APPLIED PHYSICS LETTERS 98, 081910 共2011兲
Theory of ionic polymer conductor network composite
Xiao Wang (王宵兲1 and Wei Hong (洪伟兲1,2,a兲
1
Department of Materials Science and Engineering, Iowa State University, Ames, Iowa 50011, USA
Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50011, USA
2
共Received 1 December 2010; accepted 24 January 2011; published online 23 February 2011兲
Ionic polymer conductor network composite 共IPCNC兲 is a mixed conductor consisting of a network
of loaded ionomer and another network of metallic particles. It is known that the microstructure of
the composite, especially that of the electrodes, plays a dominating role in the performance of an
IPCNC. However, the microstructure of IPCNC has seldom been addressed in theoretical models.
This letter formulates a continuum field theory for IPCNC by considering a supercapacitorlike
microstructure with a large distributed interfacial area. The theory is then applied to the study of the
equilibrium deformation and electrochemistry in a thin-sheet IPCNC actuator. © 2011 American
Institute of Physics. 关doi:10.1063/1.3555437兴
Ionic polymer conductor network composite 共IPCNC,
including the ionic polymer metal composite兲 has been developed as soft actuators and sensors in the past decade.1–4 A
widely accepted mechanism pictures IPCNC as an ionomer
thin film sandwiched between two flat electrodes and assumes it to be homogeneous and purely ionic conductive.5–7
Upon application of a low voltage, the mobile ions in the
ionomer undergo directional motions: cations migrate toward
the cathode while anions migrate toward the anode. The actuator bends as a consequence of the charge accumulation
near the electrodes.
However, such an idealization may lead to problematic
predictions. As a conductor, the homogeneous ionomer is
electroneutral in equilibrium or in a steady state, except in
the very thin boundary layer 关i.e., the electric double layer
共DL兲兴 near the ionomer–electrode interface. The DL at a
typical thickness of nanometers for aqueous solutions is too
thin to effectively bend the IPCNC with a thickness of
⬃100 ␮m. Moreover, the estimated electric capacitance of
IPCNC as a parallel capacitor is much lower than
measured.8–10 Many models circumvent these discrepancies
by introducing a dielectric constant much higher than that of
deionized water8,11,12 with an unclear physical origin. These
difficulties may be resolved by realizing that the IPCNC
electrodes are made from precipitating nanometal particles
into the bulk polymer even in the early designs.9,13 As a
result, the active domain extends deep into the bulk even
though the boundary layer near the particle–matrix interface
is still very thin. The capacitance boost can thus be explained
from the large interfacial area.14,15 Corrections have been
made on the parallel-plate model to account for the electrode
roughness.14 The dependence on specific microstructures
limits the application of these modified models, especially
when the electrodes undergo a drastic change in design.2,3
The lack of predictive power of existing model has already
hindered the further development and scale-up of IPCNC.
Following an approach similar to that used in polyelectrolyte gels,16 this letter develops an equilibrium field theory
by considering IPCNC as a mixed conductor. Due to the
complexity of the system, the discussion is limited to the
effect of electric DL on electromechanical coupling among
a兲
Electronic mail: whong@iastate.edu.
0003-6951/2011/98共8兲/081910/3/$30.00
various other processes.8,11 Instead of the commonly accepted parallel-plate-capacitor picture, let us include the effect of the electrode materials infiltrated in the ionomer matrix and consider a trilayer model as sketched in Fig. 1. Two
layers of composite consisting of both the ionic conductive
phase 共ICP, i.e., the ionomer兲 and the electronic conductive
phase 共ECP, i.e., the metal particles兲 are connected to a battery, and the middle layer is simply ionic conductive. The
spatially distributed ECP provides a large interfacial area between the two phases, similar to the structure of a supercapacitor. The length scales of both the microstructure and the
DL are assumed to be small compared to the overall size of
the device so that a homogenized model can be deduced.
