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1m LIBRARY
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University

 

 

 

This is to certify that the
dissertation entitled

Conjugated Polymer Actuators And Sensors: Modeling,
Control, And Applications

presented by

Yang Fang

has been accepted towards fulfillment
of the requirements for the

PhD. degree in Electrical Engineering

 

 

 

 

 

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CONJUGATED POLYMER ACTUATORS AND SENSORS: MODELING, CONTROL,
AND APPLICATIONS

By

Yang Fang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Electrical Engineering

2009

ABSTRACT

CONJUGATED POLYMER ACTUATORS AND SENSORS: MODELING,
CONTROL, AND APPLICATIONS

By

Yang Fang

Conjugated polymers are soft actuation and sensing materials with promising applica-
tions in biomedical devices and micromanipulation systems. However, their sophisticated
electro-chemo-mechanical dynamics and nonlinear behaviors present significant challenges
in such applications. This dissertation is focused on using systems and control tools to ad-
dress these challenges. In particular, a systems perspective is taken to model the dynamic
and nonlinear behavior of conjugated polymer actuators and sensors, an adaptive control
scheme is developed to handle model uncertainties, and a conjugated polymer actuated-
micropump is explored both as an interesting application and a platform for validating and
extending proposed models. Experimental results on trilayer polypyrrole (PPy) actuators
and sensors are presented throughout the dissertation to support the modeling and control
studies.

On the modeling aspect, a linear, partial differential equation (PDE) is first used to
capture the ion transport dynamics in actuation and sensing. With proper boundary con-
ditions, the PDE is solved to derive an infinite-dimensional transfer function model that
is geometrically scalable and amenable to model reduction. Nonlinear behaviors, in both
electrical and mechanical domains, are also modeled. In particular, on the electrochem-
ical side, a reduction-oxidation (redox) level-dependent impedance model is obtained by
applying perturbation analysis to a nonlinear PDE. On the mechanical side, a nonlinear
elasticity theory-based framework is proposed to capture the mechanics involved in large
deformation. The framework has not only been effective in modeling a bending actuator,
but also motivated the study of a novel, torsional actuator that is based on a fiber-directed

PPy tube.

On the control aspect, a robust adaptive control scheme is proposed to tackle the time-
varying behavior of conjugated polymer actuators. Based on a reduced, linear model, a
self-tuning regulator with parameter projection is designed and implemented. The robust
adaptive control scheme has shown in experiments its superiority to traditional PID control
schemes and a fixed model-following scheme.

Finally, the application of conjugated polymer actuators to micropumps is explored. A
conjugated polymer-actuated diaphragm micropump is designed and fabricated. In contrast
to a typical design of clamping an actuation membrane at all edges, a novel, petal-shaped
diaphragm design is proposed to alleviate the effect of edge constraints. Transfer function
models from the actuation voltage to the diaphragm curvature and to the flow rate are
obtained and verified experimentally. A flow rate of 1260 uL/min is achieved for the new

diaphragm design, which represents a significant improvement over the traditional design.

DEDICATION

TO MY PARENTS AND MS. FANG XIE.

iv

ACKNOWLEDGMENT

I am very grateful to my advisor Dr. Xiaobo Tan for his careful and kind guidance. His
vision and expertise have influenced me a lot during my PhD study, and will also benefit
me in the future. I also would like to appreciate his generous help on my job search and
career development.

I would like to thank Professor Hassan Khalil, Professor Nin g Xi, and Professor Thomas
Pence for serving on my advisory committee. Their insightful comments on my research
are greatly appreciated. I would like to thank Professor Thomas Pence in particular. His
class of Nonlinear Elasticity and his personal guidance enabled me to finish the work on
modeling mechanical nonlinearity in this dissertation. I would like to thank Prof. Giirsel
Alici at University of Wollongong for providing PPy samples and valuable advice on my
research, which contributed to successful completion of this dissertation.

I am grateful to my colleagues in Smart Microsystems Lab: Zheng Chen, Dr. Mart An-
ton, Stephan Shatara, Freddie Alequin, Alex Esbrook, Dawn Hedgepeth, Chris Gliniecki,
John Thon, Andrew Temme, Qingsong Hu, Ernest Mbemmo, and Alex Will. I also would
like to thank friends at Michigan State who have offered me generous help: Hua Deng,
Guokai Zeng, Li Sun, Yue Huang, Zongliang Cao, Yixin Wang and many others.

I would like to acknowledge the financial support of my research by NSF CAREER
grant (ECCS 0547131) and MSU IRGP (05-[RGP-418).

Finally I would sincerely thank my parents Xiaolong Fang and Linhua Liu, and Ms.
Fang Xie. It is their support and love that encouraged me to finish this dissertation success-

fully.

3

List of Tables .................................... viii
List of Figures ................................... ix
Introduction 1
1.1 Background on Conjugated Polymers .................... 1
1.2 Fabrication of Trilayer Conjugated Polymer ................. 5
1.3 Research Objectives .............................. 7
1.4 Contributions ................................. 8
1.4.1 Linear Models for Actuation and Sensing .............. 8
1.4.2 Modeling Nonlinearities in Conjugated Polymer Actuators ..... 10
1.4.3 Control of Conjugated Polymer Actuators .............. 12
1.4.4 Conjugated Polymer Micropump ................... 13
1.5 Chapter Descriptions ............................. 14
Linear Models for Conjugated Polymer Actuators and Sensors 16
2.1 Electrical Admittance Module ........................ 16
2.1.1 Review of Diffusive-elastic-metal Model and Adaptation to the Tri-
layer PPy Actuator .......................... l. 7 '
2.1.2 Scaling Laws for Double-layer Capacitance and Circuit Resistance . 21
2.2 Electromechanical Coupling ......................... 23
2.3 Mechanical Output .............................. 25
2.4 Complete Actuation Model and Experimental Verification .......... 29
2.4.1 Complete Actuation Model ...................... 29
2.4.2 Experimental Verification ...................... 30
2.5 Conjugated Polymer Electromechanical Sensor ............... 33
2.5.1 Full Sensing Model .......................... 36
2.5.2 Simplified Sensing Model ...................... 39
2.5.3 Experiments and Discussions .................... 42
2.6 Chapter Summary ............................... 43
Modeling Nonlinearities in Conjugated Polymer Actuators 46
3.1 Redox Level-dependent Impedance Model .................. 46
3.1.1 The Governing Partial Differential Equation ............. 46
3.1.2 Perturbation Analysis ......................... 48
3.1.3 Impedance Model ........................... 50
3.1.4 Experiments and Discussions .................... 53
3.2 Nonlinear Mechanical Model Based on a Swelling Framework ....... 58

TABLE OF CONTENTS

vi

3.2.1 Finite Strain Tensors ......................... 61
3.2.2 Stresses ................................ 67
3.2.3 Equilibrium .............................. 70
3.2.4 Experimental Verification ...................... 72
3.3 Fiber-directed Conjugated Polymer Torsional Actuator ........... 76
3.3.1 Nonlinear Mechanical Modeling Framework ............ 80
3.3.2 Boundary Conditions ......................... 85
3.3.3 Nonlinear Mechanical Model .................... 87
3.3.4 Sample Preparation and Experimental Setup ............. 88
3.3.5 Experimental Results and Discussions ................ 90
3.4 Chapter Summary ............................... 90
Control of Conjugated Polymer Actuators 102
4.1 Model Reduction ............................... 102
4.2 Design of Robust Adaptive Controller .................... 109
4.2.1 Self-Tuning Regulator ........................ 109
4.2.2 Parameter Projection ......................... l 11
4.3 Experimental Results ............................. 112
4.3.1 Measurement Setup .......................... 1 12
4.3.2 Results and Discussions ....................... 114
4.4 Chapter Summary ............................... 123
Application: Conjugated Polymer Micropump 125
5.1 Design and Fabrication of the Micropump .................. 125
5.1.1 Diaphragm Design .......................... 125
5.1.2 Fabrication and Assembly of the Micropump ............ 126
5.2 Physics-based, Control-oriented Model for the Pump ............ 128
5.2.1 Electrical Admittance Module of PPy ................ 129
5.2.2 Electromechanical Coupling of PPy ................. 132
5.2.3 Mechanical Module of the Micropump ............... 133
5.2.4 Complete Model ........................... 141
5.3 Experimental Results ............................. 142
5.3.1 Admittance .............................. 142
5.3.2 Displacement ............................. 144
5.3.3 Flow Rate ............................... 149
5.4 Chapter Summary ............................... 149
Conclusion 152
Bibliography .............................................................. 155

vii

2.1

3.1

3.2

3.3

4.1

LIST OF TABLES

Parameter values used for the model (2.53). ................. 44
Estimated values for C and R1. ....................... 55
Estimated values for C0 and R2. ....................... 56
Geometric parameters of the samples. .................... 89
Typical values of parameters in the actuation model .............. 106

viii

1.1

1.2

1.3

1.4

1.5

1.6

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

LIST OF FIGURES

The chemical structure of polypyrrole ..................... 2
The chemical structure of polyaniline. .................... 2

Illustration of the actuation mechanism of a trilayer polypyrrole actuator.
Left: the sectional view of the trilayer structure; right: bending upon appli-

cation of a voltage. .............................. 4
The setup for fabricating trilayer conjugated polymer ............. 6
Fabricated sample of trilayer conjugated polymer. .............. 6
Thickness of layers of a trilayer PPy actuator measured with a microscope. . 7
The complete model structure for conjugated polymer actuators. ...... 16

(a) Illustration of double-layer charging and diffusion for a conjugated
polymer film with one side in contact with electrolyte; (b) equivalent circuit

model for the polymer impedance. ...................... 18
Definitions of dimensional parameters used in actuation model ........ 20
Double-layer capacitance versus actuator size ................. 24
Resistance versus actuator width (length = 20 mm). ............. 24
Resistance versus actuator length (width = 3 mm) ............... 25
Standard linear solid model for capturing polymer viscoelasticity. ..... 28

Geometric relationship between the beam curvature and the tip displacement. 29
Geometry of the trilayer actuator ........................ 31

ix

2.10

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.19

2.20

3.1

3.2

3.3

3.4

3.5

Schematic of the experimental setup for joint force-displacement measure-
ment. ..................................... 32
The setup for force/displacement measurement. ............... 32
Force versus displacement under an actuation voltage of 0.4 V (width = 3.5
mm) ....................................... 33
Force versus displacement under an actuation voltage of 0.4 V (width = 6
mm) ....................................... 34
Electrical admittance spectrum (size: 30 x 5 mm) ............... 35
Electrical admittance spectrum (size: 40 x 5 mm) ............... 36
Dynamic displacement response (size: 30 x 5 mm). ............. 37
Dynamic displacement response (size: 40 x 5 mm). ............. 38
Comparison between the numerical solution of the full model and the ana-

lytical solution of the simplified model (2.50) when C0 is 0.05 M. ..... 43

Dynamic response of conjugated polymer sensors: Experimental measure-
ment (marks) versus model prediction (line). Three samples with different

widths (fixed length: 30 mm). ........................ 44
Dynamic response of conjugated polymer sensors: Experimental measure-
ment (marks) versus model prediction (line). Three samples with different
lengths (fixed width: 5 mm). ......................... 45
Equivalent circuit for the impedance model. ................. 51

The relationship between transferred charges and applied voltage for dif-
ferent samples. ................................ 56

Model predictions versus experimental impedance for Sample 1 (0.35 M

electrolyte, 0 V DC voltage). ......................... 57
Model predictions versus experimental impedance for Sample 2 (0.35 M
electrolyte, 0.5 V DC voltage). ........................ 58

Model predictions versus experimental impedance for Sample 3 (0.35 M
electrolyte, 1 V DC voltage). ......................... 59

3.6 Model predictions versus experimental impedance for Sample 4 (0.25 M
electrolyte, 0 V DC voltage). .........................

3.7 Model predictions versus experimental impedance for Sample 5 (0.25 M
electrolyte, 0.5 V DC voltage). ........................

3.8 Model predictions versus experimental impedance for Sample 6 (0.25 M
electrolyte, l V DC voltage). .........................

3.9 Illustration of the reference configuration and the deformed configuration. .
3.10 Definition of the bending radii at different locations ..............

3.11 The relationships between the input voltages and the transferred charges
for the two different samples ..........................

3.12 Quasi-static bending under different actuation voltages for Sample 1 (13 x 5
mm) .......................................

3.13 Quasi-static bending under different actuation voltages for Sample 2 (33 x 6
mm) .......................................

3.14 Computational results on the change of a versus the applied voltage for
Sample 1 (13x5 mm). ............................

3.15 Computational results on the change of (1 versus the applied voltage for
Sample 2 (33 x6 mm). ............................

3.16 Computational results on the changes of thickness of the PPy layers with
the applied voltage for Sample 1 ........................

3.17 Computational results on the changes of thickness of the PVDF layer with
the applied voltage for Sample 1 ........................

3.18 (a) Illustration of the conjugated polymer-based torsional actuator; (b) pic-
ture of the fabricated torsional actuator. ...................

3.19 The experimental setup to fabricate the fiber-directed conjugated polymer
actuator .....................................

3.20 Illustration of the actuator configuration. Left: original configuration; right:
deformed configuration .............................

xi

60

61

62

63

67

73

74

75

77

77

80

81

3.21

3.22

3.23

3.24

3.25

3.26

3.27

3.28

3.29

3.30

3.31

3.32

4.1

4.2

4.3

4.4

Experimental setup to measure It“. The c0pper film is attached perpendicu-
lar to the outer surface at the tube bottom. .................. 89

The torsional displacement of Sample 1 with 0.005 Hz sinusoidal voltage

input ..................................... 91
The change of inner radius of Sample 1 with 0.005 Hz sinusoidal voltage
input ..................................... 92
The change of tube length of Sample 1 with 0.005 H2 sinusoidal voltage
input ..................................... 93
The torsional displacement of Sample 2 with 0.005 H2 sinusoidal voltage
input. ..................................... 94
The change of inner radius of Sample 2 with 0.005 H2 sinusoidal voltage
input. ..................................... 95
The change of tube length of Sample 2 with 0.005 Hz sinusoidal voltage
input. ..................................... 96
The torsional displacement of Sample 3 with 0.005 Hz sinusoidal voltage
input. ..................................... 97
The change of tube length of Sample 3 with 0.005 H2 sinusoidal voltage
input. ..................................... 98
The torsional displacement of Sample 4 with 0.005 Hz sinusoidal voltage
input. ..................................... 99
The change of inner radius of Sample 4 with 0.005 H2 sinusoidal voltage
input. ..................................... 100
The change of tube length of Sample 4 with 0.005 Hz sinusoidal voltage

input. ..................................... 101

 

Ratio as a function of D ................. 107
maX(lp1|,|p3|, Izzl)

Comparisons between the infinite-dimensional and reduced model ...... 108

Illustration of the robust self-tuning regulator. ................ 109

Schematic of the experimental setup ...................... 113

xii

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

5.1

5.2

5.3

5.4

5.5

Experimental results on trajectory tracking (Batch One), I = 0 h. (a) Achieved
trajectories versus desired one under the three controllers; (b) instantaneous
tracking errors under the three schemes. ................... l 15

Experimental results on trajectory tracking (Batch One), I = 4 h. (a) Achieved
trajectories versus desired one under the three controllers; (b) instantaneous
tracking errors under the three schemes (note the different vertical-axis
scales). .................................... l 16

Normalized average error ea and maximum error em under the three con-
trol schemes (Batch One experiments). (a) Evolution of ea; (b) evolution
Of em. ..................................... I 17

Experimental results on trajectory tracking (Batch Two), I = 3 h. (a) Achieved
trajectories versus desired one under the three controllers; (b) instantaneous
tracking errors under the three schemes. ................... 119

Normalized average error ea and maximum error em under the three con—
trol schemes (Batch Two experiments). (a) Evolution of ea; (b) evolution
Of em. ..................................... 120

Evolution of voltage input magnitude under the three schemes. (a) Batch
One experiments; (b) Batch Two experiments ................. 121

Identified poles and zero under the robust adaptive control scheme in Batch
One experiment. (a) Evolution over four hours; (b) Evolution over 100

seconds ..................................... 122
Experimental results of tracking a non-persistently exciting signal under
the robust adaptive controller. (a) Trajectory tracking at t = 0 h and t = 4
h; (b) Tracking errors at t = 0 h and t = 4 h. ................. 124
Schematic of a petal-shape pumping diaphragm (top view). Left: before
actuation; right: upon actuation. ....................... 126
The assembly schematic of micropump. ................... 128
The mechanism of flap check valves ...................... 129
The microfabrication process to make a flap valve. ............. 130

(a) The assembled micropump (top view); (b) The assembled micropump
(bottom view) .................................. 131

xiii

5.6

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

A magnified view of the micrOpump to show the structure. ......... 131

The complete model structure for conjugated polymer actuators. ...... 132
Definition of the principal strains. ...................... 133
The coordinates and geometry definition. .................. 135
Illustration of the geometrical relationship in (5.12) and (5. 13), R = i. . . . 136

Geometric relationship in the calculation of the volume V and the work
done by the diaphragm ............................. 139

Illustration of the volume change under the actuation voltage for the petal-
shaped diaphragm. .............................. 140

Comparison of model prediction from (2.12) with the experimental results
for the whole diaphragm and the petal-shaped diaphragm ........... 143

Comparison of model prediction from (5.33) with the experimental results
for the petal-shaped diaphragm when there is no water (actuation voltage
amplitude 4 V). ................................ 145

Comparison of model prediction from (5.33) with the experimental results
for the petal-shaped diaphragm when pumping water (actuation voltage
amplitude 4 V). ................................ 146

Comparison of model prediction with the experimental results for the whole
diaphragm when there is no water by numerically solving (5.20) (actuation
voltage amplitude 4 V) ............................. 147

Comparison of model prediction with the experimental results for the whole
diaphragm when pumping water by numerically solving (5.20) (actuation

voltage amplitude 4 V) ............................. 148

Flow rate of the micropump at different frequencies when input voltage
magnitude is 3 V. ............................... 150

Flow rate of the micropump at different frequencies when input voltage
magnitude is 4 V. ............................... 151

xiv

Chapter 1

Introduction

1.1 Background on Conjugated Polymers

Electroactive polymers (EAPs) are promising actuation and sensing materials with poten-
tial applications in robotics and biomedical systems [1—6]. One class of EAP materials are
conjugated polymers or conducting polymers [7—9], which are unusual organic materials
that exhibit electronic conductivity. In 1977, Hideki Shirakawa, Alan G. MacDiarmid, Alan
J. Heeger and coworkers reported high conductivity conjugated polymer, which eventually
rewarded them with the 2000 Nobel prize in Chemistry. Polypyrrole (PPy) and polyaniline
(PAN I) are two of the most investigated conjugated polymers, because of their good chem-
ical stability and substantial strains. The chemical structures of PPy and PANI are shown
in Fig. 1.1 and Fig. 1.2. The backbones of conjugated polymers have alternating single
and double carbon-carbon bonds (conjugation), which results in positive charge carriers
and thus electrical conductivity. Those charge carriers must overcome a band gap in order
to be delocalized and thus enable conduction. The band gap is reduced by doping, which
involves the addition of charges (dopants) to or removal of charges from the polymer chain.
The doping is generally performed chemically or electrochemically. In chemical doping,

dopants are donated to or repelled from the polymer backbone through chemical reactions

1

with dopant molecules.

/\ /\
\/ \/ \/

Figure 1.1: The chemical structure of polypyrrole.

030,020,113

Figure 1.2: The chemical structure of polyaniline.

33—2

A conjugated polymer can be electrochemically doped by placing it in contact with
an electrode in the electrolyte. When a sufficiently positive potential is applied, electrons
are removed from the polymer, and negatively charged anions are incorporated into or
positively charged cations are repelled from the polymer backbone to maintain the charge
neutrality. Application of a sufficiently negative potential can reverse the process and re-
pel the trapped anions out of or return the cations back to the polymer (reduction). The
oxidation/reduction process is called redox, and these cations or anions are called dopants.
The mass transport induced by ion movement during redox is considered to be the pri-
mary mechanism responsible for volumetric change and thus the actuation of conjugated

polymers [7, 8]. The redox process of PPy involving anions can be described as

Oxidation: PPy + A" —) PPy+A— + 6“

Reduction: PPy+A"+e_ —» PPy+A—,

where PPy represents the neutral state of polypyrrole and PPy+ is the oxidized state, A‘

2

represents the dopant anions, PPy+A’ indicates that A" is incorporated into the polymer,
and e‘ denotes an electron. Note that an anion-transporting conjugate polymer will expand
during oxidation and contract during reduction, while a cation-transporting polymer will
demonstrate the opposite behavior.

Conjugated polymer actuators have the following attractive features:

0 can be actuated under very low voltages (typically 2 V or less).

0 have large strains (3% in-plane to 30% out—of-plane) and considerable stress.
0 can operate in liquid electrolytes, including body fluids.

0 lightweight and biocompatible.

Linear extenders have been made by immersing a single piece of conjugated polymer
in electrolyte [7,10]. As the potential difference is changed between the polymer and elec-
trolyte, the volume of the conjugated polymer will change, so it can be used to generate
linear motion. The bending motion can be generated with a bilayer configuration [8]. When
the conjugated polymer layer is expanded or shrunk by accepting or repelling ions, the sub-
strate layer does not change. Due to the strain mismatch within the two layers, the bilayer
structure will bend. However, these two configurations require electrolyte environments.
A trilayer bender can work without this limitation [11], as illustrated in Fig. 1.3. In the
middle is an amorphous, porous polyvinylidene fluoride (PVDF) layer that serves both as
a backing material and a storage tank for the electrolyte. On both sides of the actuator are
the PPy layers. When a voltage is applied across the actuator, the PPy on the anode side is
oxidized while that on the cathode side is reduced. The oxidized layer absorbs anions and
expands, while the reduced layer gives up anions and contracts. The differential expansion
thus leads to bending of the actuator, as shown in Fig. 1.3 (right).

Conjugated polymers can be used as electrochemical sensors, because of the significant

electrical conductivity change over several orders of magnitude in response to change in

3

Polypyrrole

 

    

PVDF Predoped anion
0+ 0 9 G o‘ ‘3
oxidized reduced anion cation electron solvent
state state

Figure 1.3: Illustration of the actuation mechanism of a trilayer polypyrrole actuator. Left:
the sectional view of the trilayer structure; right: bending upon application of a voltage.

pH, humidity, or binding to biomolecules. These electrochemical sensors are based on the
property changes of the single and double carbon-carbon structure while interacting with

chemical compounds [12].

The conjugate polymers can also be used as electromechanical sensors where an open~
circuit voltage or a short—circuit current is generated under deformation. Comparing with
the more extensive work on conjugated polymer actuators, the understanding of the elec-
tromechanical sensing property of conjugated polymer is still in its infancy, where theory

and models are needed to explain and quantify the phenomenon. We will explore a theory

4

to explain the electromechanical sensing capability of conjugated polymer in this thesis.

1.2 Fabrication of Trilayer Conjugated Polymer

Fabrication of trilayer conducting polymer is achieved by electrochemically oxidizing pyr-
role monomer from a solution to grow PPy layers on either side of a gold-coated porous
PVDF film, which acts as the working electrode. The porous PVDF films are obtained
from Sigma-Aldrich, which has the thickness of 110 pm with pore size of 0.45 pm. It is
coated with a thin layer of gold (approximately 100 nm) by using the sputtering coater in
W. M. Keck microfabrication cleanroom at Michigan State University. This ensures a good
conductivity therefore good electrochemical growth of PPy. The electrolyte is a solution
of 0.1 M pyrrole, 0.1 M Li+TFSI— in Propylene Carbonate (PC) with 0.5 w/w% water.
Note that the solution with Li+TFSI_ will dope the PPy with TFSI’. The dopant can be
changed to other ions, such as PFg, by changing Li+TFSI_ to the salt containing these
6— .

A potentiostat is used in the electrochemical deposition, which is a control and measur-

ions, such as TBA+PF

ing device that keeps the potential of the working electrode at a constant level with respect
to the reference electrode in the electrolyte. The one used in the Smart Microsystems Lab is
Omni 101B from BSA Biosciences Inc. The cell is a cubic glass container by joining glass
together with glass sealant. Two stainless steel meshes are used as the counter electrode
and the PVDF membrane is sandwiched between them to ensure even PPy deposition. A
part of the film is left protruding out of the electrolyte for electrical connection. The exper-
iment setup is shown in Fig. 1.4, and one piece of fabricated trilayer conjugated polymer is
shown in Fig. 1.5.

