Permissions for Reuse

In this paper, we study the charge dynamics in ionic polymer metal composites (IPMCs) in response to a voltage difference applied across their electrodes. We use the Poisson–Nernst–Planck equations to model the time evolution of the electric potential and the concentration of mobile counterions. We present an analytical solution of the nonlinear initial-boundary value problem by using matched asymptotic expansions. We determine the charge and electric potential distributions as functions of time in the whole IPMC region. We show that in the bulk polymer region the IPMC is approximately electroneutral; in contrast, charge distribution boundary layers arise at the polymer-electrode interfaces. Prominent charge depletion and enrichment at the polymer-electrode interface are present even at moderately low input-voltage levels. We use the proposed analytical solution to derive a physics-based circuit model of IPMCs. The equivalent circuit comprises a linear resistor in series connection with a nonlinear capacitor. We derive closed-form expressions for the resistance and the capacitance by conducting a qualitative phase-plane analysis of the inner approximation of the asymptotic expansion. The circuit conductivity is independent of the IPMC dielectric constant and is proportional to the ion diffusivity; whereas, the capacitance is proportional to the square root of the dielectric constant and is independent of the diffusivity. The conductivity depends on the polymer thickness, while the capacitance is independent of it. The capacitance nonlinearity is extremely pronounced, and dramatic capacitance reduction is observed for moderately low voltage levels. We validate the proposed analytical solution along with the derived circuit model through extensive comparisons with finite element results available in the technical literature. ©2008 American Institute of Physics

Contents

• INTRODUCTION
• MODELING
• MATCHED ASYMPTOTIC EXPANSIONS
  • A.Nondimensional equations
  • B.Outer expansion
  • C.Inner expansions
  • D.Matching
  • E.Leading order solution of the inner problems
  • F.Composite solution
• EQUIVALENT CIRCUIT
• RESULTS AND DISCUSSION
  • A.Response to a step voltage
  • B.Steady-state boundary layers' analysis
• CONCLUSIONS
• ACKNOWLEDGMENTS
• § 8
  • APPENDIX: DERIVATION OF THE LEADING ORDER SOLUTION
• REFERENCES
• FIGURES
• TABLES
• FOOTNOTES

An ionic polymer metal composite (IPMC) is an ion-conducting polymer membrane plated by two electrodes that is infused with a solvent and electrically neutralized by mobile counterions. A voltage difference across the electrodes generates an electric field that is responsible for counterions' redistribution. This redistribution of mobile ions gives rise to a number of concurrent microscopic actuation mechanisms that lead to the IPMC macroscopic deformations—see, for example, Refs. 1,2,3,4,5,6,7. Vice versa, an imposed mechanical deformation induces a redistribution of mobile counterions that leads to a charge stored at the IPMC electrodes—see, for example, Refs. 8,9.

The charge dynamics in IPMCs can be studied using the multifield chemoelectrical formulation proposed in Refs. 10,11,12. This formulation allows for characterizing the charge and electric field distribution in IPMCs in response to a voltage applied across the electrodes. The electric potential distribution is governed by the Poisson equation, in which the free charge density depends on the mobile ions' concentration. The time evolution of the mobile ions' concentration is in turn determined by the Nernst–Planck equation. Computational analysis of the resulting nonlinear initial-boundary value problem shows the existence of thin boundary layers in the vicinity of the ion-blocking electrodes. For positively charged mobile ions, the anode-polymer interface region tends to be severely depleted of counterions and the cathode-polymer interface region tends to be drastically enriched with them. The formulation is similar to the one proposed in Ref. 13 for analyzing charge dynamics in binary electrolytes. Unlike the problem studied in Ref. 13 where both the charged species are mobile, only one charged species is mobile in IPMCs. This leads to remarkably nonsymmetric charge distributions. A related physical situation arises with charged polyelectrolyte films and membranes at semi-infinite solid-liquid interfaces. For example, in the context of biomolecular DNA films,¹⁴ the surface capacitive response reflects the layer organization and can be exploited for biosensing applications. A similar problem has also been studied in Ref. 15, where ion-channels with permanent reservoirs are analyzed.