There are usually two ways of doing work to IPCNC: by
exerting a field of mechanical forces and by applying a field
of electric voltage. Associated with a change in displacement
field ␦u共x兲, the field of mechanical tractions t共x兲 on a surface
does work 兰ti␦uidA. Imagine connecting to the IPCNC a
field of batteries with potential ⌽共x兲. Through injection of
electrons the battery field does electric work 兰⌽␦qdV, where
q共x兲 is the concentration of electronic charge in the volume.
Let W be the Helmholtz free energy per unit volume of the
IPCNC. Unlike general polyelectrolyte, IPCNC usually operates in an environment from which the exchange of mobile
species is prevented. Therefore, the total number of each
mobile species is conserved. Let c+共x兲 and c−共x兲 be the concentrations of mobile cations and anions, respectively. The
conservation of species requires that 兰c⫾dV = N⫾, where N⫾
is the total number of mobile ions. We enforce this constraint
by adding to the total free energy of the system a term,
␮⫾共N⫾ − 兰c⫾dV兲, with ␮⫾ being a Lagrange multiplier. In
electronic
conductive
ionic
conductive
ionic
conductive
only
electric
double layer
FIG. 1. 共Color online兲 A trilayer model of IPCNC. The two composite
electrode layers are mixed conductors with finite thickness.
98, 081910-1
© 2011 American Institute of Physics
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081910-2
Appl. Phys. Lett. 98, 081910 共2011兲
X. Wang and W. Hong
equilibrium, the change in the Helmholtz free energy of the
IPCNC equals the work done by the external agents:
冕
␦WdV − ␮+
+
冕
冕
␦c+dV − ␮−
冕
␦c−dV =
冕
+ c− ln
ti␦uidA
⌽␦qdV,
共1兲
for arbitrary changes in displacement ␦u and electronic
charge density ␦q.
Now let us turn to a representative homogenized material particle in the IPCNC. Large volumetric change is of less
interest to most IPCNC applications since the material is
preserved from dehydration during operation. With thin-film
bending as the common mode of deformation, the material is
only slightly strained. We thus take the as-swollen stress-free
state as the reference, and use the small strain tensor ␧ij
= 共1 / 2兲共⳵ui / ⳵x j + ⳵u j / ⳵xi兲 to represent the field of deformation. While detailed microstructures of material particles may
differ from each other, the state of a material particle is fully
determined by the homogenized fields of deformation, electronic charge density, and ion concentrations. Writing the
free energy function as W共␧ , c+ , c− , q兲, we have
␦W =
⳵W
⳵W
⳵W
⳵W
␦␧ij +
␦c+ +
␦c− +
␦q.
⳵ ␧ij
⳵ c+
⳵ c−
⳵q
共2兲
冉 冊
冉
␮⫾ = ⫾ e ⌽ −
共3兲
冊
⳵W
⳵W
+
.
⳵q
⳵ c⫾
共4兲
Eq. 共3兲 recovers the familiar form of mechanical equilibrium
equation when the stress ␴ij = ⳵W / ⳵␧ij is defined. If we introduce an effective electric potential, ⌽⬘ = ⌽ − ⳵W / ⳵q, Eq. 共4兲
will have the same form as the electrochemical potentials in
an open system. However, ␮⫾ in Eq. 共4兲 is the Lagrange
multiplier to be determined from the conservation equation,
兰c⫾dV = N⫾.
Once a specific material model is prescribed in terms of
the free-energy function W, the inhomogeneous field of deformation and ion-distribution can be solved from the
algebraic-differential system, Eqs. 共3兲 and 共4兲. For the purpose of illustration, we use a free-energy function in the
form:
冊
c−
cs
+ cs ln
.
c+ + c− + cs
c+ + c− + cs
共5兲
The first two terms on the right-hand side of Eq. 共5兲 are the
elastic energy with Lamé constants G and ␭. The third term
is the contribution from the DL. Here a linear capacitor
model is adopted for the electrostatic energy of the double
layers, in which K is a material parameter defined as the
capacitance per unit area of interface, and A a microstructural parameter representing the area of the interfaces contained per unit volume of the IPCNC. The combination AK
characterizes the electronic-charge-storage capability of the
IPCNC. The last term is the energy of mixing mobile ions
and the solvent, with only the entropic contribution accounted for and kT is the temperature in the unit of energy.
