Title: **Modeling of ionic polymer-metal composite beam on human
tissues**

Authors: Lei ZHANG ^1^ and Yaowen YANG ^2\*^

Address: School of Civil and Environmental Engineering

Nanyang Technological University

50 Nanyang Avenue

Singapore 639798

^1^ Research Scholar, <pg03462806@ntu.edu.sg> Tel: +65 67906811

^2\*^ Corresponding author, Assistant Professor, <cywyang@ntu.edu.sg>
Tel: +65 67904057

**ABSTRACT**

In this paper, a dynamic model of simply supported ionic polymer-metal
composite (IPMC) beam resting on human tissue is developed. First, the
bending moment in the IPMC beam due to the alternative electric
potential is derived from Nemat-Nasser's hybrid actuation model. This
explicit bending moment expression provides an easy way to estimate the
bending capacity of IPMC. Subsequently, the bending moment expression is
incorporated in the analytical solution of beam transverse vibration to
describe the response of IPMC beam to an electric potential. Pressure
generated by IPMC beam on human tissue is then estimated by numerical
integration. Comparisons show that the results obtained are comparable
with the experimental data in the literature. Finally, to achieve the
maximum deflection and total pressure, the optimal electrode length and
location are discussed. To increase the flexibility and variety of beam
deformation, multiple electrodes are considered. Deflection curve and
generative pressure for multiple electrodes are also derived. The
developed model is useful not only for the biomedical devices that
employ IPMC materials but also for any other applications that utilize
the vibration of IPMC materials.

**Keywords:** ionic polymer metal composites (IPMCs), actuators, beam,
vibration, tissues

**1. INTRODUCTION**

Ionic polymer-metal composite (IPMC) is a type of wet electro-active
polymers (EAP). It consists of a thin polyelectrolyte membrane and a
type of noble metal, such as gold and platinum, chemically plated on
both sides of the membrane. IPMC can undergo a fast and large bending
motion when a low electric potential is applied to its electrodes.
Conversely, IPMC can generate a measurable electric potential when it is
subjected to a sudden bending. Thus, IPMC can serve as both actuators
and sensors.

Much effort has been devoted to understanding the actuation and sensing
mechanism of IPMC materials. Shahinpoor and Kim \[1-4\] presented a
series of papers on the fundamental properties and characteristics of
IPMC, manufacturing techniques, phenomenological modeling of actuation
and sensing mechanism and potential applications. Nemat-Nasser and Li
\[5\] presented a model based on electrostatic interaction of ion
transport. Later, Nemat-Nasser \[6\] studied the micromechanics of IPMC
materials and proposed a hybrid model incorporating electrostatic,
osmotic and elastic effects in the actuation process. Shahinpoor and Kim
\[7\] investigated the mass transfer induced actuation of IPMCs and
developed a technique to minimize water leakage and to increase force
density of IPMCs. Taking into account of the viscoelastic property of
polymer material, Newbury and Leo \[8\] developed a linear
electromechanical model for IPMC transducers. Nemat-Nasser and Wu \[9\]
carried out extensive experimental studies on IPMCs with different
backbone ionomers and various cation forms. Weiland and Leo \[10\]
developed a computational micromechanics model to assess the impact of
uniform ion distribution on spherical clusters of IPMC ionomer. They
also employed a Monte Carlo style approach to study the effects of
pendant chain stiffness and charge balance of the equilibrium state of a
single cluster in ionic polymer \[11\].

However, the actuation mechanism of IPMC has yet been fully understood.
Nevertheless, research on applications of IPMC has attracted
considerable attention from various disciplinaries. IPMC offers many
advantages over the conventional EAP materials, such as compliance,
light weight, low operation voltage and capability of working in aqueous
environments. These properties make it promising for numerous
applications in biomedical, naval, robotic and microelectromechanical
system (MEMS) engineering \[4,12,13\]. One of the important applications
of IPMCs is in the biomedical related instruments which contact with
human organs or tissues, such as artificial ventricular muscles,
surgical tools and active scleral bands. The configuration of IPMC
materials could be in various forms including bands, rings and beams.

Since IPMC materials are generally expected to work in a vibration
state, it is necessary and meaningful to investigate their dynamic
behaviors. However, little research has been reported in the literature
to investigate the dynamic behaviors of IPMC materials. In this paper,
an analytical model is developed to depict the vibration response of a
simply-supported IPMC beam resting on an elastic foundation under an
alternative electric field. The elastic foundation is employed to
simulate the effect of human tissues. Nemat-Nasser's \[6\] hybrid
actuation model is incorporated into the motion equation with
modifications to account for the applied alternative electric field. A
closed-form solution is obtained to describe the transverse vibration of
the IPMC beam subjected to the applied electric field. Based on this
solution, the pressure generated on human tissue is calculated by
numerical integration. To maximize the beam deflection and total
pressure generated, the optimal location and length of the single
electrode in the IPMC beam are discussed. To increase the flexibility of
the IPMC beam and the variety of beam motion, multiple-electrodes on the
IPMC beam are also considered. The deflection curve and generative
pressure of the IPMC beam with multiple electrodes are obtained. The
developed model is useful not only for the biomedical devices that
employ IPMC materials but also for any other applications that utilize
the vibration of IPMC materials.

