On Structural Theories for Ionic Polymer Metal Composites: Balancing Between Accuracy and Simplicity

Abstract

Ionic polymer metal composites (IPMCs) are soft electroactive materials that are finding increasing use as actuators in several engineering domains, where there is a need of large compliance and low activation voltage. Similar to traditional sandwich structures, an IPMC comprises a hydrated ionomer core that is sandwiched by two stiffer electrodes. The application of a voltage across the electrodes drives charge migration within the ionomer, which, in turn, contributes to the development of an eigenstress, associated with osmotic pressure and Maxwell stress. Critical to IPMC actuation is the variation of the eigenstress through the thickness of the ionomer, which is responsible for strain localization at the ionomer-electrode interfaces. Despite considerable progress in the development of reliable continuum theories and finite element tools, accurate structural theories that could beget physical insight into the inner workings of IPMC actuation are lacking. Here, we seek to bridge this gap by contributing a principled methodology to structural modeling of IPMC actuation. Our approach begins with the study of the IPMC electrochemistry through the method of matched asymptotic expansions, which yields a semi-analytical expression for the eigenstress as a function of the applied voltage. Hence, we establish a total potential energy that accounts for the strain energy of the ionomer, the strain energy of the electrodes, and the work performed by the eigenstress. By projecting the IPMC kinematics on select beam-like representations and imposing the stationarity of the total potential energy, we formulate rigorous structural theories for IPMC actuation. Not only do we examine classical low-order and higher-order beam theories, but we also propose enriched theories that account for strain localization near the electrodes. The accuracy of these theories is assessed through comparison with finite element simulations on a plane-strain problem of non-uniform bending. Our results indicate that an enriched Euler-Bernoulli beam theory, with three independent field variables, is successful in capturing the main features of IPMC actuation at a limited computational cost.

Appendix A: Electrochemistry

Here, we put forward several hypotheses that allow us to compute a semi-analytical solution for the electrochemistry through the thickness, independently of the mechanical deformation. From the knowledge of the profiles of counterions’ concentration and voltage, we can evaluate the eigenstress related to osmotic pressure in Eq. (9) and Maxwell stress in Eq. (10).

We consider a Saint-Venant solution, whereby we neglect edge effects so that the variations of the electrochemical variables along the IPMC axis and width are negligible compared to their through-the-thickness variations [29, 66]. Specifically, we suppose that the counterions’ concentration and voltage depend only on the through-the-thickness coordinate $Y$, such that $C = C(Y,t)$ and $\psi = \psi (Y,t)$, respectively.

Under these assumptions, the electrochemistry is described by a 1D system of two PDEs, commonly known as Poisson-Nernst-Planck (PNP) system [17–19, 50, 67]. The first equation is derived from mass conservation of the counterions in Eq. (2),

$$ \frac{\partial C(Y,t)}{\partial t} + \frac{\partial J_{Y}(Y,t)}{\partial Y} = 0, $$

(74)

where $J_{Y}$ indicates the counterions’ flux through the thickness of the IPMC, obtained from Eqs. (13) and (12) as

$$ J_{Y}(Y,t) = -\mathscr {D}\left (\frac{\partial C(Y,t)}{\partial Y} + \frac{\mathscr {D}C(Y,t)}{\mathscr {RT}} \frac{\partial \psi (Y,t)}{\partial Y}\right ). $$

(75)

The second equation is the 1D Gauss law, derived from Eq. (3) as

$$ \frac{\partial D(Y,t)}{\partial Y} = \mathscr {F}(C(Y,t)-C_{0}), $$

(76)

where $D$ indicates the through-the-thickness electric displacement, which from Eq. (11) reads

$$ D(Y,t) = -\epsilon \frac{\partial \psi (Y,t)}{\partial Y}. $$

(77)

