HYBRID PARTICLE-FINