Elasticity of Main Chain Liquid Crystal Elastomers and Its Relationship to Liquid Crystal Microstructure

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ELASTICITY OF MAIN CHAIN LIQUID CRYSTAL ELASTOMERS AND ITS RELATIONSHIP TO LIQUID CRYSTAL MICROSTRUCTURE

A dissertation submitted

to Kent State University in partial

fulfillment of the requirements for the

degree of Doctor of Philosophy

Sonal Dey

December, 2013

Dissertation written by

Sonal Dey

B.Sc., Cotton College, Gauhati University, India, 2004

M.Sc., Indian Institute of Technology Roorkee, India, 2006

M.A., Kent State University, USA, 2012

Ph.D., Kent State University, USA, 2013

Approved by

, Chair, Doctoral Dissertation Committee Dr. Satyendra Kumar

, Members, Doctoral Dissertation Committee Dr. Elizabeth Mann

Dr. Brett D. Ellman

Dr. Edgar Kooijman

Dr. Jonathan V. Selinger

Accepted by

, Chair, Department of Physics Dr. James T. Gleeson

, Associate Dean, College of Arts and Sciences Dr. Janis Crowther

TABLE OF CONTENTS

LIST OF FIGURES ...... VI

DEDICATION...... XVI

ACKNOWLEDGEMENTS ...... XVII

CHAPTER 1 INTRODUCTION ...... 1

1.1 Classical rubber elastic theory ...... 2

1.1.1 Entropic elasticity of a single chain of macromolecules ...... 7

1.1.2 Gaussian elasticity of a crosslinked macromolecular network ...... 12

1.1.3 Non-Gaussian chain statistics of elastomers at high extensions ...... 14

1.2 Liquid Crystals ...... 20

1.3 Liquid Crystal Elastomers ...... 26

1.3.1 Neo-classical theory of nematic rubber elasticity ...... 29

1.4 Chapter Summary ...... 33

REFERENCES ...... 34

CHAPTER 2 SOFT ELASTICITY AND SHAPE MEMORY EFFECT ...... 38

2.1 Soft elasticity of nematic elastomers ...... 38

2.1.1 Soft elasticity in monodomain nematic elastomers ...... 40

2.1.2 Semi-soft elasticity: effect of non-ideal network ...... 45

2.1.3 Soft elasticity in polydomain nematic elastomers ...... 46

2.2 Domain rotation in smectic elastomers ...... 51

2.3 Shape memory effect in polydomain main-chain smectic-C elastomers ...... 56

2.4 Chapter Summary ...... 59

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REFERENCES ...... 61

CHAPTER 3 X-RAY DIFFRACTION: THEORY AND EXPERIMENTAL

TECHNIQUE ...... 63

3.1 Bragg’s law of diffraction ...... 63

3.2 Basic theory of X-ray scattering ...... 66

3.3 X-ray diffraction pattern of common liquid crystal phases ...... 71

3.4 Elastomer Liquid Crystals ...... 76

3.5 Mechanical properties of LCE1 and LCE2 ...... 78

3.6 Experimental Set-up ...... 80

3.6.1 Synchrotron X-ray source ...... 80

3.6.2 Beam optics and the X-ray spectrometer ...... 83

3.6.3 Application of Uniaxial Strain ...... 84

3.6.4 Image plate detector ...... 86

3.6.5 Data collection, calibration, and analysis ...... 88

3.6.6 Determination of orientational order parameter ...... 89

3.6.7 Experimental Details ...... 91

3.7 Chapter Summary ...... 92

REFERENCES ...... 94

CHAPTER 4 SOFT ELASTICITY IN A MAIN CHAIN SMECTIC-C

ELASTOMER ...... 97

4.1 Discussion of Results ...... 100

4.2 Polydomain-monodomain transition by uniaxial strain ...... 101

4.3 Mechanism behind polydomain-monodomain transition ...... 109

4.4 Relaxation at constant strain ...... 112

4.5 Strain retention ...... 118

4.6 Thermal shape recovery ...... 120

4.7 Chapter Summary ...... 125

REFERENCES ...... 127

CHAPTER 5 THE EFFECT OF TRANSVERSE RIGID SEGMENT ON STRAIN

RETENTION...... 129

5.1 Effect of TR3 on the polydomain-monodomain transition ...... 129

5.2 TR3 and strain retention ...... 136

5.3 Effect of TR3 on thermal shape recovery ...... 138

5.4 Chapter Summary ...... 142

REFERENCES ...... 144

CHAPTER 6 CONCLUSIONS AND FUTURE WORK ...... 145

6.1 Conclusions ...... 145

6.2 Suggestions for future work ...... 147

REFERENCES ...... 150

LIST OF FIGURES

Figure 1.1.1 Schematic diagram showing a network of macromolecular chains cross-

linked (the black blobs) at various positions...... 3

Figure 1.1.2 A typical force-extension curve of a classical elastomer, adapted with

permission from [3]...... 5

Figure 1.1.3 (a) Freely-jointed model of a macromolecular chain. Each segment is of

length d and there are p junction points so that the maximum extension-length is

D = pd. β is the angle between two adjacent segments. (b) A particular

macromolecular chain conformation with an end-to-end distance l (l  D)...... 8

Figure 1.1.4 Non-linear force extension curve for elastomers is shown according to the

eqn. (1.12). The relation is valid until the polymer chains remain Gaussian. But it

fails to explain the upward turn in the force-extension curve at higher extensions as

depicted in Figure 1.1.2...... 15

Figure 1.1.5 Force-extension curve of an elastomer under three different models. The

upward curve in the force-extension plot (Figure 1.1.2) is explained by taking into

account the finite extensibility of the elastomer network...... 18

Figure 1.2.1 Schematic showing the orientation of a rod-like nematic mesogen in the

laboratory frame of reference. The macroscopic director n is taken along the z-axis;

m is the axis of the molecule; θ and ϕ are respectively the polar and azimuthal angles

made by the molecule in this frame. The graph to the left shows a smoothly behaving

form of the molecular distribution function where it reaches the minimum value in

the direction orthogonal to n...... 21 vi

Figure 1.2.2 A plot of the Landau-de Gennes nematic free energy density for three

different temperatures. For T  TNI, the system is in the isotropic state with free

energy minimum at S = 0. The transition takes place at T = TNI. For T  TNI, the

minimum of the free energy density moves to S  0 which is inside the nematic

phase...... 25

Figure 1.3.1 Polymer chain conformations and mesogens (dark ellipses) organization

depend on the length, flexibility, number (even/odd) of carbons in the spacer, the

type of attachment (side-on or end-on), and interaction among them. Several

possible scenarios are shown for, both, the main-chain and side-chain liquid crystal

elastomers: (a), (c), and (d): Mesogens lying parallel to the polymer backbone

favoring prolate conformation; and (b) and (e): Polymer backbone perpendicular to

mesogens adopts an oblate conformation...... 27

Figure 2.1.1 Stress-strain plots for a polydomain nematic elastomer at different

temperatures with the appearance of a plateau. Here strain is measured as change in

length / original length. Reproduced with permission from [2], © (1998) American

Chemical Society...... 39

Figure 2.1.2 Schematic representation of the soft mode of director rotation in nematic

elastomers. The chain distribution, represented by the index ellipsoid, is embedded in

the elastomer matrix. For a strain applied perpendicular to its initial direction, it can

rotate in a continuous manner without distorting the elastomer matrix. Reproduced

with permission from [9], © (2007) Oxford University Press...... 44

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Figure 2.1.3 Polarizing optical micrographs of two polydomain nematic elastomers: (a)

isotropic genesis and (b) nematic genesis of the network. The difference in domain

sizes in the two elastomers is apparent. Reproduced with permission from [4], ©

(2009) American Chemical Society...... 47

Figure 2.1.4 Nominal stress (σ) and global orientational order parameter (S) for the two

polydomain nematic elastomers in Figure 2.1.3: (a) isotropic genesis and (b) nematic

genesis of the network. Reproduced with permission from [4], © (2009) American

Chemical Society...... 49

Figure 2.2.1 (a) Figure showing two relevant deformations while a smectic elastomer is

strained perpendicular to its layers. (b) Initial direction of layer normal for two

general in-plane unit vectors r, s; (c) Direction of layer normal after a deformation.

Note that the layer is now rotated. λij and εijk are the deformation tensor and the Levi-

Civita tensor respectively...... 53

Figure 2.2.2 The broken curve is the fit of eqn. (2.3) to the experimental data in [16].

Insets show the schematic of layer rotation. Adapted with permission from [15], ©

(2005) American Physical Society...... 55

Figure 2.3.1 The complete shape memory cycle in a polydomain main-chain smectic-C

elastomer obtained from dynamic mechanical analysis. Case (a) is for the smectic-C

elastomer, (b) is the case of a classical elastomer where no “fixing” of secondary

shape is observed and (c) is the schematic pathway for the shape-memory

experiments performed on the elastomer samples studied in this dissertation work.

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Figure 3.1.1 Schematic diagram of (a) Bragg diffraction and (b) elastic scattering of X

rays resulting in a change in momentum vector by q...... 64

Figure 3.2.1 Representation of density-density correlation function Cnn for (a) long-range,

(b) quasi long-range and (c) short-range order and the corresponding X-ray intensity

profiles I(q||)...... 70

Figure 3.3.1 Two dimensional XRD patterns of some common liquid crystalline phases.

Detector is lying in the plane of the paper and the X rays are incident normal to it. 73

Figure 3.4.1 Schematic representation of elastomers LCE1 and LCE2: Here, “x” is the

mesogen component, “y” is the TR3 part which is incorporated into the main chain

of LCE2, “m” is siloxane spacer, “n” is a siloxane based crosslinker. LCE1 has only

the mesogens incorporated at the “end-on” positions where as many as four chains

could attach to the octasiloxane crosslinker. In LCE2, 20% of the “x”-blocks are

replaced by “y” blocks in LCE2. Adapted from [17]...... 77

Figure 3.5.1 Stress-strain behavior of C11(MeHQ)Si8XL10 elastomers with and without

p-terphenyl (TR3) transverse rod. Strain is measured in terms of percentage change

in elastomer’s length compared to its un-stretched length. (a) With 0 mol% of TR3

or LCE1; (b) with 10 mol% of TR3 and (c) with 20 mol% of TR3 or LCE2. The

green dashed lines are guide to the eye, separating the three elastic regions based on

the stress-strain plot of LCE1 (solid line). Adapted from [17], © (2009) John Wiley

and Sons...... 79

Figure 3.6.1 Schematic experimental setup at the 6-ID-B beamline of the Advanced

Photon Source at Argonne National Laboratory...... 82

Figure 3.6.2 Mechanism to apply uniaxial strain: (a) the motor controller used to control

the uniaxial stretching, (b) INSTEC temperature controller and (c) photograph of

custom-made stretching setup inside the heating oven HCS402...... 85

Figure 3.6.3 Silicon standard (NIST 640C) was used to calibrate the X-ray spectrometer.

The powder-like ring in the XRD pattern (left) was integrated and matched with the

value of the corresponding d-spacing (right) for powder silicon...... 87

Figure 3.6.4 (a) A representative X-ray diffraction pattern of LCE1. The scattered

intensity is first integrated (in radial direction) between the two circles and then the

data for intensity vs. the azimuthal angle χ (direction marked by the arrow on the X-

ray diffraction pattern) is generated using the FIT2D software, (b) plot of I vs. χ for

the above wide angle area of the diffraction pattern. The solid line is the fit to the

experimental data according to the eqn. (3.12) which gives the value of the fit

parameter b, (c) χ2, which is a measure of goodness of fit, is plotted against the fit-

parameter b. The values of χ2 are normalized with respect to the best fit value. The

’s correspond to the 95% confidence limits determined by the F-test:

F(356,356) ≈ 1.191. From this plot, error in the value of the fit-parameter b is

calculated...... 90

Figure 4.1.1 Schematic diagram showing the conical distribution of smectic layers around

the mechanically induced director (dotted red arrow). The green entities are

mesogens. The polymer components are omitted for clarity. The solid arrows show

the direction of the smectic-C layer-normals which are distributed on the surface of a

cone around the macroscopic director...... 98

Figure 4.2.1 (a) Representative WAXS (left images at each strain) and SAXS patterns at

strains (marked on the patterns) applied in the vertical direction (red arrow). The

WAXS patterns also show the effect of shadowing from different parts of the sample

holder, stretching mechanism, and the oven; (b) stress-strain plot for LCE1 with

vertical dashed-lines acting as guides to the eye for separating three major regions of

the curve and (c) definition of the angle α and its relation to the liquid crystal

microstructure...... 102

Figure 4.2.2 Azimuthal intensity I (arbitrary units) profiles of both SAXS and WAXS

(hydrocarbon part) reflections in Figure 4.2.1 (a). The small modulation of the SAXS

peak at λ = 1.0 is due to a small strain induced during mounting this particular

specimen. The Gaussian peak functions are fitted to calculate the angle α. At λ = 1.2,

the SAXS are formed parallel to the stretch direction with WAXS remaining more or

less uniform. At λ = 1.5, the SAXS reflections are flat at the top, implying a possible

superposition of two or more peaks eventually splitting into four peaks. The

gradually increasing separation of SAXS peaks is clear at λ > 1.6. The WAXS

reflections also continue to gradually sharpen with increasing strain...... 104

Figure 4.2.3 Schematic diagram of the smectic-C layers formed inside the elastomer

specimen (at low strains) as the elastomer film is uniaxially deformed along the z-

direction. The x rays are incident on the sample from y-direction. The red rods here

are the mesogens while the polymer chains are omitted for clarity. At low strains, the

layers are formed parallel to the stretch direction with the layer normals distributed

uniformly in the x-y plane. For simplicity, only two sets of layers are shown. The

layers, which are formed parallel to the z-axis, contribute to the SAXS reflections

appearing along the x-axis. The random distribution of the mesogens (in the x-z

plane) contributes to the WAXS reflections (corresponding to the hydrocarbons)

appearing in the x-z plane...... 106

Figure 4.2.4 (a) - (d) Uniaxial strain dependence of α, d and S (mesogen and siloxane

parts). All values are calculated after ~ 20 min of equilibration at each strain. The

equations in the insets of the panels (c) and (d) correspond to the solid line fits,

adhering to a phenomenological growth model [17]...... 108

Figure 4.3.1 (a) Schematic representation of strain applied in vertical- or z-direction to

the elastomer film. The film is squeezed in both x and y-directions for conserving the

volume of the elastomer. The x rays are incident parallel to the y-direction. (b) - (d)

Schematic illustration of the polydomain-monodomain transition with increasing

strain, associated microstructure and the corresponding wide and small angle (right

hand side image) XRD patterns...... 110

Figure 4.4.1 (a) - (d) Time dependences of α, d and S (hydrocarbon and siloxane parts)

are shown during stretching across the polydomain-monodomain transition region.

The vertical dashed lines mark different strain regions with values of strain λ shown

at top. The green and orange points correspond to λ = 1.0 and 1.2 respectively. The

red- and black-points are from measurements on specimen S1 and S3 respectively.

Both specimens were cut from the same elastomer sample. The results also show the

reproducibility of our measurements...... 113

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Figure 4.4.2 Relaxation behavior of the angle α at different constant-strains. The solid

lines are the fits to eqn. (4.1)...... 115

Figure 4.4.3 Plot of relaxation time constant, τ as function of strain, λ. The values are

calculated from the fits of eqn. (4.1) to the experimental data in Figure 4.4.2. The

colored lines are guide to the eye showing the transition clearly...... 117

Figure 4.5.1 (a) - (b) WAXS and SAXS XRD patterns at t ~ 0 and after t ~ 24 hours. The

WAXS patterns also show the effect of shadowing from different parts of the sample

holder, stretching mechanism, and the oven, (c) time dependence of S and α. The

solid curves are fits to eqn. (4.1). The steady state was achieved after ~ 90 minutes

with S = 0.68 ± 0.01 and α = 40.4° ± 0.2° such that the elastomer is locked into a

chevron-like microstructure long after removal of the strain...... 119

Figure 4.6.1 WAXS (left column) and SAXS (right column) patterns acquired during

thermal shape recovery of LCE1. The WAXS patterns also show the effect of

shadowing from different parts of the sample holder, stretching mechanism, and the

oven. The peaks gradually turn into diffused rings and lose their intensity as one

nears TI. The curved arrows show the direction in which the SAXS reflections

combine into uniform rings...... 121

Figure 4.6.2 Temperature dependence of α and S for the mesogenic part as the transition

to the isotropic phase at TI is approached. The elastomer is slowly heated from the

chevron-like monodomain state at room temperature to the isotropic state above

temperature TI where the original polydomain state is recovered...... 123

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Figure 5.1.1 (a) Stress-strain plot for LCE2. (b) WAXS (left images at each strain) and

zoomed-in SAXS patterns of LCE2 taken at different strains (marked on the

patterns) at room temperature. The WAXS patterns also have some shadows from

different parts of the sample holder, stretching mechanism, and the oven. All images

were acquired after ~ 25 minutes of equilibration time at each strain. The SAXS

pattern at λ = 1.2 already shows splitting into four-spot reflections. The red arrow

represents the stretch direction. (c) Definition of angle α and its relationship to

chevron-like LC microstructure...... 130

Figure 5.1.2 Values of the parameters d, S and α calculated for the duration of the

uniaxial experiment on LCE2. The numbers in red in the panel for the angle α

represent a particular strain λ...... 132

Figure 5.1.3 (a) Relaxation of angle α at different constant strains. The solid lines are fits

to simple exponential plots. (b) Plot of relaxation time constant, τ as a function of

strain, λ. The colored-lines are guides to eye...... 134

Figure 5.1.4 Comparison of the values of the angle α in LCE1 and LCE2 as function of

strain. The value of angle α is always smaller in LCE2 as compared to LCE1...... 135

Figure 5.2.1 Comparison of the values of α/αₒ and S/Sₒ during strain retention

experiments. Here, αₒ and Sₒ are the values of α and S at t = 0 min. Solid lines are

simple exponential fits to the data. The values of corresponding relaxation time

constants are also mentioned adjacent to the respective fits...... 137

Figure 5.3.1 LCE2: (a) SAXS patterns during strain induced polydomain-monodomain

transition (strain increase from left to right). The black arrows show the direction of

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stretching. (b) SAXS patterns during thermal length recovery (temperature increase

from right to left). During length recovery, the SAXS peaks merge into each other

forming arc like reflections in the direction orthogonal to applied strain...... 139

Figure 5.3.2 Plots of global orientational order parameter S during thermally driven shape

recovery process of LCE1 and LCE2. The value of smectic-C to isotropic transition

temperature for both the elastomers are mentioned in the figure. LCE2 has a lower TI

so the temperature range covered for it is narrower as the experimental setup did not

provide access to below room temperature. The solid curves in the plot for S serve as

guide to the eye...... 141

Figure 6.2.1 (a) Home-made experimental setup for producing perfect monodomain

structure from the strain induced chevron monodomain. The lower end is fixed and

the upper end could slide with the help of a computer controlled motorized

micrometer. The set-up is assembled on an INSTEC heating oven and thus the

experiments could be performed for a wide variety of temperature. (b) Expected x-

ray diffraction pattern based on the initial and final arrangements of the

microdomains inside the elastomer...... 148

DEDICATION

To my loving father, for teaching me the virtue of patience and perseverance...

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ACKNOWLEDGEMENTS

At this junction of my life, I look back and find some wonderful and inspiring people without whom this journey would not have felt as satisfying as it is now. During my past few years in Kent State University, I received immense support and encouragement from my advisor, Prof. Satyendra Kumar. Without his vigilant supervision, this work would never have come to light. I shall like to thank him for being the wonderful person he is and preparing me for a greater role in the academic world. Not to mention the invaluable knowledge of x-ray diffraction in soft matter which I learned from him and will stay with me forever.