The thinness of the boundary layers demands significant computational efforts in numerical analysis.¹¹ In this paper, we present an analytical solution for the chemoelectrical model proposed in Refs. 10,11,12 based on matched asymptotic expansions—see, for example, Refs. 16,17,18. Matched asymptotic expansions have been successfully used in the analysis of charge dynamics of symmetric binary electrolytes including steric effects and compact Stern layers—see, for example, Refs. 13,19,20,21,22. The proposed solution allows for a thorough understanding of boundary layers' formation and dynamics in IPMCs. At the leading order, we transform the complex chemoelectrical formulation of Refs. 10,11,12 that consists of partial differential equations together with initial and boundary conditions, into a handleable second order ordinary differential equation with only initial conditions. The charge distribution in the polymer region is not symmetric and an ad hoc treatment is presented to cope with the mixing of the boundary conditions at the two electrodes. We validate the proposed analytical solution with the numerical findings in Ref. 12 for the steady-state distribution of mobile ions under a step input voltage.

We use the proposed analytical solution to derive an equivalent circuit model of IPMCs. The equivalent circuit comprises a linear resistor and a nonlinear capacitor in series connection. We find closed-form expressions for the resistance and the capacitance of the proposed circuit model from a qualitative phase-plane analysis of the inner expansions. We show that the IPMC conductivity depends linearly on the ion diffusivity and is independent of the IPMC dielectric constant. Moreover, we show that the IPMC capacitance is independent of the ion diffusivity and is proportional to the square root of the IPMC dielectric constant. The dependence of the IPMC capacitance on the applied voltage is remarkably nonlinear. The IPMC capacitance rapidly decreases as the applied voltage increases.

The proposed circuit model can be used to analyze the IPMC response to a variety of input voltages. We specify our results to the analysis of step input voltages and we determine closed-form expressions for the peak current density flowing through the IPMC and the maximal charge stored in the IPMC electrodes. Due to the capacitance nonlinearity, the time evolution of the electric current is not a simple decaying exponential and is very much sensitive to the voltage level of the applied input. Previously developed equivalent circuits—see, for example, Refs. 23,24,25,26,27,28—do not consider nonlinearities in the IPMC capacitance. In addition, the circuit elements in the available circuit models are determined by fitting experimental data. On the other hand, the proposed lumped circuit model directly stems from the chemoelectrical distributed model of the IPMC, and all its parameters are determined in terms of the IPMC geometry and material properties. We validate the circuit model by comparing its predictions with the numerical results in Ref. 12. These numerical results include a variety of material constants and input levels and are in close agreement with experimental results.

We organize the paper as follows. In Sec. II, we recall the fundamental equations of the chemoelectrical model of Refs. 10,11,12. In Sec. III, we present our analytical solution based on matched asymptotic expansions. In Sec. IV, we derive an equivalent circuit model of IPMCs. In Sec. V, we validate the proposed analytical solution, by comparing its predictions with the numerical results of Ref. 12. Section VI is left for conclusions.

We consider an IPMC comprised of a strip of an ion-conducting polymer membrane of thickness 2h plated by two fixed and flat metallic electrodes, see Fig. 1. We model the IPMC using the multifield chemoelectrical formulation proposed in Refs. 10,11,12. Within this formulation, the IPMC kinematics is described by the concentration per unit hydrated polymer volume c of the mobile ion species and the electric potential field in the polymer region. Both these variables are assumed to vary only along the thickness coordinate x. For ease of illustration, we assume that the mobile ionic species is positively charged. We further assume that both the fixed charges anchored to the backbone polymer and the mobile charges have valency equal to 1.

Figure 1.

The distribution of the electric potential in the mixture is described by the Gauss law

where t is the time variable, F is Faraday's constant (F=96 485  C mol⁻¹), c₀ is the concentration of the fixed ions, and is the component of the electric displacement vector in the IPMC thickness direction. The electric displacement is related to the electric potential by

where ₀ is the vacuum permittivity ₀=8.8542×10⁻¹²  F m⁻¹, and _r is the hydrated polymer dielectric constant.

The mass balance for the mobile ion species is

where j is the flux of the mobile ions in the direction of the polymer thickness. The ion flux j is related to the ion concentration c and the electric potential by

where D is the diffusivity of mobile ions, R is the universal gas constant (R=8.3143  J mol⁻¹ K⁻¹), and T is the IPMC temperature. We note that in case of polyvalent charged species, Eqs. (1),(4) need to be modified to account for the ions' valency as illustrated in Refs. 10,11,12.