After the samples are fabricated, the thickness of the PPy layer can be measured by
using the digital imaging system in W. M. Keck microfabrication cleanroom at Michigan

State University, which consists of a SPOT digital color camera and an Olympus BX60

5

 

Figure 1.4: The setup for fabricating trilayer conjugated polymer.

 

Figure 1.5: Fabricated sample of trilayer conjugated polymer.

optical microscope. The sectional view of the sample is captured by the imaging system,
which also includes the measurement tool to obtain the dimensions. One image captured is

shown in Fig. 1.6.

 

Figure 1.6: Thickness of layers of a trilayer PPy actuator measured with a microscope.

1.3 Research Objectives

In this dissertation, we will focus on modeling, control, and application of conjugated poly-
mer actuators and sensors. To fully utilize conjugated polymer in bio/micro manipulation,
bio-sensing, and micro robotics, it is crucial to understand the actuation and sensing prop-
erties of conjugated polymer. Therefore, the first objective is to obtain a physics-based
but control-oriented mathematical model for conjugated polymer, which captures its ma-
jor dynamics and is useful for applications as actuator and sensor. In many applications,
large deformation of actuators are required to achieve the tasks. Therefore, the second ob-
jective is to understand the nonlinearities in conjugated polymer actuators, which become
more significant as a higher input voltage is applied to generate larger deformation. The
third objective is to utilize feedback control in conjugated polymer-based system to tackle

the uncertainties and allow precise control of the actuator, because actuator behaviors are

7

heavily influenced by the materials and processes used during fabrication as well as the
environmental conditions during operation. The fourth objective is to explore the use of
conjugated polymer in promising applications. In this thesis, the application of conjugated
polymer in micropump is explored, because rnicrofluidic devices are in significant devel-
opment in recent years, and conjugated polymer actuators provide an important potential
mechanism for realizing compact micropumps with low actuation voltages.

PPy is used throughout this study. However, we expect that most of the results are

directly applicable or can be extended to other conjugated polymer materials.

1.4 Contributions

1.4.1 Linear Models for Actuation and Sensing

Conjugated polymer actuators require low actuation voltage (about 1 V), generate consider-
able stress and large strain output, and are light and biocompatible. These advantages make
them attractive for a wide range of robotic and biomedical applications, such as micro and
biomanipulation, biomimetic systems, and biomedical devices [2—4, 8]. For all these appli-
cations, it is highly desirable to have quantitative models available that can predict quasi-
static and dynamic actuation performance in terms of intrinsic material parameters and
actuator dimensions. Such models will be useful in feasibility analysis, design optimiza-
tion, and even actuator control. Alici and coworkers investigated modeling and geometry
optimization of bending curvature and force output for trilayer PPy actuators [11, 13, 14].
Christophersen et a1. characterized and modeled the bending curvature for bilayer PPy mi-
croactuators of different dimensions [15]. All these studies were focused on quasi-static
operating conditions, where the bending curvature and/or force output were examined in

terms of material stiffness and actuator geometry.

In this thesis, we have developed a scalable, control-oriented model for trilayer conju-

8

gated polymer that captures the major electrochemomechanical dynamics. This model con-
sists of three modules: 1) electrochemical dynamics module, adapted from the diffusive-
elastic-metal model of Madden [16]; 2) stress-generation module relating transferred charges
to internal actuation stress; and 3) mechanical dynamics module. Scaling laws for key pa-
rameters of the electrical admittance model are developed, rendering the complete model
expressible in intrinsic material parameters and actuator dimensions. Experiments are con-
ducted on trilayer conjugated polymer actuators of different dimensions to validate the
scalable model for quasi-static force and displacement output, electrical admittance, and

dynamic displacement response.

Comparing with the extensive work on modeling of conjugate polymer actuators {6,16—
19], research on modeling of conjugate polymers as electromechanical sensors has been rel-
atively limited. Wu et a1. [20] investigated the sensing behavior of a trilayer PPy beam by
considering the perturbation of the Donnan equilibrium of the ion distribution by mechan-
ical stimuli as the primary sensing mechanism. A similar viewpoint was also presented
by Takashima and coworkers [21]. For better, understanding of the sensing mechanisms,

however, quantitative modeling of the sensing dynamics is desired.

We have investigated the dynamic sensing behavior of conjugated polymers. The model
derivation starts with a partial differential equation (PDE) that governs the ion redistribu—
tion dynamics subject to diffusion and migration (due to electrostatic interactions). How-
ever, for real-time sensing applications, it is desirable to have an analytical model. For
this purpose, we linearized and solved this PDE analytically in the Laplace domain, which
leads to a transfer function that relates the open-circuit sensing voltage to the applied me-
chanical deformation. Experimental measurement of the open-circuit voltage matches the
model prediction reasonably well. This work provides the first step towards fundamental

understanding of mechanical sensing mechanisms of conjugated polymers.

9

1.4.2 Modeling Nonlinearities in Conjugated Polymer Actuators

Nonlinearities in conjugated polymer actuator become significant as a relatively high input

voltage is applied. N onlinearity exist in both electromechanical and mechanical domains.

In the electromechanical domain, the redox level, i.e., the amount of ions incorpo-
rated in the conjugated polymer during reduction-oxidation, has significant impact on the
electromechanical properties and dynamics of the material and consequently the actuation
performance. While the effect of redox level on conductivity and Young’s modulus has
been documented [15.22—26], its influence on dynamics of ion transport is more subtle and

has received inadequate attention in terms of fundamental understanding and modeling.

We have developed a redox level-dependent impedance model for conjugated polymers,
which is physically based yet has a compact, explicit form. We start with a governing par-
tial differential equation (PDE) that incorporates the dynamics of ionic diffusion, ionic
migration, and redox reactions. The PDE is linearized around a given redox level via per-
turbation analysis, and an exact, analytical solution is obtained by converting the PDE into
the Laplace domain and enforcing appropriate boundary conditions. This model captures
double-layer charging, diffusion, and migration effects. Experiments have shown that the
proposed model are comparable to the diffusive-elastic-metal model [16] when the redox

level is low, but shows clear advantage in predicting the impedance at higher redox levels.

In the mechanical domain, the classical beam theory has been used to model the bending
curvature for bilayer PPy actuators [15,27] and trilayer actuators [17]. However, this linear
elastcity-based model is only valid when the ion movement-induced swelling is small so
that: l) the strain and the stress can be related linearly; 2) the geometry nonlinearities can
be neglected. Alici and coworkers modeled the bending curvature and force output for
trilayer PPy actuators by using finite element analysis in analogy with a thermally driven
beam, but no analytical model was presented [28]. The tensile strength experiments have

shown that the strain and stress relationship of PPy film becomes clearly nonlinear as the

10

strain is increased over 4% [29,30]. These results indicate that the linear model based on the
assumption that the stress varies linearly with the strain becomes invalid as the deformation
gets large. Therefore, it is imperative to develop a nonlinear model to predict the actuator

performance for the applications where large actuation is involved.

In this thesis, a nonlinear elasticity-based method is proposed to capture the mechan-
ical deformation induced by transferred ions. Neo-Hookean type strain energy functions
are used for both PPy and PVDF to capture the nonlinear stress-strain relationship, which
incorporates the effect of swelling [31]. The actuation-induced stresses are derived from
the strain energy functions. The equilibrium configuration under a quasi-static actuation
voltage is then obtained by solving the force and moment balance equations simultane-
ously. Experiments have shown that when the applied voltages are small, predictions by
both models are close to experimental data. But the nonlinear elasticity model fits the ex-
perimental data better as the input voltage increases, which shows the superiority of the

method in modeling large deformations of conjugated polymers.

Different configurations of conjugated polymer actuators have been exploited, includ-
ing bilayer and trilayer benders [8,32], linear extenders [7]. However, more complicated
actuator motions will be required. Fiber-directed material can generate torsional motion be-
cause of the anisotropy associated with the interaction between the fibers and the material

matrix [33].

We have fabricated a fiber-directed conjugated polymer tube by integrating platinum
wires into conjugated polymer during the deposition process of conjugated polymer. We
have further proposed a nonlinear elasticity-based model to capture the relationship be-
tween the actuation voltage and the torsional motion of the conjugated polymer tube. Com-
pared with a linear elasticity-based model, this model is able to capture the complicated
deformations and large deformation. The model has been verified with experimental mea-

surements by using samples with different sizes.

11

1.4.3 Control of Conjugated Polymer Actuators

It is critical to precisely control the force and/or displacement output of conjugated poly-
mer actuators in many of their intended applications such as manipulation of single cells
and micro-surgical operations. There has been extensive work on understanding the actua-
tion mechanism of conjugated polymers as well as improving their actuation performance
(strain output, strain rate, force output, work per cycle, lifetime, etc.) [7—9, 34]. However,
control and control-oriented modeling of conjugated polymers remain largely unexplored.
A proportional controller was used by Qi er al. to speed up the transient responses of an
polyaniline actuator [35]. P. Madden treated the actuation dynamics as a first-order system
and designed a PID controller for a polypyrrole actuator, where his main interest was to
demonstrate a feedback loop consisting of polypyrrole actuator and sensor [17]. Taking
again a first-order empirical model, Bowers did simulation studies on PID and adaptive
control of conjugated polymers, but no experimental results were presented [36]. The
primitive state of conjugated polymer control study is mainly dictated by the sophisticated
electrochemomechanical processes during redox reactions, which makes it challenging to
have a physical (non-empirical) model suitable for real-time control. In addition, actuator
behaviors are difficult to characterize since they are heavily influenced by the materials and
processes used during fabrication as well as the environmental conditions (temperature,

humidity, electrolyte, etc.) during operation.

In this thesis, we have developed a robust adaptive control scheme for conjugated poly-
mer actuators with demonstrated performance in trajectory tracking experiments. A key
component of the work is a simple model structure reduced from a full, infinite-dimensional
physical model through model reduction, which captures essential actuation dynamics and
is amenable to efficient real-time control. It enables compact, embedded controller imple-
mentation for various micro, robotic, or biomedical applications. The resulting model is of

second order with a zero, after we further discard a pole and a zero considering the typical

12

range of physical parameters. A recursive least-squares algorithm is then used to identify
online the parameters of the reduced model. A self-tuning regulator [37] is designed based
on the identified parameters to make the closed-loop system follow a reference model. A
parameter projection step ensures that the parameter estimates stay within the physically-
meaningful region, and thus makes the system robust against measurement noises, and
unmodeled dynamics and nonlinearities. Experiments are conducted when the conjugated
polymer actuator is operating in air, where its actuation behavior shows significant varia-
tion over time because of the solvent evaporation. Experimental results have shown that
the proposed scheme is superior to the commonly used PID scheme and to the fixed model-

following scheme in terms of both tracking accuracy and required control effort.

1.4.4 Conjugated Polymer Micropump

Microfluidic devices have been a t0pic of extensive research in recent years because of the
significant development of biomedical industry. For many microfluidic systems, microp-
umps are desired to integrate and simplify complex analytical procedures. The methods
of actuation that have been investigated mainly include electroosmotic pumps, centrifugal
pumps, and diaphragm pumps. An electroosmotic pump induces motion of liquid by ap-
plying a voltage across a capillary tube, which drags the clustered ions in the liquid and
consequently the fluid. However, its application is limited by the drawback that fluid flow
cannot be generated when the ion concentration is below 10_5 M or above 10-2 M [38].
Centrifugal pumps are not very effective for fluid flows with low Reynolds numbers and
have only been miniaturized to a limited size [39]. Diaphragm pumps have shown promise
because of their compact sizes and the ability to deliver different kinds of fluid precisely.
Different actuation mechanisms and materials have been investigated for diaphragm mi-
cropumps. A thermopneumatic micropump utilizes an electric heater to increase the cham-
ber temperature and therefore the pressure which leads to fluid flow [40], but the bandwidth

of the device is limited by the heating and cooling process. Electrostatic actuation [41],

13

piezoelectric films [42], and P(VDF—TrFE) based electroactive polymer [43] have been ex-
plored, but the high actuation voltages required may change the properties of the samples.
Shape memory alloy thin film was also proposed [44], but the possible damage to the fluid
sample due to the high transition temperature hindered its use in biological applications.
We present a novel, conjugated polymer petals-actuated diaphragm micropump, which
is in contrast to the typical whole diaphragm design [43,45,46]. We show through modeling
analysis that the new design, by alleviating the edge constraints, can provide significantly
larger diaphragm deformation and consequently the higher flow rate. The modeling of
mechanical module for the diaphragm is developed using the energy method [47], which
incorporates the elastic energy stored in the diaphragm and the work done on the fluid by the
diaphragm. The final model for the petal-shape diaphragm pump, after model reduction, is
represented as a finite-dimensional transfer function that captures the fundamental physics
of conjugated polymer actuators and their interactions with flexible diaphragm and fluid.
The rest of the pump is fabricated through PDMS-based MEMS processes. Experiments
are conducted to evaluate the pump performance and the effectiveness of the model.The
measured admittance, deformation curvature, and flow rate are found to match the models
well. The complete model predicts that there is an optimal operating frequency to generate

the largest flow rate, which is verified in experiments.

1.5 Chapter Descriptions

The electrochemomechanical dynamics of conjugated polymer actuators and sensors are
investigated in Chapter 2, where the nonlinearities are ignored and transfer function mod-
els are presented to capture the actuation and sensing dynamics. The nonlinearities in both
electromechanical and mechanical domains are modeled in Chapter 3. The redox level is
introduced to capture the nonlinearity in the electrical domain, while nonlinear elasticity is

used to model the mechanical nonlinearity. A fiber-directed conjugated polymer actuator

14

that can generate torsional motion is also proposed in Chapter 3. Its deformation is mod-
eled by using nonlinear elasticity-based method. In Chapter 4 a robust adaptive control
methodology is presented based on the proposed actuation model, and experimental results
are provided to verify the effectiveness of the control strategy. The application of conju-
gated polymer actuator in a micropump is studied in Chapter 5, where both modeling and
experiments are conducted to investigate the micropump performance. The conclusion of

this dissertation is provided in Chapter 6.

15

Chapter 2

Linear Models for Conjugated Polymer

Actuators and Sensors

The complete actuator model consists of three modules: 1) the electrical admittance mod—
ule relating the current (and thus the charge transferred) to the voltage input; 2) the elec-
tromechanical coupling module expressing the generated stress in terms of the transferred
charge; and 3) the mechanical module connecting the stress to the displacement or force
output. This structure is shown in Fig. 5.7, where V is the input voltage, I is the current, 8

is the corresponding strain, and Y is the mechanical output.

V Admittance 1 Electromechanical 5 Mechanical Y
Module _ Module f Module

Figure 2.1: The complete model structure for conjugated polymer actuators.

 

 

 

 

 

 

2.1 Electrical Admittance Module

It is desired to obtain the electrical admittance module relating the current (and thus the

charge transferred) to the voltage input in order to obtain the complete actuation model.

16

Consequently, one will be able to predict the mechanical output of the actuator under the
applied voltage. There are two possible mechanisms for the accumulated anions to enter
the polymer matrix, diffusion and migration. J. Madden proposed a diffusive-elastic-metal
model for one piece of PPy immersing in the electrolyte, where it was assumed that the
polymer matrix is perfectly conducting and the ion transport within the polymer is solely
detemiined by diffusion [16]. This model can be adapted to model the trilayer PPy actuator
in our work and shows good agreement with experimental data for a wide frequency range

from 10‘4 to 105 Hz.

2.1.1 Review of Diffusive-elastic-metal Model and Adaptation to the

Trilayer PPy Actuator

Firstly, the method in [16] is briefly described. And then we present a different and more

straightforward derivation of the final admittance expression than the one in [16].

The voltage input of the actuator is applied across the two PPy layers. This produces a
potential difference between each PPy layer and the electrolyte, as illustrated in Fig. 2.2(a).
Note that only one side of the PPy is in contact with the electrolyte. Under the potential dif-
ference, the anions in the electrolyte migrate toward the polymer, which results in double-
layer charges at the polymer/electrolyte interface - like a double-layer capacitance with an
equivalent thickness of 5. Fig. 2.2 (b) shows an equivalent circuit model of the polymer
impedance, where the faradaic current has been ignored. C denotes the double-layer capac-
itance at the polymer/electrolyte interface, R is the electrolyte and contact resistance, and

ZD represents the “diffusion impedance”, which will be clarified in the derivation below.

In the Laplace domain the total current [(3) in the circuit is the sum of the double-layer

charging current and the diffusion current into polymer:

[(5) =IC(S)+ID(S). (2.1)

17

 

 

Figure 2.2: (a) Illustration of double-layer charging and diffusion for a conjugated polymer
film with one side in contact with electrolyte; (b) equivalent circuit model for the polymer
impedance.

The Kirchhoff ’s voltage law gives
1
V(s) = [(3) -R + s—-C -IC(s). (2.2)

Let x denote the thickness direction of the polymer, with x = 0 representing the poly-
mer/electrolyte interface. Assume that the ion concentration varies only in the x-direction,

which will be denoted as c(x, s). From Fick’s law of diffusion, one has

a
Qfifi=—F¢Ldggtfiu:0 (am

where A is the surface area of the polymer, F is the Faraday constant, (1 is the diffusion
coefficient, and x = 0 represents the gradient of ion concentration at the interface. To
compute [c(s), one first calculates the charges Qc(s) stored in the double-layer capacitor.
Assume that the double layer has a thickness 5 and that the ion concentration within the
(thin) double-layer is uniform, which equals c(0,s). Then Qc(s) = F -A - 6 - c(0,s), which
leads to

[C(s) : s-Qc(s) = F-A - 5 ~s-c(0,s). (2.4)

The last equation needed for the derivation is the diffusion equation, which reads in the
time domain:
ac 32c

=d

— — 0 h —h 2.5
at 8x2, <X< 2 17 ( )

where hz — hl is the thickness of the polymer layer according to Fig. 2.3, which defines
the dimensional parameters in this chapter. The generic solution of (2.5) in the frequency

domain is obtained as [48]:

c(x,s) = y1(s)e—\/;—/—dx + 72(s)e\/S/—dx, . (2.6)

19

where

 

Y (S) ___ [C(S) ID“)
1 2F-A-5-s ”AW/(7:, (27)
72(3) : [C(S) 10(5) . .

 

2F-A-6-s—2F.A-\/.T._s

 

Figure 2.3: Definitions of dimensional parameters used in actuation model.

For the trilayer polypyrrole actuator, there is no ionic flux at the other surface of the

polypyrrole layer, which gives the boundary condition at x : h2 — h1:

 

dc(x,s) s _ _ s _
(2.8)
From (2.1) and (2.2), [C(s) and [0(3) can be written in terms of V(s) and 1(5):
[C(s) =2 s-C- (V(s) —I(s) -R), (2.9)
ID(s) = l(s)-—s-C~(V(s)—I(s)-R). (2.10)

Plugging (2.9) and (2.10) into (2.8), one gets the following equation in terms of V(s) and

20

Ce\/s/—d(h2_hl)_e-\/;/—C_1(h2—hl)+c s ]V()_
256M(h2—h1)+e—\/S‘7g(h2—hl) 2 d .—
R-CeM(h2’hl)_e-\/S/_d(h2—hl) M)“;

+
25 eM(h2—h1)+e—W(h2-h1) 2"

 

 

]l(s).(2.11)

Since

 

3M(h2—h1)_e—\/S—/E(h2—hl) _ h h h f
eM(h2—h1)+e-\/s/_d(h2_hl)“tan (( 2‘ l) :1),

the admittance model of a trilayer conjugated polymer can be derived from (2.11) as

K.) st—gtam<<h2—h1>¢s/—d>+¢sl
= . (2.12)

3§+R.s3/2+R-l/s—d-s-tanhflhz—th/S—M)

 

 

It is highly desirable to have quantitative models available that can predict quasi—static
and dynamic actuation performance in terms of intrinsic material parameters and actuator
dimensions. Such models will be useful in feasibility analysis, design optimization, and
even actuator control. A scalable electrochemomechanical model is presented for trilayer
conjugated polymer actuators, by developing the scaling laws for key parameters of the

electrical admittance model.

2.1.2 Scaling Laws for Double-layer Capacitance and Circuit Resis-

tance

Two key parameters in the electrical admittance model (and thus in the force or displace—
ment response model) are the double-layer capacitance C and the resistance R. It is of inter—
est to know how they scale with the actuator dimensions. From the analogy to parallel-plate

capacitors, one expects C to be proportional to the interface area between the PPy and the

21

electrolyte, i.e.,

C = COWL, (2.13)

where C0 is the double—layer capacitance per unit area.

The scaling of R is more involved. A transmission line model is proposed considering
the nonzero (although low) resistivity of PPy. Here the top and bottom layers represent PPy
resistance while the middle layer represents the electrolyte resistance. Let p1 and p2 denote
the resistivity of the PPy and that of the electrolyte, respectively. Then the resistances R1

and R2 per unit distance are:

2 h
R1: L, R2: 132—1.
. (hz—h1)W w

From the basic transmission line theory [49],

4P192h1
h —h w2
R: (2 1) h , (2.14)
4
tanh<L M)
(hz—h1)W2

 

which is expressed in terms of fundamental material parameters and layer dimensions.

The scaling laws (2.13) and (2.14) can then be plugged into (2.12) to obtain scalable

models for the admittance and the bending dynamics of trilayer actuators.

To measure the double-layer capacitance C, a step voltage U of 0.4 V is applied. The
transferred charge Q into the PPy layer is computed by integrating the charging current,
which is obtained by subtracting the faradaic current from the total current. It can be
shown from (2.12) (using the Final Value Theorem [50]) that

1 I22 — h]

Q: §U(1+ 5 xx (215)

22

 

and thus C is computed by
2Q

h -h '
U(1+2—5—l)

The double-layer thickness 6 is hard to measure directly and an estimate of 25 nm is used

C: (2.16)

 

based on the range reported in [51]. To measure the resistance R, a high-frequency sinu-
l
soidal input U sin(a)t) is applied. From (2.12), the admittance Y ( jw) —+ -2_R— as a) —> 00,
i.e., the polymer becomes resistive for high-frequency inputs. Thus R is obtained from
1 . . .
R = . . In ex enments an in ut of fre uenc 250 Hz rs used.
2 y( J a) I P P q 3’

Fig. 2.4 shows the comparison between the measured double-layer capacitance and the

 

predicted value from (2.13) for different actuator sizes (L x W), where C0 is identified to
be 7.86 x 10‘7F/mm2. Fig. 2.5 compares the measured resistance R and the predicted
value from (2.14) as a function of actuator width, while the length of actuators is fixed to
20 mm. The resistivity values of p1 = 9 x IO‘SQm and p2 = 2.79m are obtained by
fitting the experimental data points in Fig. 2.5, where p1 is restricted within the typical
range [52] during tuning. As an independent validation step, the identified p1 and p2 are
further used to predict the resistance for five 3mm-wide actuators with different lengths. As
shown in Fig. 2.6, the comparison between the measured resistances. and their theoretical
values based on (2. 14) confirms the prediction capability of the scaling law.

The agreement between measurement and simulation in Figs. 2.4-2.6 validates the pro-
posed laws for C and R, which will be used in the prediction of quasi-static force/displacement,

electrical admittance, and dynamic displacement response in the following subsections.

2.2 Electromechanical Coupling

As introduced in Chapter 1, the fundamental actuation mechanism of the conjugated poly-
mer actuator is the mass transport due to the ion movement during redox that further in-

duces the strain changes in the polymer. Therefore, the early works have focused on the

23

x10-4

 

 

 

 

 

 

 

 

 

1.2 4
fi.‘ ; 9 Experimental
7; 1_ 7 ---Simulatved
o - . ,I
C ’1
.‘3 x
'8 0.8 " "7'9
m j.
a. x
8 o 6 ’g’ol 7 ,,,,,,,
h- ‘ I
a '
_t! 0 4 """""" 91/" """"""""""""""""""
52 ,x’ i
g 0.2»-- °’~ --------------------
o /
a ,x . :

0O 50 100 150
Size (mm2)

Figure 2.4: Double-layer capacitance versus actuator size.

