Electro-osmotic flow of semidilute polyelectrolyte solutions
Yuki Uematsu and Takeaki Araki
Citation: J. Chem. Phys. 139, 094901 (2013); doi: 10.1063/1.4820236
View online: http://dx.doi.org/10.1063/1.4820236
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THE JOURNAL OF CHEMICAL PHYSICS 139, 094901 (2013)

Electro-osmotic flow of semidilute polyelectrolyte solutions
Yuki Uematsua) and Takeaki Araki
Department of Physics, Kyoto University, Kyoto 606-8502, Japan(Received 20 June 2013; accepted 21 August 2013; published online 5 September 2013)
We investigate electro-osmosis in aqueous solutions of polyelectrolytes using mean-field equations.
A solution of positively charged polyelectrolytes is confined between two negatively charged planar
surfaces, and an electric field is applied parallel to the surfaces. When electrostatic attraction between the polymer and the surface is strong, the polymers adhere to the surface, forming a highly
viscous adsorption layer that greatly suppresses the electro-osmosis. Conversely, electro-osmosis
is enhanced by depleting the polymers from the surfaces. We also found that the electro-osmotic
flow is invertible when the electrostatic potential decays to its bulk value with the opposite sign.
These behaviors are well explained by a simple mathematical form of the electro-osmotic coefficient.
© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4820236]
I. INTRODUCTION

Aqueous solutions of polyelectrolytes are widely seen in
many systems,1 and are used in a variety of industrial applications. Some bio-macromolecules, such as proteins andDNA, also exist in the charged state. The chemical and physical properties of polyelectrolyte solutions have been extensively studied.2–6 When a polyelectrolyte solution flows near acharged interface, electrokinetic phenomena occur. The electrokinetics of polyelectrolyte solutions are important for understanding some physiological situations such as strain andrestoration of bone7 and blood flows in capillary vessels.8
However, although electro-osmosis and electrophoresis havebeen well investigated in simple electrolyte solutions,9 theelectrokinetics in polyelectrolyte solutions are not entirely
understood.10–12
The static properties of electrically neutral polymers nearsolid walls have been described by mean-field and scaling
theories.13–15 According to prediction, the spatial decay of theconcentration profile from the surface to the bulk follows an
exponential or power law. If the chemical intermolecular interactions between the polymer and the wall surface are sufficiently strong, adsorption or depletion layers form on thewall. By contrast, in polyelectrolyte solutions, the concentration profile and conformations of the polymer chains depend on the interplay between the electrostatic and chemicalinteractions.16–21 Electrostatic interactions attract the polyelectrolytes to the oppositely charged wall, and vice versa
thereby, lead to formation of the adsorption layer. If the polyelectrolyte adsorption is strong, the electrostatic potential ishighly modified and the concentration profile becomes nonmonotonic. In the phenomenon known as charge inversion
or overcharging, the electrostatic potential decays to its bulk
value with reversed sign.
A few experimental and theoretical studies on the electrokinetics in polyelectrolyte solutions have been reported.
The first measurements of electro-osmosis in a polyelectrolyte
solution did not account for the near-surface structures.22a) Electronic mail: y_uematsu@scphys.kyoto-u.ac.jp

0021-9606/2013/139(9)/094901/8/$30.00

Later researchers noted the importance of fluid viscosity inthe electric double layer.23, 24 Electro-osmosis is also thought
to be influenced by the non-Newtonian behavior of polymeric liquids.5, 25–27 However, instead of considering electrostatic and chemical interactions, these studies assumed agiven static profile near the surface. This study aims to clarify the electro-osmosis of polyelectrolyte solutions, focusing
on the structure of electric double layers. In determining theelectro-osmotic coefficient, both viscosity near the interface
and the strength of the overcharging are considered.
II. MACROSCOPIC ELECTRO-OSMOSISWhen a charged-surface capillary is filled with an electrolyte solution and subjected to an electric field E and a pressure difference P( = −∇p), a volume flux and electric current
are induced along the external fields. Here, p is the pressure.
When E and P are sufficiently weak, the volume flux and theelectric current are given by
V = L11 P + L12 E,(1)

J = L21 P + L22 E.

(2)

V and J are the mean volume flux and mean electric current, respectively, and Lij ’s are the Onsager transport coefficients. This article focuses on L12 , known as the electroosmotic coefficient. In a solution containing many monovalentions, the electro-osmotic flow induced by the electric field is
non-Poiseuille. Instead, it presents as a plug flow that exponentially decays with the Debye screening length. When thecapillary diameter is much larger than the Debye screening
length, the electro-osmotic coefficient is independent of thesalt concentration, and is given by Smoluchowski’s formula,
L12 = −εψ S /4π η, in which ε is the dielectric constant of thesolution, ψ S is the electrostatic potential at the nonslip surfaces, and η is the viscosity of the solution. The near-surface
properties of polyelectrolyte solutions in capillaries greatly
differ from those of small-ion electrolyte solutions. For