**2. MODELING OF IPMC BEAM**

As shown in Fig.1, a Nafion based IPMC beam of length $L$, thickness $H$
and width $b$ is studied. The IPMC beam is chemically plated with
platinum electrode on both sides and $x_{1}$ and $x_{2}$ are the
coordinates of the two ends of the electrode. The thickness of Nafion
part of IPMC is denoted as $h$. The IPMC beam is assumed to be simply
supported on human tissue. The effect of tissue is modeled as a Winkler
foundation with stiffness $k$. Assume that the alternative electric
potentials at top and bottom surfaces are ![](media/image1.wmf) and
![](media/image2.wmf), respectively, where ![](media/image3.wmf) is the
time-independent magnitude of electric potential, $\Omega$ is the
angular frequency and *j* is the imaginary unit. Due to the
redistribution of cations and associated water under the applied
electric field, the IPMC beam will vibrate at a frequency the same as
that of the applied electric potential.

The motion equation of the beam can be obtained from Euler-Bernoulli's
beam theory as \[14\]

  -------------------------------------------------------------- --------
  ![](media/image4.wmf)                                          \(1\)

  -------------------------------------------------------------- --------

where $M(x,t)$ is the bending moment in the IPMC beam; $M^{e}(x,t)$ is
the bending moment due to the electric field; $\rho$ is the material
density of IPMC beam; $A$ is the cross-sectional area of beam;
![](media/image5.wmf) is the viscous damping coefficient of the IPMC
beam and $u_{3}(x,t)$ is the transverse displacement of beam.

Transferring the term due to electric field to the right hand side and
utilizing the strain-stress and strain-displacement relations, the
following equation can be obtained

  -------------------------------------------------------------- --------
  ![](media/image6.wmf)                                          \(2\)

  -------------------------------------------------------------- --------

where $Y$ and *I* are the Young's modulus and moment of inertia of the
beam, respectively. Since the IPMC beam will undergo a pure bending at
the frequency of applied electric potential, $M^{e}(x,t)$ can be
expressed as

  -------------------------------------------------------------- --------
  ![](media/image7.wmf)                                          \(3\)

  -------------------------------------------------------------- --------

where $H( \cdot )$ is the Heaviside function defined as

  -------------------------------------------------------------- --------
  $$H(x - a) = \begin{array}{r}                                  \(4\)
  \left\{ 1x \geq a \middle| \right.\                            
  \end{array}$$                                                  

  -------------------------------------------------------------- --------

$M_{0}^{e}$ is the time-independent amplitude of bending moment and
![](media/image8.wmf) is a phase difference between bending moment and
applied electric potential which will be determined in Section 3.2.

Using the method of separation of variables, solution to Eq.(2) can be
derived as

  -------------------------------------------------------------- --------
  ![](media/image9.wmf)                                          \(5\)

  -------------------------------------------------------------- --------

where
$\omega_{m} = \sqrt{(\text{YI} \cdot (m^{4}\pi^{4}\frac{)}{L^{4}} + k\frac{)}{\rho A}}$
is the natural frequency of the beam for zero damping;
![](media/image10.wmf) is a phase lag due to the viscous damping.

With the solution of Eq.(5), the total pressure generated by the IPMC
beam can be readily obtained by

  -------------------------------------------------------------- --------
  $$p = k \cdot \int_{}^{}u_{3}H(u_{3})\text{dx}$$               \(6\)

  -------------------------------------------------------------- --------

The computation of total pressure can be implemented by numerical
integration.

Once the bending moment due to electric potential $M^{e}(x,t)$ is
obtained, the transverse deflection of IPMC beam can be calculated using
Eq.(5). In Section 3, the bending moment due to electric field is
derived.

**3. BENDING MOMENT GENERATED BY APPLIED ELECTRIC POTENTIAL**

**3.1 Charge Redistribution**

When an electric potential is applied to the electrodes of IPMC, cations
will migrate to the cathode while anions remain stationary because they
are covalently fixed to the membrane. Two boundary layers will form due
to the cation movement, characterized as the anion dominant anode
boundary layer (ABL) and the cation dominant cathode boundary layer
(CBL) \[6\]. The two boundary layers will effectively balance the
electric field applied, resulting in the region between the two layers
being shielded. Stresses will develop in the two layers, leading to the
deformation of the IPMC material. It is the ABL and CBL that primarily
contribute to the bending motion of IPMC. Therefore, it is essential to
determine the thickness of the ABL and CBL prior to the calculation of
the bending moment. To achieve this, the cation redistribution under an
electric field should be considered first. The time variation of the
charge distribution is governed by the following equation \[5\],

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$\frac{\partial}{\partial x}\left\{ \frac{\partial(\kappa_{e}E)}{\partial t} - D^{+}\lbrack\frac{\partial^{2}(\kappa_{e}E)}{\partial z^{2}} - \frac{C^{-}F^{2}}{\kappa_{e}\text{RT}}(\kappa_{e}E)\rbrack \right\} = 0$$   \(7\)

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------

and the basic electrostatic equations

  -------------------------------------------------------------- --------
  $E = - \frac{\partial\phi}{\partial z}$, $D = \kappa_{e}E$,    \(8\)
  $\frac{\partial D}{\partial z} = \rho_{c}$,                    
  ![](media/image11.wmf)                                         

  -------------------------------------------------------------- --------

where $E$ is the electric field; $\phi$ is the electric potential;
![](media/image12.wmf) is the electric permittivity; $D^{+}$ is the
cation diffusivity coefficient; $R$ is the gas constant; $T$ is the
temperature; $D$ is the electric displacement; ![](media/image13.wmf) is
the charge density; ![](media/image14.wmf) and ![](media/image15.wmf)
are the valence of cations and anions, respectively; $C^{+}$ and $C^{-}$
are the cation and anion charge density, respectively; and $F$ is the
Faraday's constant.