Substituting Eq. (2) into Eq. (1) and Eq. (4) into Eq. (3), we find the Poisson equation and the Nernst–Planck equation

These equations are generally referred to as the Poisson–Nernst–Planck equations—see, for example, Refs. 13,15,19,20,21,22—and describe the charge dynamics and electric potential distribution in IPMCs. We note that in deriving Eqs. (5a),(5b), we assumed, among other hypotheses, that the dielectric constant _r and the diffusivity D are constant within the hydrated polymer.

We consider the IPMC response to a voltage difference V(t) applied across the electrodes for t0. We assume that the electrodes block the ion flux and that the IPMC is initially electroneutral. Therefore, we consider the following set of boundary and initial conditions:

By integrating Eq. (3) from −h to h and using the ion-blocking condition (6a) along with the initial IPMC electroneutrality, that is, Eq. (7a), we find that the total concentration of mobile ions per unit electrode surface area is constant over time and is equal to 2hc₀. Therefore, by integrating Eq. (1) from −h to h, we find that (h,t)=(−h,t). Upon using Eq. (2), we find

In the time independent case, Eq. (8) needs to be explicitly enforced to guarantee the overall charge conservation. This is due to the fact that in steady-state conditions, the ion flux is constant in space, see Eq. (3). Therefore, the two ion-blocking boundary conditions reported in Eq. (6a) are not independent in steady-state conditions.

The charge per unit surface area stored in the IPMC is given by the jump of the electric displacement across the polymer-electrode interfaces, that is

The resulting current per unit surface area flowing through the IPMC is

We nondimensionalize the electric potential with respect to the so-called thermal voltage RT/F and we nondimensionalize the mobile ions' concentration with respect to the fixed charges concentration c₀. Moreover, we select the polymer semithickness h as the characteristic length and the time constant

as the characteristic time. The governing Eqs. (5a),(5b) become

where the dimensionless variables are indicated with a superimposed tilde, that is, =x/h, =t/₀, =/(RT/F), and =c/c₀, and we defined

For commonly studied ionic membranes—see, for example, Refs. 11,12—c₀1000  mol m⁻³, h10⁻⁴  m, and _r100. Therefore, at room temperature, 10⁻⁵. We note that the parameter can be related to the ratio between the Debye screening length (traditionally used to analyze double-layer capacitance in binary electrolytes—see, for example, Ref. 13) and the polymer thickness. As made clear in what follows, the parameter measures the relative boundary layers' thickness. Equation (13) indicates that the boundary layer thickness, h, is independent of the polymer thickness and decreases as the concentration c₀ increases and the dielectric constant _r decreases.

The dimensionless forms of the boundary and initial conditions in Eqs. (6),(7) are

where we defined the dimensionless voltage applied across the electrodes

We seek regular asymptotic expansions of the electric potential and concentration in the outer region, that is,

where we used superscript o to label the outer solution and subscripts 1,  2,  … to identify the unknown summands in the asymptotic expansions—see, for example, Ref. 17. A consistent notation is used for the asymptotic expansions of the inner solutions.

By substituting Eq. (17) into the governing Eq. (12), using the initial condition (15), and equating terms of the same order, we obtain a hierarchy of partial differential equations for ,  ,  … and ,  ,  …. At the leading order, we find the following concentration and charge distribution in the bulk region

where the time functions A() and B() are equal to (0) and 0 at =0, respectively.

For small values of the parameter , the outer solution (17) is close to the exact solution everywhere in the ion-conducting polymer except in the neighborhood of the electrodes, where boundary layers develop. In order to determine valid expansions for the concentration and electric potential in the vicinity of the electrodes, we magnify the boundary layers' width by defining the stretching transformations

The stretching transformation (19a) is needed to magnify the boundary layer width in the vicinity of the anode, that is, for in the neighborhood of 1. Similarly, the transformation (19b) is used to analyze the charge dynamics in the cathode region. By substituting Eq. (19) into the Poisson–Nernst–Planck Eq. (12), we find

Here, we used superscripts plus and minus for the transformed functions, for example, ⁺(⁺,)(,). The stretching transformations in Eq. (19) remove the singular perturbation in the Poisson equation (12a). Thus, we seek regular asymptotic expansions of the electric potential and concentration in the inner regions, that is,

By substituting inner expansion (21) into the governing Eq. (20), we obtain a hierarchy of partial differential equations for the asymptotic sequences of ⁺, ⁻, ⁺, and ⁻. At the leading order, we find

Since the inner expansions are valid at the electrode-polymer interface, they should satisfy the boundary condition (14) that at the leading order, yields

Moreover, since the concentration rate is not present in the boundary layer inner dynamics (22), the conservation of the ion concentration due to ion-blocking electrodes is enforced by

We note that the sign inversion of Eq. (24), when compared to Eq. (8), is due to the definition of the stretched variables in Eq. (19). Further, we observe that Eq. (24) introduces a coupling between the two inner solutions.

Matching between the outer solution and the inner solutions is obtained by enforcing the following set of equalities

In addition to matching the concentration and electric potential in the bulk and the boundary layers, we need to impose the continuity of the ion flux within the IPMC region. From Eq. (12b), we have that the rate of change of the charge stored in the vicinity of the anodic or cathodic region is

where we used the ion-blocking condition in Eq. (14). We use Eq. (26) to match the inner and outer solutions. More specifically, we replace the inner solutions in the left hand side of Eq. (26) and the outer solution in its right hand side. Then, we take the limit as goes to zero while keeping fixed for the inner solutions and ^± fixed for the outer solution. By following this procedure, we obtain the following matching conditions:

By specializing the matching conditions (25) to the leading order and by using the outer solution (18), we find the following conditions on the inner solutions:

Moreover, by specializing Eq. (27) to the leading order and by accounting for Eqs. (18),(22a),(28), we find

We note that according to Eq. (10), the function A() is proportional to the current density flowing through to the IPMC at the leading order.

In the following analysis, we use the symbol to indicate the space variable of both the inner solutions, while keeping in mind that separate variables ⁺ and ⁻ have to be considered when combining the inner solutions to compute the charge distribution within the whole IPMC. By combining Eq. (22b) with the ion-blocking boundary condition (23a), we find

We introduce the functions y⁺(,) and y⁻(,) defined by

By replacing Eq. (31) to Eq. (30), we find

Equation (32) can be integrated to yield the following equation:

where K^±() are unknown functions of time yet to be determined.

By substituting Eqs. (31),(32) into Eq. (22a), we find two identical second order nonlinear ordinary differential equations in the variable for the functions y⁺(,) and y⁻(,). That is, we find

where the nonlinear function f is defined by

Boundary conditions for y⁺(,) and y⁻(,) and subsidiary conditions for the introduced functions of time, A(), B(), and K^±(), are obtained by replacing Eq. (31) into the boundary conditions (23b),(24) and into the matching conditions (28),(29). This yields

Equations (28b),(33),(36d) illustrate that −y⁺(,) represents the electric potential excess at the anode region with respect to the bulk potential. Similarly, −y⁻(,) represents the potential excess at the cathode region with respect to the bulk potential. Therefore, Eq. (31) corresponds to the Boltzmann distribution generally used to study double-layer capacitances in electrochemistry—see, for example, Ref. 29. The solution of Eq. (34) subjected to the conditions in Eq. (36) is presented in the Appendix.

The concentration and electric field distribution in the whole polymer region are determined by combining the outer solution derived in Eq. (17) with the inner solutions determined in Sec. III E and by accounting for their common limit in Eq. (28). More specifically, the concentration and electric potential in the polymer at the leading order are given by

where A() is the solution of the first order differential Eq. (A8) with initial condition A(0)=(0) and y^±(,) are the solutions of Cauchy problem (34) with initial conditions given in Eq. (A6).

The charge per unit surface stored in the IPMC can be computed through Eq. (9), where the electric field on the top electrode is computed by using Eqs. (32),(A6c). By substituting for from definition (13), we find

By taking the time derivative of Eq. (38) and using Eqs. (10),(A8), we find

where we substituted for ₀ and from definitions (11),(13).

Finally, by substituting for A() and () in Eq. (38) the expressions in Eqs. (39),(16), we obtain

where is the conductivity per unit electrode surface area that is defined by

Equations (10),(40) show that an IPMC can be represented as the series connection of a nonlinear capacitor and a linear resistor, see Fig. 2. For a unit surface area IPMC: the current flowing through the circuit is i(t); the resistance of the resistor is 1/; the voltage across the capacitor is V_C(t)=V(t)−i(t)/; and the charge stored in the capacitor is q_s(t) given in Eq. (40). Moreover, the capacitance per unit electrode surface area at the voltage _C is

From Eq. (40), we evince that the charge stored in the IPMC and consequently its equivalent capacitance is independent of the IPMC thickness. On the other hand, the polymer thickness linearly influences the equivalent resistance, as illustrated in Eq. (41).

