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\title{Finite Element Model of a Self-Oscillating IPMC}


\author{Deivid Pugal}


\email{david@ut.ee}


\affiliation{Active Materials and Processing Laboratory, Mechanical Engineering
Department,\\
 University of Nevada, Reno, U.S.A}


\affiliation{IMS Lab, Institute of Technology, Tartu University, Estonia}


\author{Kwang J. Kim}


\email{kwangkim@unr.edu}


\affiliation{Active Materials and Processing Laboratory, Mechanical Engineering
Department,\\
 University of Nevada, Reno, U.S.A}


\author{Andres Punning}


\author{Heiki Kasemägi}


\author{Maarja Kruusmaa}


\author{Alvo Aabloo}


\affiliation{IMS Lab, Institute of Technology, Tartu University, Estonia}


\date{\today}

\begin{abstract}
This paper presents a electro-mechanical model of an ionic polymer-metal
composite (IPMC) material. The technique of the modelling is Finite
Element method (FEM). An applied electric field causes the drift of
counter-ions (e.g. Na+) which in turn drags water molecules along.
The mass and charge imbalance inside the polymer is the main cause
of the bending motion of the IPMC. All foregoing physical effects
have been considered as time dependent and modelled with FEM. The
model takes into account the mechanical properties of Nafion polymer
as well as thin coating of the platinum electrodes. (Furthermore,
the water drift and the counter ion drift inside the polymer backbone
is considered separately.) As self-oscillating behavior of an IPMC
was described in our previous paper, the modelling of the electrochemical
reactions is also considered. Reactions occuring on the surface of
the platinum electrode, which is immersed into formaldehyde (HCHO)
solution, are described by partial differential equations and also
modeled using FEM. By coupling the equations to the rest of the model,
we are able to simulate self-oscillatory behavior of an IPMC sheet.
\end{abstract}
\maketitle

\section{Introduction}

Electroactive polymer (EAP) based materials are valuable for many
applications from micro robotics to military and space applications.
Some of the advantages of EAP materials are light weight, noiseless
actuation, simple mechanics and large displacement. In addition some
EAPs, such as IPMCs~\cite{Shahinpoor2001}, are able to function
in aquous environments. Those qualities make the materials possible
to use as artificial muscles. In this paper we consider simulations
of IPMC type materials with the Finite Element method.

IPMC materials are highly porous polymer materials such as Nafion$^{TM}$,
Teflon, filled with some kind of ionic conductive liquid. There are
water based IPMCs which operate in aquatic environment and current
is caused by ions such as Na+, K+ dissociated in water. Other kind
of IPMCs - ionic liquid based, do not need wet environment for operating.
A sheet of an ionic polymer is coated with a thin metal layer, usually
platinum or gold. All freely movable cations inside the polymer migrate
towards an electrode due to applied electric field, causing expansion
of the material at the one end of the sheet and contraction at the
other end, which results in bending of the sheet.

For simulating actuation of an IPMC sheet we need to solve coupled
problems due to the complex nature of bending of an IPMC. It includes
simulations in different domains such as mechanical, electrostatic
and mass transfer, and even electrochemical for more advanced models.
Some authors~\cite{Wallmersperger2004,Nemat-Nasser2006} have already
simulated mass transfer and electrostatic effects. We have used similar
approach in our model. Toi~\cite{Toi2005} has shown a Finite Element
model including viscocity terms in transportation processes explicitly.
However, the basis of the described model is a rectangular beam with
2 pairs of electrodes.  Our approach for simulating mechanical bending
is taking advantage of the numerical nature of FEM problems - we use
continuum mechanics equations instead of analytical Euler beam theroy
which is more commonly used by authors~\cite{Lee2005,Wallmersperger2007}.
By coupling equations from different domains, we get a model for an
IPMC muscle sheet, which allows us to use it as a starting point for
solving more complex problems, thus we have introduced a simulation
of electrochemical reactions on the platinum electrode of an IPMC
sheet, which leads to self-oscillating actuation. Spontaneous oscillations
are common phenomena in nature and have been studied in many experiments,
including electrochemical systems such as oxidation of organic materials
and metals~\cite{Miller2006}. Under certain conditions such system
can generate oscillations~\cite{Krischer2001}. We have conducted
a series of tests, where an IPMC sheet have been immersed into acidic
formaldehyde (HCHO) solution and exposed to a constant potential.
The measurements, however show current oscillations, which in turn
result in oscillating bending of the IPMC sheet~\cite{Kim2007}.
Hence we have introduced a FEM model in this paper to describe time
dependent bending of a self-oscillating IPMC.