139, 094901-1© 2013 AIP Publishing LLC

094901-2

Y. Uematsu and T. Araki

J. Chem. Phys. 139, 094901 (2013)

example, as mentioned above, polyelectrolyte solutions form
adsorption or depletion layers near the capillary surface.
Also, electro-osmosis in polymer solutions is likely to beaffected by the rheology of the solutions. In particular, theviscosity of such solutions is not constant but depends on
the local polymer concentration and the shear rate. Later inthis study, we account for both of these factors in determining
the electro-osmotic coefficient using mean-field equations.
III. MEAN-FIELD EQUATIONS FOR
CONCENTRATIONS, ELECTROSTATIC POTENTIAL,
AND FLOW

We consider an aqueous solution of sufficiently longpolyelectrolyte chains in a slit (see Fig. 1). A fraction f of thepolyelectrolytes is positively charged, whereas the slit wallis negatively charged. Counterions from the polyelectrolytes
and salts are also dissolved in the solution. For simplicity, we
assume that the anions from the salt and the counterions fromthe polymers are the same species and all the small ions aremonovalent. The free energy of the system is contributed bypolymer conformations, ion distributions, and electrostatic interactions as follows:
F = Fpoly + Fions + Fele .
The polymer free energy is given by3

 2

a
v
|∇φ|2 + φ 4 ,Fpoly = kB T
dr
6
2

(3)

(4)

where φ is an order parameter related√ to the local polymer
concentration c(r), given by φ(r) = c(r). kB T is the thermal energy, a is the monomer size, and v is the second virial(excluded volume) coefficient of the monomers.
The ion free energy, contributed by the translational entropy of the ions, is given by
dr[ci ln(ci a 3 ) − ci ],(5)Fion = kB T
i=±where c+ (r) and c− (r) are the concentrations of the cations
and anions, respectively. The electrostatic free energy is givenbyFele =

ε|∇ψ|2 ,
d r ρψ −8π(6)where ψ(r) is the local electrostatic potential, ε is the dielectric constant of the aqueous solution, and ρ(r) is the electriccharge density defined as
ρ = e(f c + c+ − c− ),(7)where e is the elementary electric charge.
The control parameters in this study are the bulk concentrations of the cation cb+ and the charged monomer fraction
f. These parameters should satisfy the neutral charge condition, f cb + cb+ − cb− = 0, in the bulk. Here, cb and cb− are thebulk concentrations of the monomers and anions, respectively,
whose steady profiles are obtained by minimizing the following grand potential: 2
μi ci d r,(8)
 = F − μ φ dr −i=±where μ and μ (i = ±) denote the chemical potential of eachcomponent.
The solution is confined within a slit bounded by two parallel walls. We assume that the above variables change only
along the y axis, and are homogeneous along the x and z axes.
In this scenario, the mean-field equations areia2 ∂ 2φ= v(φ 3 − cb φ) + f φβeψ,
6 ∂y 2

(9)

∂ 2ψ
4π e
8π e +f (φ 2 − cb exp[βeψ]) +c sinh(βeψ),
=−
∂y 2
ε
ε b
(10)
where β = 1/kB T. Equation (9) is the Edwards equation that
accounts for the charge effect, while Eq. (10) is the PoissonBoltzmann equation for the system containing the salts andpolyelectrolytes.
Applying a sufficiently weak electric field E in the x direction, the system evolves to steady state in which ion fluxes
are induced along E. Because E is weak and orthogonal to−∇ψ(y), we assume that it influences neither the concentration fields nor the polymer conformations (see Appendix A).
In steady state, the mechanical forces are balanced. This force
balance is expressed by the Navier-Stokes equation, whose
simplified form is


∂vx
∂
η(φ)
+ ρE = 0,
(11)
∂y
∂y
where vx (y) is the x component of the velocity field. In thiscase, because we impose no pressure difference on the system, P = 0 in Eqs. (1) and (2). η(φ) is the viscosity, which
is a function of the concentration order parameter φ. In thisstudy, we set√(12)η(φ) = η0 {1 + h(φ/ cb )α },FIG. 1. Schematic of the investigated system. A positively charged polyelectrolyte solution is confined within a negatively charged slit and an external
electric field is applied along the slit walls. The long-chain polymers are interspersed with polymer counterions and anions derived from salt.

where h and α are nondimensional parameters. Here, η0 is
the solvent viscosity and ηb = η0 (1 + h) denotes the viscosity in the bulk. Because η(φ) usually increases from η0 as φ094901-3