Figure 2.

The function defined in Eq. (A7) represents the dimensionless charge per unit surface area stored in the IPMC, whereas its derivative represents the IPMC capacitance per unit area. The function and its derivative are reported in Figs. 3,4, respectively. From Fig. 4, we note that the capacitor becomes strongly nonlinear for relatively small voltages and that the capacitance dramatically decreases as the voltage increases. At 1 V (40), the capacitance per unit surface is reduced to approximately 1/5 of its limit value at V=0  V. Loosely speaking, this means that as the IPMC stored charge increases, higher voltage increments are needed to accumulate an additional unit charge at the IPMC electrode. This may be explained by the fact that only cations are free to move in the polymer region and anions are anchored to the backbone polymer. Thus, as the applied voltage increases, the anode-polymer interface grows, thereby leading to a reduction in the IPMC capacitance. This tendency remarkably characterizes IPMC electric behavior versus symmetric binary electrolytes, in which anions and cations are equally mobile. Indeed, for symmetric binary electrolytes between ion-blocking electrodes the capacitance increases as the voltage increases according to the Gouy–Chapman theory—see, for example, Refs. 13,29.

Figure 3.

Figure 4.

Here, we compare the analytical solution derived in this paper with the fully converged finite element results in Ref. 12. The thickness of the polymer is 2h=0.2×10⁻³  m and the fixed negative charge concentration is c₀=1200  mol m⁻³. The dielectric constant _r and the diffusivity D are varied in a broad range of feasible values to characterize their effect on the IPMC response.

We study the time response of an IPMC to a step voltage of magnitude V. We compare the finite element results in Ref. 12 with the predictions of our physics-based circuit model, see Fig. 4. The circuit dynamics can be written in terms of the current density i(t) by simply replacing Eq. (39) in Eq. (A8). Alternatively, the circuit dynamics can be described in terms of the voltage across the capacitor to produce the following ordinary differential equation:

with initial condition V_C(0)=0. The current density is [V−V_C(t)]. The peak value of the current density, say î, is attained at t=0, that is,

The maximal value of the charge density, say q_max, is asymptotically reached as t, that is, under steady-state conditions. From Eq. (40), we find that q_max is given by

Equations (44),(45) show that the peak current is independent of the dielectric constant and that the maximal charge is independent of the diffusivity. In addition, the peak current is proportional to the diffusivity and the maximal charge is proportional to the square root of the dielectric constant. These findings are in agreement with the numerical evidence reported in Ref. 12. Table I compares the predictions of Eqs. (44),(45) with the finite element results in Ref. 12 computed at V=0.2  V. The predictions of the derived circuit model are in close agreement with numerical results.

Figures 5,6 compare the predicted current density time profile in response to a step input of 0.2 V to the finite element solution of Ref. 12. In Fig. 5, different values of the dielectric constant and a common diffusivity D=2×10⁻¹¹  m² s⁻¹ are considered, whereas in Fig. 6, different values of the diffusivity and a common dielectric constant _r=40 are examined. The proposed circuit model provides very accurate estimates of the time evolution of the current density flowing through the IPMC in both scenarios.

Figure 5.

Figure 6.

Figures 7,8 illustrate the dependency of the peak current and the maximal charge density on the level of the voltage input predicted by the proposed circuit model and by the finite element solution of Ref. 12. In these figures, the dielectric constant is set to _r=120 and the diffusivity to D=2.8×10⁻¹¹  m² s⁻¹. The derived closed-form solutions, see Eqs. (44),(45), are in striking agreement with numerical findings.

Figure 7.

Figure 8.

For the same choice of diffusivity and dielectric constant, we display in Fig. 9 the current density profile at different input-voltage levels in logarithmic scale to clarify the role of the capacitance nonlinearity on the current density time decay. As the input voltage is applied, the voltage across the nonlinear capacitor is small since the IPMC is initially electroneutral. Thus, the current density exponentially decays with a time-constant given by (0)/=(F/2), see Eq. (42). As time increases, the voltage across the capacitor increases toward the input voltage, thereby leading to remarkable reductions in the IPMC capacitance, see Fig. 4. As the voltage across the capacitor approaches the input-voltage level V, the current density exponentially decays to zero with a smaller time constant given by (FV/RT)/. For example, for V=1  V, the time constant is reduced from 1.7⁻³ to 0.4⁻³  s.

