[Fluid Interaction of Segmented Ionic Polymer-Metal Composites under
Water]{.smallcaps}

Deniz Dogruer^1^, Joonsoo Lee^2^, Woosoon Yim^2^, Kwang J. Kim^1,^[^1],
and Doyeon Kim^1^

^1^Active Materials and Processing Laboratory, Dept. of Mech. Eng., MS
312, University of Nevada, Reno, Reno, NV USA 89557

^2^Dept. of Mech. Eng., MS 4027, University of Nevada, Las Vegas, Las
Vegas, NV USA 89154

\*Corresponding author, <kwangkim@unr.edu>

Submitted for Review in ***Smart Materials and Structures Journal***

ABSTRACT

The numerous possible applications of the Ionic Polymer-Metal Composite
(IPMC) as an underwater propulsor have lead to the investigation of the
IPMC behavior in an aqueous environment. This study compares the
performance of the IPMC when subjected to fluid drag forces to its
performance without such forces. Both the form (i.e. pressure) drag and
the viscous (i.e. skin friction) drag forces experienced by the IPMC due
to the surrounding liquid are modeled. These forces are incorporated
into a 2D analytical model of a segmented IPMC. The model is based on
small deflection and can be conveniently used. The maximum IPMC
deflection for aqueous and non-aqueous environments is compared, both
analytically and experimentally. Using video-capturing techniques, the
deflection of the IPMC, both in air and in water, are investigated. The
experimental results are used in order to better understand the
performance of the IPMC in water. A large deflection model for the
segmented IPMC is also proposed.

1\. Introduction

The large bending motion of the Ionic Polymer-Metal Composite (IPMC)
actuator under low applied voltages and the possibility of operation in
an aqueous environment make it a viable underwater propulsor
\[1,2,3,4\]. It should be noted that the propulsive efficiency of a two
dimensional flapping foil with chord-wise flexibility has been
experimentally shown to increase significantly, up to 36%, when compared
to a rigid flapping foil \[5\]. The possibility of attaching multiple
IPMCs, which can be controlled separately, creates a segmented IMPC that
is flexible and allows the IPMC to move in an undulatory motion, similar
to the swimming motion of many marine creatures. This flexibility can
increase the efficiency of thrust, as well as significantly can improve
maneuverability. These qualities combined with the fact that the IPMC
does not have any gears or motors that create vibration and noise \[1\]
make the segmented IPMC extremely appealing, especially for underwater
applications including many naval applications.

The possibility of using linked models to increase flexibility has been
previously investigated. Recently, Tripathi has studied the hydrodynamic
forces on a model of a swimming salamander \[6\]. The model salamander
is made of rigid links, and the form drag and viscous drag forces are
modeled on each link. The affect of added appendages, such as a legged
salamander, is also examined. Nakabo and his co-workers have studied the
use of IPMCs to make small swimming robots \[3\]. Multiple IPMCs were
linked together to allow for an undulatory, i.e. snake-like, motion when
different voltages were applied. The propulsion and turning of this
small swimming robot was modeled. Since this undulatory motion requires
less area compared to a simple bending motion to obtain similar
propulsion, such designs can be easily miniaturized which opens up the
door for many more applications \[3\]. They further elaborated the model
to adequately address the snake-like underwater robot using an
Eigenfunction expansion technique \[9\].

The IPMC is subject to both *form drag* and *viscous drag* forces when
placed in an aqueous environment. In this work, these forces are modeled
and incorporated into a previously developed analytical model of a
segmented IPMC \[4\] which is based upon a small deflection. The model
also has a capability of addressing the relaxation behavior of the IPMC.
It was determined that the flow experienced by the segmented IPMC in
this simulation lies in the *Allen* flow region, i.e. 2 \< Re \< 500.
Note that at high Reynolds numbers (Re \> 10^4^), the drag coefficient
reaches an asymptotic value and becomes constant. For the case of this
work, the Reynolds number is much lower. For Allen flows, the drag
coefficient is not constant but rather a function of the Reynolds number
\[7\]. Yano *et al.* have done experimental work to determine the
relationship between drag coefficient and Reynolds number \[8\].
Circular cylinders, rectangular prisms and airfoils were pulled at a
constant velocity through a viscous medium. The dependence of drag
coefficient on Reynolds number was obtained from the experimental
results reported by Yano *et al*. \[8\].

