charge control of ionic polymers - virginia techcarbon nanotubes, and conductive polymers are ionic...

Charge Control of Ionic Polymers

by

Walter J Robinson Jr

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

Mechanical Engineering

Donald J. Leo, ChairDaniel J. Inman

Harry Robertshaw

July 2005Blacksburg, Virginia

Keywords: Ionic polymer, IPMC, charge control

Copyright by Walter J Robinson Jr, 2005

Charge Control of Ionic Polymers

Walter J Robinson Jr

Virginia Polytechnic Institute and State University, 2005

Advisor: Donald J. Leo

Abstract

Ionomeric polymer metal composites can be used as transducers characterized by high strain

and low force. They are created by bonding a thin conductive electrode to the surfaces

of an ionomeric polymer. Much of the work in the past has focused on using a voltage

across the thickness of the polymer to produce mechanical motion. That work has often

demonstrated that the mechanism of transduction within the polymer was associated with

the accumulation of charge in the polymer. This thesis will discuss the use of current as

a means to better control the accumulation of charge. Better control of the charge will

provide more reliable control of the mechanical motion of the polymer.

The data presented in this thesis demonstrates that the response of an ionomeric

polymer to a current input is repeatable. The repeatability is a desirable result; however,

using current to actuate the polymers also produces back relaxation in the response. Ex-

amination of the back relaxation reveals a low frequency non-linearity. The nonlinearity is

quantified by the fact that the gain associated with the back relaxation does not increase

linearly with an increase in input current. There is also a change in the response at certain

voltage thresholds. For example, when the voltage across the polymer exceeds 3 V, the

rate of back relaxation increases. The repeatability of the response will aid in implementing

reliable control of the polymer, but the non-linearities in the back relaxation will provide a

considerable challenge in developing a model to be used in control.

Acknowledgments

I would like to begin by thanking my adviser Dr. Don Leo for his help and for his patients

during the time spent on my research. I would also like to thank the other members of my

committee, Dr. Dan Inman and Dr. Harry Robertshaw, not only for agreeing to be on the

committee but also for their guidance in the process of finding research.

Also I would like to thank all those at CIMSS that have shared their knowledge and

experience to aid in my research. Specifically I would like to thank Barbar Akle for his

introduction to the lab equipment and for supplying some of the polymers that were used

in experiments. I would like to thank Matt Bennett who also provided polymers. Finally I

owe a thank you to all of the friends and family that have offered their encouragement and

support throughout my college career.

iii

Contents

Abstract ii

Acknowledgments iii

List of Figures vi

List of Tables ix

Chapter 1 Introduction 1

1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Ionic Liquids as Solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 The Relationship Between Charge and Strain . . . . . . . . . . . . . . . . . 6

1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Chapter 2 Charge Control 9

2.1 Charge Control Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Current Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Transconductance Amplifier Design . . . . . . . . . . . . . . . . . . 10

2.2.2 Calibration of Transconductance Amplifier . . . . . . . . . . . . . . 12

iv

Chapter 3 Testing the Electromechanical Response 15

3.1 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Presentation of Step Response Data . . . . . . . . . . . . . . . . . . 18

3.1.3 Results of Step Response Tests . . . . . . . . . . . . . . . . . . . . . 21

3.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Presentation of Frequency Response Data . . . . . . . . . . . . . . . 24

3.3 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Repeatability of Results . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.2 Discussion of Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 4 Modeling 33

4.1 Linear Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Chapter 5 Conclusions 43

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 46

Appendix A Step Response Data 48

Vita 53

v

List of Figures

1.1 Picture of an IPMC in a cantilevered configuration. . . . . . . . . . . . . . . 2

1.2 A representation of ion movement as a mechanism of bending in IPMCs. . . 4

1.3 An equivalent circuit for an IPMC. . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Voltage response of a NafionTM IPMC to an input of 0.55 mAcm2 . . . . . . . . 10

2.2 Transconductance Amplifier Circuit used in experiments to transform voltage

into current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 A schematic showing the test setup to calibrate the current source. . . . . . 12

2.4 Calibration curve for transconductance amplifier. . . . . . . . . . . . . . . . 12

2.5 FRF of the transconductance amplifier with a 150 Ω load. . . . . . . . . . . 13

2.6 The output of the circuit to a sine wave input, the solid blue line is the

voltage in, the dashed red line is the current out. The top graph is a 100 Hz

sine wave and the bottom graph is a 20 Hz sine wave. . . . . . . . . . . . . 14

3.1 A schematic showing the experimental setup for step responses. . . . . . . . 16

3.2 The effect of large deflections on the laser signal. . . . . . . . . . . . . . . . 17

3.3 Step response for a current density of 1.1 mAcm2 . These are the results from

three trials with identical test conditions. . . . . . . . . . . . . . . . . . . . 19

3.4 Step response for a current density of 4.4 mAcm2 . These are the results from

three trials with identical test conditions. . . . . . . . . . . . . . . . . . . . 20

3.5 Maximum voltage as a function of current density . . . . . . . . . . . . . . 20

3.6 Maximum strain as a function of current density . . . . . . . . . . . . . . . 21

vi

3.7 Strain plotted against Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.8 Normalized strain plotted against normalized voltage . . . . . . . . . . . . . 23

3.9 The inflection points on the voltage plot line up with the inflection points in

the strain plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.10 FRF of an ionic polymer to a current input. . . . . . . . . . . . . . . . . . . 25

3.11 Integration of the current density FRF yields a FRF in terms of charge density. 26

3.12 Impedance of an ionic polymer. . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.13 The time response to an input of 5.5 mAcm2 demonstrates repeatability. These

are the results from three trials with identical test conditions. . . . . . . . . 28

3.14 A normalized plot of strain versus voltage with overlaid lines to show the

different responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.15 Strain versus voltage for a 4.4 mAcm2 input with the voltage threshold indicated.