Figure 9.

The charge distribution and voltage profile within the polymer can be computed from composite solution (37). Here, we compare the proposed composite solution with the finite element results in Ref. 12 derived in steady-state conditions for a step input voltage equal to 0.2 V and for the relatively large dielectric constant _r=400 000.

Figures 10,11 illustrate the boundary layers of the charge distribution in the vicinity of the electrodes. We note that the composite solution is in very close agreement with the finite element solution, despite the relatively wide boundary layers. In this case, =2.9×10⁻⁴. As the dielectric constant decreases, the computational time required by the finite element numerical solution drastically increases, while the computational effort required by the asymptotic solution is unaltered and its accuracy enhanced.

Figure 10.

Figure 11.

We studied the charge dynamics in IPMCs with ion-blocking electrodes in response to time-varying voltage inputs applied across the IPMC electrodes. We used the Poisson–Nernst–Planck equations to describe the time evolution of the charge distribution and the electric potential within the IPMC region. We proposed an analytical solution to the initial value boundary value problem based on matched asymptotic expansions. The analytical solution predicts the formation of thin boundary layers in the proximity of the polymer-electrode interfaces. Even for moderately low voltage inputs, mobile cations tend to populate the cathode-polymer interface while depleting the anode-polymer region. We show that the boundary layers' thickness depends on the concentration of fixed charges and on the IPMC dielectric constant and is independent of the polymer thickness.

We used the analytical solution to derive a physics-based circuit model of IPMCs. The circuit comprises a linear resistor in series connection with a nonlinear capacitor. We derived closed-form handleable expressions for the resistance and capacitance defining the circuit model. We showed the ion diffusivity affects the circuit resistance, while the dielectric constant alters the capacitance. The circuit resistance is proportional to the polymer thickness, while the capacitance is not sensible to changes in the polymer thickness. The capacitance nonlinearity becomes prominent at extremely low voltage levels, as low as 100 mV at room temperature. As the voltage increases, the IPMC capacitance drastically decreases and reduces to approximately 1/5 of its initial value at 1 V. For a step input voltage, we determined closed-form expressions for the peak current density flowing and the maximal stored charge densities. The proposed circuit model is the first of its kind in the technical literature on IPMCs since it is directly derived from fundamental physics-based equations.

We validated the proposed analytical solution and the related circuit model with the numerical results in Ref. 12 for a variety of IPMC material constants and for different input-voltage levels. The analytical results are in very close agreement with fully converged numerical findings. Determining the current flow density through an IPMC using the proposed circuit model requires the numerical integration of a single first order differential equation. Once the current density has been computed, the charge distribution within the polymer region is computed by solving a second order differential equation describing the electric potential profile. On the other hand, the numerical solution presented in Refs. 10,11,12 requires the integration of a large set of ordinary differential equations whose size is affected by the boundary layer thickness.

This research was supported by the National Science Foundation under Grant No. CMMI-0745753. The author would like to acknowledge Ms. Nicole Abaid and Mr. Matteo Aureli for their careful review of the manuscript.

Equations (36a),(36d) can be rearranged to establish the following condition on the values of the functions y⁺(,) and y⁻(,) at =0:

The functions y⁺(,) and y⁻(,) can be determined by solving Eq. (34) completed by the four conditions (36b),(36c),(A1). We note that the function

is a first integral of Eq. (34)—see, for example, Ref. 30—that is,

By making use of the first integral U, we can transform the nonlinear boundary value problem (34) into a simpler Cauchy problem. In Fig. 12, we show the level lines of U in the phase plane. From Fig. 12, we evince that the only solutions of Eq. (34) that tend to the origin as goes to infinity are those for which U=−1 in the second and fourth quadrants of the phase plane. Therefore, we replace limit condition (36c) with

together with

Solving Eqs. (36b),(A1),(A4) for the values of y^±(,) at =0 along with their derivatives, we find

Here, we used condition (A5) to select the sign of the derivative of y(,) at =0 while assuming ()A(), and we introduced the function

Figure 12.

The function A() can be determined in terms of the prescribed voltage () by substituting Eq. (A6c) into Eq. (36e), using the chain rule of differentiation, and solving the resulting ordinary differential equation for A(),

The initial condition for the first order differential Eq. (A8) is A(0)=(0).