\section{Bending Simulation Details}

An IPMC sheet consists of a polymer host and a metal coating. In our
experiments, we have used Nafion$^{TM}$ 117, coated with thin layer
of platinum. Mass transfer and electrostatic simulations are done
in three mechanical domains: pure backbone polymer, pure platinum
coating, and mixture of polymer and platinum - some platinum tends
to diffuse into the polymer during coating process~\cite{Nemat-Nasser2003}.
Physical properties and dimensions of pure, 2~$\mu\mbox{m}$ thick
platinum coating are considered only when calculating bending. That
gives us five mechanical domains as shown in Fig.~\ref{fig:Illustration-of-domins}.
Most simulations are carried out for an IPMC strip, 2-4~cm long,
200~$\mu\mbox{m}$ thick polymer, including 10~$\mu\mbox{m}$ thick
Pt diffusion region on each side, coated with 2~$\mu\mbox{m}$ thick
platinum, in a cantilever configuration - one end of the strip is
fixed.

%
\begin{figure}
\includegraphics[scale=0.32]{IPMC}


\caption{\label{fig:Illustration-of-domins}Illustration of domains and dimensions
used in simulations. Notice that there are three different mechanical
domains - pure Nafion$^{TM}$ polymer, pure Pt coating and diffusion
layer, where Pt has diffused into the polymer.}
\end{figure}


The Nernst-Planck equation describes diffusion, convection and in
presence of electric field and charges, migration of the particles.
The general form of the equation is \begin{equation}
\frac{\partial C}{\partial t}+\nabla\cdot(-D\nabla C-z\mu FC\nabla\phi)=-\vec{u}\cdot\nabla C,\label{eq:nernst planck}\end{equation}
where C is concentration, D diffusion constant, F Faraday constant,
$\vec{u}$ velocity, z charge number, $\phi$ electric potential,
and $\mu$ mobility of species, which is found by using known relation
$\mu=D/(RT)$. The variable T is absolute temperature and R universal
gas constant. Movable counter ions are described by Eq.~\eqref{eq:nernst planck}.
As anions are fixed, they maintain constant charge density throughout
the polymer. After a voltage is applied to the electrodes of an IPMC,
all free cations will start migrating towards cathode, causing current
in the outer electric circuit. Because of the fact that ions cannot
move beyond the boundary of the polymer, charges start to accumulate,
resulting in increase of the electric field, which cancels out the
applied one. The process could be described by Gauss' Law:\begin{equation}
\nabla\cdot\vec{E}=-\Delta\phi=\frac{F\cdot\rho}{\varepsilon},\label{eq:gauss}\end{equation}
where $\rho$ is charge density, $\varepsilon$ is absolute dielectric
constant and $E$ is the strength of the electric field. A steady
state of the cations forms when electric field created by distribution
of cations cancels out the applied electric field, i.e. the strength
of the electric field inside the polymer is approximately zero, as
also shown in Fig.~\ref{fig:The-concentration-of}. The steady state
cation concentration, with average value of $1200\ mol/m^{3}$ is
also shown in the same figure. %
\begin{figure}
\includegraphics[scale=0.32]{field_concentration}