 

 

 

 

 

 

 

 

70 1
0 Experimental
605 ............................. ---Simulated ,
\‘ 2
A \
9,504; ................................................... i
a: \
0 \\
g 40 ————— -_.,\. ...............
é 30~ - '5, ------------------------------------
. \“‘~‘
20- »-~f~~- ‘3 ~~~~~~~~~~~~~ .
‘~~‘--"‘I--
102 4 6 8 10
Width (mm)

Figure 2.5: Resistance versus actuator width (length = 20 mm).

static electro-mechanical relationship between the charges transferred into the conjugated

polymer and the corresponding strain generated. It was shown that the strain is propor-

24

N
on

 

 

 

 

 

 

 

 

0 Experimental
---Simulated
26i‘” ...... : ................
A \
e. j
a: 24— \-—-- .....
2 s
'3 \s l
.3 22L ...... V,“ .....
a, \s 7 i
m ‘sa‘~~
20" i ------- '9" """""" {p
“10 2‘0 31) 40
Length (mm)

Figure 2.6: Resistance versus actuator length (width = 3 mm).

tional to the transferred charge density in the conjugated polymer [16, 53], which can be

described as:

e: 19p, (2.17)

where 19 is the strain-to-charge ratio, p is the charge density. Equivalently, the induced

stress by the transferred charges is

where E p py denotes the Young’s modulus of the PPy layer.

2.3 Mechanical Output

When a voltage is applied across the actuator, the PPy layer on the anode side is oxidized
while the other layer on the cathode side is reduced. The oxidized layer absorbs anions

and expands, while the reduced layer gives up anions and contracts. The differential ex-

25

 

pansion thus leads to bending of the actuator, as shown in Fig. 1.3. The bending of the
trilayer beam can be predicted based on the linear elasticity theory after the swelling ra-
tios are determined for different layers. One key assumption for the linear model is that
the relationship between the stress and strain is linear, given by the small strain Young’s
modulus, which is reasonable when the strain is sufficiently small. The elastic strain is
obtained through the elementary beam theory by assuming that the strain changes linearly
in the thickness direction with the distance from an appropriately located neutral line in the
thickness direction [17]:

c(x) = K-x, (2.19)

where K' is the curvature to be determined.

When there is no external load, the beam curvature can be obtained by solving the force

and moment balance equations simultaneously

_hl hl h2
—h2 Oppy1(x)dx+/_h1 O'pVDF(x)dx+/hl Oppy2(x)dx=0, (2.20)

hi [12
_h2 GPPyl (xlxdx + [_hl GPVDFCrlxdx + fhl O'ppy2(X)xdx = 0, (2.21)

where I21 and hz are as defined in Fig. 2.3. By superimposing the bending effect term upon
the swelling term, the normal stresses in the PPy and PVDF layers along the beam direction

are taken to be

Gppy1(x) = Eppy8(x) —Eppyl9p, (2.22)
GPVDF (x) = EPVDFEOC)’ (2.23)
GPPy2 (x) = Eppy£(x) + Eppyfip, (2.24)

where E PPy and E PV D F are the small strain Young’s moduli of the PPy and PVDF, and
PPyl and PPy2 represent the reduced and oxidized PPy layers, respectively. For a symmet-

26

rical trilayer actuator, with the assumption that the thickness of the two PPy layers remains
constant, it follows from (2.20) that the neutral line is the beam center line. Therefore, the
actuation-induced curvature K can be obtained as [17]:

h2—h12
l -— —l
319 ( + hl )

K:—. .
2’11 (”’72-’11)3+Epvdf_1
hr Eppy

 

p, (2.25)

where E pvd f is the Young’s modulus of the PVDF layer, and h pvd f denotes half of its

thickness.

Note that the mechanical dynamics is not considered in the derivation, which means that
the results only hold in the quasi-static condition. One needs to consider the mechanical
dynamics of the trilayer beam at high frequencies. The actuation bandwidth of a PPy
actuator is typically much lower than its natural frequencies. For example, a sample of
dimensions 20 x 5 x 0.17 mm was measured to have an actuation bandwidth of 0.5 Hz
while its natural frequency was measured to be 65 Hz. Thus the inertial dynamics will
be ignored. Therefore, only the material damping effect is considered. The standard linear
solid model, illustrated in Fig. 2.7, can capture the viscoelasticity of solid polymer [54], and
is adopted for modeling the dynamic moduli of PVDF and PPy. The equivalent modulus

E’ in Fig. 2.7 is
l

 

 

 

+ 5+ +
E’(s) :5“ 76. 11x1 76. (2.26)
X s + _
X
Therefore one can furthermore modify (2.25) to be (2.27) to capture the higher frequency
dynanucs
h -— h
(1+-l——h2—1
319 121
- d
(1+—%7—H3+—Eli—i
1 Ema

In experiments the beam tip displacement is measured by a laser distance sensor, as

27

 

 

 

Figure 2.7: Standard linear solid model for capturing polymer viscoelasticity.

illustrated in Fig. 2.8. One can relate the measured displacement y 2 do — d to the curvature

it“ via simple geometric calculations:

1 2y

K — — = —,
r y2 + 12
where l is the distance between the clamped end and the laser incident point when the beam
is at rest. For small bending (y << I), the curvature is approximately linear with respect to
the displacement:

K x —. (2.28)

Actuator

'/

 

 

Clamp

Figure 2.8: Geometric relationship between the beam curvature and the tip displacement.

2.4 Complete Actuation Model and Experimental Verifi-

cation

2.4.l Complete Actuation Model

One can obtain the complete scalable actuation model as follows by combining Eq. (2.12),

(2.18), and (2.27):

 

 

E _ 319 .
v _ 2LWh1(h2—hl)
h d
(it—2.9.4 sz[§tanh((h2—h1)\/§)+\/§l
lE’ . fl , (2.29)
(Ir—2)3+ pvdf_1 [5+R.S3/2+R.—-s-tanh((h2-h1)\/E)
hr Ebpy C 6 d

29

 

where C and R follow the scaling laws in (2.13) and (2.14). Considering Eq. (2.28), one

can obtain the actuation model with tip displacement as the output:

 

2___3'EL__.
v _Wh1(h2-h1)
h
(h—2-)2—1 sz[fitanh((h2—hl)\/E)+\/§]
1 5 d
E’ -\/§ Jr? s . (2.30)
(h_2)3+ PVdf_1 —+R-s3/2+R-—-s«tanh((h2-h1) —)
”I Elvpy C 5 d

Consider a trilayer actuator clamped at one end, as shown in Fig. 2.9. Under quasi-static
conditions, it can be shown through moment balance (2.21) that the following equation

holds to characterize the relationship between force and displacement:

F: —71Y+sz. (2.31)

where 71 and y; are defined by

 

3
4WEPPyhpvde1+ h )3 _1+ Epvdf]

1’1 =
3L} hpvd f Eppy

2

Lf hpvdf

 

)2—11.

and E pvd f denotes the Young’s modulus of the PVDF layer.

2.4.2 Experimental Verification

Quasi—static force/displacement measurement is conducted using an experimental setup il-
lustrated in Fig. 2.10. Through the slide, a PVDF micro-force sensor [55] can measure
the actuation force at different deflection levels under a given actuation voltage, from the

free-bending configuration F = 0 to the fully-blocked configuration (v = 0). A picture of

30

 

   
 

Figure 2.9: Geometry of the trilayer actuator.

the actual setup is shown in Fig. 2.11.

Two sets of PPy actuators are used with widths of 3.5 and 6 mm respectively. In each set
there are three samples with lengths of 20, 30 and 40 mm. Given a constant voltage input,
the current can be predicted by (2.12) and thus the charge density p. Therefore the force
versus displacement curve can be computed using (2.31), where E p py = 80 MPa, E pvd f =
440 MPa [14], 19 = 1.3 x 10—10 m3 -C"'1 [16]. Fig. 2.12 shows the comparison between
the measured force-displacement curve and the model prediction for the set of actuators
with width 3.5 mm under an actuation voltage of 0.4 V. Fig. 2.13 shows the results for the
set of actuators with width 6 mm. It can be seen that in both figures reasonable agreement
between measurement and simulation is achieved. Experiments are also conducted for
other actuation voltages with satisfactory agreement with model prediction. Those results
will not be presented here due to space limitation. One of the implications of the results in
Figs. 2.12 and 2.13 is that the longer the actuator is, the smaller is the force output, and
vice versa. This fact is in agreement with the results presented in [14].

In Fig. 2.14, the measured admittance is compared to the predicted one for an actuator
of size 30 x 5 mm, while in Fig. 2.15 the same is presented for an actuator of size 40 x 5
mm. In both figures the agreement in magnitude plots is excellent. In the mean time
the match for the phase plots is good except for the relatively big discrepancy at the low-

frequency end. The low-frequency phase mismatch is likely due to the change of actuation

31

 

Stepper motor

 

Laser !
sensor "

PPy actuator

   

Figure 2.10: Schematic of the experimental setup for joint force—displacement measure-
ment

 

Figure 2.11: The setup for force/displacement measurement.

behavior in air when the solvent evaporates over time (the high-frequency measurements
are done in a short period of time and the behavior change there is minimal).

The model (2.30) for the dynamic displacement response is verified for a range of actu-
ation frequencies for two different actuator sizes, 30 x 5 mm and 40 x 5 mm. The compari-
son between the experimental measurement and the model prediction is shown in Fig. 2.16
and Fig. 2.17. The coefficients related to dynamic modulus in (2.26) for PPy and PVDF
are identified based on curve fitting: 17 = 5.02 and x = 0.83 for PPy, and n = 4.68 and

x = 0.044 for PVDF. From the figures the agreement between measurement and simu-

32

 

 

2 : f . 20mm Exp.
0 30mm Exp.‘
I 40mm Exp.
—20mm Sim.
--- 30mm Sim.
---40mm Sim.

j

 

 

 

Force (N)

_________________

 

 

 

 

"o i 2 5 4 5
Tip displacement (mm)

Figure 2.12: Force versus displacement under an actuation voltage of 0.4 V (width = 3.5
mm).

lation is good for the magnitude plots. However, mismatch exists in the low-frequency
portion of the phase plots. The cause of this, other than the possible behavior change due

to solvent evaporation, requires further investigation.

2.5 Conjugated Polymer Electromechanical Sensor

Research on the possibility to use conjugate polymers as electromechanical sensors has
been relatively limited, comparing with the extensive work on modeling of conjugate poly-
mer actuators [6, 16—19]. Wu et al. [20] investigated the sensing behavior of a trilayer
PPy beam by considering the perturbation of the Donnan equilibrium of the ion distribu-

tion by mechanical stimuli as the primary sensing mechanism. A similar viewpoint was

33

 

x10

 

 

.. 20mm Exp.
0 30mm Exp.
1 a 40mm Exp.
,,,,,,, i-.. —20mm Sim.

i --- 30mm Sim.
~---40mm Sim.—

f

 

 

 

Force (N)

 

 

 

 

.3 3 4
Tip displacement (mm)

Figure 2.13: Force versus displacement under an actuation voltage of 0.4 V (width = 6
mm).

also presented by Takashima and coworkers [21]. For better understanding of the sensing
mechanisms, however, more experiments and quantitative modeling of the sensing dynam-
ics are desired.

A preliminary model is proposed to explain the sensing mechanism, which starts with
a partial differential equation (PDE) that governs the ion redistribution dynamics subject
to diffusion and migration (due to electrostatic interactions). Two boundary conditions are
postulated: l) the ion concentration at the PPy/PVDF interface is proportional to the ap-
plied mechanical strain, as the latter directly influences the pore sizes of the PPy layer;
and 2) there is no diffusion flux at the interface between the PPy layer and the air. This
PDE can be solved numerically. However, for real-time sensing applications, it is desirable

to have an analytical model. For this purpose, we ignore the migration term of the PDE,

34

 

  

 

 

 

 

 

 

A'20 I 1’ _T ’ -9- Experimental] 1
g 3 q _ L": Simulated J 1
a, '25‘ ' 1 ' ' _- "111]
u l.
B . 1 ‘ ‘
'E-30»~-» ‘ ,p l
U} j .
g _ i
51mm, .;;;. . ... L. . l
'35?” o 1 2
1 10 10 10 10
60 . 7V1 ——fi'—' “Via—WT Y 'rr—‘*—v—‘ tr'r—f rr‘fi'}
’3‘ ' l
340' '1
a '
«I 20» 4
" ‘ l
“L ‘1 3,
G 1 “‘13
10' 10

 

10 10
Frequency (rad/sec)

Figure 2.14: Electrical admittance spectrum (size: 30 x 5 mm).

which makes the equation linear. The latter can be solved analytically in the Laplace do-
main, which leads to a transfer function that relates the open-circuit sensing voltage to the
applied mechanical deformation. The obtained solution shows good approximation to the
solution of the original PDE when the ion concentration is relatively low and matches the
experimental data for conjugated polymer sensors with different lengths and widths. How-
ever, some recent experimental results by using conjugated polymer with different PPy
thickness show discrepancy from the model prediction, which implies the model has not

captured all the sensing phenomenon.

35

 

,—

.2o .. 3 . q wig-Experimentalm

 

  

 

 

% , l—Simulated ‘ j
TJ-25 -‘*-.' . , +
'u , l
.3 ‘ 'j
-E I3ol" ‘ .. “ ‘ ‘ ‘ ‘ ‘ ‘ _ w
a .«1. * * * » . ,.
2 . . l
'35-1L “no; 1 l W2 ‘1‘“ 3
10 10 10 10 10

 

   

Phase (deg)

 

0L... .L.L L 1:13.; L E . .L. .
-1 0 1 2 3
10 0

10 1
Frequency (rad/sec)

Figure 2.15: Electrical admittance spectrum (size: 40 x 5 mm).
2.5.1 Full Sensing Model

The Nemst—Planck equation is used widely to describe the flux of ions under the influence

of both an ionic concentration gradient and an electric field [56]:

CF
2 — V — . . 2
J d( C+RTV¢) (23)
The continuity equation holds:
8C
v. : ___ 2.33
J at ( )

Since the thickness of the PPy layer is much smaller than its length or width, one can
assume that dominant changes are all restricted to the thickness direction (denoted as x

direction). This enables one to drop the boldface notation for these variables. From (2.33),

36

 

  

 

 

 

 

 

 

A 40 . 4 ;.____
m ‘ ;-9-Experimenta|]
3 .1 —Simu|ated j
5.! .
“E 0» .-
Oi .
t5 . .
E 20
10" 10° 101 102
e . . 1
r ~ 1
E400. .i
e a ‘ . ‘ J
_150i:1_._k 1-..!) 1 . 1. .11 . . . i . 2
1o 10

10 10
Frequency (rad/sec)

Figure 2.16: Dynamic displacement response (size: 30 x 5 mm).

wecan derive
2 2
8C 8 C F 8C8¢+ FCa (b) (2.34)

a: (m HEB—x 1173,22

One can relate (l) to the ionic concentrations through the Gauss’s law:

D

— = ——V (2.35)
Ke 4’
F- (C+ — C‘), (2.36)
where D denotes the electric displacement, E denotes the electric field, and ice is the di—

electric permittivity of PPy. Therefore, the following equations can be derived based on

37

 

-B - Experimental 1

 

  

 

 

 

 

 

E .
3 ‘ [717; —SirrLulated ,
§ .“ . L ‘ L p.
’E' 0i i a 1
5’ ' ‘5‘ ‘L i
g i ,, J
-2o_,wA+~~+~- 1- .1 . ~ 4
10 10 1O 10
0‘ .1 7 . ., l
a i ' % ~ - l
3 -50. “t L
3 . 3
to 400‘» in
E I]
1‘ I ,‘ ‘ , . ‘ l
.150 . . - ‘1‘ . 1 11 111~~ . 1 1 -s-~
10'1 ° 01 1o2

10 1
Frequency (rad/sec)

Figure 2.17: Dynamic displacement response (size: 40 x 5 mm).

(2.35) and (2.36):
61> _ F x 9_¢
.97 — e/<)C(5=‘)d5+ax(°*’)’ (2‘37)
32¢ F
3x7 : EC(X,I). (2'38)

In mechanical sensing, one can assume that C(x, t) fluctuates about some equilibrium con-

centration C0:
C(x, r) = C0 + C1 (x, t), (2.39)
with C0 > C1 (x, t). Further assume that the electrolyte has high ionic conductivity, which

implies :—:(0,t) = 0 in (2.37). Therefore, (2.37) can be approximated by

i‘hfl,

3): ice (2.40)

38

Combining (2.34), (2.38), (2.39) and (2.40), and ignoring the C? term, one obtains

32C] F2 8C1], F2

 

3C1 ( + + (C2+2C C )) (241)
— z u x -— . .
a: 3x2 RTKe 3.. 0 RTKe 0 0 1
Two boundary conditions are imposed on (2.41):
C1 (0,1) = KC0£0(I), (2.42)
8C1
Wlxth—hl —0, (143)

where 80(t) represents the strain at the PPy/PVDF interface. Note that x = 0 denotes the
PPy/PVDF boundary while x = ’12 — hl denotes the PPy/air interface. The first boundary
condition implies that the concentration perturbation at x = 0 has a linear correlation with
the applied deformation. The second one means that there is no diffusion flux on the other

side of the PPy layer.

2.5.2 Simplified Sensing Model

Eq. (2.41) is difficult to solve analytically. However, it is desirable to obtain an analytical
model for conjugated polymer sensors. Such a model can facilitate fundamental under~
standing of the sensing mechanisms and be instrumental in sensor design and real-time
sensing applications. An analytical model from (2.41) can be obtained by further ignoring
the terms involving C0. Equivalently, this is to ignore the effect of electric field-induced
ion migration. The approximation is valid when the nominal anion concentration C0 is low.

The approximated PDE contains only the diffusion term:

 

8C1 __ dazcl

at — 8x2. (2.44)

39

To solve this PDE, (2.44) is firstly converted into the Laplace domain

32C1(x,s) _ sC1(x,s)

 

 

8 2 d (2.45)
x
One can obtain the following analytical solution to (2.45):
3
KC \/;<h2—h1) —\/Ex
C1(x,s) = 0808 e d
s s
—<h -11) qflrh —h>
e\/; 2 +6 d 2 l
5
KC qfiwz—hn .‘f
0503 d . (2.46)

 

+
(h h (h h
e)/;(2—1)+e—\/d(2— 1)

Furthermore one can obtain the potential difference across one PPy layer based on (2.35)

and (2.36):

AV. ¢(h.s) —¢(0.s)

dFKCOEO 2

S [ S S
K" (fir/:2 hl) 1012— h.)
(hz—h1)\[tanh(( ((—h2 h1:\/:)—1]-l-::e —O(—h2 h])2. (2.47)

Due to the symmetry of the trilayer structure, the potential across the other PPy layer is

 

+

dFKCOEO 2

3K6 [fithz— h.) —(/§(h2—h1)

012— h1)\/;tanh(( ((-h2 h1) ):/:)—1]+ :——:3(f12—h1)2. (2.48)

M

 

+

40

Therefore, an analytical model for the total open-circuit sensing voltage is:

v = Av, _sz
ZdFKCOSO 2

[
SKe S S
c([ng—hnH—(GMZ—h.)
(h2—h1)\/§tanh((h2 —hl)\/§) — I].

 

+

(2.49)

From (2.49), one can obtain the transfer function relating the sensing voltage V(s) to the

applied deformation 80(3) (strain at the PPy/PVDF interface):

V(S) _ 2dFKCO

- l
K
50(5) se e\/;(—)h2 h1)+ :—\/:(h2—) hl)+
(—h2 h1)\/;tanh(( ((112— hi) +:\/;)—1]. (2.50)

The following equation holds for the strain and the curvature

 

 

8(x) : K'x. (2.51)

The strain 80 at the boundary of PPy layer and PVDF layer can be further related to the tip
displacement:

2h
£0_ _ 721), (2.52)

41

Now taking the applied tip displacement y as the input, from (2.50), one can obtain the

transfer function for the sensing dynamics:

V(s) 4dFKCOh1 2

— — [
Y(S) SlzKe S S
e\/;(h2—h1)+e-\[g(h2-h1)
+(h2 —h1)\/§tanh((h2 _h1)\/§)_1]' (2.53)

We have compared the numerical solution of the full model with the analytical model

 

derived above. In the computation, CO was taken to be 0.05 M. A sequence of sinusoidal
inputs 80(t), up to 100 Hz, was used. For each input, the numerical solution of V(t) was
computed, and the gain and phase shift at that frequency were evaluated. Fig. 2.18 com-
pares the Bode plots obtained through numerical computation with those of the analytical
model (2.50). As one can see, overall the discrepancy is not significant; in particular, as the

frequency gets high, the discrepancy vanishes.

2.5.3 Experiments and Discussions

Fig. 2.19 and Fig. 2.20 show the experimental characterization of the dynamic sensing
behavior of the conjugated polymer samples. Predictions from the model (2.53) are also
shown in the figures for comparison purposes. The parameters used for the model are listed
in Table 2.1. It can be seen that the predictions from the analytical model in general fit the
experimental data. In particular, the magnitude plots show good fitting with the experimen-
tal data and predict the decaying trend as the frequency becomes higher. Although there is
some discrepancy in the phase plots, the general trend of the experimental data is predicted
by the model.

The model (2.53) predicts that the sensing voltage is independent of the sample width.
This is supported by the experimental data in Fig. 2.19, where one can see that the sensing

behaviors of three samples, with different widths, are close to each other. The model also

42

 

—|
O

 

 

 

  
   

 

   

 

 

g. 0 , 1 1. l
3 1 , ‘ 1
5 ‘ 1 . L J
-20 o f 1.2—W- 3
10 10 10 10
0 i F 1
; — Full model
33 _20 2 ‘--- Approximated model
i “‘ ‘ ‘ ‘ i
2 -40
a . 1 1 . i 1
_60 1 1 1111 1 1 1 1 111 1 1 1 .11.
10° 10‘ 1o2 103
Frequency(rad/sec)

Figure 2.18: Comparison between the numerical solution of the full model and the analyt—
ical solution of the simplified model (2.50) when C0 is 0.05 M.

predicts that the sample length will influence the magnitude but not the phase of the transfer

function, which is again verified by the experiments (Fig. 2.20).

2.6 Chapter Summary

A full electrochemomechanical model is presented for trilayer PPy actuators. The scaling
laws for two important parameters, the double-layer capacitance and the resistance, are pro-
posed and experimentally verified. The model can thus be written in terms of fundamental
material parameters and actuator dimensions. Through experiments with actuators of dif-
ferent dimensions (length and width), the following aspects of the scalability of the model
are independently validated: quasi-static force/displacement output, electrical admittance,

and dynamic displacement response. The model will be instrumental in feasibility analy-

43

 

Table 2.1: Parameter values used for the model (2.53).

 

 

 

 

 

 

 

 

 

 

 

Parameter Value
F 9.65 x 104 C/mol
K 0.072
’12 . 85 pm
h] 55 pm
d 1 x 10—8 m2/s
K'e 5.31 x 10—10 C2N_lm_2
CO 0.1 M

 

 

01

O

_'1 .
do 01____ -_fi

 

Magnitude (dB)

0
o

C:

  

 

 

 

 

 

 

73"
Lg-ZOL
a. . Il‘l
n. ‘ ‘ iA.
_50 1 i r . 1;:
100 1 2 103

Freguency (rad/$330)

Figure 2.19: Dynamic response of conjugated polymer sensors: Experimental measure—
ment (marks) versus model prediction (line). Three samples with different widths (fixed
length: 30 mm).

sis for new applications of conjugated polymer actuators and in optimization of actuator
fabrication and design.

Besides investigating sensing properties, we also proposed a model to explain the sens-
ing mechanism of conjugated polymers sensor. The model accounts for the ion transport
dynamics within the polymer. including both ion diffusion and electric field-induced mi-

44

 

-A
<3

G

‘;__"_L_j__l

 

Magnitude (dB)

 

 

 

 

-40'

Phase (deg)

_5oi We: s .i.. .;
0 1 2
1° Fre1d)uency(rad/s¢1ag)

 

Figure 2.20: Dynamic response of conjugated polymer sensors: Experimental measure-
ment (marks) versus model prediction (line). Three samples with different lengths (fixed
width: 5 mm).

gration. A key assumption of the model is that the applied deformation prescribes the
ion concentration at the polymer/electrolyte interface. The model was further simplified
by dropping the migration term. In that case, an analytical solution was obtained in the
Laplace domain, leading to an (infinite-dimensional) transfer function model for the sens—

ing dynamics. Experimental results were also reported to support the modeling effort.

45

Chapter 3

Modeling N onlinearities in Conjugated

Polymer Actuators

3.1 Redox Level-dependent Impedance Model

It is known that the redox level of conjugated polymer has significant impact on its ma-
terial properties [22, 23, 57], which will further influence the actuation performances [34].
However, understanding and modeling the influence of redox level are inadequate in the
previous work. Therefore the ions transport phenomenon is modeled as follows by a redox
level-dependent impedance model, which is developed by including both ionic diffusion
and migration effects and by using perturbation analysis. Experiments are conducted under

various redox conditions to verify the proposed model.