Y. Uematsu and T. Araki

increases, h and α are assumed positive. As described in Appendix B, h and α depend on the physical parameters N, f,
and cb+ , in which N is the polymer length. In this study, however, h is assumed as an independent parameter. According toFuoss law,28 we set α = 1. Later, we demonstrate that these
simplifications do not alter the essential results.
As shown in Fig. 1, the surfaces are placed at y = 0 and
2L, where 2L is the slit width and the electrostatic potentials
are the same at both surfaces. Because all profiles are symmetric with respect to y = L, we consider only the range [0, L].
At the bottom surface (y = 0), we assume φ(0) = 0, implyingthat the intermolecular interactions between the surfaces andpolymers are strongly repulsive. We also set vx (0) = 0 andψ(0) = ψ S . The former is the nonslip boundary condition for
the flow. ψ S is negative because the surfaces are negatively
charged and the electrostatic interaction between the polymer
and surfaces is attractive. Because the system is symmetric,
all y derivatives vanish at y = L:



∂φ 
∂ψ 
∂vx 
= 0,
= 0,
= 0.
(13)
∂y y=L∂y y=L∂y y=L
In this study, we assume that the electric field is sufficiently weak so that the flow speed is proportional to the fieldstrength. Specifying a coefficient λ12 (y), the flow profile isexpressed as vx (y) = λ12 (y)E. In other words, the solution
is Newtonian and the nonlinear dependence of the flow on E
can be ignored. Having obtained the static profiles (which are
difficult to solve analytically), λ12 is calculated as

 y
∂ψ 
dy 
ε
λ12 (y) =.
(14)
4π 0 η(φ(y  )) ∂y y This quantity is related to the macroscopic electro-osmotic
coefficient in Eq. (1) by

1 L
L12 =
λ12 (y)dy.
(15)
L 0

IV. RESULTS AND DISCUSSION

To study the effects of the near-surface polyelectrolyte
structures on electro-osmosis in this system, we numerically evaluate Eqs. (9)–(11). The parameter settings are cb= 10−6 Å−3 , v = 50 Å3 , L = 1024 Å, B = 7 Å, T = 300 K,
ψ S = −kB T/e = −25.8 mV, a = 5 Å, and η0 = 0.01 P.
Here, B is the Bjerrum length, given by B = e2 /(εkB T).
The polymer chains are assumed so long that cb > c*, wherec* is the overlap concentration of the polymer solution (seeAppendix B).
The electro-osmotic coefficient is evaluated from theL012 of a solution without polyelectrolytes, given by
L012 = −εψS /4π η0 = 1.82 × 10−4 cm2 /V s. The space discretization in the numerical calculations is d = 1 Å.
A. Electrically neutral polymer solution
with chemically repulsive surfaces

First, we assume that polymers are electrically neutral,i.e., f = 0. In this case, the mean-field equations (9) and (10)

J. Chem. Phys. 139, 094901 (2013)

are exactly solved asφ=βeψ = 2 ln√

cb tanhyξ

,

1 + e−κy tanh(βeψS /4),
1 − e−κy tanh(βeψS /4)

(16)

(17)

where √κ = (8π B cb+ )1/2 is the Debye wave number,
ξ = a/ 3vcb = 408 Å is the correlation length of the polymer concentration fluctuation, and e is the Napier’s constant.
Note that these analytical solutions are valid only when κ −1
 L and ξ  L because they are solved under the boundary
conditions at y = 0 and L. If |βeψ S |  1, Eq. (17) reduces toψ = ψS e−κy ,(18)using the Debye-Hückel approximation.
If the slit width is much larger than all other length scales
in the system, L12 is approximately equal to L12 ≈ λ12 (L).
Therefore, we write
 L
dy ∂ψ
ε
.
(19)
L12 ≈
4π η0 0 1 + η1 φ ∂y
Because ψ(y) is a monotonically increasing function of y in
Eqs. (17) and (18), the integral · · · dy in Eq. (19) can be replaced by · · · dψ, using ey/ξ = ζ −1/κξ . L12 is then calculatedas 1(ζ −1/κξ + ζ 1/κξ )dζ
L12
=
,
(20)
−1/κξ − (h − 1)ζ 1/κξ
L012
0 (h + 1)ζ
where ζ = ψ/ψ S is a reduced electrostatic potential. Aftersome calculations, Eq. (20) can be expanded as∞
2 κξ + 1
L12
1  h−1 n=+h + 1 κξ + 2 h + 1 n=1 h + 1
L012


1
1
+
. (21)
×2n/(κξ ) + 1 2(n + 1)/(κξ ) + 1

When h = 1, Eq. (21) reduces to
L12 = L012

κξ + 1.
κξ + 2(22)

Clearly, Eq. (22) is an increasing function of κξ .
The electro-osmotic coefficient calculated by Eq. (21) isplotted as a function of salt concentration in Fig. 2. Shown are
the coefficients for several values of the bulk viscosity parameter h. As the salt concentration increases, the electro-osmotic
coefficient increases and approaches L012 , regardless of h. By
contrast, in the low salt concentration regime, L12 decreases as(cb+ )1/2 to Lb12 = L012 /(1 + h), the electro-osmotic coefficientestimated at the viscosity of the bulk solution.
We interpret these results as follows. In the neutral polymer solution, the electrostatic interaction does not influence
the polymer concentration profile. The polymers are depleted
from the surface by short-ranged surface forces, and the nearsurface viscosity is smaller than that in the bulk. Only theregion near the surface, where ρ = 0, responds to the applied electric field. The charged region is characterized by the
Debye length κ −1 from the surface. If the Debye length is094901-4

