Bias-dependent model of the electrical impedance of ionic polymer-metal composites

In this paper, we analyze the charge dynamics of ionic polymer-metal composites (IPMCs) in response to voltage inputs composed of a large dc bias and a small superimposed time-varying voltage. The IPMC chemoelectrical behavior is described through the modified Poisson-Nernst-Planck framework, in which steric effects are taken into consideration. The physics of charge build-up and mass transfer in the proximity of the high surface electrodes is modeled by schematizing the IPMC as the stacked sequence of five layers, in which the ionomeric membrane is separated from the metal electrodes by two composite layers. The method of matched asymptotic expansions is used to derive a semianalytical solution for the concentration of mobile counterions and the electric potential in the IPMC, which is, in turn, used to establish an equivalent circuit model for the IPMC electrical response. The circuit model consists of the series connection of a resistor and two complex elements, each constituted by the parallel connection of a capacitor and a Warburg impedance. The resistor is associated with ion transport in the ionomeric membrane and is independent of the dc bias. The capacitors and the Warburg impedance idealize charge build-up and mass transfer in the vicinity of the electrodes and their value is controlled by the dc bias. The proposed approach is validated against experimental results on in-house fabricated IPMCs and the accuracy of the equivalent circuit is assessed through comparison with finite element results.

Figure 1

Schematics of the IPMC.

Figure 2

Permittivity on the ionomer core scaled by the permittivity at the interface $x=\pm h$ for the representative parameters ${\epsilon }_{\mathrm{i}}/{\epsilon }_{\mathrm{b}}=100$ and $\mu =100$.

Figure 3

Equivalent circuit model for small voltage inputs.

Figure 4

Equivalent circuit model under large dc bias voltage.

Figure 5

Concentration of counterions and electric potential in the ionomer and the composite layer computed using the proposed semianalytical solution (markers) and the numerical solutions from comsol (lines) for $\delta ={10}^{-5}$, ${\epsilon }^{*}=1$, $D^{*}=1$, $\varphi =0.5$, $d/h=0.01$, $\nu =0.6$, and $\alpha =0.4$. Red (gray), cyan (lightest gray), green (light gray), blue (darkest gray), and purple (dark gray) colors refer to $\tilde{t}=0.01$, $0.1$, 1, 10, and 100, respectively. (a) Counterion concentration in the ionomer; (b) counterion concentration in the composite layer; (c) electric potential in the ionomer; and (d) electric potential in the composite layer.

Figure 6

Concentration of counterions and electric potential in the ionomer and the composite layer computed using the proposed semianalytical solution (markers) and the numerical solutions from comsol (lines) for $\delta ={10}^{-5}$, ${\epsilon }^{*}=1$, $D^{*}=1$, $\varphi =0.5$, $d/h=0.01$, $\nu =0.6$, and $\alpha \left(\tilde{t}\right)=8+0.4\sin \left(2\pi \tilde{t}\right)$. Red (gray), cyan (lightest gray), green (light gray), and blue (dark gray) colors refer to $\tilde{t}=1$, $1.25$, $1.5$, and $1.75$, respectively. (a) Concentration of counterions in the composite layer near the cathode; (b) concentration of counterions in the ionomer near the cathode; (c) concentration of counterions in the ionomer near the anode; (d) concentration of counterions in the composite layer near the anode; (e) electric potential in the composite layer near the cathode; (f) electric potential in the ionomer near the cathode; (g) electric potential in the ionomer near the anode; and (h) electric potential in the composite layer near the anode.

Figure 7

(a) Capacitance at the anode (solid line) and cathode (dash-dotted line) scaled with respect to the value for null dc bias and (b) Warburg impedance at the anode (solid line) and cathode (dash-dotted line) scaled with respect to the value for null dc bias for $\delta ={10}^{-5}$, ${\epsilon }^{*}=1$, $D^{*}=1$, $\varphi =0.5$, and $d/h=0.02$. Red (gray), green (light gray), and blue (dark gray) colors refer to $\nu =0.1$, $0.6$, and $0.9$, respectively.

Figure 8

Steady-state concentration of counterions and electric potential in the ionomer and the composite layer computed for $\delta ={10}^{-5}$, ${\epsilon }^{*}=1$, $D^{*}=1$, $\varphi =0.5$, $d/h=0.02$, and $\overline{\alpha }=8$. Red (gray), green (light gray), and blue (dark gray) lines refer to $\nu =0.1$, $0.6$, and $0.9$, respectively. (a) Concentration of counterions in the composite layer near the cathode; (b) concentration of counterions in the ionomer near the cathode; (c) concentration of counterions in the ionomer near the anode; (d) concentration of counterions in the composite layer near the anode; (e) electric potential in the composite layer near the cathode; (f) electric potential in the ionomer near the cathode; (g) electric potential in the ionomer near the anode; and (h) electric potential in the composite layer near the anode.

Figure 9

(a) Capacitance at the anode (solid line) and cathode (dash-dotted line) scaled with respect to the value for null dc bias and (b) Warburg impedance at the anode (solid line) and cathode (dash-dotted line) scaled with respect to the value for null dc bias for $\delta ={10}^{-5}$, ${\epsilon }^{*}=1$, $D^{*}=1$, $\varphi =0.5$, and $\nu =0.6$. Red (gray), green (light gray), and blue (dark gray) colors refer to $d/h=0.01$, $0.02$, and $0.04$, respectively.

Figure 10

Experimental data on the impedance of a Nafion-based IPMC for dc bias equal to $0\mathrm{V}$ [red (gray) solid line], $0.2\mathrm{V}$ [green (light gray) solid line], and $0.5\mathrm{V}$ [blue (dark gray) solid line] and theoretical prediction (black dash-dotted line) of the circuit model in Fig. 3 with $R=3.106\Omega$, $C=0.136\mathrm{mF}$, and $W=0.0123{\mathrm{s}}^{1/2}{\Omega }^{-1}$. Magnitude is in (a), phase in (b), and Nyquist plot in (c).

Figure 11

Theoretical predictions of the slope of the impedance magnitude in the low frequency for different values of the dc voltage against experimental results from Table . Theoretical results are based on parameter values from the sample in Fig. 10 and are plotted for $d=2.3\times {10}^{-6}\mathrm{m}$ [red (gray) line], $d=1.0\times {10}^{-6}\mathrm{m}$ [green (light gray) line], and $d=0.5\times {10}^{-6}\mathrm{m}$ [blue (dark gray) line]. Open circles are the mean values of the experimental data and error bars refer to standard deviations.