Actuation and ion transportation of polyelectrolyte gels
Wei Hong*a b, Xiao Wangb
a
Department of Aerospace Engineering, bDepartment of Materials Science and Engineering
Iowa State University, IA USA 50011
ABSTRACT
Consisting of charged network swollen with ionic solution, polyelectrolyte gels are known for their salient characters
including ion exchange and stimuli responsiveness. The active properties of polyelectrolyte gels are mostly due to the
migration of solvent molecules and solute ions, and their interactions with the fixed charges on the network. In this
paper, we extend the recently developed nonlinear field theory of polyelectrolyte gels by assuming that the kinetic
process is limited by the rate of the transportation of mobile species. To study the coupled mechanical deformation, ion
migration, and electric field, we further specialize the model to the case of a laterally constrained gel sheet. By solving
the field equations in two limiting cases: the equilibrium state and the steady state, we calculate the mechanical
responses of the gel to the applied electric field, and study the dependency on various parameters. The results recover
the behavior observed in experiments in which polyelectrolyte gels are used as actuators, such as the ionic polymer metal
composite. In addition, the model reveals the mechanism of the selectivity in ion transportation. Although by assuming
specific material laws, the reduced system resembles those in most existing models in the literature, the theory can be
easily generalized by using more realistic free-energy functions and kinetic laws. The adaptability of the theory makes it
suitable for studying many similar material systems and phenomena.
Keywords: polyelectrolyte, hydrogel, ion transportation
*whong@iastate.edu; phone 1 515 294-8850; fax 1 515 294-3262
1.
INTRODUCTION
Polyelectrolyte gels consist of a polymer network with ionic groups that dissociate in a solvent and form fixed charges.
The resulting material is capable of both instantaneous elastic deformation and swelling/deswelling by exchanging
solvent molecules with the environment1,2. Furthermore, the mechanical response of a polyelectrolyte gel can be
modulated by environment stimulus such as pH or electric field, as the concentrations of both solvent and solute in the
gel couple strongly with its deformation. Such a correlation as well as the reversibility of the deformation makes
polyelectrolyte gel a candidate for smart sensing and actuating devices3-5. As the simplest design of this kind,
polyelectrolyte gel membranes have attracted considerable attention in the past decade as soft actuators and sensors6-8, in
addition to traditional applications such as ion-exchange media which use the selective transportation property9,10.
While empirical models exist among the efforts on studying the sensing and actuations capabilities of simple
structures11, in order to reveal more physical mechanism of the coupling behaviors and to have the ability of more
precise prediction, later theories of a polyelectrolyte gel are mostly based on the fundamentals of linear poroelasticity,
with the coupling behavior accounted for as a consequence of the electro osmosis. By adopting the Onsager relations, de
Gennes et al. assume that both the transportation of solvent and the effective ionic current are driven by the electric field
and the pressure gradient, and model the membrane as a linear elastic beam that couples with the transportation through
the pressure field12. A similar approach is taken by Doi and his coworkers, who also account for the reversible
dissociation of the fixed ionic groups on the network13,14. A widely accepted and more detailed model, introduced by
Nermat-Nessar and coworkers15,16, considers the migration of both counterions and solvent driven by the gradient of
electrochemical potential, and accounts for the electrochemical and mechanical coupling through the osmotic pressure
and a pressure induced by dipolar interactions. Due to the low concentration of coions, their transportation and effect on
the coupling behavior is usually neglected. Similar approaches have also been taken by various other groups, with the
Nernst-Planck equation directly taken as the governing equation, and specific sensing or actuation applications as the
main focus17-20. The same model has also been represented in variational forms 21. In these models, some specific
material laws are implicitly assumed and mixed with other governing equations, making the generalization of the theory
to other similar material systems rather difficult. Furthermore, the mathematical complexity of the theory has limited
most of the calculations to one-dimensional. Starting from the basic principles of thermodynamics, Hong et al. revisit
the field theory of polyelectrolyte gels, and proposed a theoretical framework to study the coupling behaviors22. With
the material description localized only in a few functions and most of the governing equations universally applicable, the
theory can be easily adapted to similar systems.
In the current paper, we extend the theory to more general non equilibrium cases. With the assumption of a specific
free-energy function, a linear kinetic relation, and the partial equilibrium in the mechanical and electrical fields, the
theory is reduced to a form similar to those in the literature 15,16. In addition to the flux of solvent and counterions, the
flux of coions is calculated, and their effect on the coupling behavior is taken into account. As timely examples, two
limiting cases are studied: a global equilibrium state without diffusion flux or electric current, and a steady state with all
transportation being time-independent. Using these examples, we study the two most important properties of a sheet of
polyelectrolyte gel: its autonomous bending under an applied electric field, and the selectiveness in ion transportations.
Figure 1. Sketch of a polyelectrolyte gel under inhomogeneous deformation. The volume element dV and the
area element dA are both taken in the reference state. In the current state, a battery of voltage  does work
Q , a surface traction does work TdA , and a source of mobile species 
2.
does work
  i dA .
A NONLINEAR FIELD THEORY OF POLYELECTROLYTE GELS
The theory described here closely follows the approach introduced by Hong et al.22. Fig. 1 illustrates a gel undergoing
inhomogeneous deformation. We take the stress-free dry network as the reference state, in which the coordinate of a
material particle is X . Also in the reference state, let dV be a volume element, dA be an area element, and N be the
unit vector normal to the area element. At time t , the particle X moves to a place with coordinate x . The mapping
x  x X, t  fully describes the deformation field. The deformation gradient is
FiK 
xi X , t 
.
X K
(1)
To characterize the state of a swollen gel in terms of its content of mobile solvent molecules and solute ions, we