3.1.1 The Governing Partial Differential Equation

The anion flux J inside the polymer is captured by the Nemst-Planck equation [56], includ-
ing both a diffusion term and an electric field-induced migration term:
C T F
= —d V _ ———V 3.1
J < c + RT m, < >

46

 

where d is the ionic diffusion coefficient, F is the Faraday’s constant, R is the gas constant,
T is the absolute temperature, 4) is the electric potential, and V denotes the gradient. The
continuity equation gives

8C’

7 = ‘V'J’

_ C‘F
= d(V-VC +V'('EFV¢))’ (3.2)

where V- denotes the divergence.

Since the thickness of the PPy layer is much smaller than its length or width, one can
assume that, inside the polymer, J, D, E, and other changes are all restricted to the thickness
direction (denoted as x direction). This enables one to drop the boldface notation for these
variables; in particular, D and E will be used to represent the electric displacement and the
electric field (along the x direction). Furthermore, (3.2) can be simplified as

aC— azc- _1: ar 34: C‘F 82¢

71': (79:,2—+RTa—x§;+ira—,2‘)' (33)

A pure delay is adopted to capture the oxidation dynamics, i.e.,
C+(x,t) = C~ (x,t — T0), (3.4)

where T0 is the time taken for an anion A’ to react with PPy. Finally, there are two
boundary conditions for (3.3): 1) C“ at x = 0 equals the concentration in the PPy/PVDF
double layer; 2) The ionic flux J (x,t) at x = hz -— h] (the other boundary of PPy layer)

vanishes, assuming no ions leaking outside.

Equations (2.35), (2.36), (3.3), (3.4), and the boundary conditions form a complete

description of the ionic dynamics inside PPy.

47

3.1.2 Perturbation Analysis

a¢ 32¢
From (2.35) and (2.36), one can express 5; and 3x? as
39’ _ F x + — £9
79.; _ ’6 O (c _c we: + ax (0,1) (3.5)
8% F
—— = —— CL -C' . 3.6
ax, x.‘ > < >

Supposing that the double layer charging process is relatively fast comparing to the ionic
dynamics in PPy (which is usually the case), the net charge inside the double layer is zero.
Further assume that the electrolyte in PVDF has good ionic conductivity, and thus the

electric field within is negligible. Under these assumptions, one can apply Gauss‘s law to

a small volume containing the double layer and conclude @(OJ) 2 O in (3.5). Plugging

3x
(3.5), (3.6), and (3.4) into (3.3), one obtains

 

aC— 82C“ F2 _ _ _ .
7 = (3x2 —RTKeC (C (x,t—T0)—C (x,t))
F2 3C‘ x __ _
—m—5;/O<c (ct—Tore (mat). (3.7)

Eq. (3.7) is a nonlinear integro-differential equation that cannot be solved analytically.
We introduce perturbation analysis to linearize it around a nominal concentration Co, which

represents the redox level of the conjugated polymer. To proceed, let
C‘ (x, t) = C0 + 8C1— (x,t), (3.8)

with 0 < 8 << 1. In (3.8), 8C1— (x,t) represents the small perturbation imposed on C0.

2

Plugging (3.8) into (3.7) and neglecting terms involving 6 , one gets the following lin—

48

earized equation that depends on the redox level CO:

 

_. 2 __
8C1 _ d 8 C1 1:2
8x2 RTKe

T _ _ com-(11,: — r0) —Cl_(x,t))) . (3.9)

To obtain an analytical solution, one can convert (3.9) from the time domain to the Laplace

domain: 2 2
a C“ F C'—
l 0 —ST —
- l—- 0 .10

 

SC] 2 d
wheres is the Laplace variable.

Eq. (3.10) can be further written as

32C]— _ S—KICOU -e_ST0)

 

 

 

_, 3.11
8x2 d C1 ( )
with the constant
9 sz
1 _ RTKe'
The solution of (3.11) has a generic form
C1—(x,s) = a1(s)efi(3)x + a2(s)e—B(s)x, (3.12)
with
— K C 1— ‘370
B(S)=\/S 1 0(d e ). (3.13)

The functions (11 (s) and a2(s) are determined by the boundary conditions. The boundary

condition at the PPy/PVDF interface, x = O, is
Cl—(O,S)=Cdl(s), (3.14)

where Cd1(s) denotes the ionic concentration in the double layer. Combining (3.12) and

49

(3.14) gives
a1 (Si-(12(8) =Cdl(s). (3.15)

The other boundary condition, at x = hz — hl , requires that the ionic flux be zero. Using

 

(3.1), (3.5), and (3.8), and ignoring terms involving 82, the zero flux condition at x =
hz — hl reads
aC‘ C F2 112 —h1
1 0 —ST —
h—h, ——1— 0/ ,d=0. 3.16
8x12 1s>+me< e )0 ales): < 1

Plugging (3.12) into (3.16), one gets

s<a1 «WWW 4’1) — a21s>e*5(5)<"2 41)) = (s —dB2(s))(a1(s) — a21s».

 

 

(3.17)
Combining (3.15) and (3.17), one can solve for on and a2:
—l3(3)(h2"h1)_( —dB2( ))
se s s _
(11(5) = A(S) Cdl(s), (3.18)
B(S)(h2-h1)_ _ 2
012(3) = 5" A(s) (3 ““6 (mean). (3.19)

with A(s) defined as

A(s) = s (eWW'z “h1)+e"[3(~‘)(h2 41)) -—2(s—dfi2(s)). (3.20)

3.1.3 Impedance Model

Fig. 3.1 shows the equivalent circuit for the proposed impedance model. The total current
1 consists of three components, the double layer charging current 1C, the diffusion current
ID, and the migration current 1M. In the figure C denotes the double layer capacitance,

R1 is the ohmic resistance of the oxidized PPy layer, and R2 denotes the resistance of the

SO

reduced PPy layer, the PVDF (electrolyte) layer, and the contacts. ZD and ZM represent
the diffusion dynamics and the migration dynamics, respectively. Each of the current com-
ponents is discussed next. We will be concerned with the voltage/current associated with

the charge concentration perturbation 8C1— .

 

 

 

 

 

Figure 3. 1: Equivalent circuit for the impedance model.

Double layer charging current 15. Double-layer charges are established at the PPy/PVDF
interface in response to an applied voltage, which can be captured equivalently through a
double layer capacitance C. The total stored charge in the double layer is EFA5Cd1(s),
where A and 6 denote the area of PPy layer and the thickness of the double layer, respec-

tively. The time derivative of the stored charge gives IC:

[C(S) = EFAast[(S). (3.21)

Difi‘usion current ID. The diffusion flux at the PPy/PVDF interface (i.e., x = 0) is given

8C
by the Fick’s law and equals —-d£—l—(0,s). But an electron is not released until the

8x
51

oxidation takes place after TO delay. 10 is thus expressed as

 

10(5) = —£FAde_ST0§a£x1:(O,s)
= —eFAde—ST0/3(s)(al(s)—a2(s)) (3.22)
5 -8T 3 [3(5)(h -h )_e—fi(5)(h -h)

where the second equality follows from (3. 12).

Migration current 1M. [M is induced by the net charges Qp inside the PPy layer. Ap-
plying the integral forrn of Gauss’s law to the PPy layer and recalling E (0,s) = 0, one can

obtain

8
Qp(s) 2‘ AD(h2 —hl,S) 214195012 —h1,5) = _AK€£(h2 _hl’S)

_ h2—111
= ——£FA(l——e 5T0)/0 cl-(§,s)d§ (3.24)

= —£FA(1 —e—ST0) ,6

 

where (3.24) follows from (3.5) and (3.25) follows from (3.12). [M is then expressed as

114(8) = SQP(S)

= —8FAS(1 —e_ST0)alefi(h2 —hl) _ (be—[3012 —h1)—(a1 T (”2)

 

 

[3
£FAds(l — 6—ST0)fi(s)(efi (5)012 - hl) _ e‘5(3)(h2 - ’11))F
2 _ A(s) ”611(5)‘
(3.26)
Since the total current is
1(5) =IC(s)+ID(s)+IM(s), (3.27)

52

and the total potential drop (see Fig. 3.1) is

[c(S)

V(s) 2 SC

 

+ R11C(s) + R210), (3.28)

the impedance of the trilayer conjugated polymer actuator can be calculated using (3.21),

(3.23), and (3.26):

5A(S)(1 +sR1C)
SC (6M3) “”3 ‘5)(23—ST0 — W" (W2 ‘ ”1) —e-B(s><h2 — m )1).
(3.29)

 

2(3) 2 R2 +

The model captures the dependence on the redox level C0 through [3(5) defined in (3.13).
The model is consistent with the diffusive-elastic-metal model [16] when the migration

dynamics and the oxidation dynamics are ignored.

3.1.4 Experiments and Discussions

The samples used in the experiments are fabricated by the Intelligent Polymer Research
Institute at the University of Wollongong, Australia, and a description of the fabrication
process can be found in, e.g., [58]. The PVDF layer is 110 pm thick, while each PPy
layer is 30 pm thick. The electrolyte used is tetrabutylammonium hexafluorophosphate
(TBA‘TFE) in the solvent propylene carbonate (PC), where PFg is the mobile anions that
can transfer into and out of the PPy layer. Each actuator is soaked in the electrolyte for two
hours before testing. The electrolyte stored in the PVDF layer enables the actuator to work
in air for several hours. Notice that both PPy layers were already doped with certain PFg
during fabrication.

A dSPACE system (DSl 104, dSPACE Inc) is used for data acquisition and processing.
A trilayer actuator is clamped on one end, where electrical contacts are made using copper
tapes coated with silver. The applied actuation voltage and the corresponding current are

measured. All the experiments are conducted at temperatures between 22 °C and 24 °C.

53

The samples were predoped with PFg during fabrication, and the nominal concentration
C0 in the absence of DC bias is estimated to be 1000 mol/m3 based on the deposition
conditions. Different C0 is obtained by applying DC biases of 0.5 V and 1 V, respectively,
where the change in C0 is calculated by dividing the transferred charges before reaching
the steady state by the PPy volume. Sinusoidal voltages of amplitude 0.05 V and frequency
0.08 - 200 Hz are superimposed on the DC voltage, as perturbations, for the measurement
of impedance spectrum. Electrolyte is prepared for two concentrations, 0.35 M and 0.25
M, for two sets of experiments. In each set, three samples of same dimensions and from
the same fabrication batch are prepared and subject to DC voltages of 0 V, 0.5 V, and l V,
respectively. The samples prepared with 0.35 M electrolyte have dimensions 15 x 5mm,
while those with 0.25 M electrolyte have dimensions 30 x 6mm. For the convenience of
referencing, the samples with 0.35 M electrolyte are named Sample 1 (0 V), Sample 2 (0.5
V) and Sample 3 (1 V). The samples with 0.25 M electrolyte are named Sample 4, 5 and 6

in the same manner.

Some parameters of the model (3.29) are physical constants or can be measured di-
rectly, including gas constant R = 8.3143 J/mol - K, Faraday’s constant F = 96487 C/mol,
absolute temperature T = 300 K, and PPy thickness h2 —— hl = 3 x 10’5 m. Based on stud-
ies on the dielectric properties of conjugated polymers [59—61], the dielectric permittivity
Ice is taken to be 6080, where so represents the permittivity in vacuum. The double layer
thickness 5 is difficult to measure, and an estimation of 25 nm is used based on the range

reported in [51].

Experiments are conducted to estimate the double-layer capacitance C, the nominal
redox level CO, and the resistances R1 and R2. If we assume that the internal capacitance of
the polymer per unit volume is close to the double-layer capacitance per unit volume [16],

C can be estimated using
5

-— 3.30
hz-hi’ ( )

_9
_U

54

where Q is the charge transferred under an applied DC voltage U. Fig. 3.2 shows the trans-
ferred charges under DC voltages from 0.2 V to 1.2 V for different samples, from which
Q/ U is estimated to be 0.13 for Sample 1, 2, 3, and 0.37 for Sample 4, 5, 6. Estimated
values of C based on (3.30) are listed in Table 3.1. Also listed in the table are the values
for the resistance R1, which are estimated with a conductivity of 200 S/cm based on the
studies on PPy doped with PFg [52, 62] under room pressure and temperature. The results
in Fig. 3.2, together with the sample dimensions, also enables us to compute the nominal
anion concentrations (redox level) Co, which are listed in Table 3.2. Finally, to measure
R2, a sinusoidal input 0.05 sin(5007tt) V is applied. Note that the small amplitude ensures
minimal influence on C0. At this high frequency, the trilayer actuator is almost resistive
and the measured impedance is approximately R1 + R2. It is observed that R] is negligible
compared with the measured resistance, thus R2 is chosen to be the measured resistance.
Since the redox level has impact on the resistance of the reduced PPy layer, values of R2
will be different for different samples, as shown in Table 3.2. The only remaining parame-
ters are the diffusion constant (I and the time delay of oxidation To, which will be identified

through curve fitting.

Table 3.]: Estimated values for C and R1.

 

 

 

 

 

 

 

Parameter Sample 1, 2 and 3 Sample 4, 5 and 6
C (F) 1.03 ><10-4 3.17 ><10—4
R1 (52) 2 x 10—5 8.3 x 10-6

 

Experimentally measured impedance spectra have been used to verify the proposed
redox level-dependent impedance model. For comparison purposes, the diffusive-elastic-
metal model [16] has also been used. Besides the parameters estimated in the previous
subsection, the only free parameter for the diffusive-elastic-metal model is the diffusion
constant d. For each set of experiments (0.35 M or 0.25 M electrolyte), the value of d is

estimated by fitting the diffusive-elastic-metal model to the measured impedance spectrum

55

Charges (Columb)

 

 

 

 

 

 

 

 

i X Sample1,2and3 i
—- Simulated 9
05* o Sample4,5and6 ~ ; X’“
--- Simulated I
l 1 ’16
0.4L" ~~—-~fk-’/~r ~J;——-— —1
: ,x’6 '
0.3~~ -, I! ‘ r,
1 ’2’6
I’I
o.2—-- -. 7(6--- -
.Iz’
3 /°
0.1 """ “ 7“ f" ......................... _.
.I‘T .
o r’ 1‘ l 1 1 ,
0 0.2 0.4 0.6 0.8 1 1.2 1.4
DC voltage (V)

Figure 3.2: The relationship between transferred charges and applied voltage for different

samples.

Table 3.2: Estimated values for CO and R2.

 

 

 

 

 

 

 

 

 

C0(mol/m3) R2152)
Sample 1 1000 34.0
Sample 2 1323 46.3
Sample 3 1646 56.7
Sample 4 1000 26.6
Sample 5 1363 30.2
Sample 6 1726 50.3

 

 

 

under a DC voltage of 0 V. The same value will be used for the proposed model as well. The

last parameter, T0, for the proposed model, is obtained by fitting the measured impedance

spectrum under a DC bias of 0.5 V. Then the measured spectra under DC voltages of 0 V

and 1 V are compared to the predictions by the redox level-dependent model, without fur-

ther tuning of any parameters, thus serving as independent checks on the proposed model.

Note that the predictions by the diffusive-elastic-metal model do not vary with the DC bias.

56

Fig. 3.3 through Fig. 3.5 show the results for Sample 1 through Sample 3. Using the
aforementioned methods, the values of d and T0 are identified as: d = 2.5 x 10' 11m2 /s,
T0 = 1.9 ns. It is found that, while the two models are comparable at a low redox level,
the proposed model shows clear advantage at higher redox levels, as can be seen in Fig. 3.4
and Fig. 3.5. Consistent results are obtained for Sample 4 - Sample 6. Here d and T0 are
estimated to be d = 1.5 x 10; 1 1m2 /s, T0 = 4 ns. It is again seen that the proposed model is
able to capture the redox-level dependent impedance, as shown in Fig. 3.6 through Fig. 3.8.

This work, for the first time, incorporates the effect of redox level into the dynamics of
conjugated polymers in an integrative way, which enhances fundamental understanding of
conjugated polymers and and facilitates the use of nonlinear control methods for effectively
controlling these materials. The proposed method applies to general conjugated polymers

as well.

 

-)(- Experiments
--- Diffusive-elastic-metal model
— Redox level-dependent model

 

J:
tn

    
 

‘1

 

 

é
O

 

 

Magnitude (dB)
9.:
-1 _sLfi___

 

 

 

 

 

 

 

10" ° 10
9 e.
’51
0 ‘L
3 .
o -2or ..
U1 .
“I
S
n.
_40 1 11‘1 1 11 1 i 1 1
10'1 10° 10“ 10‘

10‘ 102
Frequency (rad/sec)

Figure 3.3: Model predictions versus experimental impedance for Sample 1 (0.35 M elec-
trolyte, 0 V DC voltage).

57

 

“TH—7' ‘ _F‘fii -)t- Experiments 1

--- Diffusive—elastic-metal model 1
i L:— Redox level-dependent model
0. _— q—"T— ___f ”‘t‘éfif

  
  
 

h
D

 

  

 

Magnitude (dB)
_1 Cage 33?:

Phase (deg)
'8

 

10°

10‘ 102
Frequency (rad/sec)

Figure 3.4: Model predictions versus experimental impedance for Sample 2 (0.35 M elec-
trolyte, 0.5 V DC voltage).

3.2 Nonlinear Mechanical Model Based on a Swelling Frame-

work

The complete actuation model structure has been discussed in Chapter 2. Extensive works
have been done to understand the dynamic relationship between the amount of transferred
ions and the applied potential. Another important aspect is to predict the mechanical output
based on the swelling introduced by the transferred ions. The classical beam theory has
been applied to model the bending curvature for bilayer PPy actuators [15, 27] and trilayer
actuators as shown in Eq. (2.27). However, the model is only valid when the ion movement-
induced swelling is small so that the following two assumptions hold: 1) the strain and
the stress can be related linearly; 2) the geometry nonlinearities can be neglected. The
tensile strength experiments have shown that the strain and stress relationship of PPy films

becomes clearly nonlinear as the strain is increased over 4% [29,30]. These results indicate

58

 

   
  

 

 

A 45 w] -)(- Experiments
% 1 1 1 { --- Dittusive-elastic-metal model
V ,1 —- Redox level-dependent model
0 40 . 1
1: ‘ “.1 . .. i
= . .
on ‘ 1
“a 35 W
a: . . .
g “ 1 1
3o . . 1. 1 1
10'1 10° 101 1o2 103 10‘

4 _O
i

 

Phase (deg)
”e

 

at
°I

—.
_A
c i

o
_A
O

b

10 10
Frequency (rad/sec)

Figure 3.5: Model predictions versus experimental impedance for Sample 3 (0.35 M elec-
trolyte, 1 V DC voltage).

that the linear model based on the assumption that the stress varies linearly with the strain
becomes invalid as the deformation gets large. Therefore, it is imperative to develop a
nonlinear model to predict the actuator performance for applications where large actuation
is involved.

Large swelling and bending present challenges to the mechanical model based on the
linear elasticity theory. Firstly, when the deformation is large, the swelling effect and the
bending effect are coupled and thus cannot be superimposed as those in (2.22) and (2.24)
because of the geometric nonlinearities. Secondly, the strain-stress relationships of PVDF
and PPy films become nonlinear as the strain increases [29,30,63,64]. Thirdly, under large
swelling, the symmetrical trilayer arrangement generates a neutral line that is no longer at
the center line due to the geometric nonlinearity. Therefore, the nonlinear elasticity theory-
based method is introduced in this section to model large bending motions.

We use v1 and V2 to denote the swellings of the reduced and oxidized PPy layer, respec-

59

 

 

    

 

8 NEW—aw. (TE—fitments
'o --- Diffusive-elastic-metal model 1
I 35- , — Redox level-dependent model 1
g , 1 . .1
'E 30- . ~3
9 1 '-
E 25 L-A_141111.a; gAw17
-1 0
10 10

 

 

Phase (deg)

 

.,1_x_1_1_11 , 1

 

L+A+A4¢J

 

d
°-

1 ° 101
Frequency (rad/sec)

Figure 3.6: Model predictions versus experimental impedance for Sample 4 (0.25 M elec-
trolyte, O V DC voltage).

tively. The swelling is defined as the ratio between the volume after electrical actuation and
the original volume. If v,- > 1, the volume is increased; if Vi < 1, the volume is decreased.
Assuming that the volume change is proportional to the transferred charge density, one can

calculate V1 and v; at the steady state

V0 — k2PV0 :

V1 V0 l—kzp, (3.31)
v +k V
v2 : 0V—02p0:1+k2p, (3.32)

where V0 represents the original volume. The value of k2 can be determined by experimen-
tal data, which has the order of 10”10m3 -C_1 [16]. When one PPy layer is reduced and
repels dopants, the other PPy layer is oxidized and accepts dopants. Therefore, the PVDF

layer is merely an ion conducting layer, and we assume that its volume does not change.

60

 

 

 
    

 

 

 

 

 

 

 

 

 

 

 

A 40 f a . .. -)t- Experiments
g --- Diffusive-elastlo-metal model
v 35 _, — Redox level-dependent model
8 :‘..~ ‘::':::: :::::;::: :::":::
3 .3: : .4 " '1: 1: ' : ':.'
'E 30"" “"r"7"“1“*'r“itr: """" . . ."I‘
U? "z : 11': ':.':':: ' :::::
g ‘1:1 1:1 ll llll t|ll
25 .1 4 11110 1 11111 L 1L1112 11 11 3
10 10 10 10 10
0
A
a... .
V .
ca -20- j
g 1
D. 31: .. ‘f"f:'.' :.'.L§.'ill i
_30 1 .111111 : 1:11;;1'1 ; 11:11:11 1
-1 0 2 3
10 10 10

10‘ 10
Frequency (rad/sec)

Figure 3.7: Model predictions versus experimental impedance for Sample 5 (0.25 M elec-
trolyte, 0.5 V DC voltage).

3.2.1 Finite Strain Tensors

In linear elasticity the infinitesimal strain tensor is used to capture the structural deforma-
tion, where “infinitesimal” implies that the theory is only valid as the displacement gradient
is vanishingly small. However, the strain can be significant in many applications. Thus the

finite strain tensors are introduced in nonlinear elasticity, which will be adopted below.

Consider a trilayer beam that bends because of differential swellings in the two outer
layers. The reference configuration is in Cartesian coordinates, while the deformed config-
uration is in cylindrical coordinates for the convenience of modeling the bending. This is
shown in Fig. 3.9, where x represents the thickness coordinate, y represents the length co-
ordinate, and 2 stands for the width coordinate. The axes in the cylindrical coordinates are
labeled by r, 9 and g, which represent the radius, azimuth, and width coordinates, respec-

tively. Because the length of the beam is usually much larger than the width, it is assumed

61

 

 

      
 

 

 

 

 

 

 

 

 

 

 

 

a 45 TTT -)t- Experiments 1
3 :3.‘ --- lefusive-elastlc-metal model
3 40- 1 — Redox level-dependent model ~
:1 ; 1'1" “ ":':
I: l ‘ III I I
c 35' - 1 . 1..
0’ . ‘. . : . . ..
g l I'l t l l i l t 1 1
30 .1 10 441111 1 L112 1 ll 3
10 10 1O 10 10
A
i’ -1
V
3
m -2
.c
a :.:' .:1 .1.....: ':':::." ':. 1.
_30 : .:.:..11 . :1:.:.:: 1 :.:-.:::: ; 1:11;;
-1 0 1 2 3
10 10

10 1
Frequency (rad/sec) 0

Figure 3.8: Model predictions versus experimental impedance for Sample 6 (0.25 M elec-
trolyte, l V DC voltage).

that the there are no changes along the width direction 2, which means g = z. The bending
angle 9 is considered to be proportional to the position along the y axis in the reference
configuration,

9 = ay, (3.33)

where a is a proportionality constant that will be determined by the analysis. The bending
radius r is assumed to be independent of y and z,which means r = r(x). The orthonormal
basis vectors in the Cartesian coordinates are denoted by ex, ey, and ez, and they are

denoted by er, e9, and eg in the cylindrical coordinates. By definition, these vectors are

62

represented by

 

 

 

 

 

 

 

 

 

 

|/\ l/\
:11
|/\

.5‘

N
l/\ .<
P‘ |/\

|/\ N|- .3“
N

O

r=r(x)
6=ay
§=Z

Figure 3.9: Illustration of the reference configuration and the deformed configuration.