Y. Uematsu and T. Araki

J. Chem. Phys. 139, 094901 (2013)
1

1

0.04

0.8

0.8

0
0.001
0.0058
0.033
0.19
1.0

0.6

0.03

0.02

0.4
0.01

0.6

0.2
0
0

0.4

-6

10

0.5
1.0
2.0
4.0
9.0

0.2

0 -6
10

-5

10

-4

10

0.001 0.01

0.1

1

smaller than the correlation length ξ , the electro-osmosis isenhanced; otherwise it is suppressed.
Given the effective viscosity ηS , the electro-osmotic coefficient is calculated by the usual Smoluchowski’s formula, L12= −εψ S /4π ηS . As noted above, the formation of the depletion layer near the surface effectively lowers the viscosity ofthe solution. From Eq. (21), the effective viscosity decreases
with κξ as
h
h+1
ln
,
h−1
2

-4

10

-0.01
0.001 0.01

0.1

1

-6

10

-5

10

-4

10

0.001 0.01

0.1

1FIG. 3. (a) Electro-osmotic coefficients L12 , plotted as functions of salt concentration cb+ . The fraction of charged monomers in the polyelectrolyte f isvaried for fixed h = 9. At f = 0, the curve is that of the electrically neutralpolymer solution, and it increases with cb+ as shown in Fig. 2. At sufficient
salt concentrations, all curves approach L012 . (b) Magnification of the sameplots around a small range of L12 .

FIG. 2. Electro-osmotic coefficients L12 in the solution without polyelectrolytes, plotted as functions of salt concentration cb+ . In place of polyelectrolytes, electrically neutral polymers are dissolved. The viscosity parameterh is varied.

ηS ≈ η0 (1 + h) 1 − κξ

-5

10

(23)


−1 n
when κξ  1, using ∞n=1 n r = ln[1/(1 − r)]. On theother hand, when κξ
1, the viscosity approaches the solvent viscosity, ηS ≈ η0 . This phenomenon can be explained
as follows. In the high salt limit, the electrostatic interaction
between ions and walls is screened by a short length scale.
If the wall is chemically repulsive to the polymers, the polymers are depleted from the surface with a correlation length
far exceeding the Debye screening length.

B. Polyelectrolyte solution with electrically attractive
and chemically repulsive surfaces

polyelectrolytes are highly charged. Such inversion of electroosmotic flow is never observed in neutral polymer solutions.
The electro-osmotic coefficients are plotted as functionsof f in Fig. 4(a). Here the salt concentration is fixed at a low
concentration cb+ = 10−6 [mol/ ], and the bulk viscosity parameter h is changed. We observe that the electro-osmotic
flow is weakened if the polyelectrolytes are highly charged.
The mechanism of this phenomenon will be discussed later.
Figure 4(a) also shows that electro-osmosis inversion occurs only at sufficiently high h. Figure 4(b) plots the electroosmotic coefficient versus h for cb+ = 10−6 [mol/ ] and f = 1.
As discussed above, the electro-osmotic flow in neutral polymer solutions saturates at Lb12 in the low salinity limit, according to Eq. (23). However, this equation cannot explain
the curve in Fig. 4(b).
1. Relationship between electro-osmosisand static properties

Figure 5(a) plots the curves of L12 = 0 and L12 = Lb12
in a cb+ -f plane. As the bulk viscosity parameter h decreases,
the region of inverted electro-osmosis (L12 < 0) shrinks andeventually disappears as h becomes small. On the other hand,
the L12 = Lb12 curves are less sensitive to changes in h. Thisimplies that L12 around Lb12 depends more on the static thankinetic properties.
0.03

0.1

0.025

Next, we consider polyelectrolyte solutions, i.e., f = 0.
Figure 3(a) shows the electro-osmotic coefficients as functions of salt concentration. Here we fix h = 9 and vary thefraction of charged monomers f. We find that, as in neutralpolymer solutions (see Fig. 2), electro-osmosis is suppressedin the low salt regime. At high salt concentrations, the electroosmotic coefficient approaches L012 . Figure 3(a) also indicates
that, with increasing electric charge on the polyelectrolytes,
electro-osmosis becomes more suppressed and salinity exerts
a more drastic effect. In Fig. 3(b), these plots are magnified
around L12 = 0. Interestingly, the electro-osmotic coefficient
can become negative at sufficiently dilute salt and when the

0.08
0.02
0.06

0.015
0.01

0.04

0.005
0
-0.005
-0.01
0.01

0.02
2
3
4
9

0
0.1

1

0

1

2

3

4

5

6

7

8

9

FIG. 4. (a) Electro-osmotic coefficients L12 , plotted as functions of the fraction of charged monomers in the polyelectrolytes f. h is varied at fixed salt
concentration cb+ = 10−6 mol/ . (b) Electro-osmotic coefficient L12 plotted
as a function of h. We set cb+ = 10−6 mol/ and f = 1.

