but for
the FENE dumbbell with b = 50.

7408 | Soft Matter, 2015, 11, 7402--7411

ð
1 z ps ðzÞdz;
Z0 0 xz(55)

and the polymeric part of the stress tensor is obtained byaveraging the microscopic expressions in the lateral directions asspab ¼ð
1dxdy^
spab ðxÞ:L2(56)

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Soft Matter

Here the microscopic expression of the stress tensor is given by^pab ðxÞ ¼ s

1X X
Fnm;a xnm;b dsnm ðxÞ;2 n manand the polymeric stress tensor isrp ¼ chqF s iq ckB TI
0 2
1
2f 0 fB
CB
C¼ ckB T B 0
0 0 C:
@
A
f 0 0

(57)

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where Fnm is the force exerted on the n-th bead by the m-th bead,and dsnm(x) is the symmetrised delta function given by
dsnm ðxÞ ¼

ð1dsdðx  sxn  ð1  sÞxm Þ:

(58)

0

The Kirkwood diﬀusion constant can be estimated as



kB T
3a 1
qz 2
1þ 2
1þ
12pZ0 a
4 q
q


kB T
3a q2 þ qz 21þ;
12pZ0 a4 hq2 i3=2

The symmetrised delta function is integrated in the lateral
directions as

DK
zz q ¼ðð1sdnm ðzÞ ¼ dxdydsnm ðxÞ ¼ dsdðz  szn  ð1  sÞzm Þ
0

¼

(59)

yðz  zm Þ  yðz  zn Þ;
zn  zm

where y(zn  z) = 1  y(z  zn). Then we obtainðzðz  zn Þyðz  zn Þ  ðz  zm Þyðz  zm Þnm
dz0 dS ðz0 Þ ¼
zm  zn0"

1 X
Fnm;1 ½minðz; zn Þ  minðz; zm Þ
2Z0 L2 nm¼

1 Xminðz; zn ÞFn;1 :Z0 L2 ndkB T 3a 4kf2 f2 þ 3
DK:
zz q ¼
dz
12pZ0 a 4d0 ð2f2 þ 3Þ5=2(68)

It is diﬀerentiated by z as

The migration velocity can be estimated using the splitting
approximation of the averages as(61)umig ðzÞ ¼

3kB T
w qx 2 þ qy 2  2w q
64pZ0 z2



3kB T
hwi qx 2 þ qy 2  2 q
32pZ0 z2 q

¼

3kB T
hwi f2 ;32pZ0 z2 q*Eqn (54) can be rewritten in a closed form for the second
moment of the spring coordinates in a steady state with an
imposed plug flow. The solution is given by42
0
1
1 þ 2f2 0 f
C
kB T BB
C(62)
hqqiq ¼0
1 0 C;
B
A
H @
f
0 1
where

hwiq ¼

B.2

This journal is © The Royal Society of Chemistry 20155=2 +q

(70)

FENE dumbbell

The second moment of the spring coordinate for a FENE dumbbell can be obtained by pre-averaged closures of a p-FENE
model.41,42 It is given by0
hqqiq ¼(64)

qx 2 þ qy 2
1þ
4z2


5=2
f2 þ 1
 1þ
:
2z2

(63)

Therefore, we havehqF siq = Hhqqiq,

(69)

where

Hookian dumbbell

du0¼ tkm0 Eekz :
dz

(67)

(60)

B Approximated expressions for
kinetic theoryf¼t#
3a 2 f2 þ 21þ
:
4d0 ð2f2 þ 3Þ3=2

kB T
DK
zz q ¼
12pZ0 awhere min(z,zn) = zy(z)  (z  zn)y(z  zn). Finally, the velocity
increment is expressed asðz
1 XsduðzÞ ¼
Fnm;1 ðzn  zm Þ dz0 dnm ðz0 Þ
2Z0 L2 n;m
0
¼

(66)

where the second term is split into second-order moments;
thus, we obtainminðz; zn Þ  minðz; zm Þ¼;
zn  zm

B.1(65)

kB T cB
B
B
H f@

1 þ 2c2

0 c

0

1

c

0

1

C
C
0 C;
A
1

(71)

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Paper

References

and
0
B
B
hqF s iq ¼ kB T B@

2

1 þ 2c

0

0

1

c

0

c

1

C
C
0 C;
A
1

(72)

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where
(
" 
#)
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3þb
1
bf 3 þ b 3=2
c¼6:
sinh arcsinh
543
108 54(73)

The polymer stress tensor is
0

2c2

B
B
rp ¼ ckB T B 0
@
c

0 c
0
0

1

C
C
0 C:
A
0

The Kirkwood diﬀusion constant is
sﬃﬃﬃﬃ
"
#
kB T
3a f 2 c2 þ 2
K
Dzz q ¼
1þ;
12pZ0 a4d0 c ð2c2 þ 3Þ3=2

(74)

(75)

and its derivative issﬃﬃﬃﬃ
dkB T 3af
KD¼
kdz zz q 12pZ0 a 4d0
c
"

#


2 c2 þ 2
dc 4c c2 þ 3
f dc
 f
þ
;
1
df ð2c2 þ 3Þ5=2
c df
ð2c2 þ 3Þ3=2
(76)
where
(
" 
#)
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

dc
bþ3
1
bf 3 þ b 3=2
¼2
cosh arcsinh
df
543
108 54
"



b
108

 
 #1=2
bf 2
bþ3 3
þ
:
108
54

(77)

Finally the migration velocity is obtained as
3kB T
hwi c2 ;32pZ0 z2 q

(78)


5=2
c c2 þ 1
1þ
:
f 2z2

(79)

umig 
wherehwiq 

Acknowledgements
The author is grateful to O. A. Hickey, T. Sugimoto, R. R. Netz,and M. Radtke for helpful discussions. The author also thanks
M. Itami, M. R. Mozaﬀari, and T. Araki for their critical readingof the manuscript. This work was supported by the JSPS Coreto-Core Program ‘‘Nonequilibrium dynamics of soft matter andinformation’’, a Grant-in-Aid for JSPS Fellows, and KAKENHI
Grant No. 25000002.

7410 | Soft Matter, 2015, 11, 7402--7411

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Paper

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