For ease of analysis, the dyadic product operation is denoted by the symbol ®. The

meaning of ® is illustrated as follows:

, , xm X1y2 X1Y3
(x1 X2 X3 ) ®(>’1 Y2 >3 = sz1 ma W3 - (3'34)
«W1 X3>’2 x3>’3

This gives the matrix representations

CO
CO
OO

O
O
O

(3.36)

O
O

€9®ey =

O
O
O

O
O
O

(3.37)

O
O
O

eg®ez =-

O
0
fl

Suppose that the deformation takes X = (x, y,z)T in the reference configuration to ‘1’ =

(r, 6, g)T in the deformed configuration. The deformation gradient is defined as

8‘11
: — : F .
F 8X (:2) (1‘? dX, (3 38)

where

(1‘? : dr-er + rd6e9 +dgeg,

dX = dxex+dyey+dzez.

64

The deformation gradient is written as

dr
F = Eer®ex+rae9®ey+eg®ez

dr
3; 0 O

= 0 rec O - (3.39)
O O 1

The left Cauchy-Green deformation tensor is [65]

B = FF’= 12er®er+lge9®e9 +eg®eg

3,2 0 0
= 0 Ag 0 , (140)
0 O 1
where
A Ag it Are: (341)
r_dx’ 9— ' '

For incompressible materials, the deformation satisfies the constraint that the volume is not

changed by the bending after swelling, which means

det F = v, (3.42)

where V is defined to be the swelling ratio in the different layers,

v1, for the reduced PPy layer
V = 1, for the PVDF layer - (3-43)

v2, for the oxidized PPy layer

65

From (3.81), the above equation can be written as

dr v
— = — 3.44

dx ar’ ( )
implying that it is a function of r, a, and v. One can integrate (3.84) and obtain the

following equation for the reduced PPy layer:

 

x
r(x)= \/r%+£-/ h Vldx, —h2<x<—hl, (3.45)
_ 2

where r1 is defined as the radius at the inner surface of the beam, as shown in Fig. 3.10.
Since the swelling is assumed to be uniform over the reduced PPy layer, one can simplify

(3.45) to be

 

2v h
r(x)=\/r%+——l—():x:—2—), —h2<x<—h1. (3.46)

Similar equations can be obtained for the PVDF and the oxidized PPy layers:

 

 

2 h

r(x) = \/r%+—(%-l—)-, —h1<x<h1, (3.47)
2v —h

r(x) = \/r%+—-——2(; 1), h1<x<h2, (348)

where r2 is defined as the radius of the boundary between the reduced PPy layer and the
PVDF layer, r3 as the radius of the boundary between the PVDF layer and the oxidized
PPy layer, and r4 as the radius of the outer surface of the beam. The expressions for r2, r3

and r4 are thus

 

 

2v h -—h

r2 = \/r%+——‘((Zx 1), (3.49)
4h

r3 = (/r%+—a1. (350)
2v h —h

r4 = \/r§+————2(: 1). (3.51)

66

 

Figure 3.10: Definition of the bending radii at different locations.

3.2.2 Stresses

For an isotropic incompressible material, the Cauchy stress tensor depends on the elastic
energy density function W as follows [65],

_23W 618/

__ __ _ 2 _
—V[al1 + 912(1113 B )1 p1, (3.52)

where I is the identity matrix. Here p has the interpretation of unknown hydrostatic pres-

sure. 11 and 12 are the invariants defined as follows,

11 = trace(B) = 1+ 134.2% (3.53)
12 = %[(trace(B))2 — trace((B)2)] = 3,2 + 33 + 3,215. (3.54)

67

Since [1 and 12 are functions of Ar, 19, which are in turn functions of r, a, and v, it is
possible to write

W(11,12,v) = W(Ar,19,v) = W(r,a,v). (3.55)

In the development that follows, each of these is useful in different places.

Combining (3.52), (3.53), (3.54), and (3.55), one can verify that the principal Cauchy

stresses are given by

Ar 8W
2 —- —— — 3.56
O'rr v 33.,- P» ( )
19 8W
: —— — I O 7
0'99 V 82.9 p (3 5 )
In the absence of body forces, the equilibrium configuration gives
diva : 0, (358)

which holds within each of the three layers. From (3.44) and (3.56), 0'" is a function of r

only. Eq. (3.58) in the er direction then gives [31]

dO'rr
dr

 

1

Subtracting (3.57) from (3.56) and utilizing (3.55), one has

r (9W
Orr — 0'66 2 —;"W (3'60)

We will use “PPyl” to denote the reduced PPy layer, and “PPyZ” to denote the oxidized
PPy layer. Considering the boundary condition dip), 1 (r1 , a, v1) = 0 and combining (3.59)

and (3.60), one obtains the following expression for the principal Cauchy radial stress in

68

the reduced PPy layer:

PP'l 1 ,
Orr ’ = Vl-(prflvflwn— WPPyl (r1,a.v1))- (3.61)

From (3.60) and (3.61), the principal Cauchy hoop stress in the reduced PPy layer is

1 , . aWPPyl
O'PPyl = —(WPP}l(r,a,v1)—WPP)1(r1,a,V1)+r

Similar analysis can be used to derive the stresses in the PVDF and the oxidized PPy layer,

which results in

GSVDF = WPVDF(r,a)—WPVDF(r2,a)+0£VDF(r2,a), (3.63)
p 2 1 . P 2
GrrPy = v—(WPP)2(r,a, v2) — WPPy2(r3,a,v2)) + orrpy (r3,a,v2)(3.64)

2

where O£VDF(r2,a) and 05Py2(r3,a,v2) specify the interface values. Note that we

have used the assumption of no swelling for the PVDF layer in (3.63). When there are no
fractures or delamination between different layers, 0'"- on the interfaces of these layers is

continuous, which gives

VDF PPyl PPyZ
r Ir = r2

P 0
Or :orr lr:r2 :' Orrer, or,“ lr=r3:G£-V

DFlr = r3 =3 O'rrlr3

It follows from (3.61) and (3.63) that

1 PPyl

O'rrlrz = 71% ‘WPPyl

(r2,a,v1) (r1,a,vl)),

on», = WPVDF<r3,a> —— WPVDFtrz, a) +

1 .
v—1(WPP}l(r2,a,v1) —WPPy1(r1,a,V1))-
Similar to (3.62), the following equations for hoop stresses in the PVDF layer and the other

69

PPy layer can be derived,

USE/BF = C£VDF+r————ar , (3.65)
aPPyZ — oPP"z+L~—awppyz (3 66)
“96 ‘ n“ v2 ar ' °

3.2.3 Equilibrium

The force and bending moment at any cross section of the beam are zero at the equilib-

rium, which is similar to (2.31) and (2.21) but takes different expressions in the cylindrical

coordinates:
r r
[2 O'rggyld +/ 3oggDFdr+/4aggy2dr=o, (3.67)
r1 r2 r3
r r
fzo oggylrdr+/ 3 oggDFrdr+/;4 O'Ppyzrdr20. (3.68)
r1 2

To capture the nonlinear strain-stress relationship of the PVDF and PPy as the strain in-
creases [29, 30, 63,64], the energy functions for PVDF and PPy layers are assumed to be of

neo-Hookean types [66], which means that they take the general form of

w = 521—(11 ~3v2/3), (3.69)

70

where it is the shear modulus. It therefore follows from (3.41), (3.84), (3.53), and (3.54)

that
2 3
pryl(r,a,V1) = ’HPPy1(a—lv)2+ +(0‘ W121” 31/1/11 (3.70)
vamna) = it‘PVDFl‘a‘r) +1ar>2~21 (3.71)
1 v 2 31
WPP)12(rvavV2) = ‘2‘flPPy[(a_i)2+(ar)2+1”3V2/ 1. (3.72)

The shear modulus 1.1,- is related to the small strain Young’s modulus Ei by u,- = Ei/ 3,
since the constraint (3.44) implies a Poisson ratio of 0.5. It is assumed that the energy
functions for PVDF and PPy layers are of neo-Hookean types, because the neo-Hookean
model contains the quadratic terms of the mechanical deformation gradient, and it has been
adopted to model the nonlinear strain-stress relationships of different smart materials, such
as PVDF [67, 68] and ionic polymer-metal composite (IPMC) [69]. Substituting (3.70),
(3.71), and (3.72) into (3.62), (3.65) and (3.66), one obtains

V2
PP l UZPP l 1

GM = ——yvlr[a2(32— r2)——(—2+¥)], (3.73)

11 1 l
053” = ———2”VZDF1a2(3r2-r%>—-‘—,(r—2+;§)1+orr1r2, (3.74)

2
PM _ “PPy 232 2 V§1 1 375
099 _ 2V2 [a (r -—r3)—- a2(_ r2 +g)]+0'rrlr3- (- )

From (3.73), (3.74) and (3.75), the left hand side of (3.67) is expressed as

/r20_PPy1d + ’3 OPVDFdr+ [’4 (HP/32d
,1‘799 ,2 99 3 “99

r
= ;1(WPPy2('4:avV1) " WPPy2(r3, a, V1 )) + Cl'rrlr3 ' (’4 — ’3)

+r3(WPVDF(r3ia1vl)“WPVDF(r21a)V1))+Grrlr2 ' (r3 _ r2)
“Fez—(pryl (f2, 01, V1) — WPPy1(r1 , (IA/1)). (3.76)
1

71

Substituting (3.70), (3.71), and (3.72) into (3.76), one finds that the force balance (3.67)

becomes

 

2
“PPyV211222HPVDF111222
372—;(3*;§)+a (r4—r3)]+ 2 1&3<:*:)+a(r3—r2)1

4 3 - 3 2

2
1” V l l
fl[—1-(— - —) + (120% —r‘17')] = 0. (3.77)

2v 222
lar2 rl

Note that (3.77) also ensures the boundary condition 055,204, a, v2) 2 0. From (3.73),

(3.74) and (3.75), one can find that the moment balance (3.68) becomes

“P133244 V12'1'122V12 2
v1 {20‘ (rz—r1)+(—07) ln6+2(r1—r2)[(arl) +(ar1) 1}

3 v r l 1
+#PVDF{ZO‘2(T§—’g)+(';1)21n%+§(r%—r§)1(a;3)2+(a’2)2l}+

“—Z’Zéazpj — r31) + (3191113 + $63 — r2)[(%)2 + (063)21} = O.(3.78)
In summary, the problem is now formulated as solving the nonlinear equations (3.77) and
(3.78) simultaneously to obtain the two unknown variables r1 and a. These two variables
capture the deformed configuration for given swellings V1 and v2. Note that in the linear
elasticity theory-based method, the force balance is automatically satisfied for a symmet-
rical trilayer structure. However, (3.77) is required in the nonlinear elasticity theory-based
method. Newton’s method is applied to numerically solve (3.77) and (3.78) with the aid of
(3.49)—(3.51). Thus a large bending strain model is established for the trilayer conjugated

polymer actuator with actuation voltage as the input and bending radius as the output.

3.2.4 Experimental Verification

The PPy layer is 30 pm thick, and the PVDF layer is 110 pm thick, which implies that the

values of the parameters h] and I12 are 55pm and 85pm, respectively. The electrolyte used

72

is 0.05 M tetrabutylammonium hexafluorophosphate (TBA+PF'6”) in the solvent propylene
carbonate (PC). Two actuators of different dimensions are cut from the fabricated sheet.
The size of two samples are 12.8 x 5.0 mm2 (Sample 1) and 31.2 x 6.0 mm2 (Sample 2)
respectively. Each actuator is stored in the electrolyte before experiments for five hours to
ensure that the PVDF layer is sufficiently soaked with electrolyte.

The relationships between the input voltages and the transferred charges for the two
different samples are shown in Fig. 3.11. The constant k1 in (2.15) can be determined from
the experimental data, which is found to be 0.0936 C / V and 0.1801 C / V, respectively, for
Sample 1 and 2. The value of the swelling-to-charge ratio K2 is obtained through fine tuning
based on the reported value in [16], which results in k2 = 1.12 x 10—10 m3 -C—l. The
small strain Young’s moduli of PPy and PVDF are taken to be 60 MPa and 612 MPa [13],

respectively.

0.35

 

 

1 O ISample1
03-... .- .. . .. _ x Samplez

 

 

4L

 

0.25 ........ ______

I
I
I
I
1
1
1
1
1
1
1
,
1
1
I
1
1
1
I
l
1
I
l
1
l
1
1
1
l
1
t
1
1
I
I
I
I
1
1

I
d
"I

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

Charges (C)

 

 

 

0:5 1 1:5
Input voltage (V)

Figure 3.11: The relationships between the input voltages and the transferred charges for
the two different samples.

The predictions of r1 and a on the basis of (3.77) and (3.78) are shown in Fig. 3.12

— Fig. 3.15 for the two samples. The experimental data for the bending radii are also

73

compared with the predictions of r1 in Fig. 3.12 and Fig. 3.13 for the two samples. The
predictions by the linear model (2.29) are also shown in the figures for comparison pur-
poses. Note that the same parameters are used in the linear model. It can be seen that
when the applied voltage is low, (i.e., the volume changes of the PPy layers are small), the
predictions by the linear and nonlinear models can both fit the experimental data. But as
the applied voltage increases, the prediction of the linear model deviates significantly from
the experimental data. On the other hand, the nonlinear model still shows good fitting, thus

demonstrating the effectiveness of the nonlinear model over a larger deformation range.

 

0.35

 

; -e- Experimental
0.3- | — NonlinearModel
A r --- Linear Model

1

 

 

 

P
N
or

P
[0

Bending radius (m
0
b P -t
a or

O
01

 

  

 

 

l

0.5 1 1:5
Input voltage (V)

C

Figure 3.12: Quasi-static bending under different actuation voltages for Sample 1 (13x5
mm).

Based on the obtained values of r1 and a, one can calculate r2, r3 and r4 by using
(3.49), (3.50) and (3.51). Furthermore the thickness of PPy layers can be obtained by
calculating r2 — r1 and r4 — 13 at different actuation voltages. Similarly the thickness of
PVDF layer is obtained through r3 -— r2. The results for Sample 1 are shown in Fig. 3.16
and Fig. 3.17. When the voltage increases, the thickness of the reduced PPy layer will

decrease and that of the oxidized PPy layer will increase for the oxidized PPy layer. The

74

 

 

0.7 l

-9- Experimental
0.6 ~~~~~~~ ~ -~ 1-: — Nonlinear Model J
--- Linear Model

 

 

m)

 

v0.51-—-
0.41--
0.31-
0.2-~44

Bending radius

 

 

 

 

0'5Voltage (V)1

Figure 3.13: Quasi-static bending under different actuation voltages for Sample 2 (33x6
mm).

 

4o , , 1 .
35~- AAAAA

30.-- , , ....... W-

,,,,,,,,,,,,,,,

a. (rad/ m)
N
9

 

 

 

15~~- 11111111111111111111
10» 1
5» .
0o o. 2

Voltaige (V)

Figure 3.14: Computational results on the change of a versus the applied voltage for Sam-
ple 1 (13x5 mm).

75

 

25 l V fl T l 1

 

 

 

 

8.2 0:4 0:6 0:8 1 1:2 114 1.6
Voltage (V)

Figure 3.1.5: Computational results on the change of 01 versus the applied voltage for Sam-
ple 2 (33x6 mm).

change percentage at 2 V is -0.83% for the reduced PPy layer and 0.67% for the oxidized
PPy layer. The thickness of the PVDF layer is decreasing but at a slower rate when the input
voltage is increasing. The change percentage at 2 V is 0.21%. Note that these results pro-
vide interesting insight into the deformed configuration. Such information is not available

from linear models, since the latter assume fixed thickness throughout the deformation.

3.3 Fiber-directed Conjugated Polymer Torsional Actua-

tor

Different configurations of conjugated polymer actuators have been exploited. Bilayer and
trilayer benders are investigated in [8, 32], which utilize the strain difference in different
layers of a beam to generate bending motion. Linear extenders are investigated in [7],

where the conjugated polymer is immersed in electrolyte and the volume is changed as

76

 

30.2

 

 

 

 

 

 

 

 

— Reduced PPy layer ,2”
30.1 '
5 so-
3
10
.5
_2 29.9
.1:
l-
29.8~---
29'70 0 1.5 2

Volta19e (V)

Figure 3.16: Computational results on the changes of thickness of the PPy layers with the
applied voltage for Sample 1.

 

110

 

 

1 — PVDF layer 1

109.95“:

(11m)

d
in
.
l
|
1
.
.
.
I
.
l
.
.
1
.
l
l
|

 

 

 

 

 

(I!
m
35’
£109.05 ..................... - .
.C
.-
109.8 ‘ ,— ______ f .............. L ______________
109.7 1 i 1
50 0.5 1 1.5 2
Voltage (V)

Figure 3.17: Computational results on the changes of thickness of the PVDF layer with the
applied voltage for Sample 1. '

ion transfer between the polymer and electrolyte takes place under an applied potential.

However, more complicated actuator motions will be required. For example, torsional

77

motion is highly desirable in microsurgery and other robotic applications. Fiber-reinforced
elastic material can generate complicated deformation [70—72]. The anisotropy associated
with the interaction between the fibers and the material matrix makes the material resist
elongation in certain directions, which generates torsional motion when the volume of the
material changes [33]. This principle provides a potentially useful approach to fabricating
compact torsional actuators using conjugated polymer. Also conjugated polymer is an ideal
material to realize the design of fiber reinforced elastic material actuator in practice, since

the volume of the polymer can be changed easily with electrical actuation.

A conjugated polymer tube was fabricated by Ding et al. [73], where platinum wires
were integrated into conjugated polymer during fabrication process. However, the purpose
for the platinum wires in that paper was to increase the electrical conductivity along the
tube, so the torsional motion introduced by the fiber-directed structure was not investigated.
To the best of the author’s knowledge, there has been no detailed investigation on torsional

motion in a fiber-reinforced conjugated polymer actuators.

In this dissertation, torsional motion of a fiber-directed conjugated polymer actuator is
investigated from both theoretical and experimental perspectives. A nonlinear elasticity-
based model is proposed to capture the relationship between the actuation voltage and the
torsional motion of the conjugated polymer tube. Compared with a linear elasticity-based
model, this model is able to capture the large deformations. The model is verified with

experimental measurements made on a platinum fiber-directed conjugated polymer tube.

Fiber-reinforced elastic materials can generate complicated deformation when subject
to swelling. When the fibers are aligned in special ways, the material matrix containing
fibers will expand in the direction perpendicular to the fiber [33]. A specific case is studied
where the fiber is helically wound about a elastic tube with a fixed pitch angle, which
is illustrated in Fig. 3.18. The elastic material we consider here is PPy, the volume of
which can be changed through electrical activation. Theoretical analysis shows that the

tube will generate torsional motion when subject to swelling. Platinum wires are chosen to

78

be integrated into PPy tube during the fabrication with some pitch angle [3.

 

 
 
  
 

‘\\\\\\\\\\\\\\\\\‘\\\\‘\\V‘
\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\\\\\\l\\\\\\\\\\\\\\\\\\\\
1111 \l \ \\\\

PPy

Platinum

‘~_ _—

lll III/ I III III/III
Ill/llllI/II/l/I/ll/lll/l /

1/1/1[III/[11111111111111

(a)

 

(b)

Figure 3.18: (a) Illustration of the conjugated polymer-based torsional actuator; (b) picture
of the fabricated torsional actuator.

To fabricate the actuator, it is crucial to integrate the fibers into the material matrix of
PPy. This would ensure that, when the PPy tube volume changes, the torsional motion will
be generated when the motion in other directions are constrained. For this purpose, firstly a
glass cylinder with a diameter of 2.5 mm is used as the substrate and coated uniformly with
gold through sputtering. Then a platinum wire with diameter of 25 pm is wound uniformly
along the glass cylinder. The pitch angle [3 is 80°. This tube is soaked in the electrolyte
(0.05M Li+TFSI" in Propylene Carbonate) and connected to the working electrode of a
potentiostat (OMNIlOl from Cypress Systems). An electrochemical deposition process
is conducted by following the method in [13]. To maintain a constant potential on PPy
when PPy grows on the surface of glass tube, an Ag/AgCl reference electrode (Aldrich
Chemical) is used. The auxiliary electrode is a stainless steel mesh. The electrolyte used in

experiment is a mixed solution of 0.1M pyrrole, 0.1M Li+TFSI— in Propylene Carbonate

79

with 0.5 w/w% water. The current density is maintained at 2 mA/cm2 for good uniformity.
Different PPy tube wall thickness can be achieved by controlling the deposition time. The
PPy wall integrates the platinum wire during the deposition. Then because the sputtered
gold layer has poor adhesion force with the PPy inner surface, the PPy tube can be easily
taken off from the glass cylinder. The inner diameter of the tube can be controlled by

choosing different glass cylinder diameters.

      
   

1.Reference

 

electrode 1 2 3 ~
2. Working [UUHMI
electrode ’

 

  

3.Auxiliary
electrode

Figure 3.19: The experimental setup to fabricate the fiber-directed conjugated polymer
actuator.

3.3.1 Nonlinear Mechanical Modeling Framework

The original and deformed configurations are shown in Fig. 3.20, where F represents the
deformation gradient, which can map vectors (expressed in Z, R, 6) domain) in the original

configuration to vectors (expressed in z, r, 9 domain) in the deformed configuration. The

80

pitch angle is defined as {3 (0 < [i < 90") [33].

 

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’
’
’
’

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—-L__.—O’
I
r._—
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‘L
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I
r~~

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r

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x\\\s\\\\\\\s\\\~\ss\\s~\x\x~
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‘
‘

.0
‘- -’

-~‘~

‘
‘
~
‘

 

 

 

Figure 3.20: Illustration of the actuator configuration. Left: original configuration; right:

deformed configuration.

The deformation can be characterized by the following equation:

(3.79)

=®+KZ, 2:212,

r: r(R), 9

where K is the twist per unit length, and AZ is the elongation per unit length in the longitu-

dinal direction. Suppose that the deformation takes a particle at location X in the reference

81

configuration to the location x in the deformed configuration. The deformation gradient is
defined as
F = — (3.80)

where

3X 2 (1R CR +Rd®e® +dZ eZ,

8x = drer+rd9e9+dzez,

which are the orthonormal unit vectors in the original configuration, and the deformed
configuration. The deformation gradient tensor is

r

F— dre ®e +

(1]? e9 ®e® + 10139 ®ez + Azez 1862, (3.81)

where 18> is the dyadic product. We assume that PPy is mechanically incompressible both
before and after swelling. This assumption is based on relevant literature [74]. So the
deformation satisfies the constraint that the volume is not changed by the deformation after

swelling, which means

det F = v. (3.82)
This, together with (3.81), implies
l d
E32717: = v. (3.83)
Rewriting (3.83) as rdr = {ERdR and integrating, we get
2 R
r2 = r? + 1.2. [R deR, (3.84)
z

where ’1' and ,Rl- are the inner radii of the tube in the original configuration and the deformed

82

configuration, respectively. The left Cauchy-Green deformation tensor is defined by
B = FFT. (3.85)

Following (3.81) and (3.83), B can be further expressed as

v2R2

1 2
B — Wer®er+(K2+—R—2)r €9®89

+x'lzr(e9 ®ez+ez®e9) +1122ez®ez. (3.86)
The principal invariants of B are defined as
11 = tr(B), 12 = I3tr(B—1), 13 = det(B). (3.87)
From (3.81), the invariant 11 is obtained as

22 2
R

v2 +-r—+K2r2+lzz. (3.88)
2

I'lzr

 

I:
l R2

The elastic energy function of PPy can be denoted by (Pm and modeled in the neo-Hookean

form as

l
Chm = Ella] — 3v2/3), (3.89)
where u is a material elastic constant that can be taken as Eppy / 3, where Eppy is the

Young’s modulus [48].