094901-5

Y. Uematsu and T. Araki

J. Chem. Phys. 139, 094901 (2013)

1

1

0.1

0.1

0.01

0.01
2
3
4
9

0.001 -6
10

0
0.005
0.01
-5

10

-4

10

0.001 0.01

0.1

0.001

1

-6

10

-5

10

-4

10

0.001 0.01

0.1

1FIG. 5. (a) State diagram of the electro-osmotic flow in the cb+ -f plane. Because the polyelectrolyte concentration varies in the slit, the local viscosity
depends on the distance from the wall. Lb12 is the electro-osmotic coefficient, estimated from the shear viscosity of the bulk solution. (b) Contoursof the amount of excess adsorption, , for  = 0, 0.005, and 0.01. Shown are
the contour lines of L12 = 0 (solid) and L12 = Lb12 (dashed) for h = 9. The
 = 0 contour behaves similarly to the line L12 = Lb12 in (a).

To characterize the static properties, we define a quantity
 as
=

 L

dy(c − cb ).

(24)

0

 measures the amount of excess adsorption of the polyelectrolytes. Figure 5(b) plots the contour lines of  in the
cb+ -f plane. The  = 0 contour characterizes the adsorptiondepletion transition.20 In the system investigated here, thepositively charged polymers are dissolved in the slit between
the negatively charged walls. Electrostatic interaction adheres
the polymers to the oppositely charged wall surface. On theother hand, intermolecular interaction prevents the polymers
from directly contacting the surface (see Fig. 6(a)). When theelectrostatic interaction is well screened by high salt content,chemical interaction depletes the polymers from the surface
vicinity.
Interestingly, when cb+ is fixed, excess adsorption does
not continuously increase toward f = 1 but instead peaks at
an intermediate f. As shown in Fig. 5(b), the polyelectrolytes
with cb+ = 10−6 mol/ are most strongly adsorbed when f ≈ 5
× 10−3 . This nonmonotonic behavior is counterintuitive
1

1
polymer order parameterelectrostatic potential

0.5

because one expects that highly charged polyelectrolytes will
be adsorbed with greatest strength. The adsorption-depletiontransition has been intensively studied by Shafir et al.20 Comparing Figs. 5(a) and 5(b), we find that the curves L12 = Lb12
roughly coincide with that of  = 0. When the polymers areadsorbed to the surface ( > 0), the electro-osmotic coefficient is smaller than that determined by the surface potential
and bulk viscosity Lb12 , and vice versa.
Figure 6 shows profiles of the polymer concentration andelectrostatic potential at (a) high and (b) low salt concentrations. The fraction of charged monomers is f = 0.03. The
solution conditions are as indicated in Fig. 5(a). Under lowsalinity conditions, where L12 < Lb12 , a peak appears in theconcentration profile. Hereafter, the height and the position of
the peak are denoted as φ M and yφ , respectively. As shown in
Fig. 5(b), the amount of adsorption is positive (i.e., in excess).
√
Hence, we refer to the region of φ > φb (= cb ) as an adsorption layer, although the polymers themselves do not contact
the surface. The electrostatic potential also peaks at y = yψ .
We call this peak an overcharging potential and its height isdenoted as ψ M . We should note that the φ and ψ peaks appear at different positions, with yψ > yφ . We also define y0 ,which satisfies ψ(y0 ) = 0. As discussed below, the adsorption layer and the overcharging potential play essential rolesin the decrease and inversion of the electro-osmotic coefficient. Conversely, under high-salinity conditions, the profiles
monotonically increase to the bulk values without developing peaks. The dependences of φ M and ψ M on cb+ and f are
shown in Figs. 7(a) and 7(b), respectively. The contours of
φ M and ψ M are qualitatively similar to that of L12 in Fig. 5(a)but are dissimilar from that of the excess adsorption. This implies that the maximum amount of adsorption is not importantin the electro-osmotic phenomena.
The uncolored region, in which the profile does not peak,almost coincides with that of L12 > Lb12 . The gradient of
the electro-osmotic flow is localized to the range of the Debye screening length from the surface (see Fig. 8(a)). Therefore, the formation of the depletion layer effectively reduces
the solvent viscosity. Because the electro-osmotic flow is inversely proportional to the viscosity, depletion enhances theelectro-osmosis. At the adsorption-depletion transition, theincrease in L12 caused by the depletion cancels the decreasecaused by adsorption. Then, the  = 0 curve is roughly

0.5
1

18

1

0.1
0.09

16

0

0.08

14

0
0.1

12

0.07

0.1

0.06

10

0.05

-0.5

-0.5

8
0.01

polymer order parameterelectrostatic potential
-1

6

100

200

300

400

500

0.03

4

-1
0

0.04
0.01

0.02

2

0

100

200

300

400

500

FIG. 6. Profiles of the polymer order parameter φ and the electrostatic
potential ψ near the surface. The bulk concentrations of the salt are (a)cb+ = 0.0476 [mol/ ] and (b) cb+ = 2.91 × 10−5 [mol/ ]. In both cases, thefraction of charged monomer in the polyelectrolyte is f = 0.03. The profilesin (a) and (b) correspond to conditions (A) and (B) in Fig. 5(a). For a clearer
representation, we plot φ(y)/15φ b and 15ψ(y)/|ψ S | rather than φ(y) and ψ(y).