I would also like to thank my dissertation committee members Prof. Elizabeth

Mann, Prof. Brett D. Ellman, Prof. Edgar Kooijman and Prof. Jonathan V. Selinger who provided encouraging and constructive feedback. It is no easy task, reviewing a thesis, and I am grateful for their thoughtful and detailed comments.

I am thankful to Prof. Anslem C. Griffin at Georgia Institute of Technology in

Atlanta, Georgia for providing the liquid crystal elastomer samples studied in this dissertation research. The insightful discussions I had with him go a long way in strengthening my understanding of these complex systems.

Appreciation also goes to some of the current and past group members: Dr. Dena

Mae Agra-Kooijman, Dr. Gautam Singh, Dr. Jin-Tae Yuh, Dr. Hyung-Guen Yoon, Dr.

Seung Yeon Jeong, Dr. Leela Joshi, Dr. Shin-Woong Kang and Lewis Sharpnack. I shall

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always remember the numerous discussions we had over a cup of coffee in our ‘coffee- room’ at SRL 011. Thank you all for making this journey so enjoyable. Especially Dr.

Dena Mae Agra-Kooijman and Dr. Leela Joshi for the guidance and the time spent in showing me the working of different experimental setups in the lab and method of data analysis using Fit2D software.

I would also like to thank Dr. Alan Baldwin for the invaluable support regarding all kinds of instrument-troubleshooting. Thank you Alan for teaching me so many unconventional things in such simple ways. Thanks to Mr. Wade Aldhizer for the support in the machine shop and also for introducing me to quite a few techniques in the machine shop. Special thanks goes to Mr. Greg Putman, Ms. Cindy Miller, Ms. Kim Birkner, Ms.

Kelly Conley (present Graduate Secretary) and Ms. Loretta Hauser (past Graduate

Secretary) in the Physics Office. They were of immense help all the time I was at Kent

State University.

Thanks to Prof. David Allender for teaching me the basics of Liquid Crystal and

Statistical Physics. I shall always miss his wonderful lectures. Many thanks to Prof.

Michael Fisch at College of Technology, Kent State University for his various comments and encouragement during the preparation phase of my dissertation defense.

Special appreciation goes to all my friends who provided constant encouragement over the years. Especially Ayan Chakrabarty, for all those memorable discussions we had over last six years. I am very grateful to my parents and my little brother. Without their unconditional love and support, this journey would not have begun at the first place. Last, but not the least, I am forever indebted to my wife Sharmistha, who has been the most

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patient and supportive witness to this journey. There is no way I could repay the invaluable sacrifices she has made, the least I could do is thanking her from the bottom of my heart.

Sonal Dey

November 4, 2013

Kent, Ohio, USA

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CHAPTER 1

INTRODUCTION

This dissertation is focused on understanding the evolution of the liquid crystal microstructure in two main-chain smectic-C elastomers which exhibit soft-elasticity and shape-memory. The stress-strain curve [1] of these elastomers is considerably different than their classical counterparts and comprehending their structural details will give important insights into the origin of the shape-memory effect and soft-elasticity exhibited by these elastomers.

Chapter 1 begins with a discussion about the elastic theory of macromolecular systems and the nature of the force-extension curve in a classical elastomer. Next a brief overview of the physics of nematic liquid crystals is given which originate from their inherent anisotropy. These liquid crystalline components are then introduced into the chains of an elastomer, forming a complex new material, known as liquid crystal elastomer. The neo-classical elastic theory of nematic elastomers is then discussed in brief pointing out the important differences between the classical and liquid crystalline elastomers.

In Chapter 2, the prevailing theories of soft-elasticity, as observed in liquid crystal elastomer systems, are introduced. A brief introduction to the shape memory effect, as observed in smectic-C elastomers, is also given. These constitute the theoretical and experimental background for this dissertation research.

1 2

In Chapter 3, an overview of the basic theory of x-ray diffraction is given. The discussion is followed by some examples of how the technique could be employed in identifying the structural details of different liquid crystalline phases. The elastomer samples used in this study, namely, LCE1 and LCE2, are then formally introduced followed by a detailed discussion of their stress-strain behavior, soft-elastic response and strain-retention abilities, the experimental setup and protocol.

Chapters 4 and 5 have the results on LCE1 and LCE2, respectively. We [2] are specifically interested in the soft-elastic response, strain retention mechanism, and the effect of introducing a transverse segment into the main-chain backbone of the elastomer network. Chapter 6 gives a summary of the major findings of this dissertation project and ends with suggestions for future work on these systems.

1.1 Classical rubber elastic theory

A classical polymer or a classical elastomer does not have rigid anisotropic components in the polymer backbone. Rather, the backbone is relatively floppy. Many such macromolecular chains, when cross-linked at different positions, create the semi- solid matrix of an elastomer, Figure 1.1.1. The material now loses the bulk viscoelastic property exhibited by the macromolecules in a melt. On the microscopic scale, the macromolecular chains still remain viscoelastic between the cross-linking points and could undergo a plethora of conformations which accounts for high extensibility of the rubbery matrix at relatively small stress values. Figure 1.1.2 shows a typical force extension curve for an elastomer. The curve shows a characteristic high extensibility that is natural to a rubbery matrix along with low values of the Young’s modulus. Typically, a

Figure 1.1.1 Schematic diagram showing a network of macromolecular chains crosslinked (the black blobs) at various positions.

rubber can withstand reversible extensions more than five times its original length whereas solid elastic materials like steel have a maximum extension less than 1% of the initial length. Also in comparison, the Young’s modulus for rubber is approximately

106 N/m2 but it is approximately 2x1011 N/m2 for steel in the initial elastic regime at very small strain (< 1%) [3]. Elastic response of the conventional solids like steel is due to the change of the equilibrium inter-atomic distances under stress which correspond to a change in the internal energy of the material. Elastomers are composed of chains which are crosslinked into a network. External stress changes the equilibrium end-to-end distance of a chain and makes it adopt a less probable conformation. The elasticity of rubbers is thus of entropic nature. A material like steel follows linear elasticity theory or

Hooke’s law but rubber shows non-linear elastic behavior in the large extension regime it could cover. The theory of rubber elasticity is thus considerably more complex than the theory of elasticity in conventional solids like steel.

It is possible to determine the relation between the force of extension (f) and free energy (F = U − TS) of the elastomer network on purely thermodynamic grounds. The first and second law of thermodynamics can be combined to write:

(1.1)

Here, U is the internal energy, T is the temperature, S is the entropy and W is the work done on the system. The change in free energy F at constant temperature is written as:

The above two equations can be combined to write (at constant T):

Figure 1.1.2 A typical force-extension curve of a classical elastomer, adapted with permission from [3].

(1.2)

Eqn. (1.2) clearly shows that the work done on the system is equal to the change in the free energy for any isothermal reversible process. Typically, an elastomer is deformed under a constant force f which extends it by a distance r in the direction of applied force at constant atmospheric pressure P and negligible volume (V) changes. The work done by this external force is written as:

The change in volume of elastomer is negligible for all practical purposes so that .

In that scenario, the expression for the force of extension becomes (using eqn. (1.2)):

(1.3)

Eqn. (1.3) expresses the force of extension in terms of the change in free energy per unit extension/compression under constant temperature. The elastomer possesses minimum free energy in the un-deformed state. The free energy increases during extension and reduces during the recovery process. We would later use this equation to find a relation between f and the deformation, λ, which could explain the non-elastic behavior of an elastomer network as shown in Figure 1.1.2. The internal energy of a system depends on the state variables P, V, and T. At constant atmospheric pressure and neglecting the small changes in volume and temperature for all practical purposes, one could see that

for rubber deformation. Eqn. (1.1) and (1.2) then combine to give:

The change in free energy of rubbery network thus comes at the cost of decreasing entropy of the network. This is in contrast to solids like steel where the elasticity originates from a change in internal energy of the system. During extension, the free energy of the elastomer system will rise which amounts to a decrease in the entropy of the network [3]. To lower the free energy, the system will try to maximize its entropy and vice versa. This explains the entropic origin of the elasticity of an elastomer network.

1.1.1 Entropic elasticity of a single chain of macromolecules

Simple thermodynamic argument of the previous section tells us that the elasticity of an elastomer network originates from a decrease in its entropy and the system’s tendency to regain the lost entropy so that it could minimize the free energy. In this section, we shall find the entropy of a single macromolecule chain. Application of this treatment to a cross-linked network of polymer chains forms the theoretical grounds of elastic and non-elastic deformations in elastomers [3].

In statistical mechanics, a suitable model to describe a macromolecular chain with both rigid and floppy components is the freely-jointed model of chain segments. A macromolecular chain is assumed to be equivalent to freely connected rods of length d with a total of p junction points, Figure 1.1.3 (a). Each rod-like segment makes an angle β with the previous segment and it is free to rotate on the surface of a cone about the previous segment. In the ideal case, this rotation is considered to be free but in a real scenario, the interaction energy between different chain segments should be taken into account. Nevertheless, a simple modification [4] of the freely-jointed model (discussed

Figure 1.1.3 (a) Freely-jointed model of a macromolecular chain. Each segment is of length d and there are p junction points so that the maximum extension-length is D = pd. β is the angle between two adjacent segments. (b) A particular macromolecular chain conformation with an end-to-end distance l (l  D).

below) can account for the extra potential arising from those interactions. The chain would assume a random kinked conformation as shown in Figure 1.1.3 (b) with the distance l between the end points much smaller than the fully extended length .

The thermal fluctuations make it impossible to describe the actual macromolecular conformation; rather an equivalent Gaussian random walk is employed to model the situation emphatically. The idea of this freely-jointed model is to define some statistical parameters which could be used in describing the actual macromolecular chain conformation. Based on the Gaussian random walk, it is possible to show that the root mean square (rms) distance between the end points of the macromolecular chain in

Figure 1.1.3 (b) is expressed as [4]:

(1.4)

This result shows that the root mean square end-to-end distance ( ) of the single macromolecular chain is proportional to the square root of the number of junction points in the chain. The macromolecular conformation, as depicted in Figure 1.1.3 (b), remains random and it could be inferred that the structural details of the chain segments only affect the rms distance between the end points. The probability density of finding one end of the macromolecular chain within a small volume element at a distance l from the other end could be written as a Gaussian probability [5]:

(1.5)

Here, the mean square end-to-end distance is given as (also using eqn. (1.4)):

P.J. Flory showed that the probability distribution of the end-to-end distance for a macromolecular chain remains Gaussian irrespective of its structural details when the number of junction points p is relatively large [6]. Any effect of the macromolecular chain’s internal structure is contained in . This is very important as one does not have to worry about the detailed structure of the macromolecule. Instead, an equivalent random chain configuration with similar statistical properties could be chosen to describe its behavior. The mean square end-to-end distance in the three principal directions could now be written as the following statistical average:

Using Kronecker delta, the above representation can be generalized to tensor notation:

(1.6)

In this notation, l2 is written as:

(1.7)

Here, repeated indices imply summation. The tensorial notation will be useful while discussing the case of liquid crystal elastomers later in the chapter. We now use

Boltzmann statistics [7] to find the entropy of a single macromolecular chain:

Here, k is the Boltzmann’s constant and z(l) is the number of available chain conformations for a chain with end-to-end distance l. In this canonical ensemble,

The above expression for entropy of a single chain then simplifies to:

We can shift the origin to the arbitrary constant value which allows one to write the free energy of a single chain (with zero internal energy) as:

(1.8)

The above expression is differentiated with respect to the extension l to give an expression for the force (see eqn. (1.3)) acting on a single chain:

Where, Ys is the elastic modulus of the single chain. The above expression shows that the macromolecular chain elongates along the direction of applied force and the elongation is linearly proportional to the magnitude of the force. Also, the elastic modulus is inversely proportional to the total extended length of the chain. Thus, a longer chain will be more susceptible to the external force. The elastic modulus is also directly proportional to the temperature which means that the restoring force would increase with temperature. This shows the entropic nature of the elasticity of a single chain. It is to be kept in mind that the above theory is valid for chains which are well within the Gaussian limit, i.e., their extension l is much less than the total extended length D of the macromolecular chain.

1.1.2 Gaussian elasticity of a crosslinked macromolecular network

The general form of the force-extension curve of an elastomer has been shown in

Figure 1.1.2. The statistical theory described in the previous section for a single macromolecular chain could be extended for the case of large elastic deformation of a crosslinked network. The positions of the crosslinkers decide the shape of the elastomer matrix. Under any general deformation λij, the crosslinkers are also assumed to move in a proportionate manner. This is known as the affine deformation assumption [3,8] which is at the heart of rubber elasticity. It connects the macroscopic deformation of the elastomer matrix to the microscopic deformation of the polymer strands between two cross-linking points. Let us assume there are N such strands per unit volume of the elastomer. The free energy of such a strand is calculated from the ensemble average of the free energy of the ideal single macromolecular chain discussed in the previous section. Total free energy density would then be obtained by adding up the contribution of all the N strands in a unit volume. Using eqn. (1.7) and (1.8), the average free energy of a single strand in the undeformed state is given as:

The elastomer is now subjected to external force under which a single strand deforms according to the affine deformation assumption:

(1.9)

Here, the superscript T stands for transpose of a matrix. The average free energy of the single strand in the deformed state becomes:

Using eqn. (1.6) and eqn. (1.9), the above expression could be further simplified as:

As there are a total of N strands per unit volume, the free energy density becomes:

(1.10)

The quantity is known as the shear modulus of elasticity which depends on the density of polymer strands as well as the temperature. The free energy expression of eqn. (1.10) is a product of the energy term μ and the geometric modification of the elastomer matrix given by the trace of the product of the deformation matrix and its transpose. As long as the polymer shape remains Gaussian, details of the polymer chain’s structure are not important to the physics of classical elastomers.

The above free energy expression is for general deformation tensor λij. If we now consider applying a uniaxial strain (λ = 1 + ∆L/Lₒ) along one of the principal axes (say x- axis), then the consistency of volume condition would require compressions along the y and z-axes. All other values of λij would be zero. The free energy density expression now further simplifies for the case of uniaxial strain:

(1.11)

This expression for free energy density can be used in calculating the nature of the force- extension curve in the region of deformation where the polymer chains remain Gaussian in shape. Assuming A as the initial cross-sectional area of the elastomer film in the y-z plane, one could write:

(1.12)

The nature of this force-extension curve is plotted in Figure 1.1.4. We can very well see how it deviates from the linear behavior at moderate to high extensions. This Gaussian statistical theory successfully explains the nature of the force-extension curve up to moderate strains, but it fails in explaining the upward turn in the curve at higher extensions, Figure 1.1.2. At higher strains, the individual polymer strands deviate significantly from Gaussian nature, thus causing the failure of the Gaussian statistical theory. One needs to look into the non-Gaussian extension of the theory in explaining the upward turn in the force-extension curve.

1.1.3 Non-Gaussian chain statistics of elastomers at high extensions

Until this point, the end-to-end distances of the polymer strands are assumed to follow a Gaussian probability distribution. The Gaussian distribution could extend to infinity in principle whereas the polymer strands in real network cannot extend themselves beyond their total length D = pd. Thus, the Gaussian chain statistics of previous sections are valid when r  D. At higher strains, this condition is not followed anymore and the effect of finite extensibility of the network is to be incorporated into the

Figure 1.1.4 Non-linear force extension curve for elastomers is shown according to the eqn. (1.12). The relation is valid until the polymer chains remain Gaussian. But it fails to explain the upward turn in the force-extension curve at higher extensions as depicted in Figure 1.1.2.

probability distribution function. The non-Gaussian chain model [3] leads to a limited extensibility of the network and shows that the maximum extension varies approximately with the square root of the number of joints p in the chain. Thus a chain with a smaller number of joints would be better represented using a non-Gaussian model. Nevertheless, the situation becomes more complex and a considerable generality of the Gaussian description is lost in this regime. Kuhn and Grun [3] solved the problem by calculating the most probable distribution of the link-angles in a freely-jointed model. The finite extension limit is given in terms of the Langevin function [9]:

The value of b is given by a solution of the inverse Langevin function [9]:

With this definition, the probability distribution of the end-to-end distance in the non-

Gaussian regime becomes [3]:

(1.13)

The first term in the Taylor series expansion of the above probability distribution is exactly equal to the Gaussian probability distribution given by eqn. (1.5). Thus the

Gaussian statistics is based on the assumption that all the higher terms in “l/D” are negligible compared to the first term. For “ ”, the error in the Gaussian assumption is less than 10%. For still larger chain extensions, the first term

approximation of the expansion in eqn. (1.13) (or Gaussian probability distribution assumption) would deviate considerably from reality.

The probability distribution given by eqn. (1.13) could now be used to calculate the value of entropy and the free energy of a finitely extensible single polymer strand.

The formulism can be extended to the case of elastomers following the method shown in the earlier sections, giving an expression for the force of extension in term of the uniaxial deformation λ:

Keeping only the first terms in the Taylor series expansion of the above inverse Langevin functions, we will get back the result (eqn. (1.12)) corresponding to the Gaussian chain distribution.

Figure 1.1.5 shows a comparison between the linear elastic response, non-linear elasticity with Gaussian chain statistics and the more general case of non-Gaussian chain statistics with finite extensibility of the polymer chains taken into account. The upward turn in the curve comes due to the limited extensibility of the network at high strains and is well-modeled by the inverse Langevin function approximation [3,9]. For moderate strains, the Gaussian chain model is adequate in representing the essential attributes of the force-extension curve.

We have now identified the main features of the elastic response in a classical elastomer network. The basic aspects are well-explained under the periphery of Gaussian random walk. The increase in stress at higher strain is explained when the finite extensibility of the cross-linked network is taken into consideration. For most practical

Figure 1.1.5 Force-extension curve of an elastomer under three different models. The upward curve in the force-extension plot (Figure 1.1.2) is explained by taking into account the finite extensibility of the elastomer network.

purposes, the Gaussian chain model would be adequate to represent the polymer network under the first approximation [3]. We have also seen that the classical elastomers described under the statistical theory do not require the details of its polymer structure.

The situation changes dramatically when anisotropy is introduced into the polymer network by means of rigid liquid crystalline components. The microscopic structure of the polymer chains then plays an important part in describing macroscopic elastic response of the cross-linked network.

The discussion will remain incomplete without mentioning the mechanical behavior of unvulcanized and vulcanized natural rubber. No chemical crosslinking is present in unvulcanized natural rubber while vulcanization amounts to chemically crosslinking the polymer chains and forming a network [3]. The elastic part of natural rubber is composed of approximately 94 wt % polyisoprene [10]. The remaining 6 wt % contains natural impurities in the form of proteins, phospholipids, neutral lipids, carbohydrates, metal salts and metal oxides [10]. The presence of these impurities prompts the formation of microstructures [10] in natural rubber. These microstructures are well dispersed [10] inside the rubbery network, contribute toward formation of a plateau in the stress-strain curve [11], and a permanent deformation [12] upon entering the plateau region. These experimental observations suggest that a “pseudo end-linked network” [11-13] might be present in natural rubber. Synthetic rubber, which is 100 wt % polyisoprene [10], does not possess such crystalline impurities and does not show a plateau in the stress-strain curve [11]. This comparison between natural and synthetic elastomers tells us that the presence of crystalline and/or semi-crystalline components can

contribute to the observance of a plateau in the stress-strain curve [11]. In this dissertation, the focus is on synthetic liquid crystalline elastomers which do not possess any natural impurity. As such, any unconventional elastic behavior would be dominated by the presence of the liquid crystalline entities in the system [8].

1.2 Liquid Crystals

One way of incorporating anisotropy into the polymer network is by the introduction of the liquid crystalline components. Liquid crystalline phases are a state of matter where both liquid-like flow properties and the elastic properties of crystals coexist.