2\. MODELLING

## 2.1 Hydrodynamic Model

The drag forces exerted on the IPMC by the water can be classified in
two categories: form drag forces and viscous drag forces. Form drag is
created by the disruption of the fluid particles' movement as they flow
around the IPMC. Viscous drag is imposed on the IPMC as it moves though
a viscous medium due to the no-slip condition. The IPMC is assumed to
exhibit planar motion, i.e. the center of mass does not move in the
direction normal to the plane of swimming motion, therefore the
hydrodynamic model developed is considered to be a 2-dimensional
approach.

## *► Form Drag Forces*

Each segment of the IPMC is divided into multiple small 'finite
elements'. It is assumed that the tangential and normal components of
the form drag forces can be calculated separately for each 'element' of
the segmented IPMC. In the calculation, we used 50 elements for each
segment of the IPMC. The force components, *F*~i,t~ and *F*~i,n~, can be
calculated according to the formula \[6\] as:

$F_{i,t} = \lambda_{i,t}v_{i,t}^{2}$ (1)

$F_{i,n} = \lambda_{i,n}v_{i,n}^{2}$ (2)

where

$\lambda_{i,t} = \frac{1}{2}C_{i,t}S_{i}\rho$ (3)

$\lambda_{i,n} = \frac{1}{2}C_{i,n}S_{i}\rho$ (4)

Note that $v_{i,t}$and$v_{i,n}$are the relative velocity components
(i.e. tangential and normal) of the 'element' *i*, $C_{i,t}$ and
$C_{i,n}$ are the drag coefficients that are dependent on the shape of
the 'element' and the Reynolds number, $S_{i}$ is the frontal area of
the 'element', and $\rho$ is the density of the fluid (in this case,
water).

Figure 1 illustrates the segmented IPMC with 3 segments. The first
segment will experience the maximum tangential drag coefficient, while
each progressive segment's tangential drag coefficient will decrease.
This is similar to the drag reduction technique that natural creatures
often adopt. For example, birds utilize this technique when flying in
V-formation. The front bird encounters the most drag and acts as a
shield for the birds flying behind it. The rate at which the drag
coefficient decreases is dependent on the angle between segments and the
length of each segment. However, the normal drag coefficient of each
segment remains constant. The segmented IPMC shown in Figure 1 is first
fabricated by a metal precipitation technique \[10,11,12,13,14,15\] and
later a laser beam was used to cut out the metallic portion out of the
base polymeric material. It leads to the IPMC body that is mechanically
connected but electrically disconnected. The electrode was made with
gold.

![](media/image1.wmf){width="0.5980314960629921in"
height="0.16377952755905512in"}![](media/image2.wmf){width="0.5980314960629921in"
height="0.16377952755905512in"}![](media/image3.wmf){width="0.5972440944881889in"
height="0.16377952755905512in"}![](media/image5.png){width="2.44375in"
height="1.1520833333333333in"}

**Figure 1: Illustration of a segmented IPMC (left) and a photograph of
a fabricated IPMC (right).**

A laser cutting technique was used to fabricate the segmented IPMC (3
segments).

Using the finite element method, the displacement at any point along
element *i* shown in Figure 2, can be determined from the nodal
displacement and slope of the nodes *i* and *i+1*, and the shape
functions, *N*.

![](media/image6.wmf)

Figure 2: Segment of an IPMC bending in the x-z plane.

It should be noted that that the current model is based upon small
deflection; therefore the deflection in the *x*-direction is neglected.
The displacement in the z-direction at any point along the link can be
calculated from the equation below.

![](media/image7.wmf) (5)

where *N* is a row vector consisting of ![](media/image8.wmf),
![](media/image9.wmf) is the nodal coordinate for element *i* where
*w~i~* and *φ~i~* denote the nodal displacement and slope of node *i*
and *N* is a ![](media/image10.wmf) row vector with beam shape functions
![](media/image11.wmf) \[17\] that expresses the shape of the assumed
displacement function over the domain of element.