The functions K^±() and B() can be determined in terms of A() and () by replacing Eq. (A6) into Eqs. (36a),(36d). That is,

Moreover, the functions y^±(,) can be found by numerically integrating Cauchy problem (34) with initial conditions given in Eq. (A6). Figure 13 illustrates the behavior of y^±(,) for a chosen value of ()−A(). The concentration and electric potential distribution in the vicinity of the electrodes are finally computed by substituting for y(,) and K^±(t) in Eqs. (31),(33).

Figure 13.

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30. V. I. Arnol'd, Ordinary Differential Equations (Springer-Verlag, New York, 1991). first citation in article
    

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Fig. 1. Sketch of an IPMC illustrating cations enrichment in the proximity of the cathode and cations depletion in the vicinity of the anode region. First citation in article

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Fig. 2. Sketch of the equivalent circuit of an IPMC of surface area S. First citation in article

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Fig. 3. Plot of the dimensionless charge per unit surface area vs the dimensionless voltage . The dimensionless voltage is the ratio between the capacitor voltage and the thermal voltage RT/F25×10⁻³  V. First citation in article

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Fig. 4. Plot of the dimensionless capacitance per unit surface area vs the dimensionless voltage . The initial value is 1/2. First citation in article

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Fig. 5. Current density for D=2×10⁻¹¹  m² s⁻¹ and a step input of 0.2 V: lines represent the predictions of the circuit model and symbols are the numerical data from Ref. 12. Dashed line and correspond to the case _r=20; solid line and correspond to the case _r=40; and dotted line and × correspond to the case _r=80. First citation in article

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Fig. 6. Current density for _r=40 and a step input of 0.2 V: lines represent the predictions of the circuit model and symbols are the numerical data from Ref. 12. Dashed line and correspond to the case D=1×10⁻¹¹  m² s⁻¹; solid line and correspond to the case D=2×10⁻¹¹  m² s⁻¹; and dotted line and × correspond to the case D=4×10⁻¹¹  m² s⁻¹. First citation in article

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Fig. 7. Current density peak at different voltage input levels for _r=120 and D=2.8×10⁻¹¹  m² s⁻¹. Solid line refers to Eq. (44) and to numerical data from Ref. 12. First citation in article

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Fig. 8. Maximal charge density at different voltage input levels for _r=120 and D=2.8×10⁻¹¹  m² s⁻¹. Solid line refers to Eq. (45) and to numerical data from Ref. 12. First citation in article

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Fig. 9. (Color online) Current density for different input-voltage levels and for _r=120 and D=2.8×10⁻¹¹  m² s⁻¹. First citation in article

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Fig. 10. Normalized charge distribution at the cathode side. Solid line refers to the composite solution in Eq. (37) and to numerical data from Ref. 12. First citation in article

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Fig. 11. Normalized charge distribution at the anode side. Solid line refers to the composite solution in Eq. (37) and to numerical data from Ref. 12. First citation in article

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Fig. 12. (Color online) Level lines of the function U in the phase plane. First citation in article

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Fig. 13. Space evolution of y^±(,) for ()−A()=10. Solid line refers to y⁻(,) and dashed line to y⁺(,) First citation in article

Table I. Comparison between the proposed circuit model and the fully converged finite element results in Wallmersperger et al. (Ref. 12). To be consistent with the data in Ref. 12, the peak current is measured in mA cm−2 and the maximal stored charge density in µC cm−2.
D (m2 s−1)	r=20		r=40		r=80		r=4000
	Circuit model	Ref. 12	Circuit model	Ref. 12	Circuit model	Ref. 12	Circuit model	Ref. 12
1×10−11	î=4.59		î=4.59	î=4.6	î=4.59	î=4.6	î=4.59	î=4.6
	qmax=7.09		qmax=10.0	qmax=10.1	qmax=14.2	qmax=14.2	qmax=100	qmax=101
2×10−11	î=9.17	î=9.1	î=9.17	î=9.17	î=9.17	î=9.17	î=9.17	î=9.17
	qmax=7.09	qmax=7.1	qmax=10.0	qmax=10.1	qmax=14.2	qmax=14.2	qmax=100	qmax=101
1×10−10	î=45.9		î=45.9	î=45.9	î=45.9		î=45.9	î=45.9
	qmax=7.09		qmax=10.0	qmax=10.1	qmax=14.2		qmax=100	qmax=101

First citation in article

^*Electronic mail: mporfiri@poly.edu. URL: http://faculty.poly.edu/~mporfiri/index.htm.

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