They could be broadly classified as thermotropic (transitions induced by temperature) and lyotropic (transition induced by concentration) systems. Many rod-like or disc-like organic molecules, micelles or aggregate of surfactant molecules, some anisotropic solutes in a liquid solvent, main and side chain polymers and elastomers, etc. are seen to form such phases in nature [14]. Anisotropy in the shape of the constituent entities is necessary for such phases to occur but the reverse is not always true [15]. In an isotropic liquid phase, any correlation between the molecules is lost beyond the inter-molecular distance [14]. Such a phase is said to possess only short-range positional order. On the other extreme, a regular basis could be attached to the crystalline lattice [16]. The probability of finding a similar arrangement of molecules at a large distance from it stays finite and such crystals are said to exhibit long-range positional and orientational order.

Liquid crystals are intermediate or mesomorphic between these two extremes. In a liquid crystalline phase, some kind of ordering (e.g., orientational order of the anisotropic components) is always present but they could also flow in at least one direction. Because

Figure 1.2.1 Schematic showing the orientation of a rod-like nematic mesogen in the laboratory frame of reference. The macroscopic director n is taken along the z-axis; m is the axis of the molecule; θ and ϕ are respectively the polar and azimuthal angles made by the molecule in this frame. The graph to the left shows a smoothly behaving form of the molecular distribution function where it reaches the minimum value in the direction orthogonal to n.

of the anisotropy in molecular shape, liquid crystals generally show dielectric and diamagnetic anisotropy. The phases are optically birefringent and can be easily verified by placing the sample between two crossed polarizers. The constituent entities of the simplest liquid crystalline phase exhibit only orientational order. This phase shows a characteristic thread-like texture under the microscope and because of this, it is termed as

“thread-like” or the nematic phase. An externally applied magnetic field couples to the diamagnetic anisotropy of the rod-like molecules. Depending on the sign of the diamagnetic anisotropy, the molecules tend to orient parallel or perpendicular to the direction of the applied field. This macroscopically defined direction is termed as the director, n. The directions n and –n are equivalent in this phase and the system is said to possess quadrupolar symmetry [14]. In general, there exists a distribution function f(θ) for the orientational distribution of the rod-like molecules. For molecules with positive dielectric anisotropy, f(θ) has its maximum at θ = 0 and π, Figure 1.2.1. Here, θ is the angle between the long axis of the molecule and the director n. For the simplest case of uniaxial nematic discussed here, the molecules have azimuthal symmetry (no ϕ dependence) and the distribution function depends only on θ. The simple form of the distribution function shown in Figure 1.2.1 is smoothly behaved with a maximum at the direction parallel to the nematic director and a minimum at the direction orthogonal to it.

Statistically, this means the probability of finding a molecule with its orientation parallel to the director is higher than in orthogonal orientation. The Legendre polynomials are frequently used to describe a spherically symmetric function [17] and characterize the orientational order of the system. The first moment of the multipole expansion function

(or Legendre polynomial) vanishes as a result of the quadrupolar symmetry of the nematic phase and one has to resort to the second moment, to describe the orientational order of these systems. The orientational order parameter, S is then expressed as:

Here, is the second Legendre polynomial. For perfectly parallel alignment, f(θ) will peak at θ = 0 and π giving S = 1 whereas for the case of perpendicular alignment, f(θ) will peak at θ = π/2 giving S = 1/2. For the isotropic phase, orientation of the molecules will be perfectly random giving rise to a constant distribution function. This gives = 1/3 and S = 0. So, for parallel alignment, S lies between 0 and 1 and for perpendicular alignment of the molecules it could vary between 0 and 1/2. The nematic systems with positive S possess positive diamagnetic/dielectric anisotropy while those with negative value of S have negative diamagnetic/dielectric anisotropy. Thus, one needs to distinguish between these states

( ) while writing an expression for the free energy of the system in terms of its order parameter. The continuous rotational symmetry of the molecules of the isotropic phase

(S = 0) breaks down across the isotropic to nematic phase transition. The molecules in the nematic phase acquire a preference for staying parallel to the n directions. The free energy of the system now depends on the equilibrium orientational order parameter S which must also distinguish between the states of . The Landau-de Gennes free

energy density [14] near the transition temperature (TNI) could then be expanded in terms of S as:

Here, all the phenomenological constants f, T0, a, b and c are taken to be positive. Figure

1.2.2 shows a plot of the above free energy density as a function of S. In absence of an external ordering field, the coefficient f of the linear term goes to zero. At the transition, the minimum of the free energy jumps from S = 0 to a non-zero value. The transition is found to be weakly first order. The minimum at S  0 arises because of the presence of the cubic term in the free energy expansion which is necessary to distinguish between the states of . Experimental results have shown that S varies from approximately 0.4 to

0.8 in the nematic phase [18].

Decreasing the temperature further, one typically encounters the smectic-A and smectic-C phases. In the smectic phase, the molecules could arrange themselves in layers.

In addition to the orientational order of the nematic phase, the smectic phases possess quasi long-range positional order in one dimension [14]. In the plane of the layers, the molecules do not show any positional ordering but the center of mass of the molecules show a sinusoidal variation along the direction perpendicular to the layers. In the case of smectic-A phase formed by uniaxial molecules, the molecular long axis is parallel to the smectic layer normal. The phase possesses symmetry and is optically uniaxial. In case of smectic-C phase, the molecular long axis is tilted with respect to the smectic layer normal at a non-zero angle. The molecular tilt in the smectic-C phase is strongly temperature dependent. The phase exhibits symmetry and is optically biaxial.

Figure 1.2.2 A plot of the Landau-de Gennes nematic free energy density for three different temperatures. For T  TNI, the system is in the isotropic state with free energy minimum at S = 0. The transition takes place at T = TNI. For T  TNI, the minimum of the free energy density moves to S  0 which is inside the nematic phase.

1.3 Liquid Crystal Elastomers

A typical elastomer does not have any anisotropic component in the polymer backbone and assumes a spherical equilibrium conformation in the absence of external fields. In a liquid crystal elastomer, the mesogenic moieties and the interaction between them add anisotropy to the polymer network. A liquid crystal elastomer is formed when the anisotropic liquid crystalline molecules are chemically anchored to the flexible polymer chains and then randomly cross-linked into a rubbery structure. The topology of the network is fixed by the cross-linking process. A liquid crystalline core is combined with a polymer backbone in two ways: 1) as a pendent hanging from the polymer backbone by means of threadlike spacers, called side chain liquid crystal elastomers and

2) by direct incorporation of the mesogenic parts into the polymer chains, with or without spacers, called main chain liquid crystal elastomers [19]. The liquid crystalline parts are attached to the backbone in either side-on or end-on positions. The constituent polymers are liquid like in local domains and the chains can move due to thermal fluctuations but only up to a limit fixed by the topological constraints set by the crosslinkers. In short, the liquid like flow properties are still present in the microscopic domains but macroscopically it does not flow and some energy is needed to change its soft-solid shape. The interaction between the liquid crystalline components along with the length and flexibility of the linkage group play important roles in determining the type of liquid crystalline order and the equilibrium polymer conformation [20,21]. In an end-on main chain liquid crystal elastomer, the backbones may respond to the anisotropic field imposed by the liquid crystalline order, are generally parallel to the director, and adopt

Figure 1.3.1 Polymer chain conformations and mesogens (dark ellipses) organization depend on the length, flexibility, number (even/odd) of carbons in the spacer, the type of attachment (side-on or end-on), and interaction among them. Several possible scenarios are shown for, both, the main-chain and side-chain liquid crystal elastomers: (a), (c), and (d): Mesogens lying parallel to the polymer backbone favoring prolate conformation; and (b) and (e): Polymer backbone perpendicular to mesogens adopts an oblate conformation.

prolate conformation, Figure 1.3.1 (a). On the other hand, the main chain conformation is likely to become oblate in a side-on main chain liquid crystal elastomer, Figure 1.3.1 (b).

The behavior of end-on side chain liquid crystal elastomer is influenced by the length of the linkage group. For a side-on side chain liquid crystal elastomer, the mesogens may have lower energy in a nematic field of the mesogens if the linkage group is flexible, leading to a prolate conformation of the backbone, Figure 1.3.1 (c). Furthermore, an even number of carbon atoms in the linkage group tend to orient the mesogens parallel to the backbone resulting in a prolate conformation, Figure 1.3.1 (d). The mesogens would be on average perpendicular to the polymer backbone for odd numbers of carbons in the linkage group. In such cases, the polymer backbone lies in a plane perpendicular to the mesogens in an oblate conformation, Figure 1.3.1 (e). Consequently, the liquid crystal elastomers exhibit significantly different organization of the mesogens and the backbone in three spatial directions. The liquid-like mobility between two cross-linking points would allow the liquid crystalline parts to achieve some kind of orientational order. These structural details were not required in the earlier statistical description of the classical elastomers. They bring new phenomena, e.g., large-scale thermo-mechanical actuation

[22-26], coupling between external mechanical field and the inherent orientational order

[27-29], soft-elastic deformation [8,30-32], shape memory [33-36] etc.

Synthesis of the first liquid crystal elastomer was done by Finkelmann, et al., [37] in 1981. It was a nematic liquid crystal elastomer based on flexible polysiloxane backbone. No external field was applied during the synthesis process. Thus, the director variation of the ordinary nematic phase got “frozen” into the cross-linked polymer

network. The polydomain texture became a thermodynamic equilibrium state in these nematic elastomers. This polydomain is opaque to optical wavelength due to strong light scattering by the randomly oriented directors in individual microdomains and not due to director fluctuations as observed in nematic liquid crystals [8]. The elastic energy of the network is such that it usually resists the director re-orientation process in response to any externally applied electric or magnetic fields. Instead, the local directors of the individual micro-domains could be aligned toward a particular direction by application of external mechanical force. The final state becomes optically transparent due to large-scale reorientation of the local nematic directors towards the stretch direction. The elasticity of the polymer network dominates over the Frank elastic terms encountered in case of ordinary liquid crystals.

1.3.1 Neo-classical theory of nematic rubber elasticity

We are now in a position to introduce the so-called neo-classical elastic theory of nematic elastomers [8]. Let us recall eqn. (1.6) which defines the average polymer shape in classical elastomers. The structural detail of the polymer chain is not important in classical elastomers and as such, the step-length d was taken as a scalar quantity. In case of liquid crystal elastomers, it is replaced by a tensorial quantity, called the step-length tensor dij and eqn. (1.6) is re-written as:

The Gaussian probability distribution of eqn. (1.5), which represented the distribution of end-to-end distance of the polymer chain now changes to:

The inverse step-length tensor reflects the tensorial character of the nematic network.

The Gaussian probability distribution now becomes direction dependent. Physically, it means that the extent of the polymer chain could be more in direction parallel to the length of the anisotropic components in the polymer chain. The two other perpendicular directions are equivalent in case of uniaxial symmetry of nematic systems. Following the procedure outlined in section 1.1.2 (see derivation of eqn. (1.10)), the free energy density of the elastomer can now be written as:

Here, is the step-length tensor at the time of formation of the network which is different than the step-length tensor at a particular strain value during deformation.

The above expression for free energy is at the heart of nematic rubber elasticity. It is derived assuming anisotropic Gaussian chain statistics and thus would deviate from the actual behavior at very high strains where the finite extensibility of the polymer network comes into picture via the inverse Langevin function (see section 1.1.3). But for all

practical purposes dealt here, the assumption of an anisotropic Gaussian distribution for the chain segments should suffice. The above free energy expression is often referred to as the “trace formula” in the literature [8]. It was first obtained by Warner and Terentjev in the case of nematic elastomers but the general nature of the expression makes it applicable to the cases of all liquid crystal elastomer systems with some additional terms.

The structural details of the polymer chain did not appear in the free energy expression of classical elastomers, eqn.(1.10). But now, the initial and final state of the nematic director

are contained in the step-length tensors and and the structural details of the deformed polymer chain appears in the free energy expression via the distortion

term . We shall compare the case of uniaxial deformation in nematic rubber with the case of uniaxial deformation in classical rubber to amplify this point.

For a nematic elastomer network cross-linked in the isotropic phase, the initial

step length tensor is of the form . When a uniaxial deformation λ is applied along one of the principal axes, say, the x-direction, there will be associated contraction amounting to in the two perpendicular directions. The deformation tensor and the inverse step length tensor would then take the forms:

The free energy expression for nematic elastomers could be easily simplified by multiplying these diagonal matrices:

(1.14)

Here, d and d⊥ are the step sizes along and perpendicular to the nematic director.

Eqn. (1.14) clearly shows the effect of polymer chain anisotropy in the expression for free energy which was earlier absent in the case of classical elastomers, eqn. (1.11). From eqn. (1.14), one could also calculate the value of nominal stress σn in the network [8]:

The polymer shape anisotropy now leads to a new effect: spontaneous extension at zero external stress:

In the isotropic phase, the spontaneous extension term given by above equation is equal to zero as the polymer chains are spherical in shape. With decreasing temperature, the elastomer enters the nematic phase at one point. The polymer chains ordering tend to be anisotropic and the anisotropy increases with decrease in temperature [8]. Depending on the prolate or oblate chain conformation, the nematic elastomer will either elongate or shrink along the director. The effect is more prominent in monodomain systems but may also be seen in samples with partially ordered polydomains as it originates from the step length tensor in FLCE. The spontaneous shape change gives rise to the possibility of large- scale thermo-mechanical actuation in these elastomers that has been experimentally realized [22,23,38]. By controlling the temperature one could control the amount of

spontaneous elongation [39-42] which has paved the way for thermally driven micro- actuators [8,28,43-48].

1.4 Chapter Summary

While reviewing the elastic theory of classical elastomers, we have seen that the polymer chain conformation is well described by assuming a Gaussian random walk at small to moderate strains. At high extensions, the Gaussian approximation fails to explain the upward turn in the stress-strain curve which arises due to the finite extensibility of a real polymer network. In this limit, the chain statistics is approximated by the inverse

Langevin function [9] which could successfully explain the stress-strain curve of the classical elastomers for the whole range of elongation. The first order approximation of the inverse Langevin function reduces exactly to the case of the Gaussian chain. Thus, for all practical purposes, the polymer chain conformation could be assumed as Gaussian [3].

Introduction of liquid crystalline components into the rubbery network changes the shape of the polymer chain. The chain statistics is now described by an anisotropic Gaussian distribution in the purview of the neo-classical [8] elastic theory coined by Warner and

Terentjev. The step-length now becomes a tensorial quantity and the elongation of the network becomes highly direction dependent. The shape of the polymer chain conformation at the time of formation of the network also needs to be taken into account while discussing the elastic free energy of the liquid crystal elastomers. This approach will be useful while discussing the phenomenon of soft-elasticity in the next chapter.

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CHAPTER 2

SOFT ELASTICITY AND SHAPE MEMORY EFFECT

An overview of the existing theories of soft elasticity, some relevant experiments, and the shape memory effect in smectic-C elastomers are discussed in this chapter. The soft elastic response of the elastomer network is seen in that portion of the stress-strain curve where a large deformation could be achieved without any significant rise in the value of stress. The deformation seems to take place with almost zero resistance in that region. Such behavior in a liquid crystalline elastomer network is termed as “soft” and could be explained based on: (1) a soft-mode of director rotation of the mesogens; and/or

(2) reorientation of the individual domains toward the stretch direction. We shall first introduce the phenomenon as observed in nematic elastomers and then move onto the smectic systems. We shall also introduce the idea of shape-memory effect in polydomain main-chain smectic-C elastomers. These will constitute the necessary background for the experiential work on soft elasticity and shape memory in main-chain smectic-C elastomers studied in this dissertation project.

2.1 Soft elasticity of nematic elastomers

The phenomenon of soft elasticity is observed in polydomain nematic elastomers

[1-4] during uniaxial stretching and also in monodomain nematic elastomers [5] when stretched perpendicular to the initial direction of the director. Typically a plateau in the stress-strain curve exists beyond a small threshold strain which physically means that

38 39

Figure 2.1.1 Stress-strain plots for a polydomain nematic elastomer at different temperatures with the appearance of a plateau. Here strain is measured as change in length / original length. Reproduced with permission from [2], © (1998) American Chemical Society.

macroscopic shape change is possible at zero or negligible energy cost in this region of elasticity. A representative stress-strain plot for a polydomain nematic elastomer is shown in Figure 2.1.1. The curve can be divided into three regions where an initial elastic region is followed by a plateau in the middle part and then accompanied by a rise in the value of stress. The system recovers its original shape when the external strain is removed. The width of the plateau decreases with temperature and completely disappears when the elastomer is heated to the isotropic phase. It is the plateau region in the middle part where soft elastic response is encountered.

2.1.1 Soft elasticity in monodomain nematic elastomers

A monodomain nematic elastomer is prepared such that the liquid crystalline components have a permanent alignment along a macroscopic director. Such an elastomer was first synthesized in 1991 by J. Kupfer and H. Finkelmann via a two-step cross-linking process [6]. The nematic polymer was slightly cross-linked in the isotropic phase and then the temperature of the system was lowered such that it entered the nematic phase. The final cross-linking process was done in the nematic phase in the presence of an aligning mechanical force on the network. An elastomer obtained in this way has the long-range orientational order of the nematic phase imprinted into the crosslinked network. It shows highly anisotropic behavior when stretched parallel or perpendicular to the nematic director. An interesting phenomenon is observed when the stretching is performed perpendicular to the director: the material elongates without any significant rise in the value of stress. The elongation proceeds at almost zero energy cost.

This phenomenon attracted considerable attention from the physicists and was

successfully explained in 1994 by P.D. Olmsted [7] and M. Warner, et al., [8] invoking a soft mode of director rotation. This soft mode of director rotation is briefly discussed below following references [7-9].

In this model, the elastic free energy density FLCE of a nematic rubber is given by the so-called trace formula derived in Chapter 1 [9]:

Here, μ is the shear modulus, is the step length tensor at the time of crosslinking the

network, is the inverse of the step length tensor under any general deformation, is the second rank deformation tensor and the superscript T indicates transpose of a matrix.

The free energy density FLCE incorporates the effect of polymer chain conformation via the step length tensors which carry the information about the inherent nematic order present in the network. Any macroscopic deformation , if accompanied by a distortion of the polymer chain conformation, will cost energy. But there exist modes of deformation in which the shape of the polymer chain conformation is not altered; only it is rotated in a continuous manner until the nematic director aligns along the stretch direction. Such soft modes of deformations are also called Goldstone modes of the nematic elastomers [7]. A trivial solution would be body rotation, for example. For the non-trivial case, we would have to look for a compatible transformation of the deformation tensor which leaves the elastic free energy invariant. P.D. Olmsted [7] proposed such a transformation which involved rotation of the nematic director:

(2.1) .

Here, is the general rotational matrix which is orthogonal by definition, the last and the first term being respectively the square roots of the symmetric step length tensors in states before and after the deformation. The above form of the deformation tensor leaves the free energy FLCE invariant. Physically, it means that the polymer chain conformation undergoing the above rotational mode of deformation would cost no elastic free energy for the system. Macroscopically, the deformation could proceed at zero energy cost.

Let us now consider a practical geometry where the monodomain nematic elastomer is lying in the x-y plane with the director initially pointing towards the x-axis.

A uniaxial strain is applied along the y-axis such that the director rotates in a continuous manner in the x-y plane, the axis of rotation being the z-axis. Let the angle of rotation be

θ which goes from 0  π/2 upon completion of the director rotation. In this scenario, one can reduce the deformation tensor to a matrix [9]:

The above deformation can be separated into an identity matrix (symmetric spherical part) and a second matrix involving shear (with off-diagonal elements):

(2.2)

Figure 2.1.2 shows a cartoon displaying the soft modes of deformation of a nematic polymer chain distribution. It rotates continuously in the elastomer matrix and does not distort the matrix in the process. The deformation is complete at θ = π/2 or when the chain conformation is parallel to the y-axis. At this point the terms involving shear

(the off-diagonal elements in the second matrix) disappear from the equation. The final state of the elastomer does not have any residual component of shear deformation. From eqn. (2.2), the deformation started at at θ = 0 and finished at θ = π/2 with:

Only an extension along y and a proportionate contraction along x remain in the final form. Essentially, the elastomer matrix is defined by the position of the cross-linkers which are moved around in such a way that the overall distribution remains unchanged.