Let M be the unit vector in the reference configuration that defines the fiber direction

in the material matrix:
M = sinliee +cosBeZ, (3.90)
where [i is the pitch angle defined in Fig. 3.20. Given the deformation gradient F, the unit

83

vector M is mapped into
m=FM= (Kcosfi+%sinfi)ye6+zizcosliez. (3.91)
The square of the stretch of the fiber under actuation is
14 = tr(m®m) = (KCOSfi+%Sinfi)2Y2+XZ-2C082fl. (3.92)
The strain energy function of the fiber can be modeled as
crnf = $1414 —1)2, (3.93)

where 7 is another material constant that may be taken as proportional to E pl atinum' The
proportionality constant will depend on the volume fraction of fibers in the overall compos-
ite. Using the connection between the nonlinear theory and the linear theory of anisotropic
elasticity [75], we may take 7: SE platinum in this investigation. Therefore, the total

energy of the fiber-reinforced PPy actuator is

W =<I>m+<l>f. (3.94)
The Cauchy stress tensor is [33]
_1 8W
—FT — I. 3.95

Considering the expression of total energy in (3.94), the Cauchy stress can be written as
[331
=-——B+——m®m—pl. (3.96)

84

Considering (3.86) and (3.91), we can express (3.96) as

0' = arrer®er +099e9 ®e9 + 062(96 ®ez +ez ®e9)

+O'zzez 81 e3, (3.97)
where
2
(NR
0 = _p+ , (3.98)
rr Azzrz
o - — +“r2(x‘2+-—1—)
99 — P v R2
2
+3§L(Kcosa+%smm2(z4—1), (3.99)
2
022 = _p+%Azz+—VZAZZ(I4—l)coszfi, (3.100)
092 = gxlzr+gglzrcosmkcosfi
1
+§sinfi)(I4—l). (3-101)

This framework is built on strain energy functions and general deformation gradients, and

it is thus valid for both small and large deformations.

3.3.2 Boundary Conditions
The boundary conditions on the conjugated polymer actuator are specified as follows:

0rr1R=Ri =01 0rr1R_—_RO =01 (3.102)

which represent that there is no direct loading applied to the inner and outer lateral surfaces.

Notice that R is used in (3.102) since r is also a function of R. The equilibrium equation

85

diva = 0 in the directions of r, z, and 6 can be written as follows:

 

80'" ladre aGrz 1
_ _ _ _ = 3.103
3r +r 36 37. +r(0'rr 096) 0’ ( )
892.19% 804 91.2-
8r r 80 (92 r _
80,9 1 c9099 3026 20,9

Tit—:99— a. + 1 —°’ (“05’

 

 

0, (3.104)

 

where are = 0, an = 0 in this case. Considering (3.99), (3.100), and (3.101), one can

show that (3.104) and (3.105) reduce to ’89; = 0 and 3%) = 0, which means that p is a
function of r only. Given are = 0 and orz = 0, one can write (3.103) as follows by using
(3.83):

dO'rr _ VR

which can be furthermore expanded by considering (3.98)

uvd R2

dp _ VR __ __
222511892). (3.107)

d—R; — EZ'TZ'(O'rp-— (799)-1‘

Integrating (3.107) after substituting the expression of (3.98) minus (3.99), one can obtain

the expression of p by using the boundary condition Orrl R = R0 = 0:

vRZp R0 v2R3 R(R2K2+l)

 

—Zfl(xcosfi + lsirz[3)2(14 — 1))dR. (3-108)
AZ R

Applying the other boundary condition Orrl R : Ri = 0 in (3.108) gives the following con-
straint equation:

A20, (3mm

86

where

A: — —— —
fRi (2( lzzr4+x +R2)
+(xcosp + %sin[3)2'y(l4 —1))RdR. (3.110)

3.3.3 Nonlinear Mechanical Model

One can see from Fig. 3.20 that three variables quantify the deformed configuration: r, 112,
and K. Because the profile of r can be calculated based on (3.84), variable r can be replaced
by ’1' to characterize the deformation together with the other two variables. Eq. (3.110) has
provided one constraint equation. The other two require knowledge of the moment T and

axial load P, which are given by the following expressions:

r0 2
T = 27r/ O'QZr dr, (3.111)
ri ,

r0

P = 27r/ri ourdr. (3.112)
When the end of the tube is not constrained, it follows that T = 0 and P = 0. These two
equations, along with (3.109), provide three nonlinear equations for ri, K, and 42- One
finds that: the expression (3.110) is strongly dependent on ri, the expression (3.111) for
T is strongly dependent on K, and the expression (3.112) for P is strongly dependent on
AZ. In general, the numerical analysis to obtain these variables by solving three nonlinear
equations is difficult. Therefore in this thesis, AZ, representing the length change ratio in
the axial direction, will be measured and used as a given parameter. Since A and T give
the strongest dependence on ’i and K, the equations A = 0 and T = 0 are retained in the

analysis and numerically solved to obtain ’1' and K. By substituting from (3.101), one can

87

integrate (3.111) and obtain the following expression:
TzuaK+y(b0+b1K+b2K2+b3K3)=0, (3.113)

where a, b0, b1, b2 and b3 are terms involving material constants. Similarly one can

integrate (3.110) and obtain the expression for A as follows:
A=uQ+yF, (3.114)

where Q and F are also terms involving material properties and Az, K, and ri, which will
not be listed in this thesis due to the page limitation. The two unknown variables ri and
K can be obtained by numerically solving the coupled equations A = 0 and T = 0 using

(3.113) and (3.114).

3.3.4 Sample Preparation and Experimental Setup

Three factors are changed during fabrication to study their influence on the performance of
the actuators, which are the PPy thickness, the pitch angle, and the inner radius. Two PPy
thicknesses, 0.381 mm and 0.686 mm, are obtained using different deposition duration.
Two different pitch angles are obtained during the winding process, which are 60 ° and
80 0. Two glass tubes with different radii are used to change the inner radii of the tube,
which are 2.2 and 1.3 mm. Four samples are fabricated that are named Sample 1, 2, 3,
and 4. Their parameters are listed in Table 3.3. The length is kept as 33.5 mm for all four
samples.

A computer equipped with a DSl 104 R&D Controller Board (dSPACE Inc) is used for
data acquisition and processing. A voltage is applied across the PPy tube and the counter
electrode, which is a stainless steel mesh in the experiments. The PPy tube is soaked in

0.1M Li+TFSI‘ in Propylene Carbonate (PC). Sticky copper film is attached to different

88

Table 3.3: Geometric parameters of the samples.

 

 

 

 

 

 

 

 

 

thickness (mm) pitch angles (degree) radius (mm)
Sample 1 0.381 80 1.3
Sample 2 0.381 80 2.2
Sample 3 0.686 80 2.2
Sample 4 0.686 60 2.2

 

positions of the PPy tube for measurement purposes. Three motion signals are captured to
characterize the tube deformation, which are the inner radius ri, torsional displacement )1 at
the end of the tube as shown in Fig. 3.21, and the length change at the end of the tube that
equals to A; - L based on (3.79), where L is the length of the PPy tube. The model-predicted
variables ri and 42 can be correlated directly to the measured variables, while K is obtained

based on the geometric relationship in Fig. 3.21:

y = Rmtan(K-L),

where y is the measured displacement, and Rm is 4 mm in this thesis.

    

Figure 3.21: Experimental setup to measure K. The copper film is attached perpendicular

to the outer surface at the tube bottom.

89

Laser sensor

 

3.3.5 Experimental Results and Discussions

The quasi-static response is studied by applying a low frequency sinusoidal voltage on the
actuator. The frequency is chosen to be 0.005 Hz to allow the actuator to reach the quasi-
static condition. The sinusoidal voltage amplitude is 3.6 V. Because only one laser sensor is
available, measurements of inner radius, torsional displacement, and length change are per-
formed separately and then synchronized through the voltage signal. The Young’s moduli
of platinum and PPy are chosen to be 168 GPa and 80 MPa in model prediction [48,76].
The experimental results for Sample 1, 2, 3, and 4 are listed as follows. It is observed in
experiments that the inner radius and torsional displacement signals have reversed phases.
In particular, when the inner radius decreases, the torsional displacement increases. As
discussed in Sec. 5.2, the length change will be used to obtain 42 and taken as a given
parameter to reduce the difficulties in numerical analysis. Therefore the model-predicted
inner radius and torsional displacement are compared with the experimental data for dif-
ferent samples. In general, the model can predict the reversed phase of these two signals.
The model can also predict the response change from different samples. The comparison
between Fig. 3.22 and Fig. 3.25 shows the trend that a larger tube radius leads to a larger
torsional displacement, when other parameters are kept the same. The comparison between
Fig. 3.25 and Fig. 3.28 shows the trend that a thinner tube generates a larger torsional dis-
placement. The comparison between Fig. 3.28 and Fig. 3.30 shows the trend that a smaller
winding angle results in a larger torsional displacement. Therefore the experiments and
model prediction suggest that a configuration with larger. radius, thinner tube thickness,

and smaller winding angle tend to generate large torsional motion.

3.4 Chapter Summary

In the electrochemical domain, a redox level-dependent impedance model was proposed

for conjugated polymer actuators. The model was derived based on a nonlinear partial

9O

 

Voltage (V)

 

 

 

 

 

 

 

 

 

'30 50 100 150
— Experimental data
°-°‘ ~ -u- - Model prediction

   

 

 

 

0 50 100 150
Time (s)

Torsional displacement (mm)

Figure 3.22: The torsional displacement of Sample 1 with 0.005 Hz sinusoidal voltage
input.

differential equation that incorporates the dynamics of ion diffusion, ion migration, and
polymer oxidation, which was linearized using perturbation techniques. The linearized
PDE was solved analytically in the Laplace domain, with proper boundary conditions en-
forced. Comparison with experimental results supported that the proposed model is able
to capture the influence of redox level on the impedance spectrum. The model provides
an effective way to integrate the diffusion and migration effects of ion flux in conjugated
polymers, which used to be treated separately with different models [57, 77]. The pro-
posed model not only contributes to fundamental understanding of the complicated, redox
level-dependent electrochemical behavior, but also holds potential for nonlinear control of

conjugated polymer actuators.

In the mechanical domain, a nonlinear elasticity theory-based framework is proposed

91

Voltage (V)
o n is

via

 

 

 

 

 

 

 

1 '19
A N f F—J__—-_fi
E ‘ — Experlmental data
g 1 £24- Model prediction
g 1:775 """""""" '__,,,, ————— »_ if???“ ,_ 7
E
._ 1.77
o
1:
5

1.7651 1
50 100 150
Time (s)

Figure 3.23: The change of inner radius of Sample 1 with 0.005 Hz sinusoidal voltage
input.

to analyze the bending configuration of a trilayer conjugated polymer beam under different
actuation voltages. Neo-Hookean type strain energy functions are used for the PPy and
PVDF layers of the trilayer beam to capture the nonlinearity under large deformations. For
a constant actuation voltage, the swelling of each PPy layer is determined by the amount
of transferred ions. The bending configuration at the equilibrium is obtained by solving the
force and moment balance equations simultaneously. Experimental results have validated
the effectiveness of the proposed nonlinear model. The method further provides insight into
the deformation details, such as the change of layer thickness. Similar analysis can be used
for some other EAPs that involve the local volume change (swelling). This is important for
many applications involving large deformations, when the elastic modulus of the material
already becomes nonlinear and linear elasticity analysis does not hold.

A fiber-directed conjugated polymer actuator is fabricated by integrating platinum wires

92

 

Voltage (V)

 

 

 

 

 

 

 

 

 

A 0.04 F j
E
E.
O
8’
N
.c
0
‘9
Ci .
.1
'0'020 50 100 150
Time (s)

Figure 3.24: The change of tube length of Sample 1 with 0.005 Hz sinusoidal voltage input.

into conjugated polymer during fabrication. Experiments have demonstrated that the tube-
shaped conjugated polymer actuator can generate torsional deformation, when its volume
is changed by applying an actuation voltage across itself and the electrolyte. Three impor-
tant factors are changed during the fabrication process to make conjugated polymer tubes
with different configurations, including the tube thickness, the tube radius, and the pitch
angle of the platinum wires on the tube. A nonlinear elasticity-based model is utilized to
capture the actuator performance, which holds for both small and large deformations. The
model prediction is compared with the experimental data, which has verified the effective-
ness of the nonlinear elasticity-based model. The performance of actuators with different
configurations is compared, which suggests that a configuration with larger radius, thinner

tube thickness, and smaller wounding angles will generate larger torsional motion.

93

 

Voltage (V)

 

 

 

 

 

 

P
c:
a:

— Experimental data
- -t(- Model prediction 1

    

 

 

_O
O
.5

 

 

 

b b
O
A
I
I
l
47:-
_.?.
1‘?
J;

'o

100 150

Torsional displacement (mm)
ON 0
0|
C

Time (s)

Figure 3.25: The torsional displacement of Sample 2 with 0.005 Hz sinusoidal voltage
input

94

 

Voltage (V)

 

 

 

 

0 50 100 150

 

 

 

 

 

  

 

 

— Experimental data

’é‘ 2-2 -x- Model prediction
g :
m2.19~-----— —~ :
3
,5
2
._ 2.18
2
E. . .

2.17 1

0 100 150

Time (s)

Figure 3.26: The change of inner radius of Sample 2 with 0.005 H2 sinusoidal voltage
input.

95

 

Voltage (V)

 

 

 

 

 

 

 

 

Length change (mm)
:5 :5
-s o

O”

 

 

50 ’ 100 150
Time (s)

Figure 3.27: The change of tube length of Sample 2 with 0.005 H2 sinusoidal voltage input.

96

 

Voltage (V)

 

 

 

 

 

 

 

 

 

.50 50 100 150 200
x 10'3 — Experimental data
5 r i -x- Model prediction

 

 

0 50 100 150 200
Time (s)

Torsional displacement (mm)

Figure 3.28: The torsional displacement of Sample 3 with 0.005 H2 sinusoidal voltage
input.

97

 

Voltage (V)
O N

 

 

 

 

 

P
O
_L
J

o
o
o
01

l
i

.5
O
O
0|

___________________________

 

 

Length change (mm)
9

 

I
D
d

0 50 100 150 200
Time (s)

Figure 3.29: The change of tube length of Sample 3 with 0.005 Hz sinusoidal voltage input.

98

 

 

 

 

 

 

 

 

 

 

 

 

 

A 2 —

2,

3’ o

:3

3

> .2 -

-4 i . i
0 so 100 150 200

E -——Experimentaldata
E 0.01 I. . -it- Model prediction —
C ‘ ' l I
20001 _ , , ‘m 8-”.
§ L “i x w. i - l1
— o .......... ' , -- __ .......
Q
”'9 ik‘ ‘11,.) I I l :
_.o_oos .............. . ..... «I L- _
g g I :
"' -o.01 i 1 1
g 0 so 100 150 200
'- Tlme(s)

Figure 3.30: The torsional displacement of Sample 4 with 0.005 Hz sinusoidal voltage
input.

99

 

Voltage (V)
O

  

 

 

 

 

 

 

 

 

 

 

 

 

 

-4 i L e
0 50 100 150 200
2.2 i . —— Experimental data F
E . ' «- Model prediction
E l l
7; 2.1 ‘
3
'6
E
h 2.18
0
C
E i l
2.17 . 1 .
0 50 100 150 200

Time (s)

Figure 3.31: The change of inner radius of Sample 4 with 0.005 Hz sinusoidal voltage
input.

100

A

 

Voltage (V)
O M

K:

 

 

 

 

50 1 00 1 50 200

 

_L
O

0|

7... ___.--_..._—..-__._..4

 

Length change (mm)

 

 

O 50 100 150 200
Time (s)

Figure 3.32: The change of tube length of Sample 4 with 0.005 Hz sinusoidal voltage input.

101

Chapter 4

Control of Conjugated Polymer

Actuators

4.1 Model Reduction

The full actuation model (2.30) can be rewritten as

with the constant

 

Yb") _ Cm

 

 

C(1+ Eggtanhflhz ”’11) 3))

319L[(%:13)2—1]
E .
8’11(’12 *h1)W[(7}:-%)3 + 1%; -1]

|li>

Cm

 

Note that the material dynamics (2.26) is ignored because it is only significant when conju-

gated polymer is operated at high frequencies. High frequency response becomes too weak

to be useful in applications [78]. This can be observed from Fig. 2.16 and 2.17, where the

displacement magnitude drops over 20 dB as frequency increases from 0.3 to 30 rad/sec.

Eq. (4.1) is an infinite-dimensional system due to the term tanh((h2 — h1)\/§), and

102

thus is not suitable for real—time control purposes. Utilizing the equality '[16]

1 s V
tah_ _ m
“(M/z) 1

4x/s_z —n=Os+7r2(2n+ 1)2z

 

and letting z = , one converts (4.1) into

 

 

 

4M2*h02
y(S) : Cm . (4.2)
V(S) 1
sR + 2d 2 1 )
C 1 + °°_
( 30124115 n"05+7r2(2n+l)2d(2(h2—h1))_2
To understand the rationale behind model reduction, take typical parameters d = 2 x 10" 10
l

m2/s andhz—hl =30um. Thepole of is at—13.7 for

 

s+ n2(2n+1)2d(2(h2 — ’21))“2
n = 2, at —26.9 for n = 3, and at —44.4 for n = 4, etc. This indicates that one can obtain

a low-order approximation to (4.2) by ignoring terms associated with large n. In particular,

for a low-frequency input, the approximation will preserve well the behavior of (4.2).

We thus discard terms with n 2 2. This results in the following third-order system for

the actuator: 2
I I I
V(S) s3+a’lsz+a’25+a’3’

 

103

where the parameters all have explicit physical meanings:

 

 

 

 

 

 

 

 

a, _ 40 + 52:20 + 1 _

' (hz—h1)6 2022—1192 RC’

, 57:202 97:402 57:20

(12 : 35+ 4+ 2 ’

(’12—’11) 16(h2—h1) 2(’12-’11) RC

, 971:402

a = l

3 16(h2 — h1)4RC

I Cm

bl = 7?

b, _ 400,, + 57300,,

2 (hz-h1)5R 2(h2—h1)2R’

b3 __ SnzDsz 97:402Cm

+ .
(I12 -—hl)35R 16(h2 —h1)4R

For typical parameters, (4.3) has one pole and one zero which are located far to the left
of the imaginary axis comparing to other poles and zeros, and therefore the model can be

further reduced to second order with one zero. This is shown next.

The system (4.3) has three poles with explicit expressions [79]:

9 1,

2 —Qcos(§)—§a1 (4-4)
9+2 1,

p2 2\/—Qcos( 3 n)—§a1, (4.5)

9+47r 1
p3 = 2\/-QCOS( 3 )—§a’1. (4.6)

Pl

 

 

where

P

_Q3
= 30’2—0’12
9 3
I I I I3
P : 9ala2 — 27a3 —— 2al
54

6=c05_1(

 

),

i

 

104

With typical physical parameters [16], a’l , agz, and a’3 are all relatively large numbers

(> > 1). This implies 4’12 >> 3a’2 in Q, and 20,13 > > 9a'l (1:2 and 2a? >> 27a'3 in P, which

leadsto
Q ~ ill—2- P~ 0,13
9‘ “"F’
M e if"—
3,
6 z cos 1— £1,13/27)=cos l(—-l)--7r

a’ (1’

p1 z 2--3—l-cos(§)—-3—l=0,
a' 3 a'

P2 * 2'?1'°°S(—375)“3"=‘“1’
al 571' a!

p3 z 2".T1COS(-3—‘)‘——3-l-=O

Clearly I p2| is very large while [ml and |p3| are relatively small.

The analysis on the zeros is simpler. The zeros of (4.3) are

 

I I2 I I
_b2‘\/b2 ‘4'b1'b3

 

 

 

z] 2b’1 , (4.7)
22 : —b’2 + \/b2:2:’1—4cb’1 -b’3. (4.8)
With typical parameters, the following holds:
I2

4-b’1- ’3<<b2, b2>>b1,

105

which implies

I

2
21 z ——I_’

b1
zz z 0

Therefore, |zl| is very large while |22| is relatively small.
A numerical example is provided to illustrate the analysis. Table 4.1 lists the typical
values for the relevant physical parameters. The corresponding system parameters for (4.3)

are:

a’l 2.32x103,a'2=9.79x103,a’3=3.38x103,

b’l z 0.0667Cm,b’2:71.48Cm, 3:195.1Cm,

the three poles are —0.38, —23 19, —3.84, and the two zeros are —- 1069, -2.74.

Table 4.1: Typical values of parameters in the actuation model.

 

 

 

 

 

 

Parameter Value
D 2 x 10’10 mZ/s
’22 ‘hl 30 um
R 15 Q
6 25 mm
C 5.33 x 10—5 F

 

 

 

 

When operating in air, the trilayer actuator will dry up due to solvent evaporation. This
implies that the diffusion coefficient D will decay over time. It is thus of interest to see
whether the above analysis on pole/zero locations still holds when D is very small. Fig. 4.1

shows the ratio
min(|P2!.|21 I)
max(|P1l,|P3iil32l)

 

as a function of D, while other parameters are chosen as in Table 4.]. It is clear that even

106

when D is close to 0, one can safely ignore one pole and one zero of (4.3) [50]. The final
reduced model for the trilayer actuator thus has the following structure:
(5) 3(5) bls+b2

:_=———. 4.9
(5) A“ s2+a1s+a2 ( )

‘<

<

It is expected that despite its simple looking, (4.9) captures the dominant physics of the
actuator within the actuation bandwidth. In particular, all parameters of (4.9) can be related

to fundamental physical parameters for the full model (2.29).

 

33m .
320i
310 L
300 _,
290-7 » .

280~

 

 

 

270-— 16-1‘3'A‘ ‘ A l-12 “ -11

 

1 .
Diffusion coefficient (mZ/s)

minilpziiiql)

—.— as a function of D.
max(ip1iiip3i, izzl)

Figure 4.1: Ratio

Fig. 4.2 compares the Bode plot of (4.1) and (4.9) using parameters in Table 4.1. The
unmodeled dynamics is bounded in general, and the maximum discrepancy is reached
around 103 rad/sec. However, considering that the actuator is usually operated in the low
frequency range up to a few Hz, the reduced model is a good approximation for control

implementation while capturing the main dynamics.

107

 

 

 
  

 

 

 

 

 

 

 

 

'20 r i r

E 2"";
v’ .
3'30“ ~~~~~~~ ‘-—g-———-— _~ _«
m 1 . .
.340 """" ’ ' ‘ ” ' -- ~ , a _IMinite-dimensional modeI ___
'3 --'Reduced model
63-50~ ............. ..
l6
2 60
10.2 10° 102 10‘ 106
100 j I I

-——— .H_.___‘

Phase(deg)
8

 

 

 

 

 

10. 10 102 10 10
Frequency (rad/sec)

Figure 4.2: Comparisons between the infinite-dimensional and reduced model.

108

4.2 Design of Robust Adaptive Controller

4.2.1 Self-Tuning Regulator

Controller adaptation is desirable for conjugated polymers since their actuation behaviors
can vary significantly over time. Given the model structure (4.9), a self-tuning regulator is
adopted due to its simplicity. The idea is to estimate systems parameters online, and then
construct a controller based on these estimates so that the closed-loop system would behave
like a model system Gm(s) (model-following). Fig. 4.3 illustrates the major components of
a self-tuning regulator. Note that a low-pass filter is used to filter the noises in the output
signal before the output is sent to the self-tuning regulator. The bandwidth is chosen to be

30 Hz to cover the actuation frequency range in experiments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Self-tuning regulator

r“‘--—f'. ''''''' 1
Specification l

' w v Plant parameters |

l Controller 4' Estimation <5

| design |

| Controller ______ I

| Uparameters 1-
Reference I i Conductin

. ‘ Controller : : g L_.__,.

l V Input polymer Output

 

 

 

 

 

 

 

 

 

 

Figure 4.3: Illustration of the robust self-tuning regulator.

In the estimation step, the recursive least-squares algorithm is chosen to identify pa-

rameters in (4.9) based on the input into and output from the conducting polymer shown as

 

follows
§-=memm (Mm
an=im-flman am
“2:” = am) —P<t><p<r><pT(r)P<r> (4.12)

1.09

where
9 = [01 a‘2 bi 521T

d

<pT(t)=[—EL“[ny(s)i —L—‘(ny(s>) %L“‘1va(s>1 L‘11va<s>>1-

‘ A stable transfer function H f is introduced to avoid direct differentiation of the output

signal in estimation
1

 

H = —————————-. (4.13)
f 52 + ZS + 1
Note that L"1 [*] means inverse Laplace transform.
The desired transfer function is chosen as
B b + b
C(s) = 1’— : 1ms 2’" . (4.14)
Am 52 +a1ms+azm
A general linear controller is chosen as
RV(s) 2: Tr(s) — Sy(s), (4.15)

where r(s) is the reference input. R and S are determined by the Diophantine equation
AR+BS=Am (4.16)

to track the desired denominator Am, and Bm is tracked by choosing T. Finally the con-

troller is obtained as

: blms+b2mr(s _ (aim —c’1‘1)s+(a2m _62)y(s).

V(S) A ,. ,. ,. (4.17)
b1S+b2 bls+b2

 

In experiments, the desired transfer function parameters are chosen as

alm =4,a2m =4,b1m 21,172," 21.5.