These are the results from three trials with identical test conditions. . . . . 31

3.16 Strain versus voltage for a 5.5 mAcm2 input with the voltage threshold indicated.

These are the results from three trials with identical test conditions. . . . . 31

3.17 Time response for a 4.4 mAcm2 input with the increased back relaxation indi-

cated. These are the results from three trials with identical test conditions. 32

3.18 Time response for a 5.5 mAcm2 input with the increased back relaxation indi-

cated. These are the results from three trials with identical test conditions. 32

4.1 The fit of equation 4.1 to the response for a 3.3 mAcm2 input. . . . . . . . . . . 34

4.2 Frequency response and the resulting fit. . . . . . . . . . . . . . . . . . . . . 35

4.3 Normalized gains as a function of jd . . . . . . . . . . . . . . . . . . . . . . 36

4.4 The time constants used to fit a linear model to the step responses. . . . . . 36

4.5 Response of the model to an input of 3.3 mAcm2 . . . . . . . . . . . . . . . . . . 37



4.8 The response of IPMCs to a pulsed current input. . . . . . . . . . . . . . . 40

vii

4.9 The response of an IPMC to a single step of increasing magnitude and de-

creasing time (left) and the simulated response (right). . . . . . . . . . . . . 40

4.10 The predicted response (dashed) and the actual response (solid) for 3 macm2

and 6 macm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.11 The predicted response (dashed) and the actual response (solid) for 10 macm2

and 30 macm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.1 Step response to an input of 0.55 mAcm2 . These are the results from three trials

with identical test conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 49













viii

List of Tables

4.1 The gains and time constants necessary to fit a linear model to the step

responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Normalized results of the linear fit of the step results. . . . . . . . . . . . . 35

ix

Chapter 1

Introduction

A brief overview of the history of ionic polymers as active materials is presented in this

chapter. It will then discuss the characteristics of ionic polymers that are important when

using them as active materials. This discussion will be followed by an explanation of the

motivation for the work done and the objectives that are expected to be completed. The

chapter will conclude with an overview of the rest of the thesis.

1.1 History

Electroactive polymers (EAP) are a class of smart materials that exhibit large strains and

low elastic modulus. EAPs can be separated into two classes: Electronic EAP and Ionic

EAP (3). Ferroelectric Polymers, Electrets, Dielectric Elastomers and Electrostrictive Elas-

tomers are all electronic EAPs and are driven by electric fields. Ionic Polymers and Gels,

Carbon Nanotubes, and Conductive Polymers are ionic EAPs and are driven by the move-

ment of ions in the material (4). The EAP used in this research is an ionomeric polymer

metal composite (IPMC). A picture of an IPMC in a cantilever test setup is shown in

Figure 1.1.

An IPMC is an ionomeric polymer coated with a metal electrode, a typical material

used in IPMCs is NafionTM . IPMCs have their roots in the fuel cell industry, where

NafionTM has been used since its discovery in the 1960s. Because of interest in fuel cell

1

Figure 1.1: Picture of an IPMC in a cantilevered configuration.

technology, NafionTM and other similar materials became more available, and as a result

other areas of research on the materials began. This availability enabled three groups in the

early 1990s to demonstrate the electromechanical properties of ionic polymers (5). Two of

these groups showed that mechanical strain could be induced to an ionic polymer material

by an applied electrical field (6; 7). The third group also showed that the materials would

produce an electrical current to an applied mechanical strain (8).

1.2 Characteristics

This section will briefly discuss the important characteristics of NafionTM as an IPMC.

During the discussion, certain mechanical and electrical properties will be discussed inde-

pendently of each other; however, IPMCs exhibit coupling between their mechanical and

electrical characteristics and that coupling must be considered. The actual mechanisms at

work in the polymers are debated. Several models have been proposed with different mod-

els having different attributes. The models generally contain some combination of pressure

gradients, electric field, elastic deformation, and ion and solvent transport as factors driving

2

the response of IPMCs (9).

As an active material, IPMCs present advantages and disadvantages when compared

to the more traditional active materials such as piezoelectrics. Actuators that use piezo-

electric materials are characterized by large forces, small strains, and fast response times.

In contrast, IPMCs produce small forces, large strains, and have slower response times than

piezoelectric actuators. Advantages of IPMCs include a low power requirement and large

strain. The trade off for these two results is that IPMCs do not create large forces. The

power output of an IPMC is about 100 times lower than a natural muscle (9).

The mechanical motion of IPMCs is most often attributed to ion transfer within

the polymer. The physical structure within the polymer is such that the positive ions, or

cations, are able to move while the negative ions, or anions, are in fixed positions. When

an electric field is applied across the polymer, the positive cations are attracted to the

negative electrode on the polymer surface. This net movement of ions toward the positive

surface causes swelling in that surface and bending of the IPMC (5). Figure 1.2 shows this

movement (10).

Electrically IPMCs can be represented as an RC circuit as shown in Figure 1.3 (11).

This representation, while not perfect provides a good starting point for understanding how

the material reacts to an electrical stimulus. An understanding of the electrical response

allows the creation of coupled models that more accurately predict the mechanical response.

To describe the material fully, the mechanical and electrical properties have to be

combined. Newbury proposes a coupled model based on the piezoelectric model. Although

this model does not describe the motion of IPMCs in all situations, it is a good starting

point for predicting the response of an ionic polymer to an electrical input. Equation 1.1 is

a simplified form of the equations proposed by Newbury. F

V

=

Z11 Z12

Z21 Z22

x

q

(1.1)

Where F is force, V is voltage, x is velocity, q is current, Z11 is mechanical impedance,

3

Figure 1.2: A representation of ion movement as a mechanism of bending inIPMCs.

Figure 1.3: An equivalent circuit for an IPMC.

4

Z22 is electrical impedance, Z12 and Z21 are coupling terms (4). To use these relationships,

the Z terms have to be expanded to account for geometry and are allowed to be frequency

dependent (12). The expanded versions of the equations adequately predict the current

produced from a step voltage input, but the resulting voltage from a step current is not

predicted well as is demonstrated by Newbury’s results (1).

1.3 Ionic Liquids as Solvents

In order for an IPMC to be actuated, it must be saturated with a solvent. The traditional

solvent used has been water. For some applications hydrated samples work well; however,

if the application requires that the sample be exposed to the air, water will evaporate from

the material over time. Another problem with using water as a solvent is that water has a

low electrochemical stability. When applied voltages exceed the electrolysis limit of water,

the water will break down into hydrogen and oxygen which results in the dehydration of

the material (2).