The boundary and initial conditions are

  -------------------------------------------------------------- --------
  ![](media/image16.wmf), ![](media/image17.wmf),                \(9\)
  ![](media/image18.wmf), $J^{+}( \pm \frac{h}{2},t) = 0$        

  -------------------------------------------------------------- --------

where ![](media/image19.wmf) is cation flux which in a simplified form
is \[5\],

  -------------------------------------------------------------- --------
  ![](media/image20.wmf)                                         \(10\)

  -------------------------------------------------------------- --------

By using the Fourier's series and neglecting the small terms, the
normalized charge density can be obtained as \[6\].

  -------------------------------------------------------------- --------
  $$Q(z,t) = Q_{0}(z)g(t)$$                                      \(11\)

  -------------------------------------------------------------- --------

where $g(t) = e^{j\Omega t}$ and

  ----------------------------------------------------------------------------------------------------------------------------------------- --------
  $$Q_{0}(z) = \left\{ \begin{matrix}                                                                                                       \(12\)
   - 1\  - \frac{h}{2} \leq z \leq \frac{- h}{2} + \ell\ ' \\                                                                               
  \frac{F}{\text{RT}}(A_{1}\text{exp}(\frac{z}{\ell}) + A_{2}\text{exp}( - \frac{z}{\ell})\  - \frac{h}{2} + \ell\ ' < z \leq \frac{h}{2}   
  \end{matrix} \right.\ $$                                                                                                                  

  ----------------------------------------------------------------------------------------------------------------------------------------- --------

where ![](media/image21.wmf) is a natural length scale and
![](media/image22.wmf) denotes the thickness of cation-depleted region
where no cations exist; $\text{Cap}$ is the capacitance of IPMC; and
![](media/image23.wmf) and ![](media/image24.wmf) are the integration
constants.

By using Eq.(12), the thicknesses of ABL and CBL can be calculated as

  -------------------------------------------------------------- --------
  ![](media/image25.wmf)                                         \(13\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image26.wmf)                                         \(14\)

  -------------------------------------------------------------- --------

With Eqs.(13) and (14), the bending moment due to electric potential can
be derived subsequently. In the following section, an explicit bending
moment solution is obtained based on Nemat-Nasser's \[6\] hybrid
actuation model.

**3.2 Bending Moment**

Nemat-Nasser \[6\] studied the micromechanics of the IPMC, considering
that the volumetric strains of clusters and hence the axial strain of
IMPC are affected by the water uptake. The moment rate is expressed as,

  -------------------------------------------------------------- --------
  ![](media/image27.wmf)                                         \(15\)

  -------------------------------------------------------------- --------

where ![](media/image28.wmf) is the effective Young's modulus of
boundary layers in IPMC, i.e., the Young's modulus of polyelectrolyte
membrane; $\varepsilon_{x}$ is the axial strain in *x* direction;
$w(z,t)$ is the function accounting for the water uptake in clusters;
and the dot above all symbols denotes the first derivative with respect
to time.

Completing the integration at the right hand side of Eq.(15) by
considering the thickness of ABL and CBL and neglecting the small terms,
the bending moment rate can be written as

  -------------------------------------------------------------- --------
  ![](media/image29.wmf)                                         \(16\)

  -------------------------------------------------------------- --------

where ![](media/image30.wmf) and ![](media/image31.wmf) are the water
uptake at ABL and CBL, respectively.

Integrating both sides of Eq.(16) over time and considering the initial
condition $M^{e}(0) = 0$ when $w_{A}(z,0) = w_{C}(z,0) = w_{0}$, where
![](media/image32.wmf) is the initial water uptake before the
application of electric potential, the bending moment for the beam with
width ![](media/image33.wmf) can be obtained as

  -------------------------------------------------------------- --------
  ![](media/image34.wmf)                                         \(17\)

  -------------------------------------------------------------- --------

where $\beta = \frac{L_{C}}{L_{A}}$.

Since the bending strain is small, the change of water uptake in
clusters is also small. Thus, the bending moment can be approximated by
expanding the right hand side of Eq.(17) at $w_{0}$ using Taylor's
series with the higher-order terms neglected, as

  -------------------------------------------------------------- --------
  ![](media/image35.wmf)                                         \(18\)

  -------------------------------------------------------------- --------

Once the water uptakes ![](media/image36.wmf) and ![](media/image37.wmf)
are obtained, the bending moment due to electric potential is
determined. The water uptakes at ABL and CBL are governed by the
following equations \[6\],

  ------------------------------------------------------------------- --------
  $$\frac{{\overset{˙}{w}}_{A}(z,t)}{1 + w_{A}(z,t)} = D_{A}t_{A}$$   \(19\)

  ------------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image38.wmf)                                         \(20\)

  -------------------------------------------------------------- --------

where

  ---------------------------------------------------------------------------- --------
  $$t_{A} = \Pi_{A} + p_{\text{AA}} + p_{\text{ADD}} + \sigma_{r}(a_{0},t)$$   \(21\)

  ---------------------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image39.wmf)                                         \(22\)