Applying the specific form of free energy function to the
equations of state, we obtain that
␴ij = 2G␧ij + ␭␧kk␦ij −
冋 冉
␮⫾ = ⫾ e ⌽ −
Assuming molecular incompressibility, the concentration of
solvent cs is related to the volumetric strain as vscs − vsc̄s
= ␧kk, where vs is the volume of a solvent molecule and c̄s is
the reference solvent concentration.
As a conductor, IPCNC in equilibrium is electroneutral
except for the thin DL on interfaces.17 The representative
material particle contains both ICP and ECP, and the total
charge 共electronic and ionic兲 is zero. We enforce the electroneutrality by q + e共c+ − c− − c0兲 = 0, where e is the elementary
charge and c0 the concentration of fixed ions. Without losing
generality, we assume that the fixed ions on ionomer carry
negative charge, and all ions have either ⫺1 or +1 valence.
Utilizing Eqs. 共1兲 and 共2兲 and the divergence theorem, we
arrive at the following equilibrium equations:
⳵ ⳵W
= 0,
⳵ x j ⳵ ␧ij
冉
␭
1 q2
c+
W = G␧ij␧ij + ␧kk␧ jj +
+ kT c+ ln
2
2 AK
c+ + c− + cs
冉
冊
cs + c+ + c− c̄s
kT
ln
␦ij ,
vs
c̄s + c̄+ + c̄− cs
kT c+ c−
− −1
e␥ c0 c0
冊册
+ kT ln
共6兲
c+
,
c+ + c− + cs
共7兲
where c̄⫾ and c̄s are the concentrations of the mobile ions
and solvent molecules in the reference state, respectively.
The third term on the right-hand side of Eq. 共6兲 is due to the
coupling between volumetric strain and the solvent concentration, and is often referred to as the osmotic pressure. A
dimensionless parameter, ␥ = kTAK / e2c0, which characterizes the relative capacitance of the DL on the distributed
interface of IPCNC, can be identified in this model. In the
limit when ␥ → 0, the IPCNC reduces to a pure ICP, in which
the effective potential ⌽⬘ equals the macroscopic potential
⌽. At the two mixed conductor layers, ␥ takes a higher
value. Using a typical volume fraction 1% for metal particles, and the size range of the particles 1–100 nm, we have
an estimate of the interfacial area density A ⬇ 105 – 107 m−1.
Using the representative value for the DL capacitance K
⬇ 0.1– 1 Fm−2 共Ref. 18兲 and a fixed-charge concentration
c0 ⬇ 1 M, we further estimate that ␥ ⬇ 10−5 – 10−2. Inspired
by the measurement of the metal-particle distribution,8 we
assume that ␥ decreases linearly from both surfaces in the
mixed conductor layers and vanishes in the middle layer.