One solution to the problems of evaporation and electrolysis of water is to use ionic

liquid solvents. Ionic liquids are compounds with low melting points, meaning they exist

in liquid form at low temperatures (2). There are many advantages to using ionic liquids

instead of water, one is that there are over a trillion possible ionic liquids (13). Because

of the number of possible ionic liquids, it is possible to create a solvent to suit the specific

needs of many applications. Besides selection, there is a more important reason for using

ionic liquids. Ionic liquids are very stable compounds. Ionic liquids have essentially no

vapor pressure and therefore will not evaporate when left in open air. Ionic liquids also

have a higher electromechanical stability than that of water, allowing IPMCs with ionic

liquid solvents to operate at higher voltages (2).

Bennett (2) has demonstrated the long term stability of IPMCs with ionic liquid

solvents. The stability was demonstrated in two ways. The first set of tests were done

on three consecutive days with one hydrated sample and one ionic liquid sample. Bennet

demonstrated that after three days the response of the hydrated sample had decreased

5

significantly while the response of the ionic liquid sample remained essentially unchanged

(2). Another important aspect of stability is the ability of a sample to be cycled without

degradation. Bennett (2) also demonstrated this stability by showing that the response of

a hydrated sample was drastically reduced after 103 cycles while the ionic liquid samples

did not show degradation until 105 cycles.

The use of ionic liquids as solvents is not without drawbacks. The most noticeable

problem of IPMCs with ionic liquid solvents is that they have slow response speeds when

compared to hydrated samples. Also in the work done by Bennet (2), the ionic liquid

samples demonstrate small strains when compared to hydrated samples. With the large

selection of ionic liquids available it is likely that these limitations can be remedied by the

selection of an appropriate solvent.

1.4 The Relationship Between Charge and Strain

In attempts to control the actuation of an IPMC using an input voltage, the results are often

sample specific. Given multiple samples with a different compositions, consistent testing

of each sample can produce inconsistent results. Such results are not a problem with the

method testing, but with the assumption that the strain is proportional to the input voltage.

It has been shown that the strain of an IPMC is related to the transportation of charge

through the polymer (14). Also, Akle et al. (15) have shown that the strain increases

linearly as a function of capacitance, independent of the composition of the sample. As

stated by Akle (15), ”‘the low frequency capacitance of an ionomer is strongly related to

the charge accumulation at the blocking electrode.”’ Akle also noted that the εV relationship

changed by as much as 10 times between samples, while the εQ relationship only changed

by a maximum of 1.5 times between samples (15). These results demonstrate that there is

a strong correlation between the accumulated charge and the strain in an IPMC.

6

1.5 Motivation

Most of the work done with IPMCs in the past has used voltage to actuate the polymer. The

purpose of this work is to explore the possibility of using current for actuation. There are

some expected advantages of using current as opposed to voltage. One expectation is that

by controlling current, the charge in the polymer can be directly controlled. By controlling

charge, the ion transport might become more controllable and lead to more direct control

over the polymer. When using voltage to control an IPMC, there is a resulting current

spike to a step input. An advantage of using current is that the resulting voltage is a signal

that resembles the charging of a capacitor.

1.6 Objectives and Contributions

There were four main objectives for the research presented in this paper.

1. Construct a suitable current source for controlling IPMCs. The current source must

meet specifications that will allow the actuation of IPMCs as well as being stable to

allow for extended use.

2. Test and calibrate the current source. The calibration of the current source should

provide a DC gain for the amplifier as well as a frequency response.

3. Evaluate the performance of IPMCs to a current input. Step and frequency responses

will be used to gain understanding of the IPMC’s response.

4. Develop a model for the response of IPMCs to a current input. The model must also

be compared to various experimental data to verify that it is valid.

The work done in this paper contributes to the research of using IPMCs as actuators

based on the previous objectives. First a current source was constructed and calibrated.

That source was used in the experiments presented in this paper and a packaged version

was built for the use in future research.

7

The response of an IPMC to a current input was evaluated. That evaluation also

revealed nonlinearities in the response as well as confirming certain nonlinearities proposed

by Kothera (16). This paper discusses those nonlinearities and proposes explanations that

could be useful in future work where control methods are implemented.

A model was created to based on the experimental data. It was demonstrated that

a single linear model is not sufficient to represent the DC response of an IPMC to a current

input. The model was also compared to results from other experiments done specifically

for model evaluation. Finally a nonlinear model is suggested to more accurately represent

the data and methods for creating such a model are discussed.

1.7 Overview

The remainder of this thesis will discuss the equipment, tests, and analysis used to evaluated

the implementation of charge control of IPMC transducers. Chapter 2 will present the

circuitry used in the experiments. Chapter 3 will describe the tests that were done on

material. It will also present the data from those tests as well as an analysis of that data.

Chapter 4 will continue the analysis of the data past observations to an empirical model

of the data. Chapter 5 will consist of a conclusion and a presentation of suggested future

work.

8

Chapter 2

Charge Control

2.1 Charge Control Technique

Current was used to control the charge in the material by taking advantage of the capacitive

characteristics of the transducer. Using current allows the charge applied to the material

to be calculated by

q =∫

idt (2.1)

where q is charge (C), i is current (A), and t is time (s).

Using current to control charge has advantages and disadvantages. One advantage

is that for simple signals it is easy to know the amount of charge being accumulated by the

material, and for more complicated signals it can be calculated. The major disadvantage of

using current to control charge is that to achieve a fast change in charge, a large current is

needed. When the IPMCs are subjected to too high of a current they will break down and

quit working.

Because NafionTM is both capacitive and resistive, the charge does not accumulate in

the material as it would if the transducer were only capacitive. Because of this characteristic,

the charge applied to the material is not necessarily the same as the charge that accumulates

in the material. This can be seen in Figure 2.1, where the input signal is a step change in

current. If the material were purely capacitive, the voltage would rise linearly in response

9

to a step current input, and would continue to rise as long as the input did not change.

Figure 2.1: Voltage response of a NafionTM IPMC to an input of 0.55 mAcm2 .