110

4.2.2 Parameter Projection

If the model (4.9) were exact, the controller (4.17) in combination with the online param-
eter identifier would lead to asymptotic model-following [37]. However, higher-frequency
dynamics and nonlinearities are not included in (4.9). Measurement noises are not reflected
in the model either. These undesirable factors might lead to instability of the closed-loop
system if no proper steps are taken.

In this thesis parameter projection is adopted as a robustification mechanism for the
self-tuning regulator. From the expressions of the parameters in (4.3), one knows that
the two poles and the zero of (4.9) are all negative, thus the parameters a) , 02, b1, and b2
should all be positive, which are bounded by a small constant m > 0. Furthermore, based on
the given polymer parameters and the knowledge of physical parameters, an upper bound
M > 0 of these parameters can be determined. The update rule (4.10) is thus modified to

incorporate parameter projection:

0 if 19,-(t) 2M and [P(t)tp(t)e(t)],- >0

 

t) = 0 iféi(‘)=m and [P(t)<p(t)e(t)]i <0 , (4.18)
[P(t)<P(t)e(t)],- otherwise

where 0,-(t) and [P(t)(p(t)e(t)]i denote the i-th components of 0 and P(t)(p(t)e(t), respec-
tively. Clearly, if 0,-(0) E [m,M], 0,-(t) E [m,M], Vt > 0. In the experiments the values of M
and m are chosen to be M = 1x 104, m = 0.001.

One can represent the true output y(s) of the actuator under the input V(s) as

bs+b
y(s)-———-—‘ 2 v

— 2 (s)+A(s)V(s)+./V[V(s)]+w(s). (4.19)
S +GIS+02

Here A(s) represents the unmodeled higher-frequency dynamics, i.e., the difference be-
tween (4.1) and (4.9). Since both (4.1) and (4.9) are stable and have bounded frequency
responses, A(s) will be stable and bounded. JV [V(s)] denotes the influence of unmodeled

111

nonlinearity (e.g., hysteresis). The nonlinearities will be bounded due to the dissipative na-
ture of the materials. In (4. 19), w(s) denotes the measurement noise, which is also bounded.

It can be shown [80, 81] that the proposed adaptive control scheme in Section 4.2.1
with parameter projection (4.18) is robust in the presence of bounded unmodeled dynamics
and nonlinearities, measurement noise, and slow and bounded parameter variations; in

particular, all signals in the closed-loop system will be bounded.

4.3 Experimental Results

4.3.1 Measurement Setup

Tracking experiments are conducted to examine the effectiveness of the proposed robust
adaptive control scheme. A trilayer polypyrrole actuator (20 x 5 x 0.17 mm) is clamped
at one end, where the actuation voltage is applied. The tip displacement is measured by
an OADM 2016441/S 14F laser sensor from Baumer Electric Inc. with resolution of 5 um.
The controller is implemented in a PC equipped with dSPACE DS1104. The experimental
setup is shown in Fig. 4.4. Before each experiment, an actuator cut with the specified size
is soaked in the electrolyte (TBA+PF8 in propylene carbonate). The inner porous PVDF
layer thus stores electrolyte, which enables the actuator to operate in air for some time. The
time of continuous in-air operation depends on how fast the stored solvent evaporates, and
without further packaging, it is about 4-5 hours. The experimental temperature and humid-
ity are maintained as 25°C and 27% respectively. For practical applications, the packaging
issue will have to be solved so that the actuator can work in air for much longer time. On
the other hand, the current actuator demonstrates significant time-varying behavior, which
provides a good testbed for verifying the proposed adaptive scheme.

For comparison purposes, a PID controller and a fixed model-following controller are

also implemented. Actuators with same dimensions and same conditions are used for all

112

   

4— Laser sensor

___]

i“

 

Amplifier Computer and DSpace

Figure 4.4: Schematic of the experimental setup.

three controllers. The design of the PID controller and that of the model-following con-
troller are also based on the model structure (4.9), while the model parameters are identified
in separate experiments shortly before the tracking experiment starts. The model-following
controller is constructed as in (4.17), except that the controller parameters will not be up-
dated. The PID gains are carefully tuned and verified in Matlab simulation before the
experiments to ensure that the initial tracking errors are within the similar range as those

under other controllers, and the PID controller is
U 6
_ = 6 + — +0.02s. (4.20)
E s

For each continuous tracking experiment (3-4 hours long), the parameters of the PID con-
troller and the model-following controller remain constant (i.e., non-adapting). Throughout

the experiments, the reference model Gm(s) is chosen to be

s+1.5
G s:——.
m() 32+4s+4

113

4.3.2 Results and Discussions

In the first batch of experiments (Batch One), the reference input r(t) = 0.Ssin(7rt) +
0.53in(0.27tt)V, and the actuator output y(t) is required to track the desired trajectory
ym(t) = Gm(s)[r(.)](t). The latter contains two frequency components (0.1 Hz and 0.5
Hz) with peak-to-peak variation of 0.62 mm. Each experiment runs continuously for four
hours. Fig. 4.5 shows the tracking results at the beginning of the experiment (I = 0 h),

while Fig. 4.6 shows the results when approaching the end of the experiment (I = 4 h).

To better compare the control schemes, two metrics are defined for the tracking error.

Given a starting time to and a constant T > 0, we define the normalized average error ea

 

t +T
A 1,3 ly(t) -ym(t)|dt
ea = to + T , (4.21)
fro lymmid’
and the normalized maximum error em
max 0’0) "Ymmi
t e t ,t + T
em 2 [0 0 ] (4.22)

 

maxt e [t0,t0 + T] Irm(t)l

Throughout this thesis, T is chosen to be 100 3. Under the robust adaptive scheme, ea drops
from 11% at to = 0 h to 7% at t = 4 h, and em drops from 15% to 9% for the same period.
In comparison, ea increases from 7% to 28% under the PID controller, from 7% to 50%
under the fixed model-following controller, and em increases from 8% to 25% under the
PID scheme, and from 10% to 48% under the model-following scheme. Fig. 4.7 shows the
evolution of ea and em, measured and calculated every half a hour, under the three schemes.
It is clear that the robust adaptive control scheme delivers consistent tracking performance
during the four-hour continuous operation, while the tracking performance under the PID

scheme or the fixed model-following scheme deteriorates over time.

A second batch of experiments (Batch Two) is conducted to examine the effectiveness

114

 

 

Robust adaptive ___ Achieved (raj,
T T ' — Desired tra].

 

 

 

 

.5
(BOO!
OI
.a
O

15 20 25

 

P

 

 

I
9
UI

9
01°

1
l i I

1 I l l
l 1

._ _- . e.-_L~ . 4_._—7._~— ,.__. .. . -_

l i 1

t 1 l 1

I l l

. 1 , I

I - l

1 1 L 1 .

 

Bending disp. (mm)

 

 

 

 

o r-
'°'5o 5 1o 15 2o 25

Time (s)
(a)
Robust adaptive

 

 

 

 

 

   

 

 

’5‘

5

§ 0

20025 , ,

c '1 1 1 ‘

g 0 5 10 15 20 25
E

 

 

 

o 5 10 15 20 25
Time (s)

(b)

Figure 4.5: Experimental results on trajectory tracking (Batch One), I = 0 h. (a) Achieved
trajectories versus desired one under the three controllers; (b) instantaneous tracking errors
under the three schemes.

115

 

ROW“ adaptive --.-Achleved tral.
: _ I 1 — Desired tral.

 

 

 

 

  

 

 

 

I
P

PID

 

P

 

 

 

5 1‘0 1‘5 20 25
Model-following

 

I
P

Bending dlsp. (mm)
01 001 O

P

 

   

 

 

 

o 5 10 115 210 25

Time (s)
(a)

 

Robust adaptive
i l

 

 

 

 

I
P
0
UI

 

 

 

Tracking error (mm)
C)
l

 

 

 

 

..
b.--e-..-
)—

 

0 '5 10 115 20 25
Time (s)

(b)

Figure 4.6: Experimental results on trajectory tracking (Batch One), I = 4 h. (a) Achieved
trajectories versus desired one under the three controllers; (b) instantaneous tracking errors
under the three schemes (note the different vertical-axis scales).

116

 

0.5 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ Robust adaptive
: .i
0.4-~"'F"D __,..w’.
+ModeI-foliowing , .I"
o“ 2
a 0-3' """" )2", """"" ’1}
t z . ,I‘ [I
at ' ’0’ . II
< ; ,2 ‘ Ix
.I" ,x -
0.1’ ":.'-’¢ ;' ’:-a’f ,. .‘ _
L‘:::——-0"'"" : N)
0O 2 3 4
Time (h)
(a)
0.5 . . . )L
+ Robust adaptive 2 x.
o4..-.-°-P|D __,:-
' +Model-followingJ ; ,’
GE 1 .1
§
3
x'
a
E
0O 2 4
Time (h)
(b)

Figure 4.7: Normalized average error ea and maximum error em under the three control
schemes (Batch One experiments). (a) Evolution of ea; (b) evolution of em.

117

of the three schemes in tracking trajectories of much larger magnitude. With the reference
input r(t) = 2.1 sin(0.47rt) + 2.25in(0.17rt)V, the desired trajectory ym(t) has peak-to-peak
variation of 3 mm. All three schemes show good tracking performance at the beginning.
The experiment under PID control has to be stopped after 3 hours since at that time the
voltage input exceeds the limit (1.6 V). Fig. 4.8 shows the tracking results when t = 3 h.
Fig. 4.9 shows the evolution of ea and em. The trend is consistent with that in Batch One
experiments. It can be seen that the robust adaptive controller keeps ea under 3.5% and em
under 4% throughout the four-hour experiment. In the mean time ea rises from 5.5% to
8.4%, em from 5.7% to 10%, under the PID controller (in three hours), and ea rises from
5% to 80%, em from 5.5% to 81%, under the fixed model-following controller (in four

hours).

It is also important to compare the control efforts required under the different control
schemes. Low control effort is highly desirable since that leads to long working life for the
conjugated polymer actuator. Fig. 4.10 shows the evolution of the magnitude of voltage
input under each scheme, for both Batch One and Batch Two experiments. The required
voltage increases over time under every scheme, which is due to the deteriorating actuation
capability of the actuator as the solvent evaporates. However, it can be clearly seen that the
voltage input under the adaptive scheme is much lower than that under the PID scheme,
and also lower than that under the model-following scheme most of the time. In the Batch
Two/PID experiment, the polymer actuator was actually damaged and stopped functioning

after 3 hours due to continuous high-voltage (> 1.5 V) actuation.

There is another interesting observation during the experiments that supports the valid-
ity of the reduced model. The effect of solvent evaporation can be incorporated by taking
the diffusion coefficient D —> 0. This leads to p1 -+ 0, p3 —> 0, and 22 —-> 0, where p1,
p3, and 2:2 are the poles and the zero of the reduced model (5.2), as defined by (4.4), (4.6),
and (4.8). Fig. 4.11 (a) shows the evolution of the poles and the zero during the Batch One

experiment every half an hour when the adaptive control scheme is adopted. Fig. 4.11 (b)

118

2_ Robust adaptive L-Achleveditgj.

— Desired tra].

   
 

O

 

20 25

 

Ni O”
- < at
_I
o
_fi
01

 

'9 a"

1'5 20 25

Bending disp. (mm)
O

 

 

 

 

 

1‘0 1'5 20 ‘25
Time (s)
(a)

 

Robust adaptive

 

 

 

 

 

 

15 20 25

 

 

 

 

0 5 10 1‘5 20 25

Time (s)
(b)

Figure 4.8: Experimental results on trajectory tracking (Batch Two), I = 3 h. (a) Achieved
trajectories versus desired one under the three controllers; (b) instantaneous tracking errors
under the three schemes.

119

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ Robust adaptive ,
-°- PID E
0.3” , ......... ,1.
-- Model-followm 3 x’
a ; : . ’.
d) : .5
a 0-6'.” — .-:-..’0’,___.-
t E .1.“
Q I ’°’
' f ,1 .
gt 0 4 .9]...
< I.’ :
.l’
0.2” ' ‘;’*’----- .........
: ~’ , E
L‘J:rJ:L'{__—:———2-—--t-"t
OY v Y Y Y V L :
0 1 2 3 4
Time (h)
(a)
0.8 I , z .7
+ Robust adaptive § x”
-.- PID 3 X
E 05"" 4* Model-following (4 - -- -- —-
0 I 3 ’3‘:
b ‘ I
O f 1 ”ya :
E 0.4.. _ ' I.’°’E:“"“ -"
I . ’0 . 1
2 ; ,-
0.2 1' """"""" i- - l.- - - ......
z!
.’.I-.-‘ ‘0‘ _o- __.———--O
fur-1"- ;‘ - " 1 1 -
off ' j Y

Time (h)
(b)

Figure 4.9: Normalized average error ea and maximum error em under the three control
schemes (Batch Two experiments). (a) Evolution of ea; (b) evolution of em.

120

 

_L
O

 

 

 

+ Robust adaptive
1.4"” ‘9'PID ----------- 3 ------ ‘9
-I- Model-following 1 ’1’
1.2- ----------- 2. 4- ————————— --; ....... ‘ .II 1
E ,’
a 1 ' ’ ‘ ‘IIII'
.5 i
0 1’0
g 0.8 ~~~~~~~~~~~~~~~~~~~~~~ ~ ,4 ---
'6
>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)
2 7 7 v
-o- Robust adaptive
1'8“-._P|D ““““““““ 1 —————————— --.4
In... . :
1.6-- Model following: _______ .75. __________
. ’(l I
2,9" 1.4 ------------------------- 3,4" ...................... 1
3 e" :
g ",e’ .
'3 1 2 ’3’, - - ~ -, j
G ,I
2 2’
5 ‘1 1
0" 1
0.8L """"""""" ‘ ------------- :r---—.-...).¢'.z
| I ,o'1
”L """ 5 M
Em: *; e "'9 ’ 5 ' :
0. 1

Figure 4.10: Evolution of voltage input magnitude under the three schemes. (a) Batch One
experiments; (b) Batch Two experiments.

121

shows this evolution during 100 seconds after the experiment started for half an hour. It

can be seen that the poles and zero all tend to O, as predicted by the model.

 

 

1 -0—Pole1
-----Pole 2

 

 

 

 

 

 

 

Time (h)
(a)

 

° :

JyaI-aaam-émawflam
0 5<Poero~mooo§eeooooom

—¥—fiolé1 ‘
g "0" Pole 2
”9.- Zero

 

 

 

 

«WW: ;

0 5‘0 . 160 150
Time (s)

(b)

 

 

 

Figure 4.11: Identified poles and zero under the robust adaptive control scheme in Batch
One experiment. (a) Evolution over four hours; (b) Evolution over 100 seconds.

In both Batch One and Batch TWO experiments, the reference inputs used contain two
frequency components and are thus persistently exciting of order 4 [37]. This is a necessary
and sufficient condition for correctly identifying the four parameters in (4.9). In practice,
however, the persistent excitation condition may not always hold, and it is of interest to

know whether the robust adaptive scheme still works well in that case. An experiment

122

is conducted for this purpose, where the reference input r(t) = 28in(7tt) is persistently
exciting of order 2. Fig. 4.12 shows the tracking results under the robust adaptive controller,
at the beginning and the end of four-hour continuous operation, which demonstrates the

capability of the proposed scheme in tracking a non-persistently exciting signal.

4.4 Chapter Summary

A robust adaptive control scheme has been presented for conjugated polymer actuators. The
key to the success of this method is a simple actuator model that is reduced from the full,
infinite-dimensional physical model. Model reduction is based on the knowledge of the ac-
tuation bandwidth as well as typical values of the physical parameters. The reduced model
captures relevant actuation physics and it makes the control design and implementation
easy. Note that the physical-based model provides justification of the specific controller
structure. This is in contrast to pure empirical models obtained through system identifica-
tion, which requires re-identification for different samples. The parameter projection step
in the self-tuning regulator ensures the stability of the closed-loop system in the presence
of noises and unmodeled dynamics. Experimental results have shown that the proposed
scheme is superior to the commonly used PID scheme and to the fixed model-following

scheme in terms of both tracking accuracy and required control effort.

123

 

 
 
 
 
    
  

1 t = o hf ‘ 1 KW].
f : : L—Desired tra].

 

i
I

t=4h

 

Bending disp. (mm)

 

l

 

 

 

 

 

 

 

 

 

'10 5 1b 15 20 25
Time (s)
(a)
t : 0 h
0_05___. . .
0.025 , . - i , 1 «
l
o , . .
l l ' I
-0.025 « . «
E n: . i i
E, '0 ”“0 5 10 15 20 25
h
g t=4h
a, o. i . .
cu i ,
,5 0.02 ' .
X
o l l l l l l I
g o
.- i
-0.02‘[ «
L,,___-. I
'0‘040 5 1.0_ 15 20 25
Time (s)
(b)

Figure 4.12: Experimental results of tracking a non—persistently exciting signal under the
robust adaptive controller. (a) Trajectory tracking at t : 0 h and t = 4 h; (b) Tracking errors
att=0handt=4h.

124

Chapter 5

Application: Conjugated Polymer

Micropump

5.1 Design and Fabrication of the Micropump

5.1.1 Diaphragm Design

In a typical design of BAP-based pumping diaphragm, one uses a single EAP plate as the
diaphragm to seal the pump chamber directly. The edge of the plate is mechanically fixed
with electrodes on both sides. However, the strain of the middle—plane will be nonzero due
to the restriction at the edge, which implies that a significant portion of energy will be re-
quired to stretch the middle—plane. The latter constrains the displacement of the diaphragm

in the vertical direction. Detailed analysis will be shown in Section 5.2.

A new design is thus proposed here for the generation of large out-of-plane deforma-
tion, as illustrated in Fig. 5.1. A passive membrane with low stiffness is used to seal the
chamber. Then a‘conjugated polymer plate is cut into the shape of petals and bonded to
this passive layer. When the voltage is applied, the conjugated polymer petals will bend

together to move the elastic diaphragm and generate pressure changes inside the chamber.

125

Therefore, it is the passive layer instead of the conjugated polymer that is being stretched,
and one can choose a passive layer material with low Young’s modulus to significantly

reduce the energy required to stretch the middle-plane.

   

Electrode

Figure 5.1: Schematic of a petal—shape pumping diaphragm (top view). Left: before actua-
tion; right: upon actuation.

5.1.2 Fabrication and Assembly of the Micropump

Fabrication of the trilayer conjugated polymer is achieved by electrochemically oxidizing
pyrrole monomer from a solution to grow PPy layers on either side of a gold-coated porous
PVDF film, by following the similar recipe in [82]. Polydimethylsiloxane (PDMS) is cho-
sen as the material for fabricating the micropump. It is an elastomeric polymer that is much
more compliant than silicon or glass, and it is becoming popular for microfluidic devices.
The Young’s modulus of PDMS varies between 360-870 KPa [83]. Other attractive features
of PDMS include biocompatibility, low cost, and transparent view (which makes it easy to
monitor the fluid flow). The PDMS material is composed of two parts, a curing agent and
the polymer. They are mixed by a certain volume ratio and allowed to cure under a cer-
tain temperature. Therefore, once the micro patterns are fabricated, the inverse patterns of
PDMS can be obtained by applying PDMS on the substrate and peeling it off after it is
cured. SU-8 2150 photoresist is exposed with conventional UV (350-400 nm) radiation to

make micro patterns on a silicon substrate.

126

By utilizing the microfabrication method involving PDMS and SU-8, different func—
tional layers of the pump can be fabricated and eventually assembled together. The as-
sembly schematic is shown in Fig. 5.2. Notice that there are four alignment holes at the
corners of PDMS layers for the ease of final assembly. The conjugated polymer actuator
will be positioned at the top of the pump chamber and attached to the PDMS membrane by

applying uncured PDMS as glue.

Flap valves are chosen to keep the directional flow because of their ease of fabrication
and high efficiency to regulate the fluid flow. The mechanism of a flap valve is illustrated
in Fig. 5.3. The end of each flap is attached to the middle layer, which has two channels
covered by the flaps. When a pressure is applied from the top, the flap on the right will be
pushed down and allow the fluid to flow downwards. When a pressure is applied from the

bottom, the flap on the left will be lifted up, and the fluid will flow upwards.

The microfabrication process to make the flap valve layer is shown in Fig. 5.4. Firstly
an SU-8 layer is spun on the silicon substrate. Then a photo mask is used to pattern the
SU-8 through photolithography. After SU-8 is developed and forms the desired pattern,
PDMS is spun on the SU-8 pattern and baked in oven to cure. Finally the PDMS layer is
peeled off from the substrate for assembly. Other functional layers can be fabricated in the

same way.

The assembled micropump is shown in Fig. 5.6. The structure of the micropump can
be observed in the magnified view shown in Fig. 5.6. The size of the pump is 25x25 mm
(top view) and 10 mm (height). The diameter of the diaphragm is 19 mm. The depth of
the pump chamber is 9.8 mm. The micro channels have the diameter of 1 mm. The check

valve flapper is 1 mm thick, 3 mm wide, and 5 mm long.

127

Conjugated

 

 

 

 

 

 

 

 

 

 

 

 

 

polymer
0 O
5...
\ - _ __ _ _, , ’ Pump
o o chamber
0 O
\
437°
0 c O
l
O O > Check
valves
9 o o
O
l
O

 

 

 

/ Outlet

° °/ Inlet

 

 

Section view of
pump chamber

Figure 5.2: The assembly schematic of micropump.

5.2 Physics-based, Control-oriented Model for the Pump

Because PPy has sophisticated electrochemomechanical dynamics that can vary signifi-

cantly over time [6], a model that captures the complicated dynamics yet amenable to con-

128

 

L 1L 7L

. wilting

l J lfi #

¥ 1 1 |—_ 1
11mm

__

 

 

 

 

 

Figure 5.3: The mechanism of flap check valves.

trol design is important for applying advanced control algorithms to deliver accurate and
consistent flow rates for different applications. As shown in Fig. 5.7, the developed model
in this thesis consists of three cascaded modules: 1) the electrical admittance module of
PPy relating the current I (and thus the charge transferred) to the voltage input U; 2) the
electromechanical coupling module of PPy expressing the generated strain 8m in terms of
the transferred charge; and 3) the mechanical module connecting the generated strain to the
diaphragm curvature K or flow rate <1) of the pump, which captures the dynamics of both
the PPy membranes and the flap valves. In this thesis, the modules 1) and 2) follows those
in Section 2 and therefore will only be briefly reviewed. Module 3) is derived based on an

energy-based method, which will be elaborated in detail.

5.2.1 Electrical Admittance Module of PPy

A diffusive-elastic-metal model was proposed for PPy, where it was assumed that the poly-
mer matrix is perfectly conducting and the ion transport within the polymer is solely deter-

mined by diffusion [16]. It was adapted to model the ions dynamics in the trilayer beam [6].

129

-SU-8 PDMS 1:] Si

 

1 spin on SU-8

W

lpattern SU-8 with
photo mask

{51 m m E]

 

 

O O

@

 

 

 

0

 

 

sMnonPDMS

 

 

lpeel off PDMS

/97/

Figure 5.4: The microfabrication process to make a flap valve.

 

 

The admittance model of a trilayer conjugated polymer was derived as [6]

1(5) ___ s[%—5tanh(h\/S/D)+\/E]
U(S) §+Rs3/2+R3%§stanh(h\/S/D)’

 

 

(5.1)

where U (s) and l (s) are the applied voltage and the resulting current in the Laplace domain,
respectively, 3 is the Laplace variable, 6 is the double layer thickness, D is the diffusion

coefficient, R is the resistance across the trilayer polymer, C is the double-layer capacitance,

130

 

Outlet Conjugated
polymer

0’)

Figure 5 .5: (a) The assembled micropump (top View); (b) The assembled micropump (bot-
tom view).

Alignment holes

Check valves

Electrode

 

Figure 5 .6: A magnified view of the micropump to show the structure.

131

V Admittance I Electromechanical 5 Mechanical Y
Module f Module Module

Figure 5.7: The complete model structure for conjugated polymer actuators.

 

and h is the thickness of the PPy layer. This infinite-dimensional system can be reduced to

the following second—order transfer function

[(3) ~ K~s(s+Z1)
U(S) ~ (8+p1)(S+p2)’

 

 

(5.2)

where the parameters are functions of physical parameters [6], which are either known or

measurable.