\caption{\label{fig:The-concentration-of}The concentration of counter ions
and electric field strength inside the polymer according to simulations.}
\end{figure}
 It is interesting to notice that there are fluctuations in charge
distribution only in really thin boundary layers, leading to the conclusion
that there is no charge imbalance inside the polymer. General understanding
is that locally generated charge imbalance nearby platinum electrodes
is directly connected, and mainly responsible, to the bending of an
IPMC~\cite{Nemat-Nasser2000}. Therefore we define longitudinal force
per unit area at each point in the polymer of an IPMC as follows~\cite{Wallmersperger2007}:
\begin{equation}
\vec{F}=(A\ \rho+\ B\ \rho^{2})\ \hat{x},\label{eq:force}\end{equation}
where $\rho$ is charge density and $A$ and $B$ are constants which
could be found from different experiments. Values of the constants
are brought out in Table~\ref{tab:Parameter-values-used} and it
is interesting to notice that the ratio $A/B$ is close to the value
suggested by Wallmersperger~\cite{Wallmersperger2007}. Equations~\eqref{eq:nernst planck}-\eqref{eq:force}
are described only for pure Nafion$^{TM}$ and Pt diffusion domain
(see Fig.~\ref{fig:Illustration-of-domins}). There are no ion diffusion
nor migration in thin Pt coating domain.

%
\begin{table}

\caption{\label{tab:Material-parameters-used}Material parameters used in
continuum mechanics equations.}

\begin{ruledtabular}

\begin{tabular}{cccc}
Parameter&
Value&
Unit&
Domain where applied\tabularnewline
\hline
$E_{N}$&
200&
MPa&
Nafion$^{TM}$\tabularnewline
$\nu_{N}$&
0.49&
-&
Nafion$^{TM}$\tabularnewline
$E_{Pt}$&
168&
GPa&
Pt\tabularnewline
$\nu_{Pt}$&
0.38&
-&
Pt\tabularnewline
$E_{diff}$&
84&
GPa&
Pt diffusion layer (estimated)\tabularnewline
$\nu_{diff}$&
0.42&
-&
Pt diffusion layer (estimated)\tabularnewline
\end{tabular}

\end{ruledtabular}
\end{table}


To relate the force in Eq.~\ref{eq:force} to the physical bending
of an IPMC sheet, we introduce a set of continuum mechanics equations,
which are effective in all domains (Fig.~\ref{fig:Illustration-of-domins}).
These equations are described in Comsol Multiphysics structural mechanics
software package. Normal and shear strain are defined as \begin{equation}
\varepsilon_{i}=\frac{\partial u_{i}}{\partial x_{i}},\;\varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}-\frac{\partial u_{j}}{\partial x_{i}}\right),\label{eq:normal shear strain}\end{equation}
where u is the displacement vector, x denotes a coordinate and indices
i and j are from 1 to 3 and denote components correspondingly to x,
y, or z direction. The stress-strain relationship is \begin{equation}
\sigma=D\varepsilon,\label{eq:stress strain}\end{equation}
where D is $6\times6$ elacticity matrix, consisting of components
of Young's modulus and Poisson's ratio. The system is in equilibrium,
if the relation\begin{equation}
-\nabla\cdot\sigma=\vec{F},\label{eq:equilibrium,contiinum}\end{equation}
is satisfied. This is Navier's equation for displacement. The values
of Young's modulus and Poisson's ratios, which are used in the simulations,
are shown in Table~\ref{tab:Material-parameters-used}. The values
for platinum diffusion region are not measured, but estimated as an
average of values of pure Nafion$^{TM}$ and Pt region.