2.2 Current Source

2.2.1 Transconductance Amplifier Design

To use current to control charge, an appropriate source of current was identified. For the

purposes of the experiments presented in this paper, a voltage controlled current source was

desired. Using a voltage controlled current source meant that the current could be easily

controlled using the computer equipment already in place.

The current source used in the experiments described in this document was con-

structed based on the transconductance amplifier circuit shown in Figure 2.2 (17). The two

transistors used to construct the circuit were: NPN - TIP3055, PNP - TIP42. All other

components are as described in the schematic. To achieve a greater maximum current, two

identical circuits were connected in parallel.

A power source capable of supplying sufficient current to the circuit while main-

taining the required voltage to the operational amplifiers and transistors is needed. If the

10

supply cannot provide the necessary power, the voltage provided by the supply will drop

as the current requirements are increased. In this case the maximum current output of

the circuit will be dependent on the power supply and the gain of the circuit will not be

linear. With a sufficient power supply, the circuit performs as expected (current output

proportional to voltage input) as long as the voltage levels in the circuit remain below the

saturation levels of the op-amps. A power supply that supplies ±15 V at 500 mA on each

channel is sufficient to power the circuit.

With a proper power supply chosen, the limits of the circuit are based on the physical

limitations of the op-amps and the transistors. The absolute limitations of the circuit were

not explored because the current needed to drive a polymer is much less than the current

limit of the circuit. Generally for the experiments the maximum current was less than

100 mA, at this level the circuit can operate and remain stable for an extended period of

time. It should be noted that if the voltage input to the circuit is too high, the components

of the circuit could overheat and fail.

Figure 2.2: Transconductance Amplifier Circuit used in experiments to trans-form voltage into current.

11

2.2.2 Calibration of Transconductance Amplifier

Before the circuit could be used in experiments, the circuit had to be tested to verify that

it performed as expected. The first step was to calibrate the circuit to a DC input. This

was done by providing several voltage inputs and measuring the voltage across a known

load to calculate the current output of the circuit as shown in Figure 2.3. Plotting the data

from the calibration tests demonstrated that the gain of the circuit was linear as can be

seen in Figure 2.4. Repeating the process for a different load, also shown in Figure 2.4,

demonstrated that the current output from the circuit is independent of the load on the

circuit. The gain of the circuit was found to be 42 mA/V.

Figure 2.3: A schematic showing the test setup to calibrate the current source.

Figure 2.4: Calibration curve for transconductance amplifier.

12

The next step in verifying the circuit was to measure the dynamic output of the

circuit. This was done in two ways, the first was to measure Frequency Response (FRF)

using a Fast Fourier Transform (FFT) analyzer. The FRF, Figure 2.5, demonstrated a cut

off frequency of approximately 30 Hz. Even though the cut off frequency is lower than

100 Hz, the next test suggests that the amplifier is usable up to at least 100 Hz. The

second test that was done was to give the circuit various sine wave inputs to show that the

resulting current was a scaled version of that sine wave, Figure 2.6. This showed that the

current could follow a given input so that the expected output of the circuit was the same

as the actual output.

Figure 2.5: FRF of the transconductance amplifier with a 150 Ω load.

13

Figure 2.6: The output of the circuit to a sine wave input, the solid blue line isthe voltage in, the dashed red line is the current out. The top graph is a 100Hz sine wave and the bottom graph is a 20 Hz sine wave.

14

Chapter 3

Testing the Electromechanical

Response

3.1 Step Response

3.1.1 Test Setup

This section will describe the test setup and procedure for measuring the response of an

ionic polymer to a step current input. It will begin by describing the physical setup, then

discuss some precautions necessary to achieve good data, and will end by describing the

tests that were performed.

The five physical components of the experiment are:

1. dSpace DS2002/D2003 Mux DAC

2. Transconductance Amplifier

3. Polymer Sample

4. Polytec OFV3001 Laser Vibrometer

5. dSpace DS2103 ADC

15

The Digital to Analog Converter (DAC) converts the digital signal from the computer

to a continuous signal to be sent to the input of the amplifier. The transconductance

amplifier then converts the given voltage into the current that will control the polymer.

The polymer is the object of interest in this experiment and is connected in series with

the amplifier so that the current coming out of the amplifier will be the current through

the polymer. One input to the Analog to Digital Converter (ADC) is connected in parallel

with the polymer to measure the voltage across the polymer during the experiment. The

assumption that the impedance in the ADC is much greater than that of the polymer is

made so that there is negligible current flowing in that path of the circuit. The displacement

of the polymer is measured using the laser vibrometer. The laser vibrometer sends a signal

to one of the inputs on the ADC. Figure 3.1 shows the experimental setup.

Figure 3.1: A schematic showing the experimental setup for step responses.

To obtain results that were repeatable and to have a meaningful comparison between

the different input currents, the test setup was not disturbed during or between tests of a

single polymer sample. The use of polymers with ionic liquid solvents made this possible

because, unlike hydrated samples, they did not have to be removed from the clamp for

rehydration between tests. Also, the laser was positioned once at the beginning of the first

16

test and left alone during the rest of the tests. These procedures made it possible to be

certain that the laser was always aimed at the same point on the polymer. Being certain

that the point of measurement did not change on the polymer made it possible to compare

the results of consecutive tests without worrying about uncertainties related to identifying

that point.

Because the laser was not moved between tests, the expected deflection of the poly-

mer at the highest current had to be considered when positioning the laser for the lowest

current. For a good measurement, the laser should be aimed perpendicular to the polymer

surface so that the signal is reflected back into the laser. As the polymer deflects, the laser

starts to be deflected away from itself as shown in Figure 3.2. As a result, the laser will lose

tracking for the higher currents if it is aimed at the tip of the polymer. To ensure that the

laser would maintain good tracking through all of the tests, the laser was positioned close

to the root (approximately 5mm) where the deflection is small.

Figure 3.2: The effect of large deflections on the laser signal.

It is also important to pay attention to the initial conditions of the polymer. The

preferred initial conditions are zero voltage and zero current, this will ensure that the initial

charge of the polymer is also zero. If a new test is started too soon after the previous test

has ended, the polymer may not have enough time to reach this zero charge state. A shunt

17

resistor can be used to speed up the process of discharging the polymer and a multimeter can

be used to measure the remaining charge in the polymer. The connection for the shunt and

the multimeter are placed in a position that allows them to be accessed without disturbing

the polymer.