introduce the nominal concentration C X , t  at the vicinity of the material particle X , so that C X , t dV is the
number of particles of species  in a volume element. In the current paper, we will use the a simple solution that

contains only mono-valence ions as an example, and denote the concentrations of the cations and anions as C and C
s

respectively, and that of the solvent as C . In the absence of free electrons in the volume, the electric charge is
contributed by mobile solvent ions and fixed charges on the network. Here we assume the fixed ions on the network to
0
be fully dissociated with a constant nominal concentration C . Without losing generality, we also assume the fixed ions
to be negatively charged. Denoting the total charge in a volume element as X , t dV , we have
  eC   C   C 0  ,
(2)
where e is the elementary charge. Gauss’s law can also be expressed in terms of the nominal fields 23,24, and the nominal
~
electric displacement D satisfies the following equation:
~
DK X, t 
 X, t  .
X K
(3)
Following a common practice in formulating a continuum field theory, we stipulate that an inhomogeneously swollen gel
can be divided into many small volume elements, each of which is locally in a homogeneous state, characterized by the
nominal density of free energy as a function of various thermodynamic variables:


~
W  W F, D, C  , C  , C s .
(4)
Associated with small changes in the independent variables, the free-energy density changes by
W 
W
W
W
W
W
 FiK 
 DK    C     C   s  C s .
FiK
DK
C
C
C
(5)
There are three ways of doing work to a polyelectrolyte gel: applying mechanical forces, attaching batteries, and
exchanging solvent molecules or solvent ions through external sources. In this paper, we neglect all body forces and
volumetric sources, so that the three types of work are all done via the surface of a gel. In the current state, let
TX, t dA be the mechanical force applied on the area element. Associated with a small deformation x X  , the
surface force does work
 T x dA .
i
i
Similarly, through transporting a charge Q from the cathode to the anode, a

battery of voltage  does work Q . Let  be the chemical potential of the mobile species  in an external
source. The external sources do work


i dA upon injecting i dA amount of species  into the gel. The
Helmholtz free energy of the system is the sum over the parts: the change in the free energy of the gel is
 WdV , and
the change in the free energy of the mechanical forces, batteries, or external sources is the negative amounts of the work.
Here we have neglected any leakage of electric field to the space out of the gel, and such an effect can be easily
accounted for if needed. Thermodynamics dictates that the Helmholtz free energy of the system should never increase:
 WdV   T x dA   dA   
i
i

i dA  0 .
(6)
We assume that no chemical reaction takes place in the volume, so that the number of particles is conserved for each
species:
C 
J 
 K ,
t
X K
(7)
where J is the nominal flux of species  , defined with respect to the reference state such that J K N K dA is the
number of particles crossing an area element per unit time. By definition, the flux on the surface satisfies the relation