Differentiating Equation (7) with respect to time, the velocity of any
point on the segment *i* along the *z*-axis is determined as

![](media/image12.wmf) (6)

where {**^.^**} represents the time derivative. The angle of deflection,
or slope, of the link can be found from the partial derivative of the
displacement with respect to *x~i~*. Thus, the slope angle
![](media/image13.wmf) becomes,

![](media/image14.wmf) (7)

![](media/image15.wmf) (8)

The tangential and normal velocity components along the *z*-axis of the
IPMC can be found from the angle of deflection as shown in Figure 3.

From Figure 3, the velocity components (i.e., tangential and normal) are
found to be:

![](media/image16.wmf) (9)

![](media/image17.wmf) (10)

In order to simulate the segmented IPMC swimming in the *x*-direction
through the water, the segmented IPMC is modeled to be in a freestream
velocity, *U*. Then, the relative velocities of the IPMC become:

$v_{i,t} = U\text{cos}\theta - {\overset{˙}{w}}_{t}$ (11)

> $v_{i,n} = U\text{sin}\theta - {\overset{˙}{w}}_{n}$ (12)

As stated earlier, the drag coefficient is a function of Reynolds
number, Re. The relationship between the drag coefficient and Reynolds
number was determined from the experimental results of Yano *et. al*
\[8\] that describes the *Allen flow* region. In their work, for a
rectangular prism, with the dimensions of the IPMC link, the drag
coefficient values obtained at various Reynolds values were graphed. The
equation of the best-fit line was used to determine how drag coefficient
changes with Reynolds number. The following equations show the
tangential and normal drag coefficients for the range of calculation
\[8\].

$C = \text{11}\text{.}\text{312}\text{Re}^{- 0\text{.}\text{2591}}$ (13)

$C = \text{10}\text{.}\text{347}\text{Re}^{- 0\text{.}\text{3239}}$ (14)

where Reynolds number, Re, is defined as $\rho\frac{\text{Vd}_{e}}{\mu}$
and *V* is either the tangential or normal relative velocity component
for the respective drag coefficient calculations, *μ* is the dynamic
viscosity, and $d_{e}$ is defined as the equivalent diameter,
$\sqrt{4\frac{A_{c}}{\pi}}$, where *A*~c~ is the cross sectional area.

## *►Viscous Force*

Assuming that the flow over each individual segment of the IPMC is
laminar, the shear force on each surface of an object moving through a
fluid can be found from \[6\]:

$F_{i,v} = \frac{0\text{.}\text{664}{\rho v}_{{i,t}^{\frac{3}{2}}}A}{\sqrt{\frac{\rho\Delta x}{\mu}}}$
(15)

where *A* is the surface on which the shear force is acting and Δ*x* is
the element length. Equation (15) is derived from the boundary layer
flow over a flat plate.

## 2.2 Equations of Motion with Hydrodynamic Force

Based on the small deflection assumption, the equation of motion without
hydrodynamic force terms for the segmented IPMC actuator with *n*
segment is obtained as \[1\]

![](media/image18.wmf) (16)

where ![](media/image19.wmf) is the generalized coordinates,
![](media/image20.wmf) and ![](media/image21.wmf) are mass and stiffness
matrix respectively and![](media/image22.wmf) is an input moment vector
and ![](media/image23.wmf) is an input control matrix for moment vector
*m* where ![](media/image24.wmf) is a vector associated with the control
input moment *m~i~(t)* applied on the element *i*. It should be noted
that the matrix ![](media/image25.wmf) transforms a distributed moment
*m~i~* to two concentrated moments at the two nodes *i* and *i+1*.

The total hydrodynamic force,![](media/image26.wmf), exerted on the
segment *i* can be included to the equations of motion of the IPMC
actuator, Equation (16), by determining the virtual work
(![](media/image27.wmf)) done by *F~i~* by expressing the virtual
displacement ![](media/image28.wmf) in terms of ![](media/image29.wmf).
The generalized force term *Q~i~* for the *F~i~* can be determined as

![](media/image30.wmf) (17)

where ![](media/image31.wmf) is the generalized force associated with
the hydrodynamic force *F~i~* for the element located in the segment
*i*. Thus, the generalized force associated with the total hydrodynamic
force along the entire *n* segments is obtained by integrating along
each segment coordinates and add contribution from all segments as

![](media/image32.wmf) (18)

where ![](media/image33.wmf)is the generalized force associated with the
total hydrodynamic force exerted to the entire actuator and the total
virtual work becomes

![](media/image34.wmf) (19)

Now, the equations of motion, Equation (16) can be modified with the
hydrodynamic force term as

![](media/image35.wmf) (20)

where ![](media/image36.wmf). It should be noted that the total thrust
and lateral forces can ultimately be obtained by integrating the axial
and transverse directional forces along the actuators.