The numerical solution to the differential-algebraic system, Eqs. 共3兲 and 共4兲, by using the commercial finite-element
software COMSOL 3.5A, is shown in Fig. 2. Corresponding to
a Nafion 117 with 50% volume increase when swollen by a
solution of concentration 1 M, parameters used in this example are: fixed-charge density c0 = 1.7 M, reference ion
concentrations c̄− = 1.8 M and c̄− = 0.1 M, reference solvent
concentration c̄s = 28 M, and the microstructure parameter
␥ = 5 ⫻ 10−4 on the two surfaces. While the actual infiltration
depth is determined by the synthesis protocol, here as an
example, we have taken the thickness of each mixed conductor layer to be one third of the total thickness. The driving
force for bending the IPCNC is the asymmetrically distributed osmotic pressure resulting from the biased ion distributions. Due to the Donnan exclusion effect, the available
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081910-3
Appl. Phys. Lett. 98, 081910 共2011兲
X. Wang and W. Hong
(a)
0.1
0.02
c+/cs
0.05
Pvs/kT
0
c-/cs
0
−0.5
0
y/H
Osmotic pressure
Ion fraction
(b)
−0.02
0.5
FIG. 2. 共Color online兲 共a兲 Bending of ICPNC under applied electric voltage.
The shading shows the electronic charge density. 共b兲 The through-thickness
distributions of counterions, co-ions, and the resulting osmotic pressure.
counterions are more abundant than the co-ions.19 The osmotic pressure is evenly distributed through the thickness of
the IPCNC, as shown in Fig. 2共b兲, rather than being concentrated near the surfaces. The peak value of the normalized
osmotic pressure, is also at a much more reasonable order of
magnitude than those from parallel-capacitor models.
Detailed mechanisms aside, the deformation of IPCNC
is driven by the inhomogeneous fields in the DL as a result of
ion and solvent migration. When the distributed interface
area is accounted for through the microstructure parameter ␥,
the characteristic length of the effective layer is no longer the
Debye length but determined by the distribution.
Integrating the moment of the axial stress yields the
bending moment per unit width M. The normalized blocking
moments of two IPCNC sheets with different mixedconductor-layer thicknesses are plotted in Fig. 3. Here H is
the thickness of the IPCNC sheet, and h that of the mixed
conductor layer. The influence of the microstructure is also
shown with various values of the dimensionless parameter ␥.
The blocking moment increases almost linearly with the applied potential. If we neglect chemical reactions, with certain
0.014
Mvs / H2kT
0.012
h/H = 0.05
h/H = 0.5
0.01
0.008
0.006
-2
γ = 10
0.004
0.002
0
0
20
40
eΦ/k T
10-3
10-4
60
80
FIG. 3. 共Color online兲 Blocking moment as a function of the applied voltage. The microstructure parameter ␥ = 10−1, 10−2, 10−3, and decreases linearly through the mixed conductor layers.
density of fixed charges, there exists a limiting blocking moment. This limit corresponds to the maximum response of
IPCNC when the mobile counterions are depleted near the
anode. Existing designs of IPCNC often have very small ␥ so
that this limit is hardly reached. Nevertheless, the results
show that redesigning the microstructure of the mixed conductor layers for a higher ␥ value 共e.g., by using finer metal
particles or using layer-by-layer deposition兲 or simply thickening the mixed conductor layers will greatly enhance the
performance of IPCNC. The results also show that the performance of IPCNC is strongly correlated with its effective
capacitance, which is a function of ␥ as well as the thickness
of the mixed conductor layers. Such a correlation agrees
qualitatively with existing observations.20
In summary, a continuum theory is developed to answer
the question: How could the inhomogeneous fields localized
within nanometers from interfaces drive macroscopic IPCNC? While the basic physics is not altered and the DLs
remain to be nanometer thick, the microstructure of the
mixed conductive IPCNC provides a distributed interfacial
area which results in a much thicker active actuation zone.
Through a microstructure parameter that captures the distributed DL capacitance, the field theory is capable of predicting
the coupling behaviors of IPCNC, and more importantly correlating their performance with the underlying microstructures. Although the model presented couples through osmotic
effect only, the field theory can be extended to include other
mechanisms, e.g., the clustered ion pairs.8 As the interfacial
area increases, all mechanisms whose effects depend on the
inhomogeneity in the DL would also be enhanced.
The authors acknowledge the support from the NSF
through Grant No. CMMI-0900342, and Professor Q. M.
Zhang at Penn State for the valuable discussions.
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