J K N K dA 
i
dA .
t
(8)
Substituting Eqs. (2), (3), (5), (7), and (8) into inequality (6), and applying the divergence theorem, we have

~
 W ~  D
siK xi
x
dV   siK N K  Ti  i dA    ~  E K  K dV
X K t
t
 DK
 t
i 
  
        
dA   
J K dV  0
t
X K
,
(9)
where siK  W FiK
~
EK   X K is the nominal electric field, and
is the nominal stress tensor,
   ez    W C  is the electrochemical potential of species  . The integrals in (9) consist of the energy
dissipation in distinct processes: the first two terms are due to the deformation of the network, the third is due to the
redistribution of electric charges, the fourth is due to the injection of mobile species from the surface, and the last is due
to the migration of mobile species within the gel.
Following Hong et al in the study of neutral gels25, we assume that the rate of swelling is limited by the migration of
mobile species, and that all terms but the last in (7) vanish, giving rise to equalities which represent the partial
equilibrium of the elastic deformation of the network:
siK
0
X K
(10)
in the volume, and siK N K  Ti on the surface, the electric field
W
~
EK  ~
D K
(11)
   
(12)
in the volume, and the mobile species
on the surface. Only the last term in (9) remains negative for arbitrary non-vanishing fluxes. A common approach is to
introduce kinetic laws

J K   M KL
 
,
X L
(13)
with a positive-definite mobility tensor M for each species.
Given a material behavior written in terms of the free-energy function and the mobility tensors, Eqs. (3), (7), and (10) –

(13) evolves the fields of displacement xX, t  , electric potential X, t  , and nominal concentrations C X, t 
simultaneously. While the field equations are written in terms of the nominal quantities, they can also be converted into
equivalent expressions in term of the true quantities: the true stress  ij  siK FjK det F , the true electric displacement
~
Di  DK F jK det F , the true concentrations c  C  det F , and the true flux vectors ji  J K FiK det F .
3.
A SPECIFIC MATERIAL MODEL
While the field theory described in the previous section is material-independent, a commonly used material model will
be adopted in the current paper. Following Flory26, Ricka and Tanaka27, Hooper et al.28, Brannon-Peppas and Peppas29,
and many others, we idealize the polyelectrolyte gel by assuming the free energy per unit reference volume in the form:


G
1 FiK FiL ~ ~ kT  s
vC s
 
~
 vC ln

W F, D, C  , C  , C s  FiK FiK  2 ln det F  
DK DL 

s
2
2 det F
v 
1  vC 1  vC s 
 



C
C




 kT C   ln

1

C
ln

1
s


 c vC s

  cref vC
ref



,
(14)
where G is the initial modulus of the polymer network,  is the effective permittivity of the gel, kT is the temperature
in the unit of energy, v is the volume of a solvent molecule, and cref is the reference value of the concentration. The
first term in the free-energy function (14) is due to stretching the network, the second polarization, the third mixing the
solvent with the network, the last mixing the solvent with solute ions.
The typical stress in a gel is small and the swelling ratio is usually large, we therefore assume both individual polymers
and mobile particles to be incompressible. For simplicity, we neglect the volume taken by solute ions, so that the
volume increase in a gel is fully taken by solvent molecules:
1  vCs  det F .
(15)


s
Following Hong et al.25, we enforce the constraint (15) by replacing adding the term  1  vC  det F into the free-
energy function (14). Here  X, t  is a Lagrange multiplier.
Consequently, we have the constitutive relations of the true stress
 ij 
G
FiK FjK   ij   1  Di D j  1 Dm Dm ij    ij ,
det F