Throughout this process, the free energy remains invariant. This soft deformation mode starts and ends at the plateau region of the stress-strain curve. Beyond this, a force is again needed to deform the nematic elastomer. For the case of isotropic elastomers,

Figure 2.1.2 Schematic representation of the soft mode of director rotation in nematic elastomers. The chain distribution, represented by the index ellipsoid, is embedded in the elastomer matrix. For a strain applied perpendicular to its initial direction, it can rotate in a continuous manner without distorting the elastomer matrix. Reproduced with permission from [9], © (2007) Oxford University Press.

d|| = d. One can easily see from eqn. (2.2) that the anti-symmetric component of the deformation matrix vanishes in this case, leaving only the unity matrix corresponding to the spherical part. As such no shear is present and hence no soft elasticity exists in isotropic elastomers. Soft elasticity is the characteristic of anisotropic elastomers only and the anti-symmetric component of shear is necessary for such mode of deformation.

This theory of soft mode of director rotation has been very successful in explaining the experimental stress-strain results [10] in monodomain nematic elastomers.

2.1.2 Semi-soft elasticity: effect of non-ideal network

The soft mode of director rotation is a very delicate phenomenon in which the chain distribution should be able to rotate in the elastic matrix without any distortions.

This can only be realized in an ideal defect free state. In real scenarios, the structure of the elastomer is not defect free [9]. For example, the cross-linking is usually performed by rigid, rod-like objects which bring in non-ideality into the network. Also, the polymer chains have polydispersity in their length. This polydispersity hinders the process of soft mode of director rotation. The optimal distortion for the soft mode at a particular angle is not the same for all chains in the system. The presence of other topological constraints, e.g., hairpin defects could also play its role in rendering the effect non-ideal. Thus, in practice, the stress-strain plateau might not be flat but shows some slope or non-zero value of the elastic-modulus.

Another argument by Warner and Terentjev [9] for soft elasticity points toward the state of network genesis. It is argued and have been experimentally verified [4] that

the presence of anisotropy in the network genesis would render the elastomer away from

“softness”. The elastomers cross-linked in an anisotropic state would have some residual order permanently imprinted into the network. But the ones crosslinked in isotropic state would not have any such preferences. Thus, in some nematic elastomers, the director rotation would happen beyond a threshold strain λ >> 1 as the memory of imprinted network anisotropy is to be overcome first. The qualitative aspect of soft-mode of director rotation is still retained. The universal form of the director rotation given above along with the non-classical deformation behavior is always seen in these systems [9].

2.1.3 Soft elasticity in polydomain nematic elastomers

Having discussed the phenomenon of soft mode of director rotation in monodomain nematic elastomers and also pointing out the importance of network genesis state for soft elasticity, let us now focus on the behavior of polydomain nematic elastomers. A polydomain elastomer is opaque due to randomly crosslinked director orientation but becomes transparent upon stretching. This behavior is an indication of the stress-induced polydomain-monodomain transition. The state of network genesis plays an important role in describing the mechanical and structural response of these systems [4].

Figure 2.1.3 shows the polarizing optical micrographs of two polydomain nematic elastomers. The one on the left has been cross-linked in isotropic state and the one on the right is cross-linked in the nematic state. Both of them have the same degree of crosslinking and isotropic-nematic transition temperature. The textures show the effect of network genesis: the characteristic domain size in isotropic genesis one is at least an order of magnitude smaller than the nematic genesis one. The characteristic schlieren

Figure 2.1.3 Polarizing optical micrographs of two polydomain nematic elastomers: (a) isotropic genesis and (b) nematic genesis of the network. The difference in domain sizes in the two elastomers is apparent. Reproduced with permission from [4], © (2009) American Chemical Society.

texture implies that the elastomer, crosslinked in the nematic state, possesses a memory of the initial director field before crosslinking. Figure 2.1.4 shows the behavior of stress

(σ) and global orientational order parameter (S) for these systems. The elastomer crosslinked in the isotropic phase shows a plateau in the stress-strain curve. The value of

S gradually increases, ultimately saturating to a maximum value set by the crosslinking condition of the network. The case of nematic genesis elastomer is quite different. The stress increases gradually without the appearance of any plateau. The value of S ultimately saturates to almost the same value as for the isotropic genesis elastomer. The final monodomain state of both the elastomers has same degree of ordering. It is the process of monodomain formation which distinguishes these two elastomers. Both of them undergo the polydomain-monodomain transition but only the isotropic genesis elastomer shows soft behavior in the mechanical response curve. Also, both elastomers return to the initial polydomain state upon removal of the external strain. No strain retention is observed in these systems.

This fundamental difference between the behavior of isotropic and nematic genesis polydomain nematic elastomers has attracted extensive theoretical consideration

[11,12]. Theories have been mostly successful in explaining the key features of this transition. Using the neo-classical Gaussian chain model [9], it is possible to write the expression of elastic free energy as [12]:

Here, is the local nematic director at the time of network genesis, is the local nematic director after network formation and α is coefficient of non-ideality. The first

Figure 2.1.4 Nominal stress (σ) and global orientational order parameter (S) for the two polydomain nematic elastomers in Figure 2.1.3: (a) isotropic genesis and (b) nematic genesis of the network. Reproduced with permission from [4], © (2009) American Chemical Society.

term in above equation is the ideal term from the trace formula which allows for soft deformation modes. The second term arises because of the non-ideality in the network originating from (1) rigid crosslinkers, (2) polydispersity in length of polymer chains and

(3) elastic incompatibility of different domains in the polydomain state [9]. The introduction of the non-ideal term in the free energy expression accounts for a lower limit of strain as observed in the stress-strain plateau [4]. Assuming λ as the uniaxial strain, β

as the angle between and , and as the anisotropy of the polymer chain conformation, above equation could be simplified to [12]:

This expression of free energy could successfully explain [12] the origin of the stress- strain plateau [4] in case of polydomain nematic elastomers. The physical interpretation is as follows: the microscopic domain structure of the elastomer originates from a need to maintain a random stress field inside the material. The domain directors maintain uniformity in changing direction from one domain to another. Except for the ones lying parallel to the stretch direction, the mechanical field introduces a shear component to each domain. The domains thus rotate in a soft manner, with minimal distortion of the neighboring elastic matrix. A small non-ideality originates from size mismatch at domain boundaries for domains of different orientation. The microscopic semi-soft rotation of the individual domains gives rise to the macroscopic semi-soft elastic plateau in these elastomers. A nematic genesis state of the network can remember the director orientation at the time of network genesis, Figure 2.1.1 (b). This is enough to introduce considerable non-ideality into the expression for free energy which destroys macroscopic soft behavior

of the network. Thus, the nematic-genesis elastomers are mechanically harder and do not exhibit any soft behavior in the stress-strain curve.

We could conclude that a polydomain elastomer crosslinked in a random state is the ideal candidate to exhibit soft elasticity. Genesis in the anisotropic state destroys the random nature of the domains and imprints the underlying order of the particular mesophase onto the elastomer network. The above arguments hold true for other types of elastomers [12] also, e.g., polydomain smectic-C elastomers, which is the focus of this dissertation work.

2.2 Domain rotation in smectic elastomers

In the case of smectic elastomers, broken translational symmetry in one- dimension results in the formation of layered structures. Introduction of quasi long-range positional order lead to significant strain–retention [13] which is also responsible for the shape memory effect [14] observed in smectic elastomers. Consequently, it constrains the director perpendicular to the layers making so-called soft mode of director rotation impossible in the case of monodomain side-chain smectic-A elastomers [15]. J. Adams and M. Warner [15] proposed a theory which successfully explains the experimental observations [16] in layered smectic-A elastomers.

Distortion of a monodomain smectic-A elastomer parallel to the layer normal is strongly resisted by a high value of the bulk modulus of elasticity B. The value of B is as high as the values encountered in lower mass smectic-A liquid crystals. But it is still much smaller than the bulk modulus of rubber and thus, deformations in smectic elastomers is also assumed to be taking place at constant volume. While stretched parallel

to the layer normal, changes in the layer separation is not energetically favored. To avoid this energetically costly situation, the smectic layers will rotate in response to the applied strain. This rotation could be observed both by x-ray scattering and optical methods [16].

The crosslinkers are strongly attached to the smectic layers and they prefer to stay at the interface of the layers rather than being in the bulk. The pinning of crosslinkers to the smectic layers is the reason for smectic-like response (instead of a typical rubbery response as discussed in chapter 1, section 1.1) of elastomers when stretched perpendicular to the layers. On the other hand, any stretching force parallel to the layers will act directly on the polymer backbone. The decrease in the width of the smectic layers is resisted strongly, and the increase in one in-plane direction is compensated by the decrease in other in-plane direction [16] to maintain constant volume.

In Adams and Warner’s model [15], the layers are assumed to be rigidly constrained to the elastomer network. Any deformation of the network structure would also distort the layers in proportionate manner. The situation is illustrated in Figure 2.2.1.

An applied strain (λzz) perpendicular to the layers could also possess a component of shear (λxz) due to small misalignment between the smectic layer normal and the stretch direction. The initial direction of layers would change with this deformation according to:

The above expression shows that for any non-trivial deformation of the elastomer matrix, the layer normal rotates from the initial n0 direction to the final direction n. This has the effect of changing the layer spacing ( ) such that [15]:

Figure 2.2.1 (a) Figure showing two relevant deformations while a smectic elastomer is strained perpendicular to its layers. (b) Initial direction of layer normal for two general in- plane unit vectors r, s; (c) Direction of layer normal after a deformation. Note that the layer is now rotated. λij and εijk are the deformation tensor and the Levi-Civita tensor respectively.

Under the deformation , the free energy of the smectic network would also change according to [15]:

The first term in above free energy expression comes from the trace formula and the second term originates from the resistance to changing the layer spacing values in smectic systems. Under the deformation shown in Figure 2.2.1 (a), one could further simplify the free energy expression, and then from simple thermodynamical arguments

(as discussed in section 1.1), find the expression for nominal stress σ in terms of uniaxial deformation λ along the z-axis [15]:

(2.3)

Eqn. (2.3) clearly shows that layer rotation begins beyond a threshold strain λth. Value of the threshold strain is determined from the material dependent parameters B, μ and r.

Beyond the threshold strain, the stress increases linearly with strain via a reduced value of the elastic modulus and would continue to do so until the film breaks. Figure 2.2.2 shows a plot of the nominal stress of eqn. (2.3) with the experimental data from [16].

This theory of layer rotation in smectic-A elastomers agrees very well with the experimental data [16]. It also predicts that the layers rotate by an angle θ given by:

Figure 2.2.2 The broken curve is the fit of eqn. (2.3) to the experimental data in [16]. Insets show the schematic of layer rotation. Adapted with permission from [15], © (2005) American Physical Society.

(2.4)

Layer rotation can be measured from the small angle x-ray diffraction data [16].

Eqn. (2.4) fits the experimental data very well and gives another way for calculating the material dependent parameter λth. Eqn. (2.3) and (2.4) can be used to find the values of the parameters B, μ and r ( ) which gives information about the polymer chain conformation at a particular temperature.

In conclusion, we have seen that the smectic layers in monodomain smectic elastomers are strongly coupled to the elastomer matrix. Any distortion of the matrix perpendicular to the layered structure would rotate the smectic layers beyond a certain threshold. The rotation of the layers can be measured from small angle x-ray diffraction and the theory [15] agrees very well with the experimental data [16]. The apparent plateau in the stress-strain curve originates from the rotation of the layers. The linear region beyond the small threshold strain value would continue until the specimen ruptures. The appearance of a reduced stress region from domain-rotation in these systems should not be confused with the soft mode of director rotation as seen in nematic systems. Such a soft mode of director rotation is forbidden in smectic-A elastomers due to layer constraints.

2.3 Shape memory effect in polydomain main-chain smectic-C elastomers

An early report by Ortiz, et al., [17] described strain retention in a smectic main- chain elastomer. However, the pioneering work of Rousseau, et al., [14] on the shape memory effect was performed on a polydomain main-chain smectic-C elastomer prepared

Figure 2.3.1 The complete shape memory cycle in a polydomain main-chain smectic-C elastomer obtained from dynamic mechanical analysis. Case (a) is for the smectic-C elastomer, (b) is the case of a classical elastomer where no “fixing” of secondary shape is observed and (c) is the schematic pathway for the shape-memory experiments performed on the elastomer samples studied in this dissertation work. Adapted from reference [14] © 2003 American Chemical Society.

with two benzoate based rod-like mesogens (in varying concentrations), flexible poly(dimethyl siloxane) spacers, and point-like siloxane cross linkers. The shape memory cycle of this material, Figure 2.3.1, consists of heating the elastomer above the clearing temperature where it behaves like an isotropic rubber. The elastomer was then subjected to elongation, subsequently cooled to low temperature, followed by the removal of the external stress. The secondary shape induced by stretching at elevated temperature retained ~ 84% of the original strain after removal of the mechanical load at room temperature. The initial shape could be recovered by heating the elastomer above the clearing temperature and then cooling back to room temperature in absence of any external load. The process of fixing and subsequent recovery of primary shape is highly repeatable. The results revealed that the secondary shape consists of a highly ordered and well-aligned smectic-C mesophase. The elastomer network is rubbery (μ ~ 20 MPa) even at temperatures as low as − 20 °C. In contrast, a classical elastomer subjected to the same shape memory experiment recovered the original shape as soon the external stress was removed. No shape memory was observed in case of classical elastomers.

The apparent “fixing” of the secondary shape in these liquid crystalline elastomers has its origin in the strong positional ordering introduced by the presence of smectic-C mesophase. A classical elastomer lacks these structural advantages and thus exhibits no shape memory effect. Also, nematic elastomers, with orientational order but no positional order, do not show any shape memory effect. A liquid crystalline phase with underlying positional order (e.g., smectic-C) is needed for realizing the shape memory effect in elastomers.

2.4 Chapter Summary

In this chapter, an overview of the existing theories of soft elasticity in elastomers is given. The phenomenon could have two possible origins: (1) a soft mode of director rotation which leaves the free energy invariant; (2) polydomain-monodomain transition.

It is observed that the state of formation of the network plays an important role. The usual crosslinking process of monodomain elastomers always imprints the inherent anisotropic order of the mesophase into the elastomer matrix. This imprinting of the anisotropic property brings in non-ideality and works against soft-elastic behavior. An elastomer crosslinked in a phase with no preferred direction of ordering is suitable for the observance of soft elastic deformation. This argument holds true for both nematic and smectic systems [12]. In nematic monodomain systems, the inherent order of the mesophase, presence of finite-size crosslinkers, and polydispersity in polymer chains’ length render the effect semi-soft. The case of a polydomain sample with isotropic genesis is the closest to the ideal case of soft deformation. The shear component of deformation in individual domains can rotate the domains towards the stretch direction instead of elongating them. Thus, the plateau in the stress-strain curve of a polydomain sample originates from shear induced director rotation where each domain deforms

“softly” toward the stretch direction. The case of smectic-A system is completely different. The mesogens are now rigidly coupled to the layers, which in turn, is bound to the elastomer matrix. Any deformation of the smectic layer normal would produce a rotation of the layers and give rise to an apparent flat region in the stress-strain curve.

The presence of positional order is seen to introduce the reversible shape memory effect

in main-chain smectic-C elastomers. We can extend these theoretical considerations to the soft elastic behavior and the shape memory effect of the smectic polydomain elastomers studied in this dissertation.

REFERENCES

[1] J. Schatzle, W. Kaufhold, and H. Finkelmann, Makromol. Chem. 190, 3269 (1989).

[2] S. M. Clarke, E. M. Terentjev, I. Kundler, and H. Finkelmann, Macromolecules 31, 4862 (1998).

[3] A. Hotta and E. M. Terentjev, J. Phys.-Condens. Mat. 13, 11453 (2001).

[4] K. Urayama, E. Kohmon, M. Kojima, and T. Takigawa, Macromolecules 42, 4084 (2009).

[5] J. Kupfer and H. Finkelmann, Macromol. Chem. Phys. 195, 1353 (1994).

[6] J. Kupfer and H. Finkelmann, Makromol. Chem. Rapid Commun. 12, 717 (1991).

[7] P. D. Olmsted, J. Phys. II 4, 2215 (1994).

[8] M. Warner, P. Bladon, and E. M. Terentjev, J. Phys. II 4, 93 (1994).

[9] M. Warner and E. M. Terentjev, Liquid Crystal Elastomers (Oxford University Press Inc., New York, NY, USA, 2007).

[10] I. Kundler and H. Finkelmann, Macromol. Rapid Commun. 16, 679 (1995).

[11] S. V. Fridrikh and E. M. Terentjev, Phys. Rev. E 60, 1847 (1999).

[12] J. S. Biggins, M. Warner, and K. Bhattacharya, Phys. Rev. Lett. 103, 037802 (2009).

[13] C. Ortiz, C. K. Ober, and E. J. Kramer, Polymer 39, 3713 (1998).

[14] I. A. Rousseau and P. T. Mather, J. Am. Chem. Soc. 125, 15300 (2003).

[15] J. M. Adams and M. Warner, Phys. Rev. E 71, 021708 (2005).

[16] E. Nishikawa and H. Finkelmann, Macromol. Chem. Phys. 200, 312 (1999).

[17] C. Ortiz, M. Wagner, N. Bhargava, C. K. Ober, and E. J. Kramer, Macromolecules 31, 8531 (1998).

CHAPTER 3

X-RAY DIFFRACTION: THEORY AND EXPERIMENTAL TECHNIQUE

X rays are electromagnetic radiation which interacts with the electron cloud of atoms while passing through a media. The distribution of electron density in a condensed matter system depends on the underlying structure of a particular phase. The interaction between X-ray and the electronic charges has been exploited by researchers to understand the nano-scale structure of materials for decades [1].

3.1 Bragg’s law of diffraction

Let us assume an infinite set of parallel planes in a crystal separated by a distance d from each other, Figure 3.1.1 (a). If the wavelength (λ) of the incident monochromatic radiation is of comparable dimension to d, one can observe constructive interference between the waves reflected from the adjacent planes when the path difference is equal to an integral multiple of the wavelength. This condition is known as Bragg’s law:

Historically, X-ray diffraction was first observed in 1912 by William Lawrence Bragg and William Henry Bragg while performing X-ray scattering experiments with rock-salt crystals [2]. Bragg’s law tells us that a smaller periodic length scale would diffract at a larger angle and vice versa. The phenomenon of Bragg reflection and X-ray diffraction is better understood in the reciprocal space which is the Fourier transform of the real space.

Figure 3.1.1 (b) shows the schematic of such an experimental situation. The incident

63 64

Figure 3.1.1 Schematic diagram of (a) Bragg diffraction and (b) elastic scattering of X rays resulting in a change in momentum vector by q.

plane wave X rays with wave-vector interact with the electrons inside the specimen and come out with a wave-vector . The X rays are elastically scattered from the electron cloud around the atoms which only changes the direction of the incident wave vector

(magnitude is constant). The direction of the incident radiation is

changed by an angle 2θ which is a measure of the periodicity (d) of the system. When the

Bragg condition is satisfied, the scattering wave vector corresponds to one of the reciprocal lattice vectors in the momentum space. The magnitude of the momentum change vector could easily be calculated from the simple elastic scattering geometry in

Figure 3.1.1 (b) as:

This is an alternate representation of the Bragg’s condition.

Why electrons are the scattering centers is easier to understand classically. The electromagnetic field of the incident wave interacts with the charged particles in the system and accelerates them momentarily. The accelerated charges go back to their respective ground states by emitting radiation of the same frequency. The radiation rate is found to be inversely proportional to the square of the mass of each scattering centers [3].

Hence, the scattering is dominated by the light particles, i.e., electrons present in the system. The x-ray diffraction intensity could contrast the inhomogeneous electron distribution in a system where some degree of ordering is present.