3\. MOdeling Results, Experimental observation, AND REMARKS

## 3.1 Modeling Results

The dimensions of the IPMC link used in the simulations are shown in
Figure 4. The simulation results listed below are for an applied voltage
\[-2 2 2\]^T^ volts and a freestream velocity, *U*, of 0.04 m/s (in the
negative *x*-direction). Fifty finite elements for each segment were
chosen for the calculation. It should be pointed out that the largest
component of the drag force comes from the normal form drag force, which
is proportional to the square of the relative velocity of the IPMC. It
can be seen that the IPMC exhibits a fast response to the applied
voltage, approximately 1s for this simulation, and begins to slightly
relax after this time. Also note that the way that we handle the
relaxation is to incorporate a lead network of a bending moment applied
to the IPMC segment. There are three parameters that can be
experimentally fit: the gain, zero and pole of the network \[8\]. Figure
5 collectively plots the deflection profile of the calculated results.
It can be seen that the maximum deflection is reached in 1 second, and
the IPMC begins to exhibit the slight relaxation phenomenon after this
time step. It can be seen, from Figure 5 that the magnitude of the
maximum deflection is approximately 36 mm and the IPMC does not change
significantly when it is subject to hydrodynamic forces compared to when
the IPMC is not subject to these forces.

![](media/image37.jpeg){width="5.727777777777778in" height="4.3in"}

Figure 5: IPMC deflection at various time steps V = \[-2 2 2\]^T^ volts

Figure 6 captures the IPMC deflection at t=0.51 second when the largest
velocities occur during this time. Since the drag forces are functions
of the velocity, the affect of the drag forces is the greatest during
this time. Figure 6 compares the deflection of the IPMC due to the
applied voltage only and the total deflection due to applied voltage and
drag forces. As expected, the maximum deflection the IPMC achieves at
this time step is less when the effect of the drag forces of the
surrounding water is introduced.

In this model, the finite elements of each segment are considered to be
small enough to be rigid so the drag coefficient of each element is
calculated as such. But, in the broader picture, the segmented IPMC
cannot be considered to be rigid. Therefore, the drag coefficient of
flexible elements is expected to be much larger compared to rigid
elements due the partial blockage of the flow near the surface of the
element. In order to examine the affect of larger drag coefficients of
flexible elements the drag coefficient is multiplied by a correction
factor, *f*. As a first step, we plot the IPMC deflection of the IPMC at
the time step t = 0.51s for correction factors *f* = 1 and *f* = 2 in
Figure 7. As expected, the larger drag coefficients restrict the
displacement of the IPMC more compared to smaller drag coefficients for
the initial time steps.

+----------------------------------+-----------------------------------+
| ![](media/image38.j              | ![](media/image39.j               |
| peg){width="3.347916666666667in" | peg){width="3.3715277777777777in" |
| height="2.511111111111111in"}    | height="2.5284722222222222in"}    |
|                                  |                                   |
| Figure 6: IPMC deflection,       | Figure 7: IPMC deflection for     |
| t=0.51s.                         | various *f*, t=0.51s.             |
|                                  |                                   |
|                                  | \(a\) without fluid drag force,   |
|                                  | (b) with correction factor,       |
|                                  | *f*=1, (c) with correction        |
|                                  | factor, *f*=2                     |
+----------------------------------+-----------------------------------+

## 3.2 Experimental Observation

In order to check the validity of the current model, we have constructed
an experimental setup to video-capture the deformation of IPMC both in
air and in water. As shown in Figure 8, a video capturing system (B&W,
100 frames/s) is used to detect the shape of deformed IPMC and a
potentiostat/amplifier (Hokuto Denko HA-151) connected to the *d*-Space
1104 system is used to apply voltage to the IPMC.