For this experiment, the input was a step current that was held for 5 minutes. The

step time was chosen to allow the polymer to reach its maximum voltage and displacement

and in some cases begin to back relax.

3.1.2 Presentation of Step Response Data

To be useful the raw data needed to be processed. For data processing the gain of the

laser and the dimensions of the polymer are needed. Multiplying the data by the laser gain

will give the displacement, but the displacement is not meaningful from one test to another

because it is dependent on the measuring point and also on the dimensions of the polymer.

Strain is a more useful measurement that can be related between tests and between polymer

samples. The assumption is made that the sample exhibits constant curvature, meaning the

strain in the polymer is constant along its length. Under the constant curvature assumption,

strain can be calculated by

ε =xt

L2d

(3.1)

Where x is the measured displacement (m), t is the thickness of the polymer sample(m),

and Ld is the distance the measuring point is from the root of the polymer (m).

In addition to converting displacement to strain, it is also useful to change current to

current density. This change in variables allows the response to a given input to be scaled

and compared to other tests of various dimensions. This conversion is made by simply

dividing the input current by the area of one of the polymer electrodes.

jd =Iin

Ltw(3.2)

Where Iin is the input current (A), Lt is the total length of the polymer sample (m), and w

is the width of the polymer (m). The result of the equation is current density, jd, in Am2 .

18

To start the analysis of the step data, the strain and voltage data were plotted so

that each trial for a given current density was plotted on the same axis, example data sets

are shown in Figure 3.3 and Figure 3.4, the rest of the data is presented in the Appendix.

This gave a good first indication of the repeatability of the data. Also, the maximum

voltage, Figure 3.5, and maximum strain, Figure 3.6, reached during the experiment were

found and plotted against the input current density.

Figure 3.3: Step response for a current density of 1.1 mAcm2 . These are the results

from three trials with identical test conditions.

19

Figure 3.4: Step response for a current density of 4.4 mAcm2 . These are the results

from three trials with identical test conditions.

Figure 3.5: Maximum voltage as a function of current density

20

Figure 3.6: Maximum strain as a function of current density

3.1.3 Results of Step Response Tests

Because of the coupling between the mechanical and electrical characteristics of the material,

it is interesting to present the data as a plot of the strain versus voltage, Figure 3.7. This

way of representing the data shows results that are not easily seen from the time data. One

result that is shown by this plot is that the strain changes as a function of the voltage and

that change is basically repeatable for each current density. Another feature of the plot is

that it shows that the way the strain changes with increasing voltage is dependent on the

input current density. The back relaxation that was apparent in the time response graphs

also shows up as a function of voltage.

Assuming the voltage has reached steady state, the maximum voltage reached can

be assumed as the maximum voltage for that current density. The strain and voltage

can then be normalized by the maximum voltage and plotted against each other. From

Figure 3.8, the results of increasing the input current density can be seen more clearly. For

the input current densities of 1.1 mAcm2 and lower, the data falls on a line with a slope of

approximately 1 decade/decade before back relaxing. Input current densities or 4.4 mAcm2

21

Figure 3.7: Strain plotted against Voltage

and 5.5 mAcm2 demonstrate a slope of approximately 2 decades/decade before back relaxing.

Input current densities of 2.2 mAcm2 and 3.3 mA

cm2 produced a transitional curve where part of

the response fell on the first curve and part of the curve fell on the second curve. It also

appears that the back relaxation was greater for those responses that fell entirely on the

steeper curve.

Because of the resistive and capacitive characteristics of the polymers, one simple

way of viewing them from an electrical standpoint is as a resistor and a capacitor in parallel.

If the polymer exhibits the characteristics of a simple RC circuit when given a step input,

the voltage across the polymer would be an exponential curve. The voltage does look nearly

exponential for a current density of 0.55 mAcm2 ; however, as the current density is increased,

the voltage curve starts to exhibit inflections.

If the location of the inflections are compared with the locations of inflections in the

strain curves, they tend to match up as shown in Figure 3.9. The matching of the inflections

is to be expected because of coupling between the electric field and the mechanical motion

of the material. The interesting thing is that the change in voltage is always positive, but

the change in strain is not always positive.

22

Figure 3.8: Normalized strain plotted against normalized voltage

3.2 Frequency Response

3.2.1 Test Setup

This section will describe the test setup and procedure for measuring the FRF of an ionic

polymer when current is the input. It will begin by describing the physical setup, then

discuss some precautions necessary to achieve good data, and will end by describing the

tests that were performed.

The four physical components of the experiment are:

1. TekTronics FFT Analyzer

2. Transconductance Amplifier

3. Polymer Sample

4. Polytec OFV3001 Laser Vibrometer

In this experiment the TekTronics FFT Analyzer supplied an output voltage to the

Transconductance Amplifier to be transformed into a current. The current was then passed

23

Figure 3.9: The inflection points on the voltage plot line up with the inflectionpoints in the strain plot.

through the polymer and the displacement was measured by the Laser Vibrometer. The

voltage across the polymer was also measured during the process. Both the output from

the laser and the polymer voltage were returned to the FFT which calculated a frequency

response.

To obtain a clean transfer function the laser should be aimed as close to the tip

of the polymer as possible. This will increase the amplitude of the actual response with

respect to any noise present in the system. Because the high frequency amplitude is much

lower than the low frequency amplitude, the frequency response should be taken in bands

to ensure good coherence at every frequency. It is also helpful increase the resolution on

the FFT channels just to the point where they start to overload. This will produce a clean

FRF with good coherence.

3.2.2 Presentation of Frequency Response Data

Processing the FRF data was similar to processing the step data. The first step is to

assemble the FRF from the high and low frequency data to create one continuous FRF.

24

The resulting FRF then must be converted to the intended measurements. For the strain

FRF, equation 3.1 is used along with the gain of the current source and the surface area of

the polymer. The result is a FRF, Figure 3.10, with units of µεA/cm2 .

Figure 3.10: FRF of an ionic polymer to a current input.