2

(16)
of the true electric field
Di  Ei ,
(17)
of the electrochemical potentials of the mobile ions
   e  kT ln
C
,
vCs cref
(18)
and of the chemical potential of the solvent

vC s
1

C  C 



 v .
s
s
1  vCs 2 C s 
 1  vC 1  vC
 s  kT ln
(19)
The Lagrange multiplier  couples Eqs. (16) and (19), and physically represents a measure of the osmotic pressure.
To describe the kinetic property of the transportation in the gel, we assume that the particles of mobile species diffuse in
the gel25. For simplicity, we assume the coefficients of diffusion of species  , D , to be isotropic and independent of
the deformation and concentration. Thus, the true fluxes relate to the gradient of the electrochemical potentials by30
ji  
4.
c D  
.
kT xi
(20)
A SHEET OF POLYELECTROLYTE GEL
We consider a thin sheet of polyelectrolyte gel submerged in a fully-dissociated ionic solution as the model system.
Two idealized conductive electrodes are attached to the two surfaces of the gel. For simplicity, in this paper, we will
only look at a uniaxial deformation state of the gel by properly constraining the gel in the lateral direction with a lateral
stretch 0 , as shown schematically in Fig. 2. All field variables are thus one-dimensional: the axial stretch   x  , the

electric potential x  , and the nominal concentrations C  x  . To further simplify the expressions, we normalize all
lengths by the Debye length LD  kT 2e2cref and all energies by kT , and introduce the dimensionless coordinate
  x LD and the dimensionless electric potential   e / kT . We also denote the ratio between the ion


s
concentration and the solvent concentration as   C C . The field equations are specialized as follows. In
equilibrium with the stress-free external liquid solution, the axial stress vanishes and Eq. (16) becomes
2
 d  v
vG 2  1
 
 vcref 
0.
2
kT 0
 d  kT
(21)
Gauss’s law requires that
s
d 2   
C0 
 C








,
d 2
2 
cref cref 
(22)
where   1 0 is the inverse of the swelling ratio. Upon applying a voltage, the electrodes may exchange charge
with the mobile ions and result in an electrochemical reaction. Since neither the free-energy nor the kinetics of the
reactions is present in our current model, we will instead look at two limiting cases.
2
j+
Solution
j–

Solution
Gel
Permeable
electrode
Figure 2. A sheet of polyelelctrolyte gel, constrained in the vertical direction. The two surfaces of the gel are in
contact with two reservoirs of ionic solution. The electrodes are permeable to the solute ions as well as the solvent
molecules. When a voltage is applied between the two electrodes, the electric field drives the fluxes of the mobile
species.
In the first limiting case, we will neglect any charge exchange between the electrodes and the mobile ions. In
equilibrium, the chemical potentials of all mobile species are uniform and no diffusion flux is present. The
electrochemical potentials of the solute ions equal those in the external solution far from the electrodes, where the ion

concentration is taken to be the reference, cref . Thus,   0 , and from Eq. (18), we further have, in equilibrium,
   vcref exp   .
(23)
s
2
Substituting Eq. (23) into Eq. (22), and applying the incompressibility constraint, 1  vC  0 , we arrive at the
Poisson-Boltzmann equation for a polyelectrolyte gel in equilibrium:
 2
C 0



1


sinh


.
 2
2cref
(24)
Similarly, the solvent is also in equilibrium with the external solution with a chemical potential  s  2kTvcref .
Cancelling the Lagrange multiplier from Eqs. (19) and (21), we have
2
 d 
vG
  ln 1        2   2vcref cosh  1  0 .
 2  1  vcref 
kT
 d 
(25)
Equations (24) and (25) form a differential-algebraic system of the dimensionless electric potential    and the axial
stretch    . They can be solved numerically by using the electric potentials on the electrodes as boundary conditions.
As the other limiting case, we will consider the steady state of the gel in which the fluxes of the mobile species are timeindependent. Further, we assume that the solvent in the external solution on both sides of the gel layer has the same
chemical potential, so that there is no flux of solvent molecules in the steady state, and  s  2kTvcref . From Eq. (19),
we have
      2vcref  ln 1        2  
v
.
kT
(26)
A combination of Eqs. (21) and (26) yields
2
 d 
vG
  ln 1        2        2vcref  0 .
 2  1  vcref 
kT
d