5.2.2 Electromechanical Coupling of PPy

The anions transferred to the polymer cause expansion of the polymer. It can be shown that
the strain 8m introduced by the volumetric change is proportional to the density p of the
transferred charges [16]:

em = ap, (53)

where a is the strain-to-charge ratio, which varies for different anions. PPy doped with
TFSI' is used in this thesis, and the strain-to-charge ratio is estimated to be 7 x 10—10
m3 -C"1.

Because the double-layer capacitance is much smaller than the bulk capacitance of
the PPy polymer, the charges stored in the double layer at the steady state is negligible

comparing with those in the bulk. Therefore one can obtain the density p(s)

 

NS) = . (5.4)

where A is the area of PPy.

132

5.2.3 Mechanical Module of the Micropump

Given the actuation strain 8m in the PPy layers, an energy-based method is used to model
the deformation of the pump diaphragm and consequently the flow rate. In this method the
equilibrium of a mechanical structure is obtained by minimizing a properly defined total
energy, and such as approach has been taken to predict the curvature of a composite plate

generated by the strain mismatch in different layers [47, 84, 85].

In the following, we first discuss the common framework that applies to both a clamped
whole diaphragm and a petal—shape PPy actuated diaphragm (simply called petal-shaped
diaphragm hereafter), and then specialize the discussion to individual cases. There are two

relevant energy terms: the elastic energy stored and the work done to the fluid.

The elastic energy of actuation diaphragm will be calculated based on mechanics of
constitutive modeling. The analysis holds for both the whole diaphragm and the petal-
shaped diaphragm cases, because the infinitesimal elements in both cases experience radial
and transverse strains as defined in Fig. 5.8. The diaphragm bends as a result of the de-
formation of all these elements. The difference between the two cases is that the midplane
strain 80 depends on the diaphragm curvature K in the whole diaphragm case because of
the edge constrains of the edges, while in the petal-shaped diaphragm case the curvature is
not related to the middle plane strain. The latter releases the constrains on the curvature K

from the clamped edge.
Figure 5.8: Definition of the principal strains.

For small deflection, the principal strains defined in Fig. 5 .8 can be expressed as follows

133

[47]:

2
du d w
€rr — d— —Z:l'—2 + Es, (55)
r
u zdw
€99 —;—;—d-‘;+€3, (5.6)

where z is the axis in the thickness direction, 8,- is the swelling strain in different layers (8m
in the oxidized PPy layer, —8m in the reduced PPy layer, and O in the PVDF layer), u(r)
and w(r) are the radial and transverse displacements of points in the midplane. Notice that
for the case of a petal-shaped diaphragm, there is a very thin PDMS layer underneath the
conjugated polymer to seal the chamber, but the Young’s modulus of PDMS used in this
thesis is 0.5 MPa, which is much smaller than the Young’s modulus of PPy (60 MPa) and
PVDF as (612 MPa) [1 1]. Therefore, the influence of the PDMS layer on the conjugated
polymer deformation is ignored, and the midplane is taken to be the middle plane of the

PVDF layer.

The displacements u(r) and w(r) in the midplane can be expressed as

u = £0r, (57)
w = g, (5.8)

where 80 represents the strain of the midplane, and K is the curvature of the plane as shown

in Fig. 5.9.

Substituting (5.7) and (5.8) into (5.5) and (5.6), one obtains the following expression

errzeggzeo—z-K+£,. (5.9)

134

 

Figure 5.9: The coordinates and geometry definition.

The corresponding strain energy density is:

E.
‘P(r,z) :2 2—(—1Tl-—2—)-(E,2r + £39 + 2Vi€rr€96)
1+V in
—V-2

l

where E,- and v,- are Young’s modulus and Poisson ratio of the material in the i-th layer,
which are notated as E pp), and VPPy for the PPy layers, EPVDF and VPVDF for the
PVDF layer. Finally the total strain energy of the trilayer conjugated polymer is obtained

by superimposing three energy densities and integrating over the volume

—h1(1+Vp )EP 2
Welastic =_/7r ”/00 [-/h2 Py Py(£0 sin—1'70 dz

 

 

 

”1 (1+V:VDF)EPVDF(80 ___z M261,z
“hl 141%va
h 1+V E
+/ 2( PPy) PPy(£0+£m —z K)2dz]rdrd1:, (5.11)

where r0 is the undeformed diaphragm radius. Since the whole diaphragm and the petal-
shaped diaphragm have the same radius under undeformed configuration, r0 is used in both

cases.

135

In the whole diaphragm case, the curvature denoted as K is assumed to be uniform
when the diaphragm displacement is small. The geometrical relationship is illustrated in
Fig.5.10, where the solid curve represents the sectional view of the middle plane of the

deformed diaphragm. The angle 9 is obtained as
0 = arcsin(x‘r0). (5.12)
One can obtain the strain in the middle plane as

r—ro 9
=———-=———1 5.13
80 r0 KTO ’ ( )

which relates 80 and K.

‘

\
-->

C

L-----__--_--_---_-_-_-__-__

     
    
 

J

1
Figure 5.10: Illustration of the geometrical relationship in (5. 12) and (5.13), R = E.

Since the deformation of the diaphragm will generate the chamber pressure, the work
done by the chamber pressure needs to be considered in the energy based method. Consid-

ering an incompressible fluid, we have the following equation from the mass conservation

136

principle,

I
pv = p /0 (china) —<I>ow<r>)dr, (5.14)

where p is the density of the fluid, (pin and (1)0“, are the flow rates at the inlet and outlet,
respectively,with the unit of m3 /s. The relationship between the chamber pressure p and

the flow rates is modeled as follows:

dp 1
27 = 'C—CW’in —¢0ut)a (515)

where CC is the chamber capacitance that can be determined experimentally with the unit

of m3 /Pa [86]. Combining (5.14) and (5.15), one can obtain
p = —, (5.16)

where pl, : 0 is considered as 0, since there is no initial net pressure in the chamber.

Therefore, one can express the work done by the diaphragm as

’0 W
W = 275/ / p r drrdr
p 0 0 ( )

27: ’O/W
= — th'rdr. 5.17
Cc/O O ( )

. . . . . dW .
The total energy rs W = We + WP. The equrhbnum is reached when E(- = O m

lastic

the whole diaphragm case, since K is the only independent variable. In the petal-shaped

(9W 8W
diaphragm case, the equilibrium is reached when — = 0 and 5? = O.

980
In order to obtain the explicit expressions for both cases, firstly the volume change in the
chamber is calculated for the whole diaphragm and petal-shaped diaphragm respectively.
In the case of the whole diaphragm, the volume change can be characterized as a dome’s

volume

7‘ 2 2

137

where ho is the height of the dome that equals R — ‘/R2 — r3 based on the geometry.

Substituting this expression into Eq. (5.18), one has

7:
_ _ _ 2_ 2 2 2_ 2_ 2
V _ 3(R j/R r0)(R +r0 R‘/R r0)

”3 5 19
z — K .
6 , ( )
r2 2
. . . . . 0 r
where the approxrmatron 1S based on the Taylor series expansron at R—z— = 0 for 1 — 13%.

since the small deflection of the diaphragm implies that R > r0.

Substituting (5.19) into (5.17), one can obtain the expression of WI) and consequently

dW
W. The equilibrium is reached when :1? = 0 in the whole diaphragm case, since it“ is the
dW
only variable. The equation of —d—r_c_ = O can be rearranged as follows:
7
_7‘50_

32cc
+C1r0h%(h2 x — 38m) x + 3C1r0h%£mx° — 6C1r0h2]2 = 0, (5.20)

—36C12r8h% + (1 — Kzrgfl x2 + (C2 — Cl )r0h315 x2

where

 

 

C1 __ (I + VPPy)EPPy 02 _ (1 + VPVDZF)EPVDF
_ 2 a — — 9
1 “ VPPy 1 VPVDF
The curvature K can be obtained by numerically solving this polynomial and appropriately

choosing the root.

In the petal-shaped diaphragm case, the expressions of volume change V and work Wp
are different. As illustrated in Fig. 5.10, the following geometric relationship holds for the

petal-shaped diaphragm when the deflection is small,

1
2a—=m, an)
K

138

which implies

K .
sine z a = 592—. (5.22)

The volume change in the petal-shaped diaphragm case can be approximated by the shad-

owed volume (a top-off cone) in Fig. 5.12 under the assumption of (5.22)

7:
V 2 508+ (r0 —sin26or0)2) -23in29-r0

4
Mo K

 

, (5.23)

22

 

 

 

 

Figure 5.11: Geometric relationship in the calculation of the volume V and the work done
by the diaphragm.

The displacement 5 defined in Fig. 5.11 is expressed as

rgx
.5: sin6 - r0 as —2—. (5-24)

The displacement at each point of the diaphragm is defined as fix in Fig. 5.11, which can

be expressed in terms of x and 5 as

 

:: . (5.25)

Deformed
configuration

 
 
 

 

l
l

 
   

Original
configuration

Figure 5.12: Illustration of the volume change under the actuation voltage for the petal-

shaped diaphragm.

By substituting (5.23) into (5.17), one can express the work done by the diaphragm as

r0 5;;
WI) 2 271’ O A; p(t')d’t'rdr

 

 

 

2 8 2
___ 7t rOK
IZCC .
W W
The equilibrium is reached when _8__ :: O and L = 0. Since
880 8K
8W
58—0— = 47tr(2)£0[C1 (112 — h1)+C2hl],
a—K‘ = “475 0[C1€m(h2“h1)- 3 ’T1
2 8
+7! I‘OK’
6CC ’
the equilibrium is:
80 = 0,

3C1 8,7101% — h?)

m6
+

 

3
C, h2 — C1h% + C25?

(5.26)

(5.27)

(5.28)

(5.29)

(5.30)

Eq. (5.30) captures the static deformation of the diaphragm in terms of em. The material

damping effect is introduced as (2.26) to incorporate the dynamics during the deformation

process. The values of parameters in (2.26) are kept the same for both PPy and PVDF.

140

Other damping effects in the pump, such as the flap valve damping, are captured by using

the equivalent chamber capacitance that includes a damping term

Cc

C' (s) = ———————, (5.31)
C 1 + CCRCS
where RC represents the equivalent damping resistance with the unit of Pa - s/m3.
Thus one can write (5 .30) in the Laplace domain as
K=H(S)'€m, (5.32)

where 2 2
3C h —h
11(3) = l( 2 1)

 

3 3 3 33,5 ,
Clhz -C1h1+C2hl + mag
and C1, C2, and CC! are the frequency—dependent terms, because the Young’s moduli in

(5.30) are replaced by the ones in (2.26).

5.2.4 Complete Model
One can obtain the complete model for the petal-shaped diaphragm in an analytical form
by combining (5.2), (5.3) and (5.32), which is shown as follows:

K(s) : aK- (s+zl )H(s)
(1(8) 4012 —h1)-'S(S+P1)(S+P2)'

 

(5.33)

 

Note that h = h2 - hl in (5.2). However,the complete analytical model for the whole
diaphragm is not easy to obtain due to the complexity of (5.20). Therefore, the model

prediction in that case is obtained by numerically solving (5.20).

141

 

Considering (5.14) and (5.23), one gets

7LT

Oh

(bin — (bout = ' S ' K. (5.34)

5|

Combining (5.33) and (5.34), one can furthermore obtain the transfer function model from
the voltage input to the flow rate:
(13mm —¢our(5) nrgaK-(s+zl)I-I(s)

Um :4A(h2—hl)'(5+p1)(s+p2)' (5’35)

 

 

5.3 Experimental Results

A trilayer PPy circular actuator is used in the experiments, with the radius r0 of 10 mm. The
curvature is measured by a laser sensor (OADM 2OI6441/S14F, Baumer Electric Inc) with
a resolution of 5 pm. The model parameter R is identified by applying a high-frequency
(100 Hz) sinusoidal input U Sin(a)t). From (2.12), the impedance approaches R as a) —+ 0°.
To measure the parameter C, a step voltage is applied and the transferred charge into the
PPy layer is computed by integrating the charging current. The calculation details can
be found. in [6]. Finally, the resistance and capacitance are identified to be: R = 27 Q,
C = 8.69 x 10’5 F. The diffusion coefficient D is chosen to be 2 x 10’10 mz/s based
on [16]. The double-layer thickness 5 is estimated to be 25 nm based on [18].

5.3.1 Admittance

Since cutting does not influence the electrical property of conjugated polymer, (2.12) is
used to predict the admittance of both the whole diaphragm and the petal-shape diaphragm.
The comparison is shown in Fig. 5.13, where a good match of admittance is seen between

the model and the experimental results.

142

 

 

 

Magnitude (dB)

1: 1 .1
11111111 11111111 11111111 111111111 1111111

 

 

 

 

  
 

 

 

 

 
       

 

 

60
1o" 10'1 10° 101 102 103
100 i l 1 l v v vvvr r I v [V l[[ v y-vvv-v
A l 0 Unit diaphragm ,_
3 ; 6 Petal-shape diaphragm‘
3 8 ; -—Model predictlon
" 5° 7 tit: ' 1"":i - 7
m I l l 1
10 . .
.c .
IL I q
0-2 A Lil-1* LL..1.0.......1 1'2 ms
10 10 10 10 10 10

Frequency (rad/sec)

Figure 5.13: Comparison of model prediction from (2.12) with the experimental results for

the whole diaphragm and the petal-shaped diaphragm.

143

5.3.2 Displacement

In experiments the displacement at the center of the diaphragm is measured by a laser
distance sensor. The predicted curvature of the petal-shape diaphragm from (5.33) is used

to calculate the displacement based on the following equation [6]:

2
K]

where l is the distance between the clamped end and the laser incident point when the beam

is at rest, which is r0 minus 1 mm in the setup.

The predicted displacements for the petal-shape diaphragm are compared with the ex-
perimental results when there is no water in the pump (Fig. 5.14) and when pumping water
(Fig. 5.15). There the actuation voltage applied is sinusoidal with an amplitude of 4 V
and with different frequencies. All the other parameters in (5.33) are given or identified in
the previous admittance model experiments. CC and RC can be determined by fitting the
experimental data for petal-shaped diaphragm. They are identified as CC = 1.02 x 10—8
m3 / Pa and RC = 3 x 108 Pa - s/m3 . To predict the results when there is no water in pump,
the parameters CC and RC are set to be cc and 0 respectively, which sets the fluid energy
to be zero to eliminate the fluid energy term. Thus the curvature is merely determined by
the equilibrium of elastic energy. The experimental results for the whole diaphragm case
are also compared with the numerical solution of (5.2), (5.3), and (5.20). The result when
there is no water is shown in Fig. 5.16, while the result when pumping water is shown in
Fig. 5.17. The actuation voltage is 4 V. It can be seen that the petal-shaped diaphragm can
generate almost 10 times larger displacment than the whole diaphragm when there is no wa-
ter in the chamber. When pumping water, the displacement of the petal-shaped diaphragm
is almost 3 times larger than the whole diaphragm. These experimental results here proved

the effectiveness of the petal-shaped diaphragm design in alleviating the constrains from

the clamped edge.

144

 

 

 

 

Displacement (mm)
[0

 

 

Viv

3: Experimental
‘ L: Simulated

5‘31

1'1:
0
1

   

Phase (deg)
'5

 

 

 

_ ..;:..;.. 1. ;:.;;
150-2 -1 0 1 2

10 1o 10 10 10
Frequency (rad/sec)

Figure 5.14: Comparison of model prediction from (5.33) with the experimental results for
the petal-shaped diaphragm when there is no water (actuation voltage amplitude 4 V).

145

 

1

P
a)
l

 

 

 

Displacement (mm)
D
.b

 

 

   
 

 

 

0.2—--
o L
10'2
o I ' fi' ViTTrT. ' ' ‘ 1 III I I I .11
A . . “ ‘ ; 1 T 3: Experimental
9 X j j , .. 1 1 . j , — Simulated
13 ; ' 1 1.2: . 1 ; 1 . ‘ . : . .
V 1 1‘1 i 1 1
a, '50 ' ********* ‘ I i .-
ge ; z
.1: . 1 .x ;
E ; 1 1
_100 i . L .. i1 1 1 1 . .1.
10' 10'1 10° 101

Frequency (rad/sec)

Figure 5.15: Comparison of model prediction from (5.33) with the experimental results for
the petal—shaped diaphragm when pumping water (actuation voltage amplitude 4 V).

146

 

P
a:

.°
C,
1

 

 

 

Displacement (mm)
D
b

 

 

 
 

 

  

 

 

 

0.2
9
10'2
0 ' I ' i ‘ #_1__._.___‘
A {’3} Experimental
9 50 — Simulated
E ' .
8
(U -100~ ‘~
g 1
o. y :
_150 1 1 1.11111 1 1111111 1 QLJ‘IA11 . . 1 1 1
10'2 10'1 ° 01 1o2

10 1
Frequency (rad/sec)

Figure 5.16: Comparison of model prediction with the experimental results for the whole
diaphragm when there is no water by numerically solving (5.20) (actuation voltage ampli-

tude 4 V).

147

 

 

Displacement (mm)
.0

 

 

1
1 1

'20 1 ' . 1 ' '11 1 1 1 1 ' 1F_1_1 . 1 1 '_'_>: T
8 ‘ ‘ i 1 5 1 ‘ I 1 3: Experimental
_ ,1 1 —Simulated

. . ‘ ‘ ‘ r ,7 ‘ ‘ ‘ J

   
 

 

Phase (deg)

 

10'2 10'1 10
Frequency (rad/sec)

 

Figure 5.17: Comparison of model prediction with the experimental results for the whole
diaphragm when pumping water by numerically solving (5.20) (actuation voltage ampli-

tude 4 V).

148

5.3.3 Flow Rate

The flow rates of the petal-shaped diaphragm pump are measured and compared with the
model prediction (5.35) for an actuation voltages of 3 V (Fig. 5.18) and 4 V (Fig. 5.19). It
can be seen that the model can predict the experimental results well under different input
voltages. The frequency to achieve highest flow rate is 0.5 Hz, which is also predicted well
by the model. When the frequency of the input voltage is low, the flow rate is small due
to the slow movement of the pump. The flow rate will increase as operating frequency
increases. However, the flow rate will decrease as frequency becomes higher, because of
the damping effect of the flap valves and the declining response of conjugated polymer
actuator at high frequencies. Thus the model (5.35) can facilitate the design optimization
and the feedback control of the flow rate.

we have also tested the pumping performance of the whole diaphragm. However, the

flow rate is barely observable in that case.

5.4 Chapter Summary

In this thesis, circular conjugated polymer actuators are investigated for potential microp-
ump applications. A petal-shaped diaphragm design is proposed to alleviate the constrains
from the clamped edge. An analytical model is proposed that captures the relationship
between the actuation voltage and the diaphragm deformation/flow rate. Experiments are
conducted to verify the model and identify the parameters. The largest flow rate achieved
in the current experimental setup is 1260 uL/min, when the operating frequency is 0.53 Hz.
For comparison, modeling analysis and experiments are also conducted on the whole di-
aphragm case. The model predicted much smaller diaphragm deformation compared with

the petal-shaped diaphragm, which is also verified in experiments.

149

I _' )7.
.

 

 

1109 ‘ MI E 1 A

‘ 1; 1 ‘ 1 : ; .11 ‘ rnfl- Experimental
1000~ 7—1‘ ‘9; ~ 931' j — Slmulated
800—
700L-
600~

500 ~

Flow rate (pL/min)

400*”'i'1:

 

 

 

 

1
Frequency (Hz)

Figure 5.18: Flow rate of the micropump at different frequencies when input voltage mag-
nitude is 3 V.

150

 

  

 

 

 

14cc , _ f 1
:1‘1' 1 1 1 1‘1: : 1 fl Experimental
3 . l‘ I — Simulated
1200”” 3 ’1 ?‘1'”"1 ’ : ” f"? 1‘11’
A
51000 ' ‘
E ’ ’
\ 1
J l 1
1 1 1
v 1 1
q) 800” * 'E
q..- .
a
h 11. 1 I 11
5 600” ”7””7' 3' WWW"? 1‘1 11*"; (”1’115’"
4oo~ »-1 4
209 . ...:;‘:1 L ..;;;.:. 1 ..;;.... 1 :..;;.i
10'2 10'1 ° 10‘ 1o2

10
Frequency (Hz)

Figure 5.19: Flow rate of the micropump at different frequencies when input voltage mag-
nitude is 4 V.

151

Chapter 6

Conclusion

In this dissertation, we have focused on modeling, control, and application of conjugated
polymer actuators and sensors. Experiments have been conducted to validate the modeling
approach. Model-based control strategies are then proposed to control conjugated polymer
actuators. The application of conjugated polymer in micropump is investigated from both

modeling and experiment perspectives.

The contributions of this dissertation on the modeling aspect include two main parts.
The first part is the development of a scalable, low-order transfer function model for con-
jugated polymer actuators that captures the major sophisticated electrochemomechanical
dynamics, but is still control-oriented and therefore suitable for real-time control. This
model is obtained by ignoring high-frequency dynamics in an infinite-dimensional model.

The proposed model provides the theoretical basis for the adaptive control scheme.

The second part is modeling of the nonlinearities existing in electrical and mechanical
domains. The coupling of redox level change with ions migration in conjugated polymer is
considered as the main source of nonlinearity in electrical dynamics. A partial differential
equation (PDE) that governs its electrochemical dynamics is proposed and solved to ad-
dress the nonlinear electrical property of conjugated polymer. The mechanical nonlinearity

of conjugated polymer actuator is also investigated. Instead of using the elastic modulus

152

as in the linear elasticity theory, we use a nonlinear strain energy function to capture the
stored elastic energy under actuation-induced swelling, which further allows us to compute
the induced stress even under large actuation voltage. The proposed framework can also be

applied to the analysis of large deformations in some other electroactive polymers.

We have presented a fiber-directed conjugated polymer actuator that can generate tor-
sional motion, which is due to the anisotropy associated with the interaction between the
fiber and the material matrix. The nonlinear elasticity-based model is utilized to capture
the actuator performance. The effectiveness of the model is verified through comparison

with experimental results.

Besides modeling on actuation performance of conjugated polymer, the electromechan-
ical sensing behavior of conjugated polymer is experimentally characterized and mathemat-
ically modeled as well. A theory for the sensing mechanism is proposed by postulating that,
through its influence on the pore structure, mechanical deformation correlates directly to
the concentration of ions at the PPy/PVDF interface. This provides a key boundary con-
dition for the partial differential equation (PDE) governing the ion diffusion and migration
dynamics. An analytical model is then obtained in the form of a transfer function that

relates the open-circuit sensing voltage to the mechanical input.

We have proposed an effective adaptive control strategy for conjugated polymer actu-
ators based on our control-oriented model. A recursive. least—squares algorithm is used to
identify online the parameters of the transfer function model, which captures the essential
actuation dynamics that varies significantly with time. A self-tuning regulator is designed
based on the identified parameters to form the closed-loop control system. A parameter
projection step ensures that the parameter estimates stay within the physically-meaningful

region, and thus makes the system robust against noises and unmodeled dynamics.

The application of conjugated polymer as a micropump is investigated. We have pro-
posed a diaphragm micropump actuated by conjugated polymer in a petal-shape design to

alleviate the edge constrains. Transfer function models from the actuation voltage to the

153

diaphragm deflection and flow rate are obtained. The pump body is fabricated through
MEMS fabrication process to miniaturize the pump size. Experiments are conducted to
evaluate the micropump performance. The effectiveness of the transfer function models
are also verified by the experiments, which will facilitate design optimization and the use
of feedback control tools in dealing with the complicated behavior of conjugated polymer
actuators.

There are several interesting directions to expand the work in this dissertation.

Firstly, although adaptive control scheme we proposed has shown its superiority in
overcoming the time-varying effects, its application is constrained in the linear range. The
actuation voltage magnitude has been tuned small so that the change of redox level with
actuation voltage can be ignored. The nonlinearity will impair the performance of the con-
trol system based on the linear model. We have proposed nonlinear models to capture
the significant nonlinearities in conjugated polymers in both electrical and mechanical do-
mains. An interesting research direction is to apply nonlinear control schemes based on the
nonlinear models of conjugated polymer actuator to achieve better performance, when the
nonlinearities become significant, i.e., large deformation under large actuation voltage.

Another direction is to further pursue the application of conjugated polymer in mi-
crofluid medical devices that are experiencing significant development recently. A microp-
ump based on conjugated polymer actuator has been investigated in this dissertation. We
have built and tested the prototype of micropump. The model relating the flow rate to
the actuation voltage is also proposed. It is meaningful to design feedback control system
based on the proposed model to overcome the performance decaying of conjugated poly-
mer, which would allow the conjugated polymer micropump to deliver desired and accurate

fluid for a long period of time.

154

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