  -------------------------------------------------------------- --------

  --------------------------------------------------------------------------------- --------
  $\Pi_{A}(t) = \frac{{\varphi Q}_{B}^{-}K_{0}}{w_{A}(t)}\lbrack 2 - g(t)\rbrack$ , \(23\)
  ![](media/image40.wmf), ![](media/image41.wmf)                                    

  --------------------------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image42.wmf), ![](media/image43.wmf),                \(24\)
  ![](media/image44.wmf)                                         

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image45.wmf)                                         \(25\)

  -------------------------------------------------------------- --------

+--------------------------------------------------------------+-------+
| ![](media/image46.wmf), ![](media/image47.wmf),              | \     |
| ![](media/image48.wmf), ![](media/image49.wmf),              | (26\) |
| ![](media/image50.wmf), ![](media/image51.wmf)               |       |
| ![](media/image52.wmf)                                       |       |
+--------------------------------------------------------------+-------+

where ![](media/image53.wmf) and ![](media/image54.wmf) are the constant
coefficients accounting for the diffusion in ABL and CBL, respectively;
![](media/image55.wmf) is the osmotic factor; ![](media/image56.wmf) is
the anion charge density of the dry bare membrane, defined as
![](media/image57.wmf), where ![](media/image58.wmf) is the density of
bare membrane and ![](media/image59.wmf) is the equivalent weight of ion
content; ![](media/image60.wmf); ![](media/image61.wmf) is the initial
cluster size in dry condition; ![](media/image62.wmf) and
![](media/image63.wmf) are the effective electric permittivity of the
cluster in ABL and CBL respectively; ![](media/image64.wmf) and
![](media/image65.wmf) are the effective dipole length at ABL and CBL,
respectively; ![](media/image66.wmf) and ![](media/image67.wmf) are
coefficients; ![](media/image68.wmf) is the water uptake;
![](media/image69.wmf); ![](media/image70.wmf) and
![](media/image71.wmf) is the coordination number;
![](media/image72.wmf) is the initial porosity of dry membrane;
![](media/image73.wmf); ![](media/image74.wmf) is the effective
stiffness of the backbone polymer which is related to
![](media/image75.wmf) by ![](media/image76.wmf). Due to the length
limit of the paper, detailed definitions of these notations, which are
readily available in \[6\], will not be provided here.

It is difficult to solve Eqs. (19) and (20) due to their nonlinearity.
By transferring ![](media/image77.wmf) and ![](media/image78.wmf) to the
right hand sides of Eqs.(19) and (20), respectively, and expanding their
right hand sides at $w_{0}$ using Taylor's series with higher order
terms neglected, two linear partial differential equations are obtained,

  -------------------------------------------------------------- --------
  ![](media/image79.wmf)                                         \(27\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image80.wmf)                                         \(28\)

  -------------------------------------------------------------- --------

where![](media/image81.wmf); ![](media/image82.wmf);
![](media/image83.wmf); and ![](media/image84.wmf).

The solutions of ![](media/image85.wmf) and ![](media/image86.wmf) can
be obtained from Eqs.(27) and (28) as

  -------------------------------------------------------------- --------
  ![](media/image87.wmf)                                         \(29\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image88.wmf)                                         \(30\)

  -------------------------------------------------------------- --------

where $C_{1}$ and $C_{2}$ are the integration constants. After
simplification, the solutions of water uptakes are

  -------------------------------------------------------------- --------
  ![](media/image89.wmf)                                         \(31\)

  -------------------------------------------------------------- --------

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------
  $$w_{C}(z,t) = \lbrack C_{2}\hslash_{C}(t) - \frac{f_{C}}{f_{C1}} - \frac{f_{C}f_{C2}}{f_{C1}}\hslash_{C}(t)\int_{}^{}{\frac{1}{\hslash_{C}(t)}\text{dt}\rbrack} + w_{0}$$   \(32\)

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------

where

  -------------------------------------------------------------- --------
  $f_{A} =$![](media/image90.wmf)                                \(33\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image91.wmf)                                         \(34\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image92.wmf)                                         \(35\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image93.wmf)                                         \(36\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image94.wmf)                                         \(37\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image95.wmf)                                         \(38\)

  -------------------------------------------------------------- --------

  --------------------------------------------------------------------------------------------------------------- --------
  $$\hslash_{( \cdot )}(t) = \text{exp}\left\{ \int_{}^{}{\lbrack f_{1}g(t) + f_{2}\rbrack\text{dt}} \right\}$$   \(39\)

  --------------------------------------------------------------------------------------------------------------- --------

where the dot in the subscript stands for *A* or *C*.

As $f_{2}$ is negative, with the increase of time,
$\hslash_{( \cdot )}(t)$ will diminish. For the steady state, i.e., *t*
is sufficiently large, the following relation stands,

  -------------------------------------------------------------- --------
  ![](media/image96.wmf)                                         \(40\)

  -------------------------------------------------------------- --------

Using the approximation of Eq.(40), Eqs.(31) and (32) can be further
simplified as

  -------------------------------------------------------------- --------
  ![](media/image97.wmf)                                         \(41\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image98.wmf)                                         \(42\)

  -------------------------------------------------------------- --------

Considering the initial conditions,

  -------------------------------------------------------------- --------
  $w_{A}(z,0) = w_{0}$, $w_{C}(z,0) = w_{0}$, $g(0) = 0$         \(43\)