(27)
Through a combination of Eqs. (18) and (20), the true fluxes of the solute ions are obtained as, in a dimensionless form:

   1      

d d  
,

d
d 
(28)
where   vLD j D are the dimensionless fluxes of the counterions (+) and coions (–), respectively. Equation (28)
is often referred to as the Nernst-Planck equation. Here it is only a consequence of the specific material model we
adopted, and it may not hold in general, for example, when the ionic solution is not ideal or the volume of the ions is not
negligible. However, the approach introduced in this paper will still be applicable, and a different material model will
only result in an equation different from the usual form of the Nernst-Planck equation. Gauss’s law still holds in the
steady state,



d 2
     C 0



1



.
d 2
2vcref
2cref
(29)

The differential-algebraic system given by Eqs. (27-29) can be used to solve for the dimensionless fields:    ,    ,
and    , as well as the unknown parameters  . To minimize the ambiguity in the boundary conditions at the
electrodes, we assume that the electrochemical reactions happen relatively fast and the rate of the kinetic process is
limited only by the transportation of ions through the gel. Therefore, the electrodes are in local equilibrium with the
surrounding liquid solution. Assuming that the ion concentration of the external solution to be the same as the reference
concentration on both sides of the gel, we have the electrochemical potential of the ions close to the electrodes

  electrode  e . The electrochemical potentials, together with the electric potential at the electrodes, are sufficient for
the boundary value problem. Although the electrochemical-potential boundary condition appears to be identical to the
electric-potential boundary condition, they are in fact independent. The similarity is due to our choice of the reference
ion concentrations.
5.
RESULTS AND DISCUSSION
As a first example, we consider the equilibrium state of a laterally constrained gel sheet by solving Eqs. (24) and (25)
simultaneously. The numerical solutions of the inhomogeneous fields in the gel are plotted in Figure 3. Neglecting the
electrochemical reactions at the electrodes, we assume that there is no electric current through the gel. A voltage V is
applied between the two electrodes. To avoid the asymmetry introduced by the bias of the electric potential, following19,
we apply symmetric boundary conditions on the two sides of the gel, i.e. with electric potential    V 2 on the two
electrodes, relative to the ionic solution remote from the gel. We set the dimensionless initial stiffness of the polymer
network vG kT  0.01 , which corresponds to a modulus on the order of ~100 kPa at room temperature when the
solvent is water. The fixed ions are assumed to be negatively charged with concentration vC  0.05 , corresponding to
a 5% molar fraction of the monomers on the network. The mobile ions are in equilibrium with those in the external
solution, which is also grounded and have a concentration of vcref  103 . The gel is constrained laterally at the free0
swelling state.
Figure 3 plots the solution of the fields in the gel, under three different voltages, eV kT  0 , 4 and 8. Here we have
chosen a gel with the thickness in the current state being 20 times of the Debye length. Therefore, the major part of the
gel is electroneutral with a uniform electric potential, except the thin layers near the surfaces, as shown in Figure 3 (a).
When no voltage is applied, eV kT  0 , the results recover the free swelling case, i.e. the Donnan equilibrium. Since
the network carries negative immobile charges, the equilibrium potential deep inside the gel is lower than that in the
external solution. The Donnan equilibrium deep inside the gel also causes the true concentration of the mobile
counterions to be close to that of the fixed ions, and the concentration of the coions to be significantly lower, as shown in
Figure 3 (b). When a finite voltage is applied, the major part of the gel is still in Donnan equilibrium with the external
solution, with both the electric potential and the ion concentrations remain unchanged. As a result, except in the surface
layers, the gel is in a stress-free state. In the surface layers with thicknesses comparable to the Debye length, the fields
are inhomogeneous. Near an anode, i.e. an electrode of positive voltage, due to the electrostatic interaction, the anions
have a relatively high concentration while the cation concentration is extremely low; for the same reason, cations
accumulate locally near the cathode. However, due to the huge difference in the base concentrations of the two types of
mobile ions in the Donnan equilibrium, the concentration of the coins (–) is still relatively low compared to that of the
counterions (+) even near the anode. Consequently, the osmotic pressure, which is directly related to the total
concentration of both types of mobile ions, is lower near the anode, and higher near the cathode.
a
b
4
10
10
C  /C s
eV/kT
2
0
-2
-4
-10
10
10
-5
0
5
10
10
-1
-2
+
C /C
-
-4
C /C
-10
-5
5
10
5
10
D
5
d
0
 yy v/kT
4