3.2 Basic theory of X-ray scattering

A detector will typically count the number of photons scattered from the sample into the solid angle subtended by the detector onto the sample. The amount of scattered photons per unit time (N) depends on the incident beam intensity (I), the solid angle (Ω) subtended by the detector, and the static differential scattering cross-section ( )

[4]. The static differential cross-section is related to properties of the sample, namely, the form factor and the structure factor which are related to the density-density correlation function of the material. Mathematically, this can be represented as:

(3.1)

In the subsequent discussion, a quantum mechanical treatment, after E. Merzbacher [5], is followed to find a relationship between these quantities. The polarization effect is neglected for simplicity. We shall further assume the interaction between the atoms of the system and the incident X rays to be weak. Strictly speaking, we assume a single scattering event for each X-ray photon. Experimentally, this assumption requires the thickness of the sample to be smaller than the absorption length of X rays in the sample.

Fermi’s golden rule [6] now relates the differential cross-section to the transfer matrix of scattering. For a scattering event such as depicted in Figure 3.1.1:

(3.2)

Here, U(r) is the total scattering potential of the system which is approximated by adding the scattering potential of individual atoms:

(3.3)

th where, un and rn are the potential and position vector of the n atom in the system. Using eqn. (3.3) and doing a coordinate transformation , the transfer matrix in eqn. (3.2) can be re-written in integral form as:

(3.4)

th Here, is the Fourier transform of the scattering potential of the n atom and V is the total volume of the system. Assuming all the atoms are identical, we can pull out the

Fourier transformed term outside the summation and also remove the subscript from it for notational simplicity. Under this condition, eqn. (3.2) and (3.4) combines to give:

(3.5)

If the atoms are rigidly fixed to the lattice, above equation is sufficient to represent the static differential scattering cross-section. As it turns out, scattering data is collected over a period of time which is large compared to the thermal equilibration time for the atoms in the sample. The situation would be properly represented if we replace the right hand side of the above equation by its ensemble average in the phase space [7]:

(3.6)

The brackets represent the ensemble average in the phase space. The first term in eqn. (3.6) is called the atomic form factor and the second term is termed as the structure factor S(q). The X-ray scattering intensity given by eqn. (3.1) is proportional to the product of both of these terms. The atomic form factor is a measure of the scattering potential of the individual atoms and depends on the type of scattering, on the shape and character of the individual scatterers, and also on the nature of the incident radiation. The structure factor S(q) contains information about the atomic arrangement inside the material. S(q) is a direct source of information about the structure of the material being probed by X rays. The structure factor is connected to the density-density correlation function Cnn of a statistical system. Cnn is defined as the ensemble average of the product of the number density at two different positions in the system [5,8]:

(3.7)

This definition of the density-density correlation function takes into account the spatial variation of system’s density. The number density operator is defined as the number of atoms per unit volume at a given position:

In this definition, the Fourier transform of becomes:

(3.8)

Using eqn. (3.8), the expression for structure factor is re-written as:

(3.9)

The above equation relates the number density operator with the structure factor in reciprocal space or, q-space. Using the definition of the Fourier transformation and the density-density correlation function from eqn. (3.7), the right hand side of eqn. (3.9) can be further simplified to show the direct dependence of the structure factor on the density- density correlation function of the system:

(3.10)

If the density-density correlation is only a function of the distance ( ) between the two points, the right hand side of the above equation can simply be replaced by the

Fourier transformation of the density-density correlation function:

(3.11)

Thus in reciprocal space, the structure factor is directly related to the density-density correlation function of a statistical system.

To recapitulate, the measured intensity of Bragg scattered X rays is proportional to the product of the atomic form factor and the structure factor. If the change in the form factor is negligible over the observed range of q, the measured X-ray intensity would depend only on the structure factor which is the Fourier transform of the density-density correlation function Cnn (as shown above). Thus the intensity of the X rays scattered at the Bragg condition is also directly related to the density-density correlations.

Figure 3.2.1 Representation of density-density correlation function Cnn for (a) long-range, (b) quasi long-range and (c) short-range order and the corresponding X-ray intensity profiles I(q||).

From Figure 3.2.1, one can gain some idea about the Bragg-reflected X-ray intensity for different kinds of ordering in a statistical system [9]. In case of long-range order, the mean positions of atoms are fixed. Any instantaneous displacement of the atoms from the mean-position is due to their thermal energy. The density-density correlation function would be highly localized in this case, Figure 3.2.1 (a). The Bragg reflections at each reciprocal space vector would look like a δ-function accompanied by sharp tails (not shown) due to diffuse thermal scattering [1]. Many higher harmonics would be present in the X-ray intensity pattern.

Short-range order, Figure 3.2.1 (c), is represented by exponentially decaying positional correlation function which physically means that the correlations die in the system over a finite distance. The resulting X-ray scattering line-shape from an exponentially decaying correlation in the system is Lorentzian.

The case of quasi long-range order (Figure 3.2.1 (b)) is observed in lower dimensional systems, e.g., smectic systems where only one-dimensional positional order is present [10]. The strength of the correlation function decreases slowly with increasing distance between the atoms. The resulting line-shape lies in between the two extreme cases and are explained based on algebraical decay of positional order by various researcher [11-14].

3.3 X-ray diffraction pattern of common liquid crystal phases

In the preceding section, we have seen how the line-shape and maximum intensity of scattered X rays at Bragg condition depends on the underlying order of a particular phase. So-called q -scans or θ-2θ scans [9,15] are performed to obtain the peak line-

shape and extract useful information about the underlying order of the system. Usually, one would use a point detector mounted on the arm of a goniometer and rotate both the sample and the detector in a coordinated manner to scan parallel to the direction of q in the reciprocal space. But this process takes a long time and many resources are needed to obtain the complete X-ray diffraction patterns.

The recording of X-ray diffraction patterns has conventionally been done on curved film surfaces which are now replaced by image plate detectors. The X-ray diffraction patterns thus obtained provide important information about the underlying mesophase and give some quantitative information about, e.g., positions of the peak maxima, orientation of the diffraction peaks etc. Any attempt to extract information pertaining to the nature of the underlying density-density correlations from these patterns must be done with caution [16].

In Figure 3.3.1, some common liquid crystalline phases and the corresponding X- ray diffraction patterns on a two dimensional detector are shown. The Bragg condition already tells us that the scattering observed at any angle is reciprocally related to the separation between the entities forming the particular phase. Thus, the diffraction patterns recorded toward the outer edges of an image plate detector correspond to a shorter dimension (typically the lateral separation between the molecules) and the ones seen near the center of the image plate correspond to longer dimension (typically the length of the molecules). The sharpness of the peaks (or, their full widths at half maximum) is qualitatively related to the spatial range of periodic order prevailing in the system. In the isotropic phase, the liquid crystalline moieties are all randomly distributed in space. Ther-

Figure 3.3.1 Two dimensional XRD patterns of some common liquid crystalline phases. Detector is lying in the plane of the paper and the X rays are incident normal to it.

mal fluctuations are dominant and molecular motions remain largely un-correlated.

Isotropic phase possesses continuous rotational and translational symmetry and the corresponding short-range positional order gives rise to two rings. A large and a small angle ring correspond respectively to the lateral and longitudinal separation between the mesogens.

In going from the isotropic to the nematic phase, the continuous rotational symmetry is lost about one rotational axis. This results in the nematic phase with long- range orientational order and short-range positional order between the molecules. The molecules now prefer to align along a macroscopic direction, specified by the director n but the centers of mass of the molecules are still randomly distributed. In the unaligned state, the X-ray diffraction pattern might still look like the isotropic phase but even a small applied ordering field (e.g., electric or magnetic field) couples to the electric or magnetic anisotropy of the system and brings in a dramatic change in the macroscopic uniformity of the nematic phase that is easily detected in the X-ray diffraction experiments, Figure 3.3.1 (b).

In smectic phases, the molecules’ centers of mass are restricted in their motion in at least one direction. They typically lie in a planar arrangement and the mass density varies in a sinusoidal manner in the direction perpendicular to the layers. Within the layers, the molecular motion is liquid-like but molecular movement perpendicular to the layers is restricted. In the case of the smectic-A phase, the X-ray beam probing the layers gives rise to an X-ray pattern as shown in Figure 3.3.1 (c). Due to the presence of one- dimensional translational order, the small-angle peaks become dramatically sharper as

compared to the nematic phase. In some cases higher order small-angle peaks are seen at

1: 2: 3… times the position of the first reflection.

A smectic-C (SmC) phase has the same translational symmetry of the smectic-A

(SmA) phase but differs in rotational symmetry. Molecules in the SmC phase are tilted with respect to the layer normal, Figure 3.3.1 (d-f). In an aligned sample, the distinction between the SmA and SmC phases can be made from the angular separation between the large and small angle reflections. The large angle peaks now rotate relative to the small angle peaks by an amount equal to the tilt angle (θ), Figure 3.3.1 (d). As the molecules tilt, the layer spacing will also decrease and the small angle reflections move away from the center by a factor . The value of smectic spacing d calculated from the position of the quasi-Bragg peak can also be used to estimate θ using the simple formula:

. The molecules in successive layers may also tilt in opposite directions giving rise to the anticlinic smectic-CA structure, Figure 3.3.1 (e). The large angle reflections now separate into four diffused spots and their angular separation could be used to calculate the value of the molecular tilt. In another scenario, the molecules could remain largely along their original orientation but the layers tilt, forming a chevron-like structure,

Figure 3.3.1 (f). In this case, the small angle reflection would separate into four spots and the angular separation gives an estimate of the molecular tilt. In a SmC phase, with decreasing temperature, the molecular tilt relative to the smectic layers increases and the layer spacing decreases accordingly.

3.4 Elastomer Liquid Crystals

The main chain smectic-C elastomers used in this study have been synthesized by

W. Ren, et al., [17] at Georgia Institute of Technology, Atlanta, GA, USA. For simplicity, we call the two elastomers LCE1 and LCE2, as shown in Figure 3.4.1. Both elastomers showed extensive strain-retention ability along with a plateau in the stress- strain curve which pertained to soft-elastic behavior. The monomers related to LCE1 and

LCE2 are also shown in Figure 3.4.1. LCE1 contained only mesogens x attached to the main-chain in the end-on position via eleven methylene groups on both sides. The polymer was crosslinked with 10 mol % of 2,4,6,8-tetra-methyl-cyclo-tetra-siloxane crosslinker n and it was given a chemical name: C11(MeHQ)Si8XL10 [17]. In LCE2,

20 mol % of the mesogens x were replaced by the transverse component y. The chain extension and cross-linking reactions for preparing the elastomers were performed at room temperature. LCE1 was chosen as the parent elastomer because its cross linking concentration is optimum for efficient liquid crystalline and rubbery network properties, and presence of the plateau [17] in the stress-strain curve indicated soft-elastic behavior.

The materials were prepared as films of thickness ~ 0.3 mm from which samples of appropriate sizes were cut for the experiments. These systems were initially opaque but could be transformed into a stable transparent state (or an optically monodomain state) by an external mechanical force even at room temperature. The materials were studied during a complete shape-memory cycle where an optically monodomain state was first formed by uniaxial stretching. The specimens were then allowed to stay at the optically monodomain state for over a day before investigating the shape recovery process by

Figure 3.4.1 Schematic representation of elastomers LCE1 and LCE2: Here, “x” is the mesogen component, “y” is the TR3 part which is incorporated into the main chain of LCE2, “m” is siloxane spacer, “n” is a siloxane based crosslinker. LCE1 has only the mesogens incorporated at the “end-on” positions where as many as four chains could attach to the octasiloxane crosslinker. In LCE2, 20% of the “x”-blocks are replaced by “y” blocks in LCE2. Adapted from [17].

supplying thermal energy to the system.

3.5 Mechanical properties of LCE1 and LCE2

The real-time stress-strain plots of LCE1 and LCE2 are shown in Figure 3.5.1.

The stress-strain curves are composed of three distinct regions. The first region is elastic as the system reverts back to its original state after removal of the strain. In the second region, the behavior is non-elastic and soft. It does not immediately fully recover the initial state upon removal of the strain. The third region is also non-elastic but there is a marked increase in the slope of the stress-strain curve. When the strain is removed, the elastomer samples attempt to revert back to the pre-strain state but do not fully go back to original shape in a finite time, cases (a) and (c) in Figure 3.5.1. In the second and third regions, the system is only partially reversible [18], and can only be brought back to the original pre-stretched state by raising temperature above the clearing point or putting the sample in appropriate solvent (e.g., acetone). The second and third elastic regions are termed, respectively, as anelastic and plastic in earlier reports [18-21] but to avoid any ambiguity we shall use the terms “non-elastic reversible” and “non-elastic irreversible” following Ricco and Pegoretti [22]. Here, “non-elastic” means that stress and strain are not linearly proportional and “non-reversible” implies that the specimen does not instantaneously return to strain-free state upon removal of stress. At small strains, the elastomers are composed of polydomains and show reversible (elastic) stress-strain behavior. At intermediate strains, they enter a plateau region with non-elastic reversible behavior where the strain increases with practically small applied stress. The polydomain-monodomain transition occurs in this region of soft-elasticity. Energy is

Figure 3.5.1 Stress-strain behavior of C11(MeHQ)Si8XL10 elastomers with and without p-terphenyl (TR3) transverse rod. Strain is measured in terms of percentage change in elastomer’s length compared to its un-stretched length. (a) With 0 mol% of TR3 or LCE1; (b) with 10 mol% of TR3 and (c) with 20 mol% of TR3 or LCE2. The green dashed lines are guide to the eye, separating the three elastic regions based on the stress- strain plot of LCE1 (solid line). Adapted from [17], © (2009) John Wiley and Sons.

again needed to stretch the elastomers beyond the plateau region. In addition, LCE2 has a lower value of the threshold-strain to the plateau region, a narrower plateau and also, it breaks at a lower strain as compared to LCE1.

3.6 Experimental Set-up

The X-ray scattering experiments were carried out at the 6-ID-B beamline of the

Advance Photon Source (APS) at Argonne National Laboratory, Argonne, IL, USA. The experimental set-up has the following key components:

1. Synchrotron source

2. Beam optics and the X-ray spectrometer

3. Uniaxial stretching mechanism

4. Two dimensional image plate detector

5. Data collection and analysis

3.6.1 Synchrotron X-ray source

Synchrotron radiation is notable for its high brilliance, high level of polarization, pulsed emission of radiation, very small angular divergence of the beam and wide adjustability of the energy/wavelength of radiation [23]. The APS generates the brightest storage ring-based X-ray beams in the western hemisphere [24]. Figure 3.6.1 shows a schematic of the experimental setup used at the 6-ID-B beamline of APS. The following key components constitute the majority of the synchrotron source for generating high intensity X rays [25]:

1. Electron gun or cathode

2. Linear accelerator or Linac

3. Booster synchrotron

4. Storage ring

5. Insertion device

6. Experimental hall

The synchrotron source starts with an electron gun. The cathode is heated to a high temperature (~ 1100 °C) at which point it emits electrons that are pulled towards the other end of the gun by a powerful electric field. This process can produce electron beams which are about the width of a human-hair. The electron beam is then fed into a linear accelerator, commonly known as a Linac. A very high-voltage alternating electric field is now applied to the electrons and by selective phasing, it can accelerate them to

~ 450 MeV. At this energy, the electrons are already travelling at the relativistic speed of

~ 99.999% of the speed of light. These high energy electrons are now allowed to enter the booster synchrotron which further accelerates the electrons from 450 MeV to 7 GeV in half a second. There are four radio frequency cavities which supply the energy needed for such high acceleration taking the speed of electrons to ~ 99.999999% of the speed of light. Next, these electrons are injected into the storage ring which has a much bigger circumference. High vacuum is maintained in the storage ring where the electron beam is focused into a very narrow size by the large number of the electromagnets present around the beam path. The speed of the electrons around the circular path is kept constant, but the electrons are constantly forced to change their direction by these electromagnets. This gives rise to highly accelerating charges which emit intense radiation. The emitted

Figure 3.6.1 Schematic experimental setup at the 6-ID-B beamline of the Advanced Photon Source at Argonne National Laboratory.

radiation is similar to a dipole radiation in the frame of reference of the electron, but in the laboratory frame, it appears as a highly intense source of forward-moving radiation.

The cone angle of radiation [23] diverges in a very small angle (in radians) and depends only on the electrons’ speed:

Here, v is the speed of electrons in the laboratory frame and c is the speed of light. The radiation cone emerges tangentially to the beam path. From time to time, more electrons are injected into the storage ring during a particular run of the synchrotron to restore the beam current which gradually decays.

The insertion device (specifically an undulator) used at sector 6 maximizes the flux and brilliance of the beam [25]. Brilliance, i.e., the number of photons per second incident on a unit area, is important for the study of very small samples or weakly scattering systems.

3.6.2 Beam optics and the X-ray spectrometer

The X-ray beam produced by the undulator at the beamline 6-ID-B consists of continuous spectrum of wavelengths over a range determined by the electron beam energy. The beam was focused with a doubly bent platinum mirror and then a monochromatic beam was chosen using a Kohzu double crystal multiple bounce monochromator made of a channel cut silicon single crystal, Figure 3.6.1. The motion of the monochromator was computer controlled to tune to the desired wavelength from the white radiation. The final beam size was determined by slits 1 and 2. The intensity of the incident beam was attenuated with copper and aluminum filters to make sure that the

sample does not suffer from radiation damage. The beamline operated in the energy- range 3.2 - 38 keV [25]. We used X rays of wavelength 0.765335 Å (or 16.2 keV in energy scale) in our experiments. The X-ray beam was defined to a cross-sectional area of 100 μm x 100 μm. This enabled us to study a very small area of the sample with very high resolution X-ray beam.

3.6.3 Application of Uniaxial Strain

For the purpose of uniaxial stretching, the elastomer films were mounted on a modified INSTEC heating stage (model no. HCS402), Figure 3.6.2 (c). HCS402 has an operating temperature range from − 20 °C to 400 °C with a precision of 0.1 °C. For precise control of stretching, a motorized micrometer (model no. TRA25CC, Newport

Corporation) was used. The micrometer had a travel range of 25 mm with a minimum incremental motion of 0.2 μm. The adjustable micrometer speed had a maximum value of

0.4 mm/s. The micrometer was controlled via a home-made controller (Figure 3.6.2 (a)) which was connected to a Windows based computer via a RS232 port.

One end of the elastomer film was clamped to the heating stage and the other end was attached to a movable aluminum strip, connected to the motorized micrometer via a mechanical extension. During stretching, the X-ray illuminated sample volume moved up and the heating stage was subsequently lowered by half of the length extension, to keep the same scattering volume in the beam. Temperature of the sample was controlled by a programmable INSTEC controller (model no. STC200), Figure 3.6.2 (b).

Figure 3.6.2 Mechanism to apply uniaxial strain: (a) the motor controller used to control the uniaxial stretching, (b) INSTEC temperature controller and (c) photograph of custom- made stretching setup inside the heating oven HCS402.

3.6.4 Image plate detector

The scattered X-ray photons were recorded on a two-dimensional image plate detector MAR345 [26] with 345 mm diameter of active imaging circular surface area and

100 μm pixel sizes. MAR345 is capable of recording low intensity X rays and the signal to noise ratio stays at a very low value (< 1 photon equivalent). The downside is long read-out times in higher resolution mode.

An image plate detector typically possesses BaFBr:Eu or BaFI:Eu as active photo-stimulable phosphor material [27]. The incident X rays interact with the Eu2+ ions and excite them to Eu3+ ions. The highly energetic electrons generated in the process are pumped from the valence band into the conduction band. They can stay there for a long time until a second illumination by a laser source of visible wavelength reads the pixels.