Substituting the constitutive relations in Eqs. (75), (77) in the PDEs in Eqs. (74), (76) and assuming that material properties are homogeneous in the ionomer, we obtain the PNP system

$$\begin{aligned} &\frac{\partial C(Y,t)}{\partial t} = \mathscr {D} \frac{\partial }{\partial Y}\left (\frac{\partial C(Y,t)}{\partial Y} + \frac{\mathscr {F}C(Y,t)}{\mathscr {RT}} \frac{\partial \psi (Y,t)}{\partial Y}\right ), \end{aligned}$$

(78a)

$$\begin{aligned} &-\epsilon \frac{\partial ^{2} \psi (Y,t)}{\partial Y^{2}} = \mathscr {F}(C(Y,t)-C_{0}), \end{aligned}$$

(78b)

which should be complemented by appropriate boundary conditions at the interface with the electrodes and initial conditions. Consistent with hypotheses in Sect. 2.1, we consider ion-blocking conditions at the ionomer-electrode interfaces,

$$ J(-h,t) = J(h,t) = 0, $$

(79)

and we assume that there is no drop of the external voltage $\bar{V}(t)$ across the electrodes, such that

$$\begin{aligned} \psi (-h,t) &= -\frac{\bar{V}(t)}{2}, \end{aligned}$$

(80a)

$$\begin{aligned} \psi (h,t) &= \frac{\bar{V}(t)}{2}. \end{aligned}$$

(80b)

Furthermore, we consider the IPMC to be initially electroneutral, that is,

$$\begin{aligned} C(Y,0) &= C_{0}, \end{aligned}$$

(81a)

$$\begin{aligned} \psi (Y,0) &= 0. \end{aligned}$$

(81b)

The system of PDEs in Eq. (78a), (78b) with boundary and initial conditions in Eqs. (79), (80a), (80b), and (81a), (81b), represents a singularly perturbed BVP [17], due to the small value of the dielectric constant multiplying the highest order derivative in the Poisson equation in Eq. (78b). In fact, should one neglect this term, it would not be possible to satisfy both boundary conditions in Eq. (80a), (80b). In this family of differential problems, boundary layers typically develop at the boundaries of the domain, challenging the application of standard numerical techniques based on the discretization of the domain, such as finite differences or FE methods [68].

The need to accurately resolve boundary layers to ensure a precise quantification of the eigenstress, along with the limitations on the aspect ratio of the elements to guarantee stability of the numerical scheme, require the use of fine meshes that drastically increase the computational burden. Specifically, the thickness of boundary layers is of the order of the so-called Debye screening length [50], which is given by

$$ \lambda = \frac{1}{\mathscr {F}}\sqrt{ \frac{\epsilon \mathscr {RT}}{C_{0}}}. $$

(82)

For common ionomers, the Debye screening length is a few Angstrom [17], thereby hindering the feasibility of numerical simulations on millimeter- and centimeter-sized domains and calling for alternative methods to solve the problem, such as the one proposed in this paper.

Singularly perturbed problems can be solved analytically with the method of matched asymptotic expansions [32]. Specifically, we divide our computational domain into three subdomains: an “outer” region in the bulk of the ionomer, and two “inner” subdomains near the interfaces with electrodes, where we define a magnified spatial coordinate to describe the formation of boundary layers. In each of these three subdomains, we expand each variable in a power series of $\delta = \lambda /h$ quantifying the ratio of the Debye screening length and the semi-thickness of the ionomer. By considering different orders, we obtain a series of simpler systems of PDEs in each subdomain, coupled through matching conditions in the overlapping region between inner and outer subdomains, where both PDE systems should be valid. A composite solution, valid in the entire computational domain, can be assembled by summing the solutions for each subdomain and subtracting their value in the overlapping regions. A detailed solution of this mathematical problem is presented in [17].