0
x/L
D
3
2
1
-10
s
-5
x/L
c
s
-3
-0.02
-0.04
-5
0
x/L
D
5
10
-10
-5
0
x/L
D
Figure 3. The spatial profile of the inhomogeneous fields, when the gel is in equilibrium with the voltages applied
between the two surfaces: a) the normalized electric potential; b) the molar fraction of the mobile counterions (+)
and coions (–) in the gel; c) the axial stretch in the thickness direction; d) the dimensionless lateral stress. Three
voltages are applied symmetrically between the two sides of the gel, with dimensionless voltage eV kT being 0,
4, and 8, signified by solid, dashed, and dotted curves, respectively. The external solution is grounded. In
dependent of the voltage applied, the major part of the gel is always electroneutral, and in the free-swelling state.
The axial stretch and the lateral stress, as shown in Figure 3 (c) and (d), are the consequence of two contributions
combined: the osmotic pressure due to the concentration of ions and the Maxwell stress due to the electric field. Unlike
the osmotic pressure, the Maxwell stress is independent of the polarity of the applied voltage, and has the same effect on
both sides of the gels – compressive along the axial direction, and tensile in the lateral direction. These two effects
combined, the lateral stress distribution is asymmetric. The asymmetry in the lateral stress will induce an effective
bending moment on the gel sheet, and causing it to bend towards the anode. Since the asymmetry in the stress
distribution is mainly due to the concentration difference of the mobile ions, which in turn is due to the existence of the
fixed charges in the gel, it is expected to be less pronounced under a higher electric field, i.e. when a higher voltage is
applied. To further quantify the bending effect of the gel sheet, we calculate the bending moment by integrating the
resulting lateral stress, and plotting it in Figure 4 (a) as a function of the applied voltage. A positive value of the
horizontal axis means a higher electric potential is applied on the right side, and a positive value of the moment indicates
that the gel will bend towards the left. As expected, when the applied voltage is relatively low, the bending moment
over the gel sheet almost increases linearly with the voltage; under a relatively high voltage, the bending moment
plateaus at a finite value. The equilibrium case studied here corresponds to a theoretical limiting case which may be
very difficult to achieve in practice, and we have not found any direct experimental evidence.
b 0.5
c
0
-1
-10
d/LD=20
Mv/kTL2D
Mv/kTL2D
1
d/LD=20
0
200
-5
0
5
40
100
200
-0.5
10
-20
0
eV/kT
Mv/kTL2D
a
0
1
5
10
eV/kT=20
-0.5
-1
-6
-5
-4
-3
-2
-1
10 10 10vc 10 10 10
20
eV/kT
ref
Figure 4. The normalized bending moment of a gel sheet as a function of the dimensionless voltage applied
between the two surfaces, in two different cases: a) The gel, including ions and solvent molecules, is in
equilibrium with the voltage applied and the external solution, so that there is no electric current or flux of mobile
species through the gel. b) The ion transportation in the gel reaches a steady state, so that the electric current is
time independent. Different curves represent gel sheets of different thicknesses, which are measured in the current
state in terms of the Debye length. c) The equilibrium bending moment at various concentrations of external
solution. The bending moment switches its direction when the external solution is very dilute.
a
b
10
-2
+
C  /C s
5
eV/kT
10
0
10
C /C
-3
-
-5
-10
-5
0
5
10
10
C /C
-4
-10
-5
x/L
s
5
10
5
10
x/L
D
c
0
s
D
x 10
d
3
-3
2
 yy v/kT