%
\begin{table}

\caption{\label{tab:Parameter-values-used}Parameter values used in bending
simulations.}

\begin{ruledtabular}

\begin{tabular}{ccc}
Parameter&
Value&
Unit\tabularnewline
\hline
$D$&
$1\times10^{-6}$&
$cm^{2}/s$\tabularnewline
$R$&
$8.31$&
$J/(K\ mol)$\tabularnewline
$T$&
$293$&
$K$\tabularnewline
$z$&
$1$&
$-$\tabularnewline
$F$&
$96.5\times10^{6}$&
$mC/mol$\tabularnewline
$\varepsilon$&
$25$&
$mF/m$\tabularnewline
$A$&
$110$&
$N\ m/mol$\tabularnewline
$B$&
$12$&
$N\ m^{4}/mol^{2}$\tabularnewline
$\alpha_{polymer}$&
$0$&
$s^{-1}$\tabularnewline
$\beta_{polymer}$&
$1.5$&
$s$\tabularnewline
\end{tabular}

\end{ruledtabular}
\end{table}


As our simulations are rather dynamic than static, we have to introduce
an equation to describe the motion of an IPMC sheet. To do that, we
use Newton's Second law \begin{equation}
\rho\frac{\partial^{2}\vec{u}}{\partial t^{2}}-\nabla\cdot c\nabla\vec{u}=\vec{F},\label{eq:newton II}\end{equation}
where the second term is the static Navier's equation. The first term
in Eq.~\eqref{eq:newton II} introduces the dynamic part. Several
authors have reached to the conclusion that IPMC materials exhibit
viscoelastic behavior~\cite{Newbury2003,Yagasaki2004}, which is
especially noticable for high frequency movements~\cite{Newbury2003II}.
However, we include the viscoelastic term in our equations by means
of using Rayleigh damping~\cite{lord1945ts} model, which is described
for a system of one degree of freedom as follows: \begin{equation}
m\frac{d^{2}u}{dt^{2}}+\xi\frac{du}{dt}+ku=f(t),\label{eq:single degree of freedom}\end{equation}
 where the damping parameter $\xi$ could be expressed as $\xi=\alpha m+\beta k$.
The parameter~m is a mass, k is a stiffness and $\alpha$ and $\beta$
are correspondingly damping coefficients. The equation for the multiple
degrees of freedom is \begin{equation}
\rho\frac{\partial^{2}\vec{u}}{\partial t^{2}}-\nabla\cdot\left[c\nabla\vec{u}+c\beta\nabla\frac{\partial\vec{u}}{\partial t}\right]+\alpha\rho\frac{\partial\vec{u}}{\partial t}=\vec{F}.\label{eq:damping,multifreedom}\end{equation}
By coupling Eq.~\eqref{eq:damping,multifreedom} to the previously
described equations, a good basic model for IPMC actuation has been
got. The damping equation turned out to be very necessary to describe
correct movement of an IPMC strip. Though the values of the parameters
$\alpha$ and $\beta$ are empirical (see Table~\ref{tab:Parameter-values-used}),
they have an important role of improving the dynamical behavior of
the model for non-constant applied voltages. 

All values, which are used in the simulations, have been brought out
in Table~\ref{tab:Parameter-values-used}. Fig.~\ref{fig:Experimental-and-simulation}
shows a comparison between the simulation and an experiment~\cite{Punning2007}.
More comparative figures are introduced in the next section of the
paper.

%
\begin{figure}
\includegraphics[scale=0.32]{punn_d}


\caption{\label{fig:Experimental-and-simulation}Experimental and simulation
results of tip displacement. Although there is a slight difference
of graphs in large displacement region, the model gives precise estimation
for smaller displacement.}
\end{figure}



\section{Modeling Self-oscillations}

We have conducted a series of tests with IPMCs in a constant electric
field in formaldehyde (HCHO) solution. Measurements show that current
oscillations begin from applied potential of ca. 0.75~V. More information
about experiments and conclusions could be read from our previous
letter~\cite{Kim2007}. Studies show that there are sequential electrochemical
reactions, which take place on the platinum cathode. The initial burst
of the current is caused by the reaction \begin{equation}
CO_{ads}+OH_{ads}\rightarrow CO_{2}+H^{+}+e^{-}+2*,\label{eq:inital current reaction}\end{equation}
where subscript $ads$ denotes species adsorbed to the platinum and
$*$ denotes an active platinum site. The result of the reaction~\eqref{eq:inital current reaction}
is clearing up 2 platinum sites, which causes CO to adsorb again.
Chronopotentiomentry scans show that before reaction~\eqref{eq:inital current reaction},
the following reactions occur:\begin{eqnarray}
HCHO+* & \rightarrow & Pt-CHO_{ads}+H^{+}+e^{-},\label{eq:HCHO adsorb}\\
Pt-CHO & \rightarrow & Pt-CO_{ads}+H^{+}+e^{-},\label{eq:CO formation}\\
H_{2}O+* & \leftrightarrow & Pt-OH_{ads}+H^{+}+e^{-}.\label{eq:OH adsorb}\end{eqnarray}
HCHO is dissociated on the electrode surface at lower anodic potentials.
Higher anodic potentials cause dehydrogenation of water which results
in water oxidation with intermediate Pt-OH formation. We believe that
these reaction lead to oscillating potentials, which in turn lead
to self-oscillating motion of the IPMC sheet.

Series of chronopotentiometry scans were conducted to characterize
oscillations for different HCHO concentration and current densities.
As was brought out in our previous paper~\cite{Kim2007}, the oscillations
start at approximately $7\ mA/cm^{2}.$ The experiments were conducted
up to current density values of $40\ mA/cm^{2}$. Tests with HCHO
concentrations of 1~M, 2~M, and 3~M show that oscillations frequencies
remain constant up to the current density of $14\ mA/cm^{2}$, but
after further increasing the current, in 1~M HCHO, the frequency
decreases, in 2~M HCHO, the frequency remains constant, and in 3~M
HCHO solution, the frequency starts to increase, as also shown in
Fig.~\ref{fig:Experimental-and-simulated frequency}. %
\begin{figure}
\includegraphics[scale=0.32]{frequency}


