# Electrode effect on actuation¶

For the electrodes, the Ohm’s law for the current density is

$\sigma\nabla V=-\boldsymbol{j},$

where $$\sigma \$$ is electric conductivity of the electrodes, $$V \$$ and $$\boldsymbol{j} \$$ are electric potential and current density in the electrodes, respectively. It must be noted that the electric potential $$\phi \$$ inside the polymer and the electric potential $$V \$$ in the electrodes are different.

One way to relate the ionic current in the polymer to the electric current in the electrodes is with the Ramo-Shockley theorem:

$I=\frac{1}{\varphi}\sum_{n}q_{n}\boldsymbol{W}\left(\boldsymbol{r}_{n}\right)\cdot\boldsymbol{v}_{n},$

where $$\boldsymbol{r}_n \$$, $$\boldsymbol{v}_n \$$, and $$q_n \$$ are the position vector, instantaneous velocity, and charge of a particle $$n \$$, respectively. $$\boldsymbol{W} \$$ is the electric field that would be caused by $$1\ V \$$ applied potential without any charges, mobile nor fixed, being present. $$I \$$ is the current in the external circuit and $$\varphi \$$ is a constant with value of 1 volt.

The integral form of this theorem for parallel plate approximation is

$j_y=\frac{1}{h}\int_{-h/2}^{h/2} f_{y}\ \mathrm{d}l,$

where $$f_{y} \$$ is the ionic flux in $$y \$$ direction and has a unit of $$\mathrm{\frac{C}{m^{2}s}} \$$ and $$j_y \$$ is the local current density at an electrode boundary. The term $$\mathrm{d}l \$$ is an integration element along the path where the particle moves. The variable $$j_y \$$ depends on all the spatial coordinates except $$\hat{y} \$$. Namely, in the model, it is assumed that the electrodes are on $$x-z \$$ plane. This is illustrated in the following figure.

The model was solved with with uniform electrode conductivity of $$\sigma=150\times10^2\ \mathrm{S/m} \$$. The potentials in the cathode and anode at different times are as follows

Copyright 2012 American Institute of Physics. Taken from the Journal of Applied Physics article (see below).

As expected, there is a noticeable negative potential gradient $$\nabla V_{\Psi_a} \$$ along the anode and positive gradient $$\nabla V_{\Psi_b} \$$ along the cathode at small time values. As the ionic current saturates, the potential gradients become zero in both of the electrodes. Overall, the calculation results are reasonable in terms of what is expected based on the physics of the IPMC.

The potential in the polymer domain boundary depends on the electrode potentials $$V_{\Psi_b} \$$ and $$V_{\Psi_a} \$$. At the same time, the potentials in the electrode domains depend on the current fluxes on the polymer-electrode boundaries This dependency makes the system of equation non-linear and conceptually is similar to a system with a circular dependency:

• When a potential $$V_{\partial\Psi_a}=V_{appl} \$$ is applied, it instantaneously propagates in the entire domain. The same is true for $$\Psi_b \$$ when $$V_{\partial\Psi_b}=0\ \mathrm{V} \$$ is applied.
• At $$t=0\ \mathrm{s} \$$, the potentials $$V_{appl} \$$ and $$0\ \mathrm{V} \$$ become boundary conditions for the Poisson’s equation in domain $$\Omega \$$.
• This in turn creates an electric field gradient in $$\Omega \$$ and induces the ionic flux.
• The ionic flux induces electric current in the electrodes.
• The circular dependency: induced potential gradients in the electrodes change the underlying ionic current dynamics.

This is illustrated:

Copyright 2012 American Institute of Physics. Taken from the Journal of Applied Physics article (see below)

The effect can be also seen in the actuation dynamics. The following measurement and calculation results illustrate that the model with the electrode feedback results in more accurate tip displacement compared to the simple model where the electrodes are not considered.

Copyright 2012 American Institute of Physics. Taken from the Journal of Applied Physics article (see below).

PDF file of the full article An explicit physics-based model of ionic polymer-metal composite actuators can be downloaded here