2.5
2
0
-2
1.5
-4
1
-10
-5
0
x/L
D
5
10
-10
-5
0
x/L
D
Figure 5: The spatial profile of the inhomogeneous fields, when the transportation of mobile species in the gel
reaches a steady state: a) the normalized electric potential; b) the molar fraction of the mobile counterions (+) and
coions (–) in the gel; c) the axial stretch in the thickness direction; d) the lateral stress distribution. Three voltages
are applied between the two sides of the gel, eV kT  0 , 4, and 10, signified by solid, dashed, and dotted
curves, respectively. A local equilibrium is assumed at the electrodes, so that the concentrations of the mobile
ions equals those in the external solution while the electric potential equals to that prescribed on the electrode.
As the second example, we consider the steady state of ion transportation in a gel sheet. Similar as in the equilibrium
case, the lateral stretch is constrained at the value of free swelling. The numerical solution to Eqs. (27-29) is plotted as
the solid curves in Figure 5. All material parameters are taken to be the same as in the previous example. For the
governing equations of the steady state, one needs to supply the electrochemical potential of the mobile ions (or its
concentration) on both sides of the gel, in addition to the electric potential boundary conditions. Here we imagine a state
in which the external solution is being actively refreshed continuously, so that the concentrations of the mobile ions near
the electrodes remain at the same level of that in the remote solution, namely, the reference concentration. In local
equilibrium with the electrodes, the dimensionless electrochemical potential of the anions and cations are given by Eq.
(23). The spatial profile of the electric potential is shown in Figure 5 (a). Except the thin layers near the surface, the
electric field is almost uniform in the gel. Therefore, just as in the equilibrium case, the gel is still approximately
electroneutral, with the concentration of counterions approximately equals that of the fixed charges and the
concentration of coions at a much lower level.
In the surface layers near the electrodes, the concentrations of both types of the mobile ions approach those in the
external solution, as prescribed by the boundary conditions. Since the total concentration of mobile ions in a
polyelectrolyte gel is usually higher than that in the solution, the osmotic pressure is lower in the surface layers, and the
unbalanced osmotic pressure would further induce a tensile lateral stress. Such an effect is similar on both sides of the
gel, while the contributions from the Maxwell stress are very different, because of the difference in the magnitudes of the
electric field near the two electrodes. As shown in Figure 5 (a), the magnitude of the electric field is much higher near
the anode, where the intrinsic potential drop due to the fixed charges in the gel is in the same direction as the applied
electric field. The asymmetrically distributed lateral stress will also result in an effective bending moment in the gel.
The moment is plotted in Figure 4 as the broken curve. As a secondary effect, the slight inhomogeneity in the
distribution of mobile ions through the thickness of the bulk will also cause the sheet to bend. Although the asymmetry
in stress distribution caused by this effect is relatively small, the resultant moment after integration can be appreciable,
especially for relatively thick gel sheets. As shown in Figure 4 (b), such an effect will induce a backward bending
moment under a relatively low applied voltage. Figure 4 (c) also shows the same effect: at a relatively low concentration
of the external solution, the contribution from the osmotic pressure dominates and the gel sheet tends to bend backward.
The electric-field induced bending behavior described here agrees qualitatively with various experimental observations
on ionic polymer-metal composites31-33, although a more rigorous quantitative comparison would require the knowledge
of many material and environmental parameters.
On the other hand, the large difference between the concentrations of coions and counterions in the gel will cause
dissimilar transportation properties of the two types of ions. Even if they have a same diffusion constant D independent
of the concentration, according to Eq. (20), they would have a huge difference in the mobility – the counterions can be
transported much more effectively than the coions. Since ancient times, polyelectrolyte gels have be used as ionexchange membranes because of this unique feature10. Our model also shows such an effect. Figure 6 shows the
dimensionless flux of mobile ions as a function of the applied voltage. Despite the existence of the surface layers, the
fluxes are almost linear in the applied voltage. To show explicitly the selectivity in different ions, we also plot the