\caption{\label{fig:Experimental-and-simulated frequency}Experimental\cite{Kim2007}
and simulated frequency dependence on concentration of HCHO and applied
current density. Simulations for 1 M HCHO concentration does not go
past $30\ mA/cm^{2}$, because given equation system did not give
reasonable results beyond that current density.}
\end{figure}
 Our goal is to develope a model for describing the frequency behavior
depending on HCHO concentration and the current density. The basic
model and concepts are taken from PhD thesis of D.~Kim~\cite{kim2006ecb}
and work of P. Strasser~\cite{Strasser1997}. To describe the oscillations,
four dynamic parameters, therefore four differential equations, must
be observed: concentration of adsorbed OH, CO, the change of the double
layer potential due to electrochemical reactions, and the change of
the concentration of HCHO near the surface of platinum. First two
variables could be expressed for a certain current density and HCHO
concentration as~\cite{kim2006ecb}:\begin{eqnarray}
\dot{\theta}_{CO} & = & k_{2}M-k_{4}\theta_{CO}\theta_{OH},\label{eq:osc_CO}\\
\dot{\theta}_{OH} & = & k_{3}\theta_{CO}M-k_{-3}\theta_{OH}-k_{4}\theta_{CO}\theta_{OH},\label{eq:osc_OH}\end{eqnarray}
where $\theta_{CO}$ and $\theta_{OH}$ are normed adsorbtion coverages
of CO and OH. Variables $k_{i}$ and M are described by equations\begin{eqnarray}
k_{i}\left(\phi\right) & = & exp\left[s_{i}\left(\phi-\phi_{i}\right)\right],\label{eq:osc_k}\\
M & = & \left(1-\theta_{CO}-\theta_{OH}\right),\label{eq:osc_D}\end{eqnarray}
where $s_{i}$ are modeling coefficients and $\phi_{i}$ are potentials
of the reactions~\cite{kim2006ecb}. As our model is highly dynamic,
the double layer with thickness $\delta$ near the platinum electrode
is introduced. At the far end of the layer, the concentration of the
formic acid is considered constant, and due to the adsorbtion of HCHO
on Pt, the concentration of the solution is changing in time near
the electrode. There are two components responsible of decrease of
the concentration. The first one is direct oxidation of the formic
acid to $CO_{2}$ and $2H^{+}$, the second one is adsorbtion of CO
on the platinum surface due to electrochemical reactions~\cite{Strasser1997}.
The mechanism, which restores the HCHO concentration near the surface
is diffusion. So the amount of the formic acid is decreasing significantly
while the adsorption rate is high and increasing due to the diffusion
during the low adsorbtion period. The equation describing the diffusion
process is\begin{equation}
\frac{\partial c_{FA}}{\partial t}=\nabla\cdot\left(D_{FA}\nabla c_{FA}\right)\label{eq:hcho_D}\end{equation}
with constant concentration at the far end of the double layer and
flux\begin{equation}
f=k_{2}M\left(1+kc_{r}\right)\times S_{tot},\label{eq:hcho_F}\end{equation}
as boundary condition on the electrode. Here $S_{tot}$ denotes the
total number of platinum sites per surface area, $c_{r}$ is normed
concentration near boundary layer and equals $c_{r}=c_{FA}/c_{0}$,
where $c_{0}$ is an initial concentration. Variable k is a simulation
constant. The second term of Eq.~\eqref{eq:hcho_F} represents simplified
version of the direct oxidation path~\cite{Strasser1997}. Considering
those equations and interesting nature of frequency characteristic
for different amounts of HCHO (Fig.~\ref{fig:Experimental-and-simulated frequency}),
we can now describe somewhat empirical, gray box~\cite{Shahinpoor2004}
equation for the last dynamic variable - the double layer potential:\begin{eqnarray}
\dot{\phi} & = & \frac{1}{C_{dl}}\left[j_{th}-j_{d}+A\times B\times j^{2}\times\left(j-j_{th}\right)\times c_{r}-\right.\nonumber \\
 &  & \left.-S_{tot}F\left(k_{1}M+k_{4}\theta_{CO}\theta_{OH}\right)\right],\label{eq:osc_phi}\end{eqnarray}
where j is applied current density, $j_{th}$ is threshold current
density with approximate value of $10\ mA/cm^{2}$, $j_{d}$ is direct
current density and is proportional to the second term of Eq.~\eqref{eq:hcho_F}.
The variable B could be explicitly written as $B=c_{0}-c_{neutral}$,
where $c_{netural}$ corresponds to the concentration of 2~M. This
is denoted {}``netural'' concentration, because as could be seen
in Fig.~\ref{fig:Experimental-and-simulated frequency}, the oscillation
frequency for the case $c_{0}=c_{neutral}=2\ M$ does not depend on
the applied current density. The numeric data could be found in Table~\ref{tab:Variables-and-values}.
The third term in Eq.~\eqref{eq:osc_phi} is empirical and reflects
the interesting behavior of the oscillations frequency for different
formic acid concentrations. Other terms are similar to the ones described
by Strasser~\cite{Strasser1997}. However, some values are adjusted
to get realistic simulation results. Measured voltage oscillation
comparison to simulation data could be seen in Fig.~\ref{fig:Potential-oscillations.}.

%
\begin{table}

\caption{\label{tab:Variables-and-values}Variables and values used in the
simulation of electrochemical oscillations.}

\begin{ruledtabular}

\begin{tabular}{ccc}
Parameter&
Value&
Unit\tabularnewline
\hline
$S_{tot}$&
$0.5\times10^{-6}$&
$mol/cm^{2}$\tabularnewline
$C_{dl}$&
$1$&
$mF/cm^{2}$\tabularnewline
$A$&
$1.2$&
$cm^{7}/\left(mA^{2}\times mol\right)$\tabularnewline
$k$&
$100$ &
-\tabularnewline
$\delta$\footnotemark[1]&
$3\times10^{-2}$&
$cm$\tabularnewline
$D_{FA}$\footnotemark[1]&
$2.5\times10^{-5}$&
$cm^{2}/s$\tabularnewline
$\phi_{1,2,3,-3,4}$\footnotemark[2]&
$\left[0.2,\ 0.3,\ 0.01,\ 0.512,\ 0.77\right]$&
$V$\tabularnewline
$s_{1,2,3,-3,4}$\footnotemark[2]&
$\left[10,\ -11,\ 9,\ -9,\ 20\right]$&
$V^{-1}$\tabularnewline
\end{tabular}