0.001
-6
10

-5

10

-4

10

0
0.001 0.01

0.01

"figure/figure17.dat" u 2:3:7

phiM
0.1

1

0.001
-6
10

-5

10

-4

10

0
0.001 0.01

0.1

1FIG. 7. Effect of cb+ and f on the peak heights of (a) concentration profile
φ M and (b) electrostatic potential ψ M . Shown are the contour lines of L12= 0 (dashed) and L12 = Lb12 (solid) for h = 9. The dotted line is L12 = 0
estimated by Eq. (30). Uncolored regions indicate where no peaks appear in
φ and ψ (i.e., where φ M and ψ M are undefined).

094901-6

Y. Uematsu and T. Araki

0.31

J. Chem. Phys. 139, 094901 (2013)
1

1

1

0.8

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.305
0.3
0.295
0.29

0.6

0.285

0.4
0.015
0.01

0.2

0.005
0

0

-0.005
-0.01
0

100

200

300

400

500

-1

-0.8

-0.6

-0.4

-0.2

0

FIG. 8. (a) Profiles of the electro-osmotic flow λ12 in three states: (A)depletion state: cb+ = 0.0476 mol/ and f = 0.03; (B) adsorption state:cb+ = 2.91 × 10−5 mol/ and f = 0.03; (C) flow inversion state: cb+ = 2.91
× 10−5 mol/ and f = 0.46. These conditions are marked in Fig. 5(a). In
condition (C), the overcharging potential is ψ M /|ψ S | ≈ 0.1. (b) Parametricrepresentations of ψ(y) and 1/η(φ(y)) for the three states. Points (ψ/|ψ S |,
η0 /η) = ( − 1, 1) and (ψ/|ψ S |, η0 /η) = (0, 1/(1 + h)) correspond to the surface (y = 0) and bulk (y = L), respectively. The bulk viscosity parameter isfixed at h = 9.

consistent with that of L12 = Lb12 . Because neutral polymers
in solution do not adhere to the surface, electro-osmosis is
more strongly suppressed in polyelectrolyte solutions than inneutral polymer solutions.
The uncolored region in Fig. 7(b), where ψ M develops
no peak, is slightly wider than that in Fig. 7(a), where φ Mdevelops no peak. This difference is delicate because theDebye screening length becomes comparable to the systemsize when cb+ and f are very small.

2. Relationship between electro-osmosisand dynamical properties

As shown in Fig. 7, φ M and ψ M are large in the regimeof large f and small cb+ , where the electro-osmotic coefficient
becomes negative. We emphasize that these large values of
φ M and ψ M are essentially important for the sign reversal ofL12 .
Figure 8(a) shows the profiles of the flow field near
the surface under three conditions. Here we note that vx (y)= λ12 (y)E. Conditions (A) and (B) correspond to the adsorption and depletion states, respectively. The global electroosmotic coefficient L12 becomes negative under condition (C).
These conditions are marked in Fig. 5(a). In all cases, the gradient of the flow field is localized to the vicinity of the surface;that is, the flow macroscopically behaves as a plug flow. Whilecurve (A) varies almost monotonically with y, curve (B) is
nonmonotonic, and curve (C) is more complex. Under condition (C), the flow direction is positive near the surface, butchanges at some distance from the wall, saturating at a negative value. The saturation value gives the macroscopic electroosmotic coefficient from Eq. (15). By contrast, curve (B)remains positive across the range. If the viscosity is homogenous and independent of the polymer concentration, the flow
field is easily calculated from Eq. (14) as
λ12 (y) =ε{ψ(y) − ψS } .
4π η

(25)

0

0
-1

-0.8

-0.6

-0.4

-0.2

0

-1

-0.8

-0.6

-0.4

-0.2

0

FIG. 9. (a) Schematic for calculating L12 in adsorption states from Eq. (27).
ψ and 1/η are parameterized with respect to y. (b) Approximate representation of ψ-η−1 in (a). This approximation gives a simple form of L12 ,
Eq. (30).