From the FRF in Figure 3.10 it can be seen that in response to a current input,

there is no quasi-static region for an IMPC given a current input. The quasi-static region,

or a flat FRF at low frequencies, would be useful in control because models of the sample

in that region could be frequency independent. Using Equation 2.1 to convert the current

input to a charge input will yield a FRF, Figure 3.11 with units of µεC/cm2 representing a

relationship between strain and charge density which exhibits the quasi-static region that

is desirable for control.

Another important FRF is the impedance, Figure 3.12. The impedance FRF shows

how the relationship between voltage and current changes with frequency. It is interesting

to note that the impedance of an IPMC is high at low frequencies and is much lower at

high frequencies. This indicates that for a given current input there will be a lower voltage

and therefore a lower power consumption at high frequencies than at low frequencies.

25

Figure 3.11: Integration of the current density FRF yields a FRF in terms ofcharge density.

Figure 3.12: Impedance of an ionic polymer.

26

3.3 Analysis of Results

This section will discuss data from the previous two sections in greater detail. The various

response characteristics will be discussed, including the attributes that contribute to or take

away from the controllability of IMPC’s under a current input.

3.3.1 Repeatability of Results

To be useful in application, the method of control has to produce repeatable results.

Figure 3.13 demonstrates that a step input of current is capable of producing a repeat-

able response. The voltage response is very repeatable from trial to trial, and the strain

response is quite repeatable as well. Even with the repeatable nature of the responses,

there were some slight deviations between different trials of the same current input. These

differences occurred in both the voltage response and the strain response. As was shown

in Figure 3.9, inflection points in the voltage response line up directly with inflections in

the strain response. Because the strain curve changes response shape at the inflections in

the voltage response, a change in the voltage response will cause the strain curve to change

shape at a different time. In Figure 3.13, this occurs twice, once at approximately 40 sec

and once at approximately 130 sec. In each situation, the inflection of the voltage curve

occurs slightly early for one trial meaning that the strain changes response sooner. Because

of the early change, the back relaxation occurs sooner as does the increased rate of back

relaxation. This leads to a lower maximum strain and a larger back relaxation for the same

time period. The differences in the voltage response are most likely due to non-zero initial

conditions. Although a shunt was connected across the polymer, it could not always be

discharged completely and the initial voltage was sometimes between 20 mV and 30 mV.

The degree of repeatability of the voltage response is demonstrated further in Figure 3.5.

While there were slight differences between trials of the same current in the actual response,

this plot shows that the final voltage reached for a given current is approximately the same.

27

Figure 3.13: The time response to an input of 5.5 mAcm2 demonstrates repeatability.

These are the results from three trials with identical test conditions.

3.3.2 Discussion of Nonlinearities

Figure 3.5 shows that the voltage response is not linear as a function of current density.

For an input current density of 2.2 mAcm2 , the maximum voltage is 2.17 V. For an input of

4.4 mAcm2 the expected voltage would be 4.34 V if the voltage increased linearly with current

density. In the experiments the maximum voltage reached for a 4.4 mAcm2 input was 3.27 V.

Another nonlinearity that is shown by Figure 3.5 is that the rate of change in max-

imum voltage as a function of current density is not constant. There are three regions

that are shown in the plot: current densities less than 1.1 mAcm2 , current densities between

1.1 mAcm2 and 4.4 mA

cm2 , and current densities between 4.4 mAcm2 and 5.5 mA

cm2 . For each region

of the plot, there is a different relationship between maximum voltage and current density.

This nonlinearity presents more difficulty than the previous example because not only do

the maximum voltages not scale from one input to another, but they also do not follow

the same trends. This means that unless the response for the region of interest is known,

the maximum voltage cannot be accurately predicted. These nonlinearities are important

because they will directly affect the strain response.

28

Figure 3.14 revisits the normalized plot of strain versus voltage that was presented

in the previous chapter. The two darker black lines overlaid on the plot are a representation

of the two response curves that occur with increasing current density. As mentioned before,

there are two main response curves. The first curve represents the response to low current

density inputs and the second curve represents the response to higher current density inputs.

There is a transition that occurs where the response begins on the low current density curve

and ends on the high current density curve. This is consistent with the three regions present

in the previous plot.

Figure 3.14: A normalized plot of strain versus voltage with overlaid lines toshow the different responses.

There are two main differences between the high current density response and the

low current density response. The first difference is that the slope of the high current density

response is twice that of the low current density response. The second difference is that the

back relaxation of the high current density response is much greater than that of the low

current density response.

Although there are differences in the two responses there are also similarities. The

most obvious similarity is the occurrence of the back relaxation. Each response exhibits back

29

relaxation to some degree and that relaxation occurs as the voltage reaches the maximum

voltage for that current density.

For control reasons, Figure 3.5 and Figure 3.14 are important. Figure 3.14 demon-

strates that if the voltage response can be calculated the strain response can be predicted.

And Figure 3.5 provides a way to determine the maximum voltage response. While these

two pieces of information are important, they are not enough to create a working model

of the polymer’s response. The problem lies in the way the polymer reacts as the voltage

increases and in the transient voltage response. The first issue will be discussed, but for

the purpose of this paper the voltage transient will be considered exponential.

Figure 3.15 and figure 3.16 are both plots of strain as a function of voltage. The

vertical and horizontal lines on the plot represent the point where an increased rate of back

relaxation occurs in the response. In each case the increased rate occurs at approximately

3 V and a strain from 1600 to 1800 microstrain. Although the strains are in a small range

when the increased rate occurs, the fact that the voltage is nearly fixed indicates that the

change is a result of crossing a voltage threshold. This result can also be seen in Figures 3.17

and Figure 3.18, where the vertical and horizontal lines were drawn to indicate the time,

strain, and voltage at the point of increased back relaxation. These plots demonstrate more

clearly that the increased back relaxation occurs at a threshold of 3 V.

30

Figure 3.15: Strain versus voltage for a 4.4 mAcm2 input with the voltage threshold

indicated. These are the results from three trials with identical test conditions.