  -------------------------------------------------------------- --------

the coefficients ![](media/image99.wmf) and ![](media/image100.wmf) can
be obtained from Eqs.(31) and (32) as

  -------------------------------------------------------------- --------
  ![](media/image101.wmf)                                        \(44\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image102.wmf)                                        \(45\)

  -------------------------------------------------------------- --------

As $\hslash_{( \cdot )}(t)$ will diminish with time, for the steady
state vibration, ![](media/image103.wmf) and ![](media/image104.wmf)
will eventually vanish. Finally, the solutions of water uptake functions
are

  -------------------------------------------------------------- --------
  ![](media/image105.wmf)                                        \(46\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image106.wmf)                                        \(47\)

  -------------------------------------------------------------- --------

Since the time variation of ![](media/image107.wmf) and
![](media/image108.wmf)should be ![](media/image109.wmf), Eqs.(46) and
(47) can be approximated as

  -------------------------------------------------------------- --------
  ![](media/image110.wmf)                                        \(48\)

  -------------------------------------------------------------- --------

  -------------------------------------------------------------- --------
  ![](media/image111.wmf)                                        \(49\)

  -------------------------------------------------------------- --------

Using the solutions of water uptake in Eqs.(48) and (49), the bending
moment generated by the IPMC can be obtained from Eq.(18) as

  -------------------------------------------------------------- --------
  ![](media/image112.wmf)                                        \(50\)

  -------------------------------------------------------------- --------

It is worth noting that Eq. (50) is an explicit expression of bending
moment including the effects of osmotic, electrostatic and elastic
stresses of IPMC. This equation provides a convenient way to estimate
the bending moment of IPMC at a given hydration rate. The frequency of
the applied electric potential $\Omega$ appears in the expression of
bending moment. It is evident that with the increase of driving
frequency $\Omega$, the generative bending moment and therefore the
vibration amplitude of IPMC decreases. This conclusion agrees with the
experimental observations in \[1\]. In addition, a phase lag between the
bending moment and the input signal is observed,
i.e.,![](media/image113.wmf). It is noted that this phase lag is
obtained without considering the viscous properties of solvent. For
different solvent, the phase lag should vary. To include the effect of
viscous property of solvent in the IPMC, Eqs.(12), (19) and (20) need
modifications. The effect of viscosity of solvent can also be reflected
in the overall damping coefficient ![](media/image114.wmf) in Eq.(5),
which can be evaluated through experiments.

Since the thicknesses of the ABL and the CBL are derived from the
condition of static electric potential and multiplied by the time
variation function, they may be inaccurate for the high frequency
electric field, especially for the CBL. This may result in an
overestimation of bending moment from Eq.(50) since the time required
for cation redistribution is neglected. However, as the time required
for cation redistribution is very short, for low driving frequency,
cations should have ample time to achieve a quasi-equilibrium state.
Thus, Eq. (50) is more accurate for low frequency electric fields.

When ![](media/image115.wmf) is very small, e.g., the DC signal is
applied, Eq.(50) is not applicable anymore. However, the bending moment
for this case can be easily obtained by setting the time variation
function ![](media/image116.wmf) in Eqs.(27) and (28) to unity. The
bending moment under DC signal is

  -------------------------------------------------------------- --------
  ![](media/image117.wmf)                                        \(51\)

  -------------------------------------------------------------- --------

Substituting Eq.(50) into Eq.(5), the transverse vibration of the IPMC
beam can be calculated. For the static signal input, the bending moment
is calculated from Eq.(51).

**4. ILLUSTRATIVE EXAPMLES AND DISCUSSIONS**

**4.1 Illustrative Examples**

Before calculating the vibration of IPMC beam, the bending moment
solution obtained in this paper will be validated by comparing the tip
displacement of a cantilever beam with the results obtained by
Nemat-Nasser \[6\]. Consider a Nafion based IPMC beam in Li^+^ form with
length 18mm, width 2 mm and thickness 224μm. The plating metal is
platinum and the thickness of electrode is 6μm for both top and bottom
surfaces of the IPMC beam. Assume that the initial water uptake of IPMC
is $w_{0} = 0\text{.}\text{533}$ and the electric potential is a 1-volt
DC signal. For the simplicity, the viscous damping of the IPMC is set as
zero. Other material properties used in calculation are listed in
Table 1. All the above parameters are the same as in \[6\].

Since the signal is DC, the bending moment can be obtained from Eq.(51).
The normalized tip displacement of the cantilever beam can be expressed
as

  -------------------------------------------------------------- --------
  ![](media/image118.wmf)                                        \(52\)

  -------------------------------------------------------------- --------

Fig.2 shows the comparison of normalized tip displacement from Eqs.
(51)-(52) and the results in \[6\]. It can be found that the present
model is in good accord with the data in \[6\] at the initial
deformation stage. Since Eq.(51) does not take into account the
relaxation of IMPC beam, the later deformation stage does not match
well. But this does not affect the accuracy of Eq.(50) since it accounts
for the continuous vibration of IPMC beam where the relaxation effect is
negligible.

The blocked force for the beam can also be estimated by considering a
cantilever IPMC beam subjected to a point force at the tip. The blocked
force can balance the bending moment caused by the electric field such
that the tip displacement is zero, which can be determined by using the
Carstigliano's theorem as ![](media/image119.wmf).