The overcharging potential is necessary the nonmonotonic
variations in (B) and (C). However, because ψ S < ψ(y) everywhere, the overcharging potential alone cannot explain thenegative L12 given that η is constant.
If the electrostatic potential monotonically changes withy as in condition (A), ψ = ψ(y) is uniquely expressed by its
inverse function y = y(ψ). Then, Eq. (14) is given by 0εη(ψ  )−1 dψ  ,(26)
λ12 (L) =4π ψS
where η(ψ) = η(φ(y(ψ))) is also a unique function of ψ. The
curves of η(ψ) are plotted in Fig. 8(b). Since η(ψ) is positive,
λ12 (L) is also positive, indicating that the flow toward E ismaintained.
When the overcharging potential arises, as in conditions(B) and (C), y is a multivalued function of ψ, which invalidates Eq. (26). In this case, Eq. (14) becomes


 yψ
 L
dy  ∂ψ dy  ∂ψ 4π
λ12 (L) =+εη(y  ) ∂y y yψ η(y ) ∂y y 
0
 ψM
 ψM
dψ 
dψ 
−
.
(27)
=


ψS |y<yψ η(ψ )0|y>yψ η(ψ )Here we should note that the paths of the two integrals in
Eq. (27) differ from each other.
According to linear analysis, the electrostatic potential
profile may have multiple peaks.17 The intensities of the peaks
decay with increasing distance from the wall. We assume thatthe highest peak (nearest the wall) plays a dominant role in
the electrokinetic flow and ignore the contributions of the remaining peaks.
Figure 9 is a schematic of Eq. (27). When the electrostatic
potential overcharges, the curve of 1/η(ψ) is divisible into
three segments. These segments delineate three realms, withareas denoted by S1 , S2 , and S3 . Within the slit, the realms
correspond to the ranges S1 : 0 < y < y0 , S2 : y0 < y < yψ , andS3 : yψ < y < L (see Fig. 6(b)). The first and second terms in
Eq. (27) are given by S1 + S2 and S2 + S3 , respectively. In
terms of these areas, the electro-osmotic coefficient is given
by L12 = (S1 + S2 ) − (S2 + S3 ) = S1 − S3 . If S1 < S3 , themacroscopic flow is inverted.
Using Eq. (27), we devise a simple method for estimating the electro-osmotic coefficient in adsorption states.

094901-7

Y. Uematsu and T. Araki

J. Chem. Phys. 139, 094901 (2013)

The viscosity is assumed constant within each realm. Moreprecisely, we assume that polymer concentration is fixed at
φ = φ M within the range 0 < y < yψ and at φ = φ b in yψ < y< L. These approximations are schematically represented inFig. 9(b). S1 and S3 are then approximated as

S3 ≈

S1 ≈

ε −ψS
,
4π ηS

ε
4π

1
1
−
ηb
ηS

(28)

ψM ,

(29)

where ηS = η0 (1 + hφ M /φ b ) and ηb = η0 (1 + h). Finally, we
obtain
L12 ≈

ηb b
ηb
L + 1−
ηS 12
ηS

LM
12 ,

(30)

where LM
12 = −εψM /(4π ηb ) is the electro-osmotic coefficientestimated by the overcharging potential. The L12 = 0 curve
estimated by Eq. (30) is drawn in Fig. 7. This curve is qualitatively consistent with the numerical solutions. In this estimation, the overcharging potential does not directly cause
the inversion of electro-osmotic flow; formation of the highly
viscous layer is also important.
V. SUMMARY AND REMARKS

Applying a continuum model, we study electro-osmosisin polymer solutions. From numerical calculations and theoretical estimations, we elucidated the behaviors of the electroosmosis in polymer solutions. The dependence of viscosity
on the polymer concentration plays an important role in ourmodel. Our main results are summarized below.
(i) Even if the polymer solution sandwiched between chemically repulsive walls is electrically neutral, electroosmosis depends on the salt concentration. Decreasing
the salinity suppresses the electro-osmosis.
(ii) In polyelectrolyte solutions, the formed adsorption layer
effectively enlarges the viscosity in the vicinity of thesurfaces. Consequently, electro-osmosis is suppressedmuch more strongly in polyelectrolyte than in neutralpolymer solutions. If a sufficiently high proportion of
the monomers are charged and if the salt concentration is
sufficiently low, the electro-osmotic flow can be inverted.
(iii) We propose a simple equation for estimating the electroosmotic coefficient in adsorption states (Eq. (30)). Thisequation captures the essential features of the inversionof the electro-osmotic coefficient, shown in Fig. 7. According to this expression, inversion is caused by two
factors: enhancement of the viscosity by the near-surface
adsorption layer and overshoot of the electrostatic
potential.
We conclude this paper with the following remarks.
(1) Charge inversion and mobility reversal induced bymultivalent electrolytes has been frequently reported.29
Grosberg et al.29 concluded that such phenomena depend on fluctuation correlations among the multivalentions, which are excluded in usual Poisson-Boltzmann