Figure 3.16: Strain versus voltage for a 5.5 mAcm2 input with the voltage threshold


31

Figure 3.17: Time response for a 4.4 mAcm2 input with the increased back relaxation


Figure 3.18: Time response for a 5.5 mAcm2 input with the increased back relaxation


32

Chapter 4

Modeling

4.1 Linear Model Construction

The data from the frequency responce was used to create a model of the polymer for a

current input. The INVFREQS command in MATLAB was used to determine the poles

and zeros for the model. This method produced a good fit for the high frequency response,

but it did not account for the back relaxation that was observed in the step responses.

To add the back relaxation into the frequency response, a fit using two exponentials

was performed in the time domain using the form

y = A(1− e− 1

τ1t)−B(1− e

− 1τ2

t) (4.1)

where y is the measured response, A and B are gains, τ1 is the first time constant, τ2 is the

second time constant, and t is time. In computing the fit, τ1 and τ2 are estimated from the

step response, and A and B are then adjusted to provide a good fit. The result of this fit

for a 3.3 mAcm2 input is shown in Figure 4.1 where the blue line is the actual data and the red

line is the fit.The original fit done by Matlab was amended to include the two poles that

were found from the time response fit. The resulting fit for the FRF is shown in Figure 4.2.

As the time data showed, the low frequency response was not linear with respect

to current. This nonlinearity meant that a fit for one current input would not necessarily

33

Figure 4.1: The fit of equation 4.1 to the response for a 3.3 mAcm2 input.

accurately predict the response at all levels of current input. Because of the nonlinearity, the

model was created for one current at a time and the results of each model were compared.

Table 4.1 shows the gains and time constants for fitting Equation 4.1 to the step response

at each current input. Each separate model will work for currents inputs near the current

for the model. Outside of a small range of currents near the original current for a particular

model, that model will not accurately predict the response.

Table 4.1: The gains and time constants necessary to fit a linear model to thestep responses.

jd ( mAcm2 ) A B τ1 τ2

0.55 300 240 18 2000.83 400 220 18 1501.1 490 220 18 2002.2 1375 800 25 2003.3 1975 1600 25 7004.4 2530 3500 22 7005.5 3300 8500 20 600

34

Figure 4.2: Frequency response and the resulting fit.

The gains presented in Table 4.1 can be normalized by jd, this is shown in Table 4.2.

Normalizing the gains demonstrates clearly the manner in which the gains change as jd is

increased. Figure 4.3 shows that the normalized gain Ajd

is relatively constant when com-

pared to Bjd

which changes dramatically over the range of jd. This shows that the gain

associated with τ1 is close to linear and that the gain associated with τ2 is very nonlinear.

This results in the rise of the strain response being relatively easy to predict and the back

relaxation being difficult to predict. Another reason that the back relaxation is difficult to

predict is that the time constant associated with the back relaxation changes as the input

jd is increased, Figure 4.4.

Table 4.2: Normalized results of the linear fit of the step results.jd ( mA

cm2 ) A/jd B/jd τ1 τ20.55 545 436 18 2000.83 482 265 18 1501.1 445 200 18 2002.2 625 364 25 2003.3 598 485 25 7004.4 575 795 22 7005.5 600 1545 20 600

35

Figure 4.3: Normalized gains as a function of jd

Figure 4.4: The time constants used to fit a linear model to the step responses.

36

Figure 4.5: Response of the model to an input of 3.3 mAcm2 .

4.2 Model Verification

Once the model was created and shown to work for the case that it was created around,

it was tested against other inputs. Simulations were run in Simulink on the predicted

model and then repeated in the lab on the actual polymer. The first test of the model was

against data that had already been collected. To compare the model to the step response

that had already been collected, the step response was plotted with the predicted response.

Figure 4.5 shows that the model created around an input of 3.3 mAcm2 can predict the correct

response for that current density.

When the current density is decreased to 2.2 mAcm2 , the 3.3 mA

cm2 model over predicts

the actual response as shown in Figure 4.6. This is evidence of a non-linearity, because the

output is not scaled with the input. It is interesting to note that the model does accurately

predict the rising portion of the response, this indicates a possibility that only a portion of

the response is non-linear.

If the current density is increased to 5.5 mAcm2 , the output from the 3.3 mA

cm2 model over

predicts the response as it did for the previous case. But the cause of the over prediction

37


for this case is not purely because of an overall scaling factor. For the previous case, the

back relaxation occurred at the same rate as the 3.3 mAcm2 case so the model fit the shape of

the response but not the magnitude. For the 5.5 mAcm2 response there are two stages of back

relaxation, a first stage similar to the previous two cases and a second more steep relaxation

that does not show up in the previous two cases. This second stage of back relaxation cases

the polymer to relax more quickly than in the previous cases and contributes to the 3.3 mAcm2

model over predicting the response.

After the model was compared to the step responses, two more tests were done. The

first was a high frequency test where the current was pulsed at 75 Hz and the total charge

imparted to the polymer was held constant. This meant that for current densities higher

than the one used for the model, the duty-cycle of the pulse was reduced. The results of

these tests are shown in Figure 4.8. When given the same inputs as were used in the tests,

the model predicted that all of the responses would be the same with the exception that the

shorter pulses would produce a noisier response. In Figure 4.8 it is adequate to say that for

current densities of 33 mAcm2 and less, the response is the same. The variation in responses

is no more than what was seen in the step responses. This indicates that the charge is the

38


determining factor in the strain response.

For the current densities of 83 mAcm2 and 110 mA

cm2 , the response does not match the

response of the other inputs. The most likely reason for the deviation of these two responses

from the previous responses is that the actual frequency seen by the polymer was actually

much higher than 75 Hz resulting in a lower response as predicted by the FRF in Figure 3.11

than the previous results. Because the pulses of these two inputs are much shorter in time

than the off time between pulses the response of the polymer to each pulse has time to

degrade significantly before the next input.

The second test also used the same total charge for each test, but as the current was

increased the length of a single step was decreased. This allowed the low frequency response

to be examined independent of charge. Figure 4.9 shows the results of these tests as well as a

simulation of the tests using the model. As the figure shows, a single step of a defined charge

does not create a single response independent of current density. This is contrasted to the

previous example where the response of the sample was dependent on the charge density

and not the current density. If the low frequency response were dependent on charge density

then the responses in Figure 4.9 would have approximately the same maximum value but

39

Figure 4.8: The response of IPMCs to a pulsed current input.

decreasing rise time as the current is increased. The model predicted the initial rise of the

strain response for each input as can be seen in Figure 4.10 and Figure 4.11. However, as

the figures show, the model does not accurately predict the back relaxation which results

in errors of the predicted maximum response also.