transportation number t  j
j


 j  on the same figure. A lower value in the transportation number indicates
more efficient selectivity in ion transportation. Even though the ion fluxes are approximately linear, the transportation
number still weakly depends on the applied voltage, which in this case is a measure of electrochemical-potential drop
between the two sides of the gel. The ion selectivity is less effective when there is a higher potential drop.
-3
+
vL D j  /D
2
-vL j /D
D
0
-
vL j /D
0.11
0.14
0.1
d /d =0.5
0.12
0.09
0.2
0.1
0.01
D
0.1
-2
-4
-20
-10
0
10
eV/k T
Figure 6 a) The dimensionless ion fluxes,

b
0.16
t-
4
x 10
t-
a
0.08
20
0.08
0.02
0.04
0.06
0.08
0.1
L /d
D
vj LD D , of counterions (+) and coions (–), and the transportation

number t , plotted as functions of the voltage applied. b) The transportation number t plotted against the
inverse of the gel thickness. Different curves correspond to different true electric field in the middle of the gel.
Due to the edge effect, the concentration difference between coions and counterions is smaller near the surface of a gel
sheet. Therefore, the transportation number is naturally dependent on the thickness of the gel sheet. As shown in Figure
6 (b), under the same true electric field, the transportation number is smaller when the gel sheet is thicker. Such a result
suggests that a thicker gel is more effective as an ion-exchange membrane. When the thickness of the gel sheet is much
larger than the Debye length, the transportation number approaches a constant independent of the thickness.
6.
CONCLUSIONS
We have investigated the actuation and ion transportation of a polyelectrolyte gel sheet under external electric stimulus
by applying a continuum theory of large deformation and electrochemistry. Subject to the coupling electric field, elastic
deformation and the mixing of different species, mobile ions inside gel migrate towards places of lower electrochemical
potential. Since both the mobile ions and the solvent are redistributed asymmetrically, a gel sheet tends to change its
shape and bend towards either electrode. The mechanisms are examined through studying the initial bending moment by
fixing the lateral deformation of the sheet. The actuation of the gel sheet is mainly driven by the combined effect of the
asymmetrically distributed Maxwell stress and osmotic pressure. The inhomogeneous fields in the system are studied
for two cases: the global equilibrium state and the steady state with local equilibrium at the electrodes. In the
equilibrium case, the external stimulus is shielded by the electric double layers, and the bending of a gel sheet is only
driven by the contributions from thin layers near the surfaces. In the case of a steady state, although the strong surface
effect persists, an asymmetric distribution of osmotic pressure also builds up in the bulk of the gel. The competition
between the bulk and the surface effects leads to the variation in both the magnitude and the direction of the bending
moment, depending on various parameters such as the thickness of the sheet, the relative concentration of the external
solution, the applied electric field, etc. The prediction of the model agrees qualitatively with existing experimental
observations. In addition to the actuation mechanism of polyelectrolyte gels, we also use the model to investigate the
selective ion-transportation property: a network with fixed charges tends to impede the migration of coions due to their
low concentration in equilibrium and transient states. The dependences of the ion selectivity on the electrochemicalpotential drop and on the gel thickness are studied using the model. Nonlinear behaviors are obtained even with the
assumption of constant diffusion coefficients.
The model developed here requires no empirical assumptions other than the material specific free-energy function and
the kinetic equations. Therefore, by assuming other types of material laws, it can easily be generalized and used to
describe similar systems with multiphysics coupling and mass transportation.
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