For the MAR345 detector, this second illumination is performed by a single 85 mW red laser which delivers more than 0.8 µJ per pixel at the plate [26]. The incident energy (per pixel) is enough to return the trapped electrons into the ground state while emitting a blue photon of wavelength 390 nm. The intensity of this blue luminescence is proportional to the number of excited electrons which in turn is proportional to the intensity of the incident X-ray photons at the particular pixel. A photomultiplier reads the intensity of the blue luminescence light from each pixel. The output signal is digitized in a dynamic range of real 16 bits to store the intensity data collected from one pixel. It takes ~ 108 seconds to read the whole data collected by all the pixels in a 345 mm diameter active area.

Figure 3.6.3 Silicon standard (NIST 640C) was used to calibrate the X-ray spectrometer. The powder-like ring in the XRD pattern (left) was integrated and matched with the value of the corresponding d-spacing (right) for powder silicon.

During data collection, a beam-stop was placed in front of the detector at the point of impact of the direct X-ray beam. We placed the detector at a distance of ~ 0.5 m from the sample which enabled us to cover diffraction data from ~ 3 Å to several hundred

Angstroms.

3.6.5 Data collection, calibration, and analysis

A Sun Microsystems workstation with SPARC operating system ran the software package SPEC at the 6-ID-B beamline. The hardware interface was managed over the

Computer Automated Measurement and Control (CAMAC) bus which was controlled by

SPEC. Another computer running Linux operating system used the MAR345 software to collect the data from the MAR345 image plate detector. Both of these computers communicated with each other during the data acquisition process. A separate windows computer was used to control sample temperature via the software available from

INSTEC and the motion of the micrometer for uniaxial stretching. The experimental data were calibrated using a NIST 640C silicon standard (d = 3.135 Å), Figure 3.6.3, and analyzed using FIT2D software [28]. The data was first corrected for the background.

Scans were generated to yield intensity vs. q and intensity vs. angle χ (azimuthal scans) data. The integrated data was then taken to Origin 8.5.1 and Mathematica software for further data analysis.

3.6.6 Determination of orientational order parameter

The orientational order parameter, S, for the siloxane and hydrocarbon segments were calculated from the wide-angle scattering data following the method of P. Davidson, et al., [29]:

Here, β is the angle made by the molecular segment with respect to the stretch direction or the macroscopic director. It is possible to parameterize the wide-angle reflections with respect to the azimuthal angle χ on the detector plane [29]:

(3.12)

A numerical fit of the above equation was performed on the I vs. χ profiles of the wide- angle reflections, Figure 3.6.4 (b). The best fit values of the fit-parameters Iₒ, a, K and b were determined by minimizing the value of χ2, which is a measure of goodness of fit, defined as [30]:

where, σj is the experimental uncertainty in Ij, n is the number of data points and c is the number of fitting-constraints.

Using the value of the fit-parameter b, the average was first calculated, using [29]:

Figure 3.6.4 (a) A representative X-ray diffraction pattern of LCE1. The scattered intensity is first integrated (in radial direction) between the two circles and then the data for intensity vs. the azimuthal angle χ (direction marked by the arrow on the X-ray diffraction pattern) is generated using the FIT2D software, (b) plot of I vs. χ for the above wide angle area of the diffraction pattern. The solid line is the fit to the experimental data according to the eqn. (3.12) which gives the value of the fit parameter b, (c) χ2, which is a measure of goodness of fit, is plotted against the fit-parameter b. The values of χ2 are normalized with respect to the best fit value. The ’s correspond to the 95% confidence limits determined by the F-test: F(356,356) ≈ 1.191. From this plot, error in the value of the fit-parameter b is calculated.

where

and then the orientational order parameter, S.

Only the fit-parameter b contribute in determining the value of S. Uncertainties in the value of b were calculated by fixing all the other parameters at the best-fit value and then changing the value of b on either side of the best-fit value while the data were refitted according to eqn. (3.12). The values of χ2, calculated from these fits, are plotted against the corresponding value of b, Figure 3.6.4 (c). These values form a parabola, with minimum at the best-fit value of b. Next, the F-test [30] was employed to determine the probable uncertainty in the value of b within the 95% confidence level. Then, using the method of error-propagation [30], error in the measured quantity S was calculated.

3.6.7 Experimental Details

At room temperature, the elastomer specimen was stretched along its length

(uniaxial strain λ = 1 + ΔL/Lₒ where Lₒ is the initial length and ΔL is the elongation of the elastomer film) to different strains at a rate of 0.4 mm/s. At each strain the film was held for ~ 20 - 25 minutes and 7 sets of X-ray diffraction data were collected. This allowed us to study the relaxation behavior of the elastomer.

The experiment on LCE1 was done with 3 strips, viz., S1, S2 and S3, all cut from the same elastomer sample. S1 was strained at an interval of ∆λ = 0.4 from λ = 1 to 4 and a few X-ray diffraction patterns were collected before moving to next higher strain. These experiments reveal that the small angle diffraction pattern separate into four spots in the strain range λ = 1.4 to 1.8 for LCE1. The film was removed after reaching λ = 4.0 and

kept aside for length recovery measurements after ~ 24 hours. Similar experiment was also performed on LCE2.

Film S2 was studied for the strain-retention almost immediately after the removal of external stress. It was strained at steps of Δλ = 1 up to λ = 4 at room temperature. Then the film was released at the position of lower clamp. A small weight (~ 0.3 g) was kept attached to the bottom part to keep the film straight. Within ~ 2 minutes of releasing the clamp, we started data collection until no change could be ascertained (~ 90 minutes) in the X-ray diffraction patterns.

Next, length recovery measurements were carried out on S1 after it had equilibrated for ~ 24 hours. This was accomplished by taking X-ray diffraction scans while the temperature of S1 was raised at a rate of 1 °C/min above the clearing temperature (TI), recovering the original length of the elastomer film.

The experiment was repeated on S3 after six months but at smaller strain intervals

(∆λ = 0.1), close to the polydomain-monodomain transition region at room temperature to confirm reproducibility and to obtain additional details of the structural changes closed to and at the polydomain-monodomain transition. The experiments on S3 were done almost six months after the experiments on S1 and S2.

3.7 Chapter Summary

We have given a brief summary of the basic X-ray scattering theory followed by a description of the typical two dimensional X-ray diffraction patterns observed in some common liquid crystalline phases. We also described the two elastomer samples studied

in this dissertation work, their mechanical behavior, and details of the experimental setup and procedures for X-ray diffraction studies.

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CHAPTER 4

SOFT ELASTICITY IN A MAIN CHAIN SMECTIC-C ELASTOMER

Polydomain smectic main-chain elastomers have been extensively studied [1-10].

Sánchez-Ferrer, et al., [7] reported the polydomain to monodomain transition caused by uniaxial strain in main-chain smectic-C elastomers. The monodomain obtained is termed a pseudo-monodomain [11] because of the conical distribution of the layers around the mechanically induced director. The situation is explained graphically in Figure 4.1.1 where the smectic-C layer-normals are shown to be distributed on the surface of a cone around the mechanically induced director. For brevity, we shall use the term monodomain to imply sample with a well-defined direction for the director, which accompanies conical distribution of smectic layers around the director.

The materials, LCE1 and LCE2, prepared by W. Ren, et al., [2] are important for two reasons: (1) they have remarkable strain retention ability at room temperature and thus a potential for use as shape memory materials [12], and (2) a plateau in the stress- strain curve points toward a region of soft elastic response in these polydomain main- chain smectic-C elastomers. Both of these phenomena are absent in a conventional elastomer which does not possess any micro-structural detail. The appearance of soft- elasticity and shape memory effect in LCE1 and LCE2 could only be attributed to the underlying liquid crystal (LC) microstructure. In this chapter, I report results of our investigation [13] on the relationship of macroscopic elastic properties to the changes in microscopic structure in the parent elastomer LCE1 and address the following important

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Figure 4.1.1 Schematic diagram showing the conical distribution of smectic layers around the mechanically induced director (dotted red arrow). The green entities are mesogens. The polymer components are omitted for clarity. The solid arrows show the direction of the smectic-C layer-normals which are distributed on the surface of a cone around the macroscopic director.

questions:

1. How are the three elastic regimes in the stress-strain curve related to the structure of

the elastomer and the SmC phase?

2. How do the mesophase and elastomer respond to applied strain. What is the

associated relaxation dynamics, particularly in the soft-elastic deformation region?

Are the relaxation dynamics different in different elastic regimes?

3. How is the applied strain retained and released by LCE1 with time and upon heating?

We proceed by collecting x-ray diffraction (XRD) data at regular intervals of time after the changes in applied strain during a complete shape-memory cycle. Three specimens, called, S1, S2 and S3, all cut from the same elastomer film, were used for these experiments. The shape-memory cycle and the XRD data collection process consisted of the following steps:

1. Uniaxial deformation and relaxation at room temperature: LCE1 was gradually

deformed by application of strain at room temperature. This uniaxial strain was

measured as λ = 1 + ΔL/Lₒ where, Lₒ = 6.0 mm was the original length of the

elastomer sample and ∆L was the amount of uniaxial deformation. The rate of

elongation was 0.4 mm/s. After reaching the target strain value, we let the specimen

relax for ~ 20 - 30 minutes which allowed it to approach steady state value (discussed

later), continually collected XRD data during equilibration, then moved to the next

higher strain and repeated the procedure again. The specimen S1 was deformed in

strain increments of Δλ = 0.4 until we reach a high strain value of λ = 4.0. After pin-

pointing the region of polydomain-monodomain transition by these initial

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measurements, experiments on the specimen S3 was performed in smaller strain steps

(Δλ = 0.1) from λ = 1.0 to 2.1 and the formation of liquid crystal microstructure was

studied in greater details.

2. Strain retention at room temperature: The specimen S2 was studied for the strain

retention process. After stretching the specimen in steps of Δλ = 1.0 from

λ = 1.0 to 4.0 at room temperature, the lower clamp holding one end of the elastomer

was removed. The elastomer was found to retain most of the imparted strain [1]. A

small weight (~ 0.3 g) was kept attached to the bottom part to keep the film straight

during XRD measurements. Within ~ 2 minutes of releasing the clamp, we started

collecting data until no changes in the XRD patterns could be ascertained (~ 90

minutes).

3. Thermal shape recovery: The specimen S1 was kept aside after reaching a strain of

λ = 4.0 for studying the shape recovery after a full day. This was the final step in the

shape-memory cycle where the initial structure was recovered by gradually heating it

beyond the SmC-isotropic transition temperature TI. Several XRD images were taken

which completed our data collection process for LCE1.

4.1 Discussion of Results

Figure 4.2.1 (a) shows representative XRD patterns for a number of strains taken approximately twenty minutes after imparting the strain. At each strain, we observe three sets of reflections. The innermost reflection is shown on an expanded scale on the right hand side at each strain. The nominal stress-strain curve [2] of LCE1 is shown in Figure

4.2.1 (b). It shows three regimes: elastic, non-elastic reversible and non-elastic

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irreversible. The non-elastic reversible plateau regime, where the elongation of the elastomer takes place at negligible energy cost, is the main focus of this study.

The two wide angle (WAXS) reflections correspond to the length-scale of ~ 4.2 Å and ~ 7.2 Å and arise respectively from the hydrocarbon and siloxane segments of the molecule which do not mix well [1] at the molecular level. The smaller dimension arises from the lateral separation between the hydrocarbon mesogenic components while a similar separation between the siloxane parts gives rise to the 7.2 Å reflections. Previous studies using x-ray diffraction and molecular simulation [14] have indicated that the siloxane segments prefer staying in a highly-coiled state. The diffused nature of the consonant WAXS reflection points toward a random orientation of these segments. Only after λ > 1.7 do they gain noticeable ordering, as reflected in circular reflections transforming into arc like reflections. The smallest angle reflections (at ~ 46 Å) correspond to the smectic layer thickness. The second harmonic peaks are also present at all strains (Figure 4.2.1 (a): more noticeable at higher strains), originating from well- developed smectic density wave in this elastomer.

4.2 Polydomain-monodomain transition by uniaxial strain

In this section, we shall see how the liquid crystal microstructure develops gradually from a random orientation at the state of no-deformation toward a chevron-like optically monodomain state at higher strains. Figure 4.2.1 (a) shows XRD patterns as function of strain. The measurements were started at λ = 1.0, i.e., with no strain applied to the crosslinked polymer network. All the reflections remained as diffused rings. At a

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Figure 4.2.1 (a) Representative WAXS (left images at each strain) and SAXS patterns at strains (marked on the patterns) applied in the vertical direction (red arrow). The WAXS patterns also show the effect of shadowing from different parts of the sample holder, stretching mechanism, and the oven; (b) stress-strain plot for LCE1 with vertical dashed- lines acting as guides to the eye for separating three major regions of the curve and (c) definition of the angle α and its relation to the liquid crystal microstructure.

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small strain (λ ~ 1.2), the SAXS reflections concentrated into two broad, arc-like reflections perpendicular to the stretch direction. The WAXS reflections remained almost uniformly-diffused rings with very little modulation perpendicular to the stretch direction, Figure 4.2.1 (a). At larger strains, the SAXS reflections were widespread in the azimuthal direction, finally splitting into four spots at λ = 1.7. At the same time, the

WAXS reflections corresponding to the hydrocarbon parts became more concentrated perpendicular to the stretch direction. The wide-angle siloxane ring also became marginally more intense perpendicular to the stretch direction, ultimately separating into two vertical arc-like reflections. It is to be noted that at low strains, the SAXS reflections were perpendicular to the stretch direction while the WAXS reflections remained more or less uniform rings. With increasing strain, the SAXS peaks gradually became oblique to both the WAXS peaks and the stretch direction.

To quantitatively determine the degree of orientation from x-ray reflections, we plotted their azimuthal intensity distribution in Figure 4.2.2. The azimuthal distribution of the SAXS reflections give information about the formation of smectic-C microstructure while the azimuthal distribution of the WAXS hydrocarbon and siloxane reflections depends on how well oriented these components are. The slight modulation of SAXS reflection at λ = 1.0 is likely coming from a small strain induced during mounting this particular specimen. The azimuthal intensity distributions could be considered uniformly diffused which is the characteristic of a predominantly polydomain nature of the network, or a powder sample. At λ = 1.2, the WAXS reflection remain more or less uniform. The

SAXS reflections are fit with two Gaussian line-shapes and the peak positions are found

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Figure 4.2.2 Azimuthal intensity I (arbitrary units) profiles of both SAXS and WAXS (hydrocarbon part) reflections in Figure 4.2.1 (a). The small modulation of the SAXS peak at λ = 1.0 is due to a small strain induced during mounting this particular specimen. The Gaussian peak functions are fitted to calculate the angle α. At λ = 1.2, the SAXS are formed parallel to the stretch direction with WAXS remaining more or less uniform. At λ = 1.5, the SAXS reflections are flat at the top, implying a possible superposition of two or more peaks eventually splitting into four peaks. The gradually increasing separation of SAXS peaks is clear at λ > 1.6. The WAXS reflections also continue to gradually sharpen with increasing strain.

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to be around 0° and 180°. This means the orientational distribution of the mesogens forming the layers is uniform in the plane perpendicular to the x rays while the smectic layers are formed parallel to the stretch direction. This can only be possible if the smectic layers are predominantly vertical, with the layer-normals distributed uniformly in the horizontal plane. The situation is elaborated in the schematic diagram of Figure 4.2.3. At higher strains, the WAXS reflections peak around 0° and 180° which tells us that the mesogenic parts are eventually becoming parallel to the stretch direction. This is accompanied by the SAXS reflections first becoming broad (for strains λ = 1.4 and 1.5), and then beginning to split around the peak positions of the WAXS reflections. For strain

λ > 1.2, the SAXS reflections are well fitted to four Gaussian functions which determine the positions of the four-spot reflections. The splitting of the SAXS reflections into four- spots in oblique direction to the WAXS reflections indicates the presence of the SmC phase in chevron-like configuration. Evidently, the mesogens have become parallel to the stretch direction and the smectic layers are tilted with respect to them, forming a chevron- like arrangement. The smectic layer-normals also incline in an azimuthally degenerate manner about the stretch direction and establish a chevron-like microstructure in the planes containing the stretch direction.

The apex of the chevron structure lies along the stretch direction, Figure 4.2.1 (c).

The separation 2α between the SAXS reflections in Figure 4.2.1 (a) is related to the chevron-like microstructure. Here, α is the angle between the smectic layer normals and the stretch direction or the direction in which long-axis of the mesogens are eventually aligned, Figure 4.2.1 (c). Usually, in a monomer liquid crystal exhibiting the smectic-C

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Figure 4.2.3 Schematic diagram of the smectic-C layers formed inside the elastomer specimen (at low strains) as the elastomer film is uniaxially deformed along the z- direction. The x rays are incident on the sample from y-direction. The red rods here are the mesogens while the polymer chains are omitted for clarity. At low strains, the layers are formed parallel to the stretch direction with the layer normals distributed uniformly in the x-y plane. For simplicity, only two sets of layers are shown. The layers, which are formed parallel to the z-axis, contribute to the SAXS reflections appearing along the x- axis. The random distribution of the mesogens (in the x-z plane) contributes to the WAXS reflections (corresponding to the hydrocarbons) appearing in the x-z plane.

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phase, this angle α is equal to the tilt of the mesogens with respect to the smectic layer normal [15]. This elastomer does not exhibit an untilted (smectic-A) phase which makes the calculation of molecular-tilt from layer spacing values [15] difficult.

The angle α demonstrates a gradual decrease with increasing strain, Figure

4.2.4 (a), saturating at a value close to ~ 50° in the chevron-like monodomain state. After the chevron microstructure is formed, the layer spacing d also becomes smaller with increasing λ, Figure 4.2.4 (b). A declining value of d indicates corresponding increments in molecular tilt. Thus, the angle α and the tilt of the mesogens operate in contrasting ways in this elastomer. This is contrary to what is commonly observed in monomer liquid crystals where the molecular-tilt and the angle α, are directly related.

The global orientational order parameter S of the hydrocarbon and siloxane parts is determined from the wide angle azimuthal distribution of the hydrocarbon and siloxane segments, following the method of P. Davidson, et al., [16]. The method was also discussed in Chapter 3, section 3.6.6. The value of S calculated this way is global, because it depends on both the orientational order of the directors of microdomains and the orientational order of the mesogens within each microdomain. The order parameter within a microdomain may be high, but on a global scale, nearly random orientational distribution of the microdomains will lead to a value of S close to zero.

At low strains, a collection of disordered microdomains in the plane perpendicular to x rays gives rise to a small value of S. In the middle region, these domains rotate toward the stretch direction and the rotation slows down beyond the value of the critical strain λ ~ 1.7. Order parameters for the hydrocarbon and the siloxane segments at small

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Figure 4.2.4 (a) - (d) Uniaxial strain dependence of α, d and S (mesogen and siloxane parts). All values are calculated after ~ 20 min of equilibration at each strain. The equations in the insets of the panels (c) and (d) correspond to the solid line fits, adhering to a phenomenological growth model [17].

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strains are shown in Figure 4.2.4 (c) and (d). With increasing strain, the microdomains rotate toward the stretch direction which also ensures an increase in the global S of both segments of the polymer chain. The rise in orientational orders conforms well to a phenomenological growth model [17], described by the equations shown in the insets of

Figure 4.2.4 (c) and (d). The model describes the changes in the parameter S whose rapid growth (across the soft-elastic plateau) is arrested as it approaches the maximum value.

The topological constraints [18] of the network prevent a perfect alignment and the system saturates to a final orientation given by S ~ 0.83 and 0.4 for the hydrocarbon and siloxane parts, respectively. The model prescribes that the secondary chevron-like monodomain be uniform above λ = 1.7. The results shown in Figure 4.2.4 (c) and (d) also support this claim.