\end{ruledtabular}

\footnotetext[1]{Ref. \cite{Strasser1997}}\footnotetext[2]{Ref.
\cite{kim2006ecb}}
\end{table}


By using the obtained voltage output in the base Finite Element Model,
which is described in the previous section, we can simulate oscillating
deflection of an IPMC muscle. Two sample results for different HCHO
concentrations and current densities are shown in Figures~\ref{fig:Oscillating-tip-displacement.1M}~and~\ref{fig:Oscillating-tip-displacement.3M}.
As it could be seen, the amplitude, frequency and for the most part,
shape of the deflection show reasonable agreement between modeling
and experimental data. However, there is some disharmony at the areas
of maximum deflection, where the experiments show really sharp deflections
in comparison to rather smooth simulation results.

%
\begin{figure}
\includegraphics[scale=0.32]{potential_osc}


\caption{\label{fig:Potential-oscillations.}Potential oscillations. Measured
data\cite{kim2006ecb} and simulated data for 3M HCHO solution, with
applied current density of $20\ mA/cm^{2}.$}
\end{figure}


%
\begin{figure}
\includegraphics[scale=0.32]{disp_osc1m20ma}


\caption{\label{fig:Oscillating-tip-displacement.1M}Oscillating tip displacement.
Experimental\cite{kim2006ecb} data and simulation data for 1M solution,
applied current density of $20\ mA/cm^{2}.$}
\end{figure}


%
\begin{figure}
\includegraphics[scale=0.32]{disp_osc3m10ma}


\caption{\label{fig:Oscillating-tip-displacement.3M}Oscillating tip displacement.
Experimental\cite{kim2006ecb} data and simulation data for 3M solution,
applied current density of $10\ mA/cm^{2}.$}
\end{figure}



\section{Summary And Conclusions}

We have developed a Finite Element model for simulating actuation
of an IPMC. The model is largely based on physical quantities and
well known or measurable variables. The migration and diffusion of
the counter ions inside the Nafion$^{TM}$ polymer is described along
the electric field change due to the charge imbalance. This in turn
is tied in to continuum mechanics and dynamics equations, forming
a complete system of equations to describe the bending of an IPMC
sheet.

The comparison of experimental and simulated tip displacement in time
shows reasonable agreement, especially for smaller deflections. The
future work is getting simulations more precise for large displacements,
possibly including equations, which describe voltage distribution
on the surface of the electrdodes depending on the curvature of the
IPMC. Taking into account the surrounding environment of the IPMC
could also improve the results.

The second part of the work describes the extended model for self-oscillating
IPMCs. The oscillations occur when a platinum coated IPMC is immersed
into formic acid solution and subjected to a constant potential or
current. The extended model takes into account formic acid concentration
changes near the electrode and poisoning level of the platinum sites.
This in turn results in the oscillating double layer potential, which
is used in the base model for calculating time dependent tip displacement
of the IPMC muscle. For the most part, the model follows simulation
data closely. However, the experimental deflection shows some very
sharp movements at certain regions, but the simulation gives rather
smooth displacement profiles. The future work includes studying further
this interesting behavior and possibly improving the model.

\begin{acknowledgments}
We want to acknowledge the support of {}``some naval institution''
and {}``some estonian insitution, grant no. 12345''. Also we acknowledge
Estonian Archimedes Foundation for travel support of Deivid Pugal
to University of Nevada, Reno.
\end{acknowledgments}
\bibliographystyle{apsrev}
\bibliography{jap_femmodel}

\end{document}