![](media/image40.wmf)![](media/image41.jpeg){width="2.4694444444444446in"
height="3.29375in"}

**Figure 8: Experimental setup for detecting the IPMC deformation.**

Experimental validation of the segmented IPMC is currently under
investigation, but the non-segmented IPMC can also conveniently be used
to validate the proposed model. Figure 9 presents two experimentally
measured deflection-plots for the sample IPMC under a 1.5V in air and in
water, respectively. We are finding that in air operation the deflection
of the IPMC is larger than in water operation. In this particular
experiment (V=1.5 volts), the tip displacement of the IPMC is cut nearly
40% when it is operated underwater compared in air operation. Also, it
can be noticed that the IPMC's displacement continuously increases for
the experimental time of 40s for in air operation, but this is not the
case for the in water operation. The IPMC's displacement increases in
the first 10s of the experiment but then remains constant for the
following 30s of the experiment. Also, the relaxation of the IPMC is not
noticeable in this experiment. The detail information regarding how to
fabricate relaxation-less IPMCs can be found in elsewhere \[16\].

![](media/image42.wmf)![](media/image43.wmf)

**Figure 9: Experimental results for the IPMC under 1.5V in air (left)
and in water (right).**

Note that the IPMC sample size is 0.280 mm thickness, 1 cm width, and 5
cm length.

## 3.3 Remarks

i\) It should be noted that the current hydrodynamic model incorporates
the fluid drag forces experienced by the IPMC underwater, but does not
include the inertial affects of the surrounding water. As the IPMC
deflects, it displaces a certain volume of 'fluid' (air/water). Due to
the higher density of water compared to air, this volume of 'fluid'
translates into a larger mass the IPMC must oppose in order to deflect
when operated in water compared to in air. Due to this simplification in
the current hydrodynamic model, it is expected that the affects of the
operation underwater will be underestimated when compared to
experimental results. Our approach to solve this issue is to incorporate
an added mass term to the equation of motion.

ii\) In order to more accurately describe the motion of the IPMC, we are
currently developing a large deflection model. Despite the complexity of
the model, the advantage of using a large deflection model is to
adequately handle the non-linearity of the IPMC bending motion. This
model will also be incorporated with fluid interaction terms. Herein, we
would like to briefly discuss a framework of how to handle the
situation. Referring to Figure 10 we can approximate the axial
displacement, ![](media/image44.wmf), of an arbitrary point *P* located
on the segment *i* by considering an infinitesimal deformation *du* in
the axial direction. Using the geometrical relationship between *du*,
*dw*, and *ds*, ![](media/image45.wmf) can be approximated as
![](media/image46.wmf) and the axial deformation ![](media/image47.wmf)
at distance *x~i~* in the element *i* can be determined by
integrating![](media/image48.wmf).

![](media/image49.wmf) (21)

where ![](media/image50.wmf) and ![](media/image51.wmf). Inclusion of
this axial displacement term will affect the kinetic energy of the
system and normal and tangential velocity terms used for fluid
interaction force calculation. However, the final form of the equations
of motion remains the same as Equation (20).

![](media/image52.wmf)

**Figure 10: Axial displacement of an arbitrary point P on segment
*i.***

4\. Conclusions AND FUTURE WORK

Both form drag and viscous drag forces experienced by the IPMC when it
is subjected in an aqueous environment have been modeled. Based upon the
results, it appears that, under low velocity conditions of interest,
these hydrodynamic forces don't significantly affect the performance of
the IPMC. The maximum deflection and response time did not change
considerably when the affect of the fluid drag forces were included. A
model experiment was performed to adequately investigate the fluid
interaction of IPMCs in water. Experimental results showed a decrease in
the overall tip deflection of the IPMC when operated in water compared
to in air. As discussed in the section 3.3i, the inertia affect of the
surrounding water is not included in the current model and is being
studied. An approach for an analytical model of the segmented based on
large deflections has been presented. We believe the large deflection
model will more accurately model the displacement of the IPMC
underwater.

5\. Acknowledgments

The authors acknowledge National Science Foundation who provided partial
financial support for this work. Also, the authors from University of
Nevada, Reno (UNR) thank the partial financial support from Office of
Naval Research and NV Space Grant Consortium.