Figure 4.9: The response of an IPMC to a single step of increasing magnitudeand decreasing time (left) and the simulated response (right).

A comparison of the previous tests yields an important result concerning the non-

linearities of the samples. Figure 3.11 suggests that the important factor in actuation is

40

Figure 4.10: The predicted response (dashed) and the actual response (solid)for 3 ma

cm2 and 6 macm2 .

Figure 4.11: The predicted response (dashed) and the actual response (solid)for 10 ma

cm2 and 30 macm2 .

41

charge density while Figure 4.9 suggests that current density, or the rate at which charge is

applied, is more important. The current densities used in each test were approximately the

same and the same charge density was used for each test. So the only difference between

the two tests was frequency. Examining the FRF given in Figure 3.11 reveals that there

is a low frequency roll off before the flat portion of the FRF. Assuming that the roll off

continues with lower frequencies the result presented by these two tests can be explained.

The first test is done at a frequency that is much higher than the low frequency roll off and

is therefor not affected. The increasing magnitude of the inputs for the second test results

in charge being applied to the material at increasing rates. This increasing rate of charge

effectively pushes the response to a higher frequency resulting in higher maximum strains

for higher current densities.

Another indication that the nonlinearities have greater effect at lower frequencies

(16) can be seen in Table 4.1. This table shows that the gain associated with τ1 does not

increase linearly with current density. Also τ2 shifts as the input is increased. Both τ1 and

τ2 are low frequency, this fact combined with the results of the two tests presented point to

the possibility that the nonlinearities present in the system are low frequency phenomena. If

this is the case a more complex model for predicting back relaxation would only be required

for the DC case and linear methods of modeling can be used for higher frequencies.

42

Chapter 5

Conclusions

5.1 Conclusions

The work presented in this thesis has demonstrated that the transconductance amplifier

presented in Chapter 2 is sufficient to actuate an IPMC. The strengths and weakness of

using that circuit to control an IPMC were discussed. The characteristics of the response of

an IPMC to various input current densities were investigated. Also, this thesis has shown

the inability of a single linear model to represent the response of an IPMC to a current input.

Finally, the possibility of implementing a comprehensive non-linear model to represent the

system was discussed.

Chapter 3 presented the response of an IPMC to step inputs of several current

densities and also the response, in the form of FRF, to a random content signal. The data

was shown to be repeatable but non-linear. An analysis of the data showed that there was

a change in the response of the polymer as the jd was increased. This change was most

clearly demonstrated in the plot of normalized strain versus normalized voltage, 3.14. Also,

the severity of back relaxation of the polymer increased as jd was increased. Examining

the back relaxation portion of the response revealed that as the input jd drove the voltage

beyond a certain threshold the nature of the response changed. The most obvious threshold

occurred at 3 V and caused the rate of back relaxation to increase. This increased rate

accounted for the increase in overall back relaxation over a given period of time.

43

In Chapter 4 the modeling of the polymer response was discussed. A model was

created based on a fit of the frequency response data that was modified to account for the

back relaxation of the polymer. This was a linear model that further demonstrated the

non-linear nature of the IPMCs. Because the model was only able to fit the response for

the current density for which it was created, a fit was performed for each current density.

The results of these fits were shown in Table 4.1 and Table 4.2, and demonstrated that the

majority of the non-linearity was a result of the back relaxation.

The model was also compared to results from experiments delivered a set amount

of charge to the polymer over the time period of the test. There were two tests done,

one at high frequency and one at low frequency. The high frequency test showed that the

model could predict the response of an IPMC at high frequencies. The low frequency test

demonstrated the ability of the model to predict the rise of the response, but it did not

predict the back relaxation or the maximum value. These results showed that the non-

linearity in the polymer was a low frequency characteristic, which corresponds to the back

relaxation being caused by a low frequency pole.

5.2 Future Work

In the process of completing this work, several additional questions were asked. Those

questions are presented in the following paragraphs.

• As has been shown, there are voltage thresholds that must be crossed when charging

an IPMC. When a voltage threshold is crossed, the nature of response changes. A

study to identify each threshold that exists and to determine parameters that affect

the threshold level would be useful to the modeling process.

• From this work it is apparent that a linear model is not sufficient to accurately describe

the response of an IPMC to a current input. A mathematical model should be derived

that includes terms to describe the nonlinear phenomena demonstrated in this paper.

• In this work the IPMCs were able to accept higher DC voltages without failure than

44

in other tests where voltage was the input. On the other hand, the maximum current

before failure was much lower than the maximum current with a step voltage input.

A method of control that uses both current and voltage input to avoid these limits

could prove useful.

45

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47

Appendix A

Step Response Data

48

Figure A.1: Step response to an input of 0.55 mAcm2 . These are the results from

three trials with identical test conditions.



49





50





51



52

Vita

Walter J Robinson Jr was born on July 27, 1981 in Abingdon, VA to Jane and Walter

Robinson. Walter was raised on a farm in Glade Spring, VA where he learned the value of

a hard days work and developed a love for the outdoors. He graduated from Patrick Henry

High School in 1999 and soon moved to Blacksburg, VA to study Mechanical Engineering

at Virginia Tech. While there he found time to pursue his interests including fishing,

kayaking, mountain biking, and skiing. Despite spending more time playing than studying

he graduated Magna Cum Laude and continued his education at Virginia Tech. Walter

worked as a Graduate Teaching Assistant for two semesters at the beginning of his graduate

career. During his second semester, Dr. Don Leo agreed to be his adviser and gave him a

project involving ionic polymers.

This thesis was typeset with LATEX2ε1 by the author.

1LATEX2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademark of theAmerican Mathematical Society. The macros used in formatting this thesis were written by Greg Walker,Department of Mechanical Engineering, Virginia Tech.

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charge control of ionic polymers - virginia techcarbon nanotubes, and conductive polymers are ionic...

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