4.3 Mechanism behind polydomain-monodomain transition

Based on the results and discussions in previous section, we explain the evolution of liquid crystal microstructure in these systems, Figure 4.3.1, as follows. The elastomer film is stretched in the vertical- or z-direction and since the volume of the elastomer is conserved, the film shrinks in both x- and y-directions, Figure 4.3.1 (a). The x-ray beam is incident parallel to the y-direction and probes the distribution of the microstructure in the x-z plane. At zero strain, the smectic-C microdomains are distributed randomly in the

3-dimensional space inside the elastomer film. The x rays encounters a random distribution of smectic layers, mesogens and the siloxane segments, leading to uniform diffused rings at both small and wide angles, Figure 4.3.1 (b).

Upon stretching the elastomer film slightly in the vertical z-direction, smectic

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Figure 4.3.1 (a) Schematic representation of strain applied in vertical- or z-direction to the elastomer film. The film is squeezed in both x and y-directions for conserving the volume of the elastomer. The x rays are incident parallel to the y-direction. (b) - (d) Schematic illustration of the polydomain-monodomain transition with increasing strain, associated microstructure and the corresponding wide and small angle (right hand side image) XRD patterns.

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layers get squeezed in the two horizontal directions (due to constancy of the volume of elastomer film). The squeezing of the layers would cause the smectic line defects to move out to the microdomain-boundaries via a flow [19] of the smectic layers parallel to the z- direction. Uniform layers would become parallel to the strain direction in the horizontal plane. Scattering of the x-ray beam from smectic layers predominantly in the vertical direction would give rise to vertical arc-like reflections at small angle, Figure 4.3.1 (c).

This being a smectic-C system, the mesogens remain tilted with respect to the layer normals. At low strain, the polymer chains are not appreciably influenced by the mesogens, thus the mesogens can arrange themselves randomly along the azimuthal direction inside the smectic layers maintaining a constant polar tilt with respect to the smectic layer normals. Furthermore, the hydrocarbon linkage groups will possess little orientational preference at low strains. The system appears to have a near random distribution of hydrocarbon segment with little orientational bias giving rise to diffused wide angle rings, Figure 4.3.1 (c).

At high strains, the polymer chains become vertically stretched and the mesogens become vertical as well. This is accompanied by redistribution of reflected intensity from uniform ring to two arc-like reflections at wide angle, Figure 4.3.1 (d). The SmC microstructure requires a non-zero angle between the final alignment direction of the mesogens and the smectic layer normal. The layers rotate in a continuous fashion inside the elastomer matrix, eventually forming a 3-dimensional chevron-like microstructure at high strains and four diffraction spots at small angle.

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At low strains, the coupling between the polymer network and liquid crystalline components is very weak and the liquid crystal microstructure is dominated by the in- plane flow-properties of the smectic phase [19]. At higher strains, the polymer chains strongly influence the orientation of mesogens leading to the formation of a chevron-like microstructure. The system finally attains an optically monodomain state, with the smectic layer normals distributed in a chevron-like fashion around the stretch direction.

4.4 Relaxation at constant strain

In initial test measurements, we collected XRD data for a long time after the application of a specific strain while the sample equilibrated. From the time dependence of the angle α, we determined the equilibration time to be approximately 4 minutes in the soft elasticity regime. In all subsequent measurements, we acquired diffraction patterns continually for at least 5 time constants, i.e., 20 - 25 minutes, to gain insight into the equilibration process and to ensure that the sample had equilibrated before acquiring the final data set and changing the strain to the higher level. The data was calibrated against the Si-standard and the I vs. q and I vs. -scans generated with the help of the FIT2D software [20]. These scans were then analyzed to obtain the values of α, d, and the two values of S shown in Figure 4.4.1. The red points in Figure 4.4.1 are from a different run on specimen S1 where larger strain steps were taken to pin-point the onset and end of the polydomain-monodomain transition. The results from the two scans overlap very well showing the reproducibility of our results. It is clear that α, d, and S relax in continuous fashion, but their values jump suddenly when strain is increased. Surprisingly, it appears that increase in strain increases the molecular tilt as reflected in the drop in d which then

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Figure 4.4.1 (a) - (d) Time dependences of α, d and S (hydrocarbon and siloxane parts) are shown during stretching across the polydomain-monodomain transition region. The vertical dashed lines mark different strain regions with values of strain λ shown at top. The green and orange points correspond to λ = 1.0 and 1.2 respectively. The red- and black-points are from measurements on specimen S1 and S3 respectively. Both specimens were cut from the same elastomer sample. The results also show the reproducibility of our measurements.

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recovers and d attempts to reach a steady state value. The angle α shows contrasting behavior which confirms that the molecular tilt and the angle α in LCEs are not directly and as simply related as they are in the smectic-C phase of monomer liquid crystals. The changes in these physical quantities become increasingly more pronounced at higher strains. The values of the global orientational order parameter calculated for the siloxane segments stay almost constant during relaxation. The increases at each strain appear to be related to the sudden changes in smectic ordering, reorientation, and deformations of the liquid crystal microstructure.

In the preceding sections, we identified two contrasting relaxation mechanisms in

LCE1. These are the flow property of the smectic liquid crystals and the elastic behavior of the polymer network and its coupling to the mesogenic parts, which dominate at the low and high strains respectively. Examining the changes in the chevron microstructure, or in this case, the angle α, provides an insight into these two phenomena. The relaxation of angle α is modeled using a simple exponential fit of the form:

(4.1)

Here, and are the initial and saturation values of the parameter y at a particular strain and τ is the relaxation time constant. A higher value of τ points toward a slow relaxation process and vice versa. Results of the fits to eqn. (4.1) are shown in Figure

4.4.2. These fits make a clear distinction between the relaxation behavior in polydomain- monodomain transition (the second) region and the non-elastic irreversible (the third) region. The values of τ calculated from these fits are plotted in Figure 4.4.3 as function of increasing strain. At small strains (1.3 ≤ λ < 1.7), the relaxation of α is relatively slow

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Figure 4.4.2 Relaxation behavior of the angle α at different constant-strains. The solid lines are the fits to eqn. (4.1).

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with τ ~ 45 min. As the strain is increased (λ ≥ 1.7), the relaxation becomes faster and τ attains the value of ~ 5 min. The system clearly responds differently to low and high applied strains with different values of τ. In the polydomain-monodomain transition regime, the slow reformation or realignment of liquid crystal microdomains seems to be dominated by the LC flow properties of the SmC phase. At these low strains, higher flexibility of the polymer chains would allow the layers to move almost freely inside the elastomer network. The layers possibly anneal via slow movements of edge dislocations

[21-24] giving rise to a higher relaxation time constant.

At strains higher than λ = 1.7, the system relaxes with an order of magnitude smaller relaxation time (τ ~ 4 - 8 minutes). In this regime, the reorientation of the smectic domains is completed and the local directors of the individual microdomains, the polymer chains, and the mesogens are all pointing toward the stretch direction. This being a main- chain system, the relaxation properties would be dominated by the properties of the polymer components at high strains. This is substantiated by the fact that stress-relaxation typically occurs over a time scale of ~ 7 minutes (at λ ~ 2.75) [25] in a conventional elastic network of polyisoprene rubber lacking any LC microstructure. Also, earlier stress-relaxation studies [26] on LCE1 have indicated relaxation times of the order of

~ 4 - 5 minutes for strains λ ~ 1.5 to 4.0. Thus, at high strains, the elastic properties of the polymer network seem to be the dominant mechanism leading to faster relaxation times.

To our knowledge, this is the first time that the relaxation time constants of liquid crystal microstructure are measured and shown to be associated with the two components of the liquid crystal elastomer systems. In other words, we have been able to separate out

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Figure 4.4.3 Plot of relaxation time constant, τ as function of strain, λ. The values are calculated from the fits of eqn. (4.1) to the experimental data in Figure 4.4.2. The colored lines are guide to the eye showing the transition clearly.

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the role of two basic components of the liquid crystal elastomer systems and their effect on the macroscopic property of the elastomer network.

4.5 Strain retention

A sample of LCE1 was elongated to strain value of λ = 4.0 establishing an optically monodomain structure consisting of chevron-like microdomains. Previous stress-strain measurements [1] indicate that a considerable amount of strain (λ ≥ 2.0) is retained even after the removal of the external load and the elastomer remains in this state for a very long time. Approximately two minutes after removing the lower clamp, we started collecting the XRD data and continued for approximately ninety minutes. Figure

4.5.1 (a) shows the WAXS and SAXS patterns at t ~ 0 which correspond to a chevron- like arrangement of the liquid crystal microstructure. We also collected one XRD data after ~ 24 hours to determine the steady state structure after the sample had fully equilibrated, Figure 4.5.1 (b). The images show that the azimuthal distribution of the hydrocarbon reflections becomes slightly broader with time, while the second harmonic is present even after ~ 24 hours of removing the clamp. This behavior demonstrates that a well-aligned smectic structure is retained by the elastomer for a considerable time period.

Figure 4.5.1 (c) shows that S and α decrease at different rates and finally saturates to their respective steady state values at room temperature. Simple exponential fits to the data reveal that S levels off relatively quickly ( ). During the same time, relaxation of the angle α is approximately two times slower: .

The difference in these time constants is consistent with our previous inference that LC orientation depends on the LC component of the system while the changes in chevron

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Figure 4.5.1 (a) - (b) WAXS and SAXS XRD patterns at t ~ 0 and after t ~ 24 hours. The WAXS patterns also show the effect of shadowing from different parts of the sample holder, stretching mechanism, and the oven, (c) time dependence of S and α. The solid curves are fits to eqn. (4.1). The steady state was achieved after ~ 90 minutes with S = 0.68 ± 0.01 and α = 40.4° ± 0.2° such that the elastomer is locked into a chevron-like microstructure long after removal of the strain.

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structure are governed by elastic properties of the polymer component. The persistence of angle α implies that the layer normals remain oriented with respect to the stretch direction even after the removal of the external strain. The elastomer is locked into a chevron-like optically monodomain state (S (hydrocarbon) ~ 0.65, α ~ 41°) even after relaxing for one full day.

4.6 Thermal shape recovery

The secondary shape induced by the application of strain is stable for a long period of time. It is thought that the presence of unfolded hairpins in the stretched state and trapping of the siloxane based cross-linkers in the siloxane rich regions of the sample

[10] lead to the stability of the secondary shape. The elastomer is gradually heated above

TI which disrupts the underlying lamellar order and supplies enough energy to the crosslinkers to move out of the siloxane rich regions. Figure 4.6.1 shows the representative XRD patterns as functions of temperature change, i.e., ∆T = T − TI. The

WAXS hydrocarbon reflections (left column, Figure 4.6.1) gradually transform into a diffused ring as the temperature is increased. The intensity of the SAXS reflections (right column, Figure 4.6.1) diminishes with temperature, ultimately merging into a uniform ring at and above TI. The disappearance of the second harmonic reflections points toward a partial loss of the macroscopic smectic ordering with heating. At and above TI, the smectic-C mesophase transforms into the isotropic phase and all reflections turn into uniform rings.

Figure 4.6.2 shows temperature dependence of S and α during the thermal shape recovery process. The value of S gradually decreases with increasing temperature and

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Figure 4.6.1 WAXS (left column) and SAXS (right column) patterns acquired during thermal shape recovery of LCE1. The WAXS patterns also show the effect of shadowing from different parts of the sample holder, stretching mechanism, and the oven. The peaks gradually turn into diffused rings and lose their intensity as one nears TI. The curved arrows show the direction in which the SAXS reflections combine into uniform rings.

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rapidly goes to zero at the first order SmC-I phase transition at TI. The curved arrows in

Figure 4.6.1 shows that the SAXS reflections are coming closer to each other in the horizontal direction before condensing into uniform rings at and above TI. This means, the value of the angle α continues to decrease with increasing temperature until ~ 20 °C below TI, Figure 4.6.2. However, it should be noted that the net change over the entire temperature range is very small (~ 2°) compared to changes observed during the application of strain. This shows that the chevron structure and S are stable over most of the smectic-C range, or that the system possesses significant shape memory. We determined the value of TI from SAXS results and it is found to be ~ 10 °C lower than the value measured by differential scanning calorimetry (DSC) [1]. The DSC curve is broad and previous stress-strain measurements on this material [10] have related this shape recovery temperature TI to the onset of the isotropic phase. Our results are consistent with that observation.

It is evident that in the chevron-like optically monodomain state, the long axes of the mesogens and the director are well-aligned in the vertical direction of stretching. The smectic layer normals are distributed on the surface of a cone around the vertical direction. The stability of this secondary structure has been explained by W. Ren, et al.,

[1,10] based on nano-segregation of the hydrocarbon and the siloxane segments, unfolding of the polymer chains at high strains, and trapping of the siloxane based crosslinkers inside the siloxane rich regions. They have argued that the immiscibility of the hydrocarbon and siloxane segments of the polymer chain would lead to nanosegregation and promote the formation of lamellar structure in these systems. Initially, a

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Figure 4.6.2 Temperature dependence of α and S for the mesogenic part as the transition to the isotropic phase at TI is approached. The elastomer is slowly heated from the chevron-like monodomain state at room temperature to the isotropic state above temperature TI where the original polydomain state is recovered.

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smectic mesophase would form with local order but with no macroscopic order of the smectic domains. Application of high uniaxial strain unfolds the polymer chains and also redistributes the positions of the crosslinkers. The siloxane based crosslinkers would prefer to remain inside the siloxane rich regions of the sample which amounts to

“energetically” trapping them inside the siloxane-rich regions. With increased thermal energy, the siloxane crosslinkers move out of the segregated regions, thus disrupting the smectic-C ordering. This is reflected by a continuous decrease in the value of S with heating.

The behavior of the angle α with increasing temperature, Figure 4.6.2, needs special consideration. The decrease in the value of α with heating tells us that the SAXS reflections, originating from the smectic layers, tend to recombine in the direction perpendicular to the applied strain. This means the layers now tend to form horizontally with an azimuthally degenerate distribution of the mesogens in the plane of the smectic layers. This is somewhat different than what we observed during the polydomain- monodomain transition by uniaxial strain where the layers were formed vertically, parallel to the stretch direction. This contrasting behavior during shape recovery can be explained by considering orientational preference of the mesogens parallel to the stretch direction in the monodomain state. The chevrons re-merge, forming poorly defined layers that are statistically perpendicular to the vertical stretch direction, giving rise to a decrease in the angle α until the elastomer enters the isotropic phase above TI.

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4.7 Chapter Summary

We have investigated detailed structural changes of the liquid crystal component of the SmC LCE1 across the shape memory cycle using x-ray diffraction and identified new structural change, their association with the elastic behavior, as well as new relaxation dynamics. Initially, the local directors of the smectic domains, the polymer chains, and siloxane segments are all randomly oriented leading to uniform diffused rings in the x-ray diffraction pattern. At slightly higher strain applied in one direction, these smectic domains are effectively squeezed in the other two perpendicular directions as required by conservation of the volume of the elastomer film. The smectic layers are effectively oriented parallel to the stretch direction with a uniform distribution of the smectic layer normals in the plane perpendicular to the stretch direction. The relatively higher relaxation time constant of ~ 45 minutes is governed by the in-plane flow property

[19] of the liquid crystal phase constrained and modified by the elastomer network. With increasing strain, the polymer chains and consequently the mesogens and the local directors of the smectic domains all effectively align parallel to the stretch direction and the system ultimately attains an optically monodomain chevron-like structure. The system relaxes with about an order of magnitude smaller relaxation rate (~ 4 - 8 minutes) due to strong coupling between the polymer chains. The smectic layers in chevron-like microstructure rotate toward the stretch direction to accommodate the alignment of the mesogens and the polymer chains parallel to the stretch direction. This gradual transformation to chevron-like microstructure completes at a critical strain of λ = 1.7. The chevron-like monodomain structure is enhanced at higher strains approaching λ = 4.0.

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The measured global orientational order parameter S for the mesogens is initially close to zero but it rapidly increases across the polydomain-monodomain transition and levels off at a maximum value of ~ 0.83 at λ = 4.0. The value of S for the siloxane parts becomes noticeably higher than zero only beyond λ ~ 1.7 and reaches a maximum of ~ 0.4 at

λ = 4.0.

We have clearly confirmed nanosegregation of siloxane segments [10] as postulated in previous studies. This nanosegregation is expected to help in the formation of well-defined smectic layers. The energy required to disrupt this lamellar order is much higher than the restoring elastic energy of the network [10]. The elastomer thus gets

“locked” into this monodomain state and prefers to remain in this state probably forever until some external stimuli cause it to change. The relaxation of the chevron-like structure is slower than that of the orientational order parameter during the first ninety minutes of strain retention. An external stimulus in the form of thermal energy gradually disrupts the smectic order and the entropic elasticity of the polymer network takes over above the clearing point as the elastomer transitions to the isotropic phase and recovers its original shape.

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REFERENCES

[1] W. Ren and A. C. Griffin, Phys. Status Solidi B 249, 1379 (2012).

[2] W. T. Ren, P. J. McMullan, and A. C. Griffin, Phys. Status Solidi B 246, 2124 (2009).

[3] A. Sanchez-Ferrer and H. Finkelmann, Macromolecules 41, 970 (2008).

[4] A. Sanchez-Ferrer and H. Finkelmann, Mol. Cryst. Liq. Cryst. 508, 348 (2009).

[5] G. Cordoyiannis, A. Sanchez-Ferrer, H. Finkelmann, B. Rozic, S. Zumer, and Z. Kutnjak, Liq. Cryst. 37, 349 (2010).

[6] A. Sanchez-Ferrer and H. Finkelmann, Solid State Sci. 12, 1849 (2010).

[7] A. Sanchez-Ferrer and H. Finkelmann, Macromol. Rapid Commun. 32, 309 (2011).

[8] W. Ren, P. J. McMullan, and A. C. Griffin, Macromol. Chem. Phys. 209, 1896 (2008).

[9] W. Ren, P. J. McMullan, H. Guo, S. Kumar, and A. C. Griffin, Macromol. Chem. Phys. 209, 272 (2008).

[10] W. T. Ren, W. M. Kline, P. J. McMullan, and A. C. Griffin, Phys. Status Solidi B 248, 105 (2011).

[11] P. Heinze and H. Finkelmann, Macromolecules 43, 6655 (2010).

[12] I. A. Rousseau and P. T. Mather, J. Am. Chem. Soc. 125, 15300 (2003).

[13] S. Dey, D. M. Agra-Kooijman, W. Ren, P. J. McMullan, A. C. Griffin, and S. Kumar (to be published).

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[14] D. Guillon, M. A. Osipov, S. Mery, M. Siffert, J. F. Nicoud, C. Bourgogne, and P. Sebastiao, J. Mater. Chem. 11, 2700 (2001).

[15] S. Kumar, Liquid crystals: Experimental Study of Physical Properties and Phase Transitions (Cambridge University Press, New York, 2001).

[16] P. Davidson, D. Petermann, and A. M. Levelut, J. Phys. II 5, 113 (1995).

[17] A. Tsoularis and J. Wallace, Math. Biosci. 179, 21 (2002).

[18] M. Warner and E. M. Terentjev, Liquid Crystal Elastomers (Oxford University Press Inc., New York, NY, USA, 2007).

[19] M. Tokita, T. Takahashi, M. Hayashi, K. Inomata, and J. Watanabe, Macromolecules 29, 1345 (1996).

[20] A. P. Hammersley, S. O. Svensson, M. Hanfland, A. N. Fitch, and D. Hausermann, High Pressure Res. 14, 235 (1996).

[21] R. B. Meyer, B. Stebler, and S. T. Lagerwall, Phys. Rev. Lett. 41, 1393 (1978).

[22] P. L. Chen and C. Y. D. Lu, J. Phys. Soc. Jpn. 80, 094802 (2011).

[23] R. Bruinsma and Y. Rabin, Phys. Rev. A 45, 994 (1992).

[24] R. G. Larson, The structure and rheology of complex fluids (Oxford University Press, New York, 1999).

[25] C. Ortiz, C. K. Ober, and E. J. Kramer, Polymer 39, 3713 (1998).