For the 1-volt DC potential, the maximum blocked force is equivalent to
0.033 gram, which is about 1.211 times the self-weight of IMPC beam.
When the electric potential increases to 5 volts, the ratio of the
blocked force to the beam weight is 4.37. This ratio is comparable with
the experimental result in \[4\] for the thin gold electrode IMPC. In
\[4\], a 2mm thick IPMC strip of weight 0.24 gram developed 1.2 gram
force under 5 volts potential, where the force-weight ratio is 5.0. This
comparison indicates that the obtained results are reasonable.

Subsequently, the DC potential is replaced by a 1-volt sinusoidal
potential with frequency 0.25 Hz. The stiffness of tissue is set as 15
kPa. By using Eqs.(5) and (50), the beam deflection curve at any time
can be obtained. Figs. 3 and 4 respectively illustrate the IPMC beam
deflection and the pressure distribution on human tissue when the
electric potential is at its maximum. It can be observed that the
maximum deflection of IPMC beam is in the order of 0.01 mm and the
maximum pressure is around 120 Pa. The total pressure can be evaluated
by Eq.(6), which is 0.2298 gram for the pressure distribution in Fig.4.
This pressure is equivalent to 0.4693 mmHg on the human tissue beneath
the beam.

**4.2 Optimal Electrode Length**

It is evident from Eq.(5) that for different values of
![](media/image120.wmf) and ![](media/image121.wmf), which determine the
position and the length of the electrode, the maximum transverse
displacement and therefore the generative pressure is different. For
biomedical applications of IPMCs, maximum deflection and maximum
generative pressure may be required for different purposes. For
examples, maximum pressure is needed for heart compression bands and
maximum deflection is sometime necessary for surgical tools. To generate
the maximum deflection or pressure, optimal values of coordinates of
![](media/image122.wmf) and ![](media/image123.wmf) can be calculated
using the developed model. For the ease of computation, we assume the
stiffness of beam does not change due to the changes of electrode
location and length. Yang and Zhang \[15\] have studied the optimal
placement of a PZT actuator on a rectangular plate in terms of
maximizing vibration amplitude. They used the natural frequency as a
measure to determine the optimal location of PZT actuator. As IPMC is
generally actuated at low frequency, far below the first natural
frequency of the beam, the predominant vibration mode is the first mode.
Therefore, to maximize the first vibration mode, the values of
![](media/image124.wmf) and ![](media/image125.wmf) should maximize the
absolute value of ![](media/image126.wmf), where
![](media/image127.wmf), ![](media/image128.wmf). It is obvious that the
solutions are ![](media/image129.wmf) and ![](media/image130.wmf) or
vice versa.

However, when the elastic foundation beneath the IMPC bema is stiff, all
vibration modes will have similar and small contributions to the overall
vibration. In this condition, the first mode may not appear to be
dominant, such as the deflection curve in Fig. 3. Fig.5 shows the
influence of the electrode coordinates ![](media/image131.wmf) and
![](media/image132.wmf) on the maximum beam deflection for the case of
Fig.3. The maximum beam deflection of any
(![](media/image133.wmf),![](media/image134.wmf)) combination is denoted
by the color of the point which can be read from the color bar. It is
observed that the optimal location of the electrode is at either end of
the beam with the electrode length of ![](media/image135.wmf) to
![](media/image136.wmf), i.e., ![](media/image137.wmf)mm to
![](media/image138.wmf)mm. For higher foundation stiffness, say 40 kPa,
the value of optimal length reduces. Fig.6 shows the relationship
between the electrode coordinates and the maximum beam deflection for
this case. The optimal electrode position is also close to either end of
beam with the electrode length of ![](media/image139.wmf) to
![](media/image140.wmf). Fig. 7 shows the optimal electrode coordinates
for the IPMC beam with foundation stiffness 0.9 kPa. It is obvious that
the optimal length of electrode is equal to the length of beam.

Therefore it can be concluded that for soft foundations, the electrode
should cover the entire beam surface, while for stiff foundations, the
electrode should be located near either end of the beam and the length
of the electrode does not necessarily cover the entire beam.

Another issue is the optimal location and length of the electrode in
terms of maximizing total pressure. Figs. 8 to 10 illustrate the
relationship between the electrode coordinates and the generative
pressure for foundation stiffness of 0.9 kPa, 15 kPa and 40 kPa,
respectively. All the results indicate that the optimal electrode length
is equal to the length of beam. Thus, IPMC beam with fully covered
electrode will generate the largest force compared to other electrode
configuration for the simply supported condition.

**4.3 Multiple Electrodes**

If the IPMC beam is discretely plated with multiple electrodes and the
applied electric potentials can be different for each electrode, the
IPMC beam will behave more flexibly and possibly render higher pressure
or displacement on human tissue. For this case, Eq.(5) should be
modified as

  -------------------------------------------------------------- --------
  ![](media/image141.wmf)                                        \(53\)

  -------------------------------------------------------------- --------

where ![](media/image142.wmf) is the total number of discrete electrodes
and $f_{n}(t)$ is the time variation function of the electric potential
applied to the *n*th electrode. From Eq.(53), it can be deduced that the
more electrodes, the larger the transverse displacement could be.
However, more electrodes imply complexity and difficulty in operation
and control system.

A two-electrode IPMC beam is studied as illustrated in Fig.11. The two
electrodes with the same length of 0.35*L* are located at the two ends
of the beam. To avoid computational difficulty due to the change of beam
stiffness, the region besides the electrodes is also considered to be
plated with platinum but is isolated from electrodes by a negligible
small interval. This can be achieved by chemical etching. The foundation
stiffness is set as 15 kPa. Figs. 12 and 13 illustrate the deflections
of beam under identical and opposite electric potentials, respectively.
It is found that the opposite electric potential can produce larger
transverse displacement. However, calculation shows that the generative
pressure for Figs. 12 and 13 are 0.2935 gram and 0.1895 gram,
respectively. This means that when the two electrodes are controlled by
the identical potential, greater pressure will be generated on human
tissue.

**5. CONCLUSIONS**

In this paper, a dynamic IMPC beam model on human tissue was developed.
Analytical solution was obtained to describe the vibration response of
the IPMC beam to a command of electric field. Explicit bending moment
expressions were derived for both dynamic and static electric
potentials, which provide an easy way to estimate the bending capacity
of IPMC strips. Based on the bending moment expressions, the beam
deflection curve and the pressure distribution generated on human tissue
were obtained. The optimal electrode location and length in terms of
maximizing deflection and generative pressure were discussed. It is
found that to achieve the maximum deflection, the electrode should be
located at either end of the beam, and to achieve higher total pressure,
full length electrode should be used. The deflection curve and
generative pressure for multiple discrete electrodes IPMC beam were also
obtained. The developed model is useful not only for IPMC related
biomedical instruments interacting with human tissues but also for any
other devices that utilize IPMC materials.

**REFERENCES**

  --- -------------------------------------------------------------------
  1   Shahinpoor M and Kim K J (2001), "Ionic polymer-metal composites:
      I. fundamentals", *Smart Materials and Structures*, 10(4), 819-833.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  2   Kim K J and Shahinpoor M (2003), "Ionic polymer-metal composites:
      II. manufacturing techniques", *Smart Materials and Structures*,
      12(1), 65-79.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  3   Shahinpoor M and Kim K J (2004), "Ionic polymer-metal composites:
      III. Modeling and simulation as biomimetic sensors, actuators,
      trnasducers, and artificial muscles", *Smart Materials and
      Structures*, 13(6), 1362-1388.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  4   Shahinpoor M and Kim K J (2005), "Ionic polymer-metal composites:
      IV. Industrial and medical applications", *Smart Materials and
      Structures*, 14(1), 197-214.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  5   Nemat-Nasser S and Li J Y (2000), "Electromechanical response of
      ionic polymer-metal composites" *Journal of Applied Physics*,
      87(7), 3321-3331.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  6   Nemat-Nasser S (2002), "Micromechanics of actuation of ionic
      polymer-metal composites" *Journal of Applied Physics*, 92(5),
      2899-2915.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  7   Shahinpoor M and Kim K J (2002), "Mass transfer induced hydraulic
      actuation in ionic polymer-metal composites", *Journal of
      Intelligent Material Systems and Structures*, 13(6), 369-376.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  8   Newbury K M and Leo J D (2003), "Linear electromechanical model of
      ionic polymer transducers-part I: model development", *Journal of
      Intelligent Material Systems and Structures*, 14(6), 333-342.

  --- -------------------------------------------------------------------

  --- -------------------------------------------------------------------
  9   Nemat-Nasser S and Wu Y (2003), "Comparative experimental study of
      the ionic polymer-metal composites with different backbone ionomers
      and in various cation forms, *Journal of Applied Physics*, 93(9),
      5255-5267.

  --- -------------------------------------------------------------------

  ---- -------------------------------------------------------------------
  10   Weiland L M and Leo D J (2005a) "Computational analysis of ionic
       polymer cluster energetics*", Journal of Applied physics*, 97,
       013541.

  ---- -------------------------------------------------------------------

  ---- -------------------------------------------------------------------
  11   Weiland L M and Leo D J (2005b) "Ionic polymer cluster energetics:
       Computational analysis of pedant chain stiffness and charge
       imballance*", Journal of Applied physics*, 97, 123530.

  ---- -------------------------------------------------------------------

  ---- -------------------------------------------------------------------
  12   Paquette J W and Kim K J (2004), "Ionomeric electroactive polymer
       artificial muscle for naval applications*", IEEE Journal of Ocean
       Engineering*, 29(3), 729-737.

  ---- -------------------------------------------------------------------

  ---- -------------------------------------------------------------------
  13   Bar-Cohen Y (Editor) (2004), *Electroactive polymer (EAP) Actuators
       as Artificial Muscles, Reality, Potential, and Challenges,* 2^nd^
       Edition, SPIE Press, Washington.

  ---- -------------------------------------------------------------------

  ---- -------------------------------------------------------------------
  14   Soedel W (2004), *Vibrations of shells and plates*, 3^rd^ Edition,
       Marcel Dekker, New York.

  ---- -------------------------------------------------------------------

  ---- -------------------------------------------------------------------
  15   Yang Y W and Zhang L (2006), "Optimal excitation of a rectangular
       plate resting on an elastic foundation by a piezoelectric
       actuator", *Smart Materials and Structures,* 15(4), 1063-1078.

  ---- -------------------------------------------------------------------

**LIST OF TABLES AND FIGURES**

**Table 1 Parameters, symbols and values**

**Figure 1. Configuration of an IPMC beam on human tissues**

**Figure 2. Comparison between calculated displacement and data in
\[6\]**

**Figure 3 Deflection of IPMC beam under 1 volt sinusoidal potential
(*k*=15 kPa)**

**Figure 4 Pressure distribution on human tissue under 1 volt sinusoidal
potential (*k*=15kPa)**

**Figure 5 Relationship between electrode coordinates and maximum beam
deflection (*k*=15 kPa)**

**Figure 6 Relationship between electrode coordinates and maximum beam
deflection (*k*=40 kPa)**

**Figure 7 Relationship between electrode coordinates and maximum beam
deflection (*k*=0.9 kPa)**

**Figure 8 Relationship between electrode coordinates and pressure
(*k*=0.9 kPa)**

**Figure 9 Relationship between electrode coordinates and pressure
(*k*=15 kPa)**

**Figure 10 Relationship between electrode coordinates and pressure
(*k*=40 kPa)**

**Figure 11 Configuration of two discrete electrodes**

**Figure 12 Deflection of beam with two electrodes under identical
potentials**

**Figure 13 Deflection of beam with two electrodes under opposite
potentials**

**Table 1 Parameters, symbols and values**

  --------------- ------------------------- -------------------------------- ------------------------- ------------------------- --------------------------------------
  **Parameter**   **Symbol**                **value**                        **Parameter**             **Symbol**                **value**

  IPMC Density    $$\rho$$                  3385 kg/m^3^                     Temperature               $$T$$                     300 K

  Gas constant    $$R$$                     8.3143 J/mol/K                   Initial porosity          $$n_{0}$$                 0.01

  Faraday's       $$F$$                     96487 C/mol                      Inner radius of cluster   $$a$$                     $1\text{.}6 \times \text{10}^{- 9}$m
  constant                                                                   at                                                  
                                                                             ![](media/image143.wmf)                             

  Bare membrane   $$\rho_{B}$$              ![](media/image144.wmf)kg/m^3^   Capacitance               $$\text{Cap}$$            12 F/m
  density                                                                                                                        

  Platinum        $$\rho_{p}$$              ![](media/image145.wmf) kg/m^3^  Effective permittivity at $$\kappa_{A}$$            ![](media/image146.wmf) F/m
  density                                                                    ABL                                                 

  Effective       ![](media/image147.wmf)   ![](media/image148.wmf) F/m      Osmotic coefficient       $$\varphi$$               1
  permittivity at                                                                                                                
  CBL                                                                                                                            

  Coordination    ![](media/image149.wmf)   6                                Equivalent weight of bare ![](media/image150.wmf)   1100 g/mol
  Number of Li^+^                                                            Nafion in proton form                               

  Formula weight  ![](media/image151.wmf)   6.94 g/mol                       Effective dipole length   $$a_{1}$$                 ![](media/image152.wmf)
  of Li^+^                                                                   coefficient $a_{1}$                                 

  Effective       $$a_{2}$$                 ![](media/image153.wmf)          Concentration factor      ![](media/image154.wmf)   0.5
  dipole length                                                                                                                  
  coefficient                                                                                                                    
  $a_{2}$                                                                                                                        

  Coefficient     $$D_{A}$$                 ![](media/image155.wmf)          Dynamic solvation number  ![](media/image156.wmf)   0
  $D_{A}$                                                                                                                        

  Young's modulus ![](media/image157.wmf)   ![](media/image158.wmf) N/m^2^   Mass scaling factor       ![](media/image159.wmf)   0.623
  of platinum                                                                                                                    
  --------------- ------------------------- -------------------------------- ------------------------- ------------------------- --------------------------------------

![](media/image160.wmf)

**Figure 1 Configuration of an IPMC beam on human tissues**

![](media/image161.jpeg){width="5.833333333333333in" height="4.375in"}

**Figure 2 Comparison between calculated displacement and data in
\[6\]**

![](media/image162.jpeg){width="5.25in" height="3.9375in"}

**Figure 3 Deflection of IPMC beam under 1 volt sinusoidal potential
(*k*=15 kPa)**

![](media/image163.jpeg){width="5.25in" height="3.9375in"}

**Figure 4 Pressure distribution on human tissue under 1 volt sinusoidal
potential (*k*=15kPa)**

![](media/image164.jpeg){width="5.25in" height="3.9375in"}

**Figure 5 Relationship between electrode coordinates and maximum beam
deflection (*k*=15 kPa)**

![](media/image165.jpeg){width="5.25in" height="3.9375in"}

**Figure 6 Relationship between electrode coordinates and maximum beam
deflection (*k*=40 kPa)**

![](media/image166.jpeg){width="5.25in" height="3.9375in"}

**Figure 7 Relationship between electrode coordinates and maximum beam
deflection (*k*=0.9 kPa)**

![](media/image167.jpeg){width="5.25in" height="3.9375in"}

**Figure 8 Relationship between electrode coordinates and pressure
(*k*=0.9 kPa)**

![](media/image168.jpeg){width="5.25in" height="3.9375in"}

**Figure 9 Relationship between electrode coordinates and pressure
(*k*=15 kPa)**

![](media/image169.jpeg){width="5.25in" height="3.9375in"}

**Figure 10 Relationship between electrode coordinates and pressure
(*k*=40 kPa)**

![](media/image170.wmf)

**Figure 11 Configuration of two discrete electrodes**

![](media/image171.jpeg){width="5.25in" height="3.9375in"}

**Figure 12 Deflection of beam with two electrodes under identical
potentials**

![](media/image172.jpeg){width="5.25in" height="3.9375in"}

**Figure 13 Deflection of beam with two electrodes under opposite
potentials**