The matched asymptotic expansion reveals that IPMC electrochemistry is determined by the solution of an $RC$ circuit, excited by the voltage $\bar{V}(t)$ applied across the electrodes [17]. The conductivity per unit area of the resistor is given by [17]

$$ \varsigma = \frac{\mathscr {D}C_{0}\mathscr {F}^{2}}{2h\mathscr {RT}}, $$

(83)

while the nonlinear constitutive behavior of the capacitor, representing the charge stored in the boundary layers, is described by [17]

$$ q_{\mathrm{S}}(t) = \sqrt{\epsilon \mathscr {RT}C_{0}}\vartheta \left ( \frac{V(t)}{V_{\mathrm{th}}}\right ), $$

(84)

where $q_{\mathrm{S}}$ is the charge stored per unit surface of electrodes, $V(t)$ is the voltage drop across the capacitor, and

$$ \vartheta (\alpha ) = \sqrt{2}\sqrt{ \frac{\alpha }{\mathrm{exp}(\alpha )-1}-\ln \frac{\alpha }{\mathrm{exp}(\alpha )-1}-1}. $$

(85)

The circuit can be solved by applying Kirchhoff law, such that

$$ \bar{V}(t) = V(t) + \frac{i(t)}{\varsigma }, $$

(86)

where $i(t) = \mathrm{d}q_{\mathrm{S}}(t)/\mathrm{d}t$ is the current through the circuit.

From the solution of the ODE of the circuit in Eq. (86), with initially discharged capacitor ($V(0)=0$), we obtain the time evolution of the voltage drop across the capacitor $\alpha (t) = V(t)/V_{\mathrm{th}}$, which completely defines the first order composite solution through the thickness as [17]

$$ C(Y,\alpha (t)) = C_{0}\left [-1+\exp \left (y^{+}\left ( \frac{1-\frac{Y}{h}}{\delta },\alpha (t)\right )\right )+\exp \left (y^{-} \left (\frac{1+\frac{Y}{h}}{\delta },\alpha (t)\right )\right ) \right ], $$

(87a)

$$ \begin{aligned} \psi (Y,\alpha (t)) &= \frac{\bar{V}(t)}{2} + \frac{i(t)}{2\varsigma } \left (\frac{Y}{h}-1\right )\\&+ V_{\mathrm{th}}\left [\ln \left ( \frac{\alpha (t)}{\exp \left (\alpha (t)\right )-1}\right )-y^{+} \left (\frac{1-\frac{Y}{h}}{\delta },\alpha (t)\right )-y^{-}\left ( \frac{1+\frac{Y}{h}}{\delta },\alpha (t)\right )\right ], \end{aligned} $$

(87b)

where $y^{\pm }(\xi ^{\pm })$ are functions of the magnified variables $\xi ^{\pm }=(1\mp Y)/\delta$ near the electrodes, describing the formation and development of boundary layers. These functions are obtained by solving the following second order differential problems [17]:

$$\begin{aligned} \frac{\partial ^{2} y^{\pm }(\xi ^{\pm },\alpha )}{(\partial \xi ^{\pm })^{2}} &= \exp (y^{\pm }(\xi ^{\pm },\alpha )) - 1, \end{aligned}$$

(88a)

$$\begin{aligned} y^{+}(0,\alpha ) &= \ln \frac{\alpha }{\exp (\alpha )-1}, \end{aligned}$$

(88b)

$$\begin{aligned} y^{-}(0,\alpha ) &= \ln \frac{\alpha }{\exp (\alpha )-1}+\alpha , \end{aligned}$$

(88c)

$$\begin{aligned} \frac{\partial y^{\pm }(0,\alpha )}{\partial \xi ^{\pm }} &= \pm \vartheta (\alpha ). \end{aligned}$$

(88d)

In summary, from the solution of the circuit model we compute the voltage across the capacitor, where Eq. (86) is used, and counterions’ concentration and voltage profiles through the thickness are obtained with Eqs. (87a), (87b) and (88a)–(88d). Since these problems are independent of the deformation, the distribution of the electrochemical variables over time can be found once for all and then used to compute the eigenstress in the IPMC according to Eqs. (9) and (10).