approaches. Our mean-field approach predicts that similar inversion phenomena occur in polyelectrolyte solutions. According to a molecular dynamics simulation,
the phenomena occurs even in monovalent ions solutions
confined within nanochannels.30 The flow profiles obtained in the earlier study are quite similar to ours; near
the surface, the flow is directed toward the electric field,
but in the bulk, it is against the field.
(2) This article considers only limited situations. The surfaces are assumed to chemically and electrostaticallyrepel the polymers. If the surfaces are chemically attractive, the adsorption is much enhanced by chemical
forces.19, 21 The electro-osmotic properties of these surfaces are equally interesting and important.
(3) From the Onsager reciprocal relations, the electroosmotic coefficient L12 should equal L21 in Eq. (2). The
latter represents the electric current due to the mechanical pressure difference. Interestingly, the Onsager coefficient L21 is inverted when f is large and cb+ is sufficiently
small.
(4) In the above numerical and theoretical analyses, theviscosity parameter h is assumed constant, although inpractice it depends on the fraction of charged monomers
f and the salt concentration cb+ . When f is large and cb+ issmall, the solution viscosity increases (see Appendix B).
Our studies indicate that large f and small cb favor flow
inversion. The same trends were observed for large h. If
we set h as a function of f and cb , more dramatic changes
would appear in the curves of L12 against f and cb+ . Although the L12 and the phase diagrams would quantitatively alter, the qualitative trends, i.e., suppression of theelectro-osmotic flow and inversion at large f and a small
cb+ , should remain intact.
ACKNOWLEDGMENTS

Y.U. is grateful to J. F. Joanny for helpful discussions
since the start of this investigation, and to D. Andelman for his
valuable comments. Y.U. is supported by a Grand-in-Aid for
JSPS fellowship. This work was financially supported by the
JSPS Core-to-Core Program “Non-equilibrium dynamics ofsoft matter and information” and KAKENHI (Nos. 23244088,
24540433, and 25000002). The computational work was carried out using the facilities at the Supercomputer Center,
Institute for Solid State Physics, University of Tokyo.
APPENDIX A: LOCAL EQUILIBRIUM CONDITIONSFOR THE COMPONENTS

Because we apply an external field E along the x direction (see Fig. 1), the total electrostatic potential is not ψ(y)in Eq. (6), but instead is (x, y) = ψ(y) − Ex. Assuming the
local equilibrium condition, the chemical potential of the ith
species is given by
μi = kB T ln(ci a 3 ) + ezi ,(A1)where zi is the charge of the ith ion. In the geometry of theinvestigated system, the diffusion flux of the ion, given by094901-8

Y. Uematsu and T. Araki

J. Chem. Phys. 139, 094901 (2013)

j i = −D i ci ∇μi , is divided into two components:

1/2

j i = jyi ey , +jxi ex ,(A2)∂
[kB T ln(ci a 3 ) + ezi ψ(y)],
∂y

(A3)

jxi = D i zi ci E.

(A4)

jyi = −D i c+

Here, Di is the diffusion constant of the ith ion, and ex andey are the unit vectors along the x and y axes, respectively.
Because the system is confined by the walls at y = 0 and 2L,
the diffusion flux along the y direction vanishes at steady state.
Thus, we obtain the Boltzmann distribution along the y axis as
ci a3 ∝exp { − zi eψ/kB T}. On the other hand, the diffusion flux
remains along the x axis. Because the applied electric field is
sufficiently weak and orthogonal to −∇ψ, it influences neither the concentration fields nor the polymer conformation.
APPENDIX B: SCALING BEHAVIORSIN POLYELECTROLYTE SOLUTIONSThe scaling behaviors of polyelectrolyte solutions are
known to widely differ from those of uncharged polymer solutions. At the overlap concentration c∗ in a polyelectrolyte
solution, the monomer density inside the coil equals the overall monomer density in the solution.28 In our notation, theoverlap concentration in a theta solvent is determined by−2.
c∗ (1 + 2cb+ /c∗ f )−3/2 ≈ N −2 a −2 −1
B f
In the low-salt or salt-free regime, the overlap concentration becomes c* ≈ (a2 B Nf)−1 . Conversely, it approaches−7 −4 1/5
N }in the high-salt regime.
c∗ ∼= {8(cb+ )3 a −4 −2
B f
Between these two extremes, the overlap concentration
decreases as f increases. Given the same polymer length
N, polyelectrolyte chains expand more than their uncharged
counterparts.
The viscosity of polyelectrolyte solutions also obeys
scaling behaviors, which depend on the solvent quality andthe polymer concentration regime. For example, the viscosity of a semidilute solution in a theta solvent is given by1/2η ≈ η0 Na B f c1/2 (1 + 2cb+ /f c)−3/4 . If the salt is not dissolved or is insufficiently dilute, this expression approachesη ≈ η0 N a B f c1/2 ; that is, the viscosity is proportional toc1/2 (Fuoss law). On the other hand, in highly saline conditions
1/2
the viscosity behaves as η ≈ η0 N a B (cb+ )−3/4 f 7/4 c5/4 . The
viscosity depends on the polymer concentration as c5/4 , identical to that of an uncharged polymer solution in a theta solvent,
namely η ≈ η0 N(ca3 )1/(3ν−1) with ν = 3/5. Physically, this result implies that electrostatic interactions in a polyelectrolyte
solution are well screened by the salt.
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