[26] W. Ren, Ph.D. Dissertation, Georgia Institute of Technology, USA, 2007.

CHAPTER 5

THE EFFECT OF TRANSVERSE RIGID SEGMENT ON STRAIN RETENTION

In this chapter, we discuss the effect of incorporating a transverse segment in the main-chain of the parent elastomer LCE1. We call the new elastomer as LCE2, whose schematic has been shown in Figure 3.4.1. LCE2 was obtained by replacing twenty molar percent of the end-on mesogens in LCE1 by the p-terphenyl transverse rod TR3 [1]. The stress-strain plot for LCE2 is shown in Figure 5.1.1 (a). LCE2 also possesses a soft- elastic plateau, albeit the plateau starts at a smaller strain as compared to LCE1. The elastomer also shows superior strain-retention ability [2] which appears to be related to the incorporation of the transverse component into the main-chain. After examining the changes in liquid crystal microstructure with strain in LCE2, we shall compare the properties of LCE2 with LCE1.

5.1 Effect of TR3 on the polydomain-monodomain transition

Varying degrees of uniaxial strains were applied to a strip of LCE2 at room temperature. After reaching a specific strain, we waited for approximately 25 minutes to allow it to equilibrate and continually collected x-ray diffraction images before moving to a higher strain. The x-ray diffraction images thus obtained are shown in Figure 5.1.1 (b).

At zero strain (λ = 1.0), LCE2 is in a polydomain state as expected. The corresponding x- ray diffraction images, Figure 5.1.1 (b), show three diffused rings corresponding to (1) the width of the hydrocarbon parts, at ~ 4.2 Å, (2) the width of the siloxane segments

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Figure 5.1.1 (a) Stress-strain plot for LCE2. (b) WAXS (left images at each strain) and zoomed-in SAXS patterns of LCE2 taken at different strains (marked on the patterns) at room temperature. The WAXS patterns also have some shadows from different parts of the sample holder, stretching mechanism, and the oven. All images were acquired after ~ 25 minutes of equilibration time at each strain. The SAXS pattern at λ = 1.2 already shows splitting into four-spot reflections. The red arrow represents the stretch direction. (c) Definition of angle α and its relationship to chevron-like LC microstructure.

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(~ 7.2 Å), and (3) smectic layer spacing shown in top panel of Figure 5.1.2. At slightly higher strain (λ = 1.2), the WAXS reflections corresponding to the hydrocarbon parts become vertical crescents which lie on the equator. The SAXS pattern also becomes vertically elongated, albeit a closer inspection reveals that they are split into four-spots as the polymer chains and mesogens undergo orientational alignment along the strain direction. The situation is similar to LCE1 at λ = 1.6, as described in Chapter 4. The polydomain-monodomain transition in LCE2 starts at a lower strain than LCE1, and in accordance with the nature of its stress-strain curve of LCE2, Figure 5.1.1 (a), where the

λ = 1.2 point marks the beginning of the soft-elastic plateau. The splitting of the SAXS pattern into four-spots is clear at strains λ ≥ 1.4. The WAXS reflections become concentrated on the equator, perpendicular to the stretch direction and the degree of macroscopic alignment of the mesogenic parts improves with strain. The SAXS reflections become oblique to them with formation of well-aligned chevron-like monodomain structure. The second order Bragg peaks are also clearly visible at λ ≥ 1.4.

As already discussed in Chapter 4, the angular separation between the SAXS peaks in first and second quadrant is related to the chevron-like microstructure, see

Figure 5.1.1 (b) and (c). Values of the global orientational order parameter S determined from the azimuthal intensity distribution of the WAXS peaks are plotted in Figure 5.1.2 along with the layer spacing d. The latter stays almost constant at λ = 1.2 and then decreases with increasing strain. During relaxation, the values of d increase gradually before jumping down at the application of higher strain. Similar jumps are also observed in the values of S and α which are related to instantaneous changes in the orientation of

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Figure 5.1.2 Values of the parameters d, S and α calculated for the duration of the uniaxial experiment on LCE2. The numbers in red in the panel for the angle α represent a particular strain λ.

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the mesogens and the deformation of the chevron-like microstructure. The values of S and α ultimately saturates to ~ 0.75 and ~ 45°, respectively at a high λ = 3.0, well within the chevron-like monodomain state.

The evolution of angle α (at constant λ) was investigated to attain a better understanding of the relaxation dynamics in this LCE. The relaxation of angle α is re- plotted in Figure 5.1.3 (a) and fitted to a single exponential function, eqn. (4.1). We observe a fast relaxation time (~ 4 - 5 minutes, see Figure 5.1.3 (b)) at high strains, dominated by the properties of the polymer chains. In case of LCE2, the splitting of the

SAXS patterns started at λ = 1.2. We do not have any data points between λ = 1.0 and 1.2, or in the so-called polydomain-rich region. Still the trend is apparent, i.e., a slow relaxation is observed at low strains apparently governed by the LC flow-properties [3] inside the polymer network.

A comparison of strain dependence of angle α for LCE1 and LCE2 is presented in

Figure 5.1.4. Angle α is lower for LCE2 than for LCE1 at the same strain. Also, the chevron-like monodomain begins to form at a smaller strain (λ ~ 1.2) in LCE2 than in

LCE1 (λ ~ 1.6). The values ultimately saturate to ~ 45° and ~ 50° in LCE2 and LCE1, respectively. The only difference in the structures of these two elastomers lies in the introduction of the transverse component in LCE2 which leads us to conclude that the presence of the transverse component in the main-chain facilitates better formation of chevron-like monodomain structure and at lower strains.

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Figure 5.1.3 (a) Relaxation of angle α at different constant strains. The solid lines are fits to simple exponential plots. (b) Plot of relaxation time constant, τ as a function of strain, λ. The colored-lines are guides to eye.

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Figure 5.1.4 Comparison of the values of the angle α in LCE1 and LCE2 as function of strain. The value of angle α is always smaller in LCE2 as compared to LCE1.

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5.2 TR3 and strain retention

In the previous section, it was concluded that the formation of the chevron-like monodomain begins at a lower strain in LCE2 (λ = 1.2) and completes in the vicinity of λ

= 1.4. For LCE1, it starts at λ ~ 1.5 and completes at λ ~ 1.7. Typically, more than fifty percent of the applied strain is retained in both of these elastomers after the monodomain formation is complete [4]. Deep in the third region, i.e., in the chevron-like monodomain region, the strain retention abilities of these elastomers were determined by removing the lower clamp after λ = 3.0 for LCE2 and λ = 4.0 for LCE1. A negligible mass (~ 0.3 g) was attached at the lower end to hold the elastomer films straight. We collected several x- ray diffraction patterns while both the samples were relaxing. We then calculated the values of α and S from the diffraction patterns as a function of time. The relaxation behaviors were monitored for ~ 90 minutes in the case of LCE1 and ~ 40 minutes in the case of LCE2. For the purpose of comparing the relative changes, the values of these parameters were normalized by dividing them with their corresponding value at t = 0 min. The results are plotted in Figure 5.2.1. Both S and α relax toward their respective steady state values, albeit with different relaxation times. Single exponential fits to the respective data reveal that both the parameters approach their equilibrium values at a faster rate in LCE2 than in LCE1. The relaxation time constants of S are found to be

~ 7.5 minutes in LCE2 and ~ 12.1 minutes in LCE1. The time constants for relaxation of the angle α are measured to be ~ 9.1 minutes in LCE2 and ~ 23.5 minutes in LCE1.

Clearly, the introduction of TR3 seems to enhance retention of the strain-induced orientational alignment of microdomains and the angle α. The introduction of TR3 has

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Figure 5.2.1 Comparison of the values of α/αₒ and S/Sₒ during strain retention experiments. Here, αₒ and Sₒ are the values of α and S at t = 0 min. Solid lines are simple exponential fits to the data. The values of corresponding relaxation time constants are also mentioned adjacent to the respective fits.

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thus lead to a more stable chevron structure of LCE2 and “locking” of the arrangement of the domain distribution with respect to the elastomer network.

5.3 Effect of TR3 on thermal shape recovery

By first affecting the polydomain-monodomain transition with uniaxial stretching, allowing the samples to retain strain under negligible load, and then recovering the original polydomain state by external stimulus, one completes the full shape-memory cycle of these materials. As discussed above, the introduction of TR3 aids LCE2 in retaining more than 95% of the macroscopic S compared to LCE1 which retains only about 80%, Figure 5.2.1. The initial polydomain state of such crosslinked networks could be restored by either swelling the network in acetone or by raising the sample temperature above the clearing point. We performed thermally driven shape recovery where the elastomer films were heated above TI.

Figure 5.3.1 (a) shows the SAXS patterns during the chevron-like monodomain formation by stretching. Subsequent restoration of random distribution of the smectic-C domains by heating is shown in Figure 5.3.1 (b). During the polydomain-monodomain transition, the SAXS peaks appear first as two vertical arcs perpendicular to the stretch direction. They then split into two pairs of peaks at high strain forming SmC chevron-like monodomain. The presence of the second order smectic layer peaks at higher strains establishes highly-condensed smectic density wave [5] in the system. As the thermal recovery process begins, the intensity of the four SAXS peaks decreases, then they merge into each other forming arc-like reflections in the direction parallel to applied strain, and finally transform into an isotropic ring at temperatures above TI. The process of

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Figure 5.3.1 LCE2: (a) SAXS patterns during strain induced polydomain-monodomain transition (strain increase from left to right). The black arrows show the direction of stretching. (b) SAXS patterns during thermal length recovery (temperature increase from right to left). During length recovery, the SAXS peaks merge into each other forming arc like reflections in the direction orthogonal to applied strain.

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recovering the polydomain structure is somewhat different than during stretching when the arcs are formed in the vertical direction and perpendicular to the stretch direction, and then they split into four spots. This behavior is observed in both LCE1 and LCE2, the extent of the change being more apparent in LCE2, shown in Figure 5.3.1 (a) and (b).

One can explain the changes in smectic layer orientation and, thus the small angle reflections, observed during stretching and thermal annealing as follows. Initially, the as-mounted sample has not been subjected to strain and is expected to be made of microdomains with random direction of local directors and smectic layers. The diffraction pattern consisting of uniform rings of such a virgin sample at  = 1.0 resembles that of an isotropic (or, powder) sample. The stretching of polymer chains causes internal flow requiring smectic layers to slide past each other. The layers will begin to orient and become parallel to the strain. Any further increase in strain renders the polymer chains increasingly parallel to direction of elongation. Since the mesogens form an integral part of the polymer main-chains, the mesogens and the local directors gradually become parallel to the stretch direction and deplete the population of perpendicularly oriented mesogens. Smectic layers are oriented symmetrically with respect to the mesogens’ long axis and form layers in a manner that will resemble chevron-like geometry. As the mesogen distribution becomes increasingly parallel to the stretch direction, the angle α decreases ultimately saturating at ~ 45° in the secondary monodomain state with the apex of the chevron pointing towards the stretch direction.

During the thermal recovery, the distribution of polymer chains and the mesogens become increasingly random and the chevron monodomain begins to relax back to the

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Figure 5.3.2 Plots of global orientational order parameter S during thermally driven shape recovery process of LCE1 and LCE2. The value of smectic-C to isotropic transition temperature for both the elastomers are mentioned in the figure. LCE2 has a lower TI so the temperature range covered for it is narrower as the experimental setup did not provide access to below room temperature. The solid curves in the plot for S serve as guide to the eye.

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polydomain state. Since at high strain, mesogens have become predominantly parallel to the stretch direction, the chevrons will begin to unfold with smectic planes generally perpendicular to the vertical direction. Since the population of domains with smectic layers in the vertical orientation is very small, the diffraction ring develops above and below the equator as seen in Figure 5.3.1 (b) at ΔT = − 2.5 °C. The smectic layers continue to exhibit orientational preference perpendicular to the stretch direction before the sample eventually transitions to the isotropic phase.

Figure 5.3.2 shows the changes in global orientational order parameter S during the thermal shape recovery for the two elastomers. TI is ~ 96.1 °C and 70.6 °C for LCE1 and LCE2, respectively, which coincide with the onset of isotropic phase in these systems

[6]. With rising temperature, S for LCE1 and LCE2 show a smooth transition to the polydomain value (S = 0). The value of S in LCE2 falls to zero more abruptly across the first order [7,8] transition at TI than in LCE1. The gradual change of S in these elastomer systems arises from randomization of liquid crystal microdomains at high temperatures.

The change in S across the transition is relatively sharper in case of LCE2 (Figure 5.3.2) which again points toward a greater tendency for strain retention in LCE2 due to the incorporation of TR3.

5.4 Chapter Summary

The as-formed shape of the SmC polydomain main-chain elastomer systems has been deformed mechanically into a stable secondary shape of chevron-like optically monodomain structure. External mechanical force couples directly to the main chain networks and in turn to the underlying liquid crystalline order. The subsequent effect is a

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continuous rotation of microdomains and the layers relative to the stretch direction accompanied by changes in the global orientational order parameter. Incorporation of

TR3 into the main chain network lowers the value of threshold strain for the formation of the secondary chevron-like monodomain. Presence of TR3 also leads to greater strain- retention. The two elastomers do not quite follow an entirely reverse path while the initial polydomain state is recovered by heating from the mechanically induced chevron state. A relatively smooth behavior of S across the SmC to I phase transition arises from two reasons: (1) reorientation of microdomains at higher temperatures as the elastic strengths weaken, and (2) a variation in transition temperatures induced by embedded defects and entanglements whereby different regions of the sample melt into the isotropic phase at slightly different temperatures thus smoothing out the transition.

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REFERENCES

[1] W. T. Ren, P. J. McMullan, and A. C. Griffin, Phys. Status Solidi B 246, 2124 (2009).

[2] W. Ren, P. J. McMullan, H. Guo, S. Kumar, and A. C. Griffin, Macromol. Chem. Phys. 209, 272 (2008).

[3] M. Tokita, T. Takahashi, M. Hayashi, K. Inomata, and J. Watanabe, Macromolecules 29, 1345 (1996).

[4] W. Ren and A. C. Griffin, Phys. Status Solidi B 249, 1379 (2012).

[5] H. Yoon, D. M. Agra-Kooijman, K. Ayub, R. P. Lemieux, and S. Kumar, Phys. Rev. Lett. 106, 087801 (2011).

[6] W. T. Ren, W. M. Kline, P. J. McMullan, and A. C. Griffin, Phys. Status Solidi B 248, 105 (2011).

[7] P. K. Mukherjee, H. Pleiner, and H. R. Brand, J. Chem. Phys. 117, 7788 (2002).

[8] P. K. Mukherjee, J. Chem. Phys. 136, 144902 (2012).

CHAPTER 6

CONCLUSIONS AND FUTURE WORK

6.1 Conclusions

We have determined the structural changes that occur during the application of strain, annealing, and shape recovery processes of two SmC main-chain elastomers using x-ray diffraction. The stress-strain curves of these materials showed a plateau which is the region of soft-elastic deformations. We have identified the relaxation dynamics in these systems pertaining to the polydomain-monodomain transition in this regime of the stress-strain curve. Initial random director and smectic layer distribution of SmC microdomains inside the elastomer network is found to gradually change into a chevron- like optically monodomain configuration with the application of strain. The strain- induced alignment and re-orientation of liquid crystal microstructure proceeds slowly

(time constant τ ~ 45 minutes) at low strains. The constant τ decreases by at least an order of magnitude (τ ~ 4 minutes) as the polydomain-monodomain transition is completed at higher strains. We attribute the slow relaxation process at low strains to flow properties

[1] of the liquid crystal layers embedded in the elastomer network. At higher strains, the elastic response of the polymer component becomes dominant, leading to a faster relaxation process for the liquid crystal microstructure.

The value of the global orientational order parameter S is initially close to zero due to the misaligned microdomains. Across the polydomain-monodomain transition, the domains attain a preferential smectic layer and director alignment in the stretch direction

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and the value of S rapidly increases from ~ 0.15 to 0.83. An appreciable increase in S for the siloxane parts is noticeable only after the formation of chevron-like monodomain is completed and reaches a maximum value of ~ 0.4 for λ ~ 4.0 for LCE1.

In the final high-strain state, various components of the system, i.e., the polymer chains, the mesogens, and the local microdomain directors, all align parallel to the stretch direction. The layers form oblique to the stretch direction conforming to the structural property of the SmC phase. This chevron-like monodomain structure is enhanced at high strains and both elastomers are found to be “locked-in” this secondary state even after removal of the external stress.

The polydomain-monodomain transition and transformation into chevron-like microstructure is completed at strains λ ~ 1.7 and 1.2, respectively, for LCE1 and LCE2.

The appearance of a lower threshold strain value in LCE2 is attributed to the presence of the transverse component in its main-chain.

Previous work on these systems predicted nanosegregation of the hydrocarbon and siloxane segments [2] as the mechanism responsible for the stability of the secondary state long after removal of the external stress. The observation of the siloxane rings shows that the siloxane and hydrocarbon components of the systems aggregate in different regions. The chevron-like microstructure is found to remain well-formed even after a full day of relaxation. The relaxation of the chevron-structure toward the equilibrium state is found to be faster in LCE2, most likely due to presence of TR3.

Thermal energy, as an external stimulus, gradually disrupts the smectic order and both elastomers recover their initial polydomain state above their respective smectic-C to

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isotropic transition temperatures. A preference for the orientation of the smectic layer normals toward the stretch direction persists in the monodomain state until the initial polydomain state is achieved. The introduction of the TR3 component enhances the strain-retention ability in LCE2.

6.2 Suggestions for future work

In this dissertation, the structural changes during soft-elastic response and shape memory effect of two main-chain smectic-C elastomers have been investigated by x-ray diffraction. The next logical step is to measure the stress and strain simultaneously during the structural studies by x-ray diffraction. Simultaneous study of the relaxation mechanism of the chevron-like microstructure and the relaxation at constant strain will be helpful in establishing a quantitative relationship between the microscopic structural changes and macroscopic elastic response of the network.

The polymer strands between two cross-linking points in the elastomer network is viscoelastic [3] and previous elasticity measurements [4] have indicated that rate at which strain is applied is an important parameter in the resultant macroscopic behavior of these systems. It will be interesting to perform “shear experiments” [5,6] on these systems. The strain-induced secondary monodomain state is stable with a chevron-like microstructure.

It should be possible to rotate these layer normals further by applying external shear perpendicular to the stretch direction. A home-made experimental setup shown in Figure

6.2.1 (a) has been developed to perform these experiments. The idea [6] is to see if a combination of external shear and stress couples in a way to give rise to a truly monodomain state shown in Figure 6.2.1 (b) . Such experiments may lead to a novel way

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Figure 6.2.1 (a) Home-made experimental setup for producing perfect monodomain structure from the strain induced chevron monodomain. The lower end is fixed and the upper end could slide with the help of a computer controlled motorized micrometer. The set-up is assembled on an INSTEC heating oven and thus the experiments could be performed for a wide variety of temperature. (b) Expected x-ray diffraction pattern based on the initial and final arrangements of the microdomains inside the elastomer.

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of generating uniform monodomains structure in initially polydomain main-chain smectic-C elastomers.

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REFERENCES

[1] M. Tokita, T. Takahashi, M. Hayashi, K. Inomata, and J. Watanabe, Macromolecules 29, 1345 (1996).

[2] W. T. Ren, W. M. Kline, P. J. McMullan, and A. C. Griffin, Phys. Status Solidi B 248, 105 (2011).

[3] M. Warner and E. M. Terentjev, Liquid Crystal Elastomers (Oxford University Press Inc., New York, NY, USA, 2007).

[4] W. Ren, Ph.D. Dissertation, Georgia Institute of Technology, USA, 2007.

[5] K. Hiraoka, W. Sagano, T. Nose, and H. Finkelmann, Macromolecules 38, 7352 (2005).

[6] A. Sanchez-Ferrer and H. Finkelmann, Macromolecules 41, 970 (2008).