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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.