6\. REFERENCES

\[1\] Yim, W., Kim, K., Paquette, J., Kim, D., "Operation of Ionic
Polymer-Metal Composites in Water," *Proceedings of SPIE: Smart
Structures and Materials*, vol. 5759, pp.22-33, 2005.

\[2\] Paquette, J., Kim, K., "Ionomeric Electroactive Polymer Artificial
Muscle for Naval Applications," *IEEE Journal of Oceanic Engineering*,
vol. 29, no. 3, pp. 729-737, 2004.

\[3\] Nakabo, Y., Mukai, T., Asaka, K., "Bio-mimetic Soft Robots using
IPMC," *a chapter in Electroactive Polymers in Robotic Applications,
Springer*, to appear in 2006.

\[4\] Kim, K. J., Yim, W., Paquette, J. W., and Kim, D., "Ionic
Polymer-Metal Composites for Underwater Operation," *Journal of
Intelligent Materials Systems and Structures (JIMSS),* 2006 (in print).

\[5\] Prempraneerach, P., Hover, F., Triantafyllou, M., "The Effect of
Chordwise Flexibility on the Thrust and Efficiency of a Flapping Foil,"
*CD- Proceedings of the 13th Int. Symp. Unmanned Untethered Submersible
Technology, Durham, New Hampshire*, August 2000.

\[6\] Tripathi, A., "Analysis of Hydrodynamic Forces on the Model of a
Swimming Salamander," EPFL Report/Biologically Inspired Robotics Group,
<http://birg.epfl.ch/webdav/site/birg/shared/atripath/Report.pdf>.
(2003).

\[7\] Douglas, J., Gasiorek, J., Swaffield, J. (2001) *Fluid Mechanics*,
4th ed., Prentice Hall.

\[8\] Yano, H., Hirata, K., Komori, M., "An Approximate Method for the
Drags of Two-Dimensional Obstacles and Low Reynolds Numbers," *Journal
of Applied Mechanics*, vol. 63, pp. 990-995, 1996.

\[9\] Takagi, K., Nakabo, Y., Luo, Z.-W., Mukai, T., "An Analysis of
Increase of Bending Response in IPMC Dynamics Given Uniform Input," Vo.
6168, paper number 6168-14, *Proc. of SPIE* (March, 2006).

\[10\] Shahinpoor, M., Kim, K. J., "Ionic Polymer-Metal Composites -- I.
Fundamentals," (Review Paper) *Smart Materials and Structure*s, vol. 10,
pp. 819-833, 2001.

\[11\] Kim, K. J., Shahinpoor, M., "Ionic Polymer-Metal Composites - II.
Manufacturing Techniques," *Smart Materials and Structures*, vol. 12,
no. 1, pp. 65-79, 2003.

\[12\] Kim, K. J., Tadokoro, S., (ed.), *Electroactive Polymers for
Robotic Application: Artificial Muscles and Sensors*, Springer, London,
2006 (to appear).

\[13\] Shahinpoor, M., Kim, K. J., "A Novel Physically-Loaded and
Interlocked Electrode Developed for Ionic Polymer-Metal Composites
(IPMCs)," *Sensors and Actuator: A. Physical*, vol. 96, no. 2/3, pp.
125-132, 2002.

\[14\] Kim. K. J., Shahinpoor, M., "Development of Three Dimensional
Ionic Polymer-Metal Composites as Artificial Muscles," *Polymer*, vol.
43, no. 3, pp. 797-802, 2002.

\[15\] Kim. D., Kim, K. J., "Experimental Investigation on
Electrochemical Properties of Ionic Polymer-Metal Composite," *Journal
of Intelligent Materials Systems and Structures (JIMSS)*, vol. 17, pp.
449-454, 2006.

\[16\] Kim, D., Kim, K. J., "Electro-chemo-mechanical Interpretation of
Pt and Au-electroded Relaxationless Ionic Polymer-Metal Composites,"
vol. 6168, paper number 6168-11, *Proc. of SPIE* (March 2006).

\[17\] Logan, Daryl L. (1997), "A First Course in the Finite Element
Method using Algor," PES Publishing Co.

[^1]: ^^ e-mail: <kwangkim@unr.edu>; voice: +1 (775) 784-7522; fax: +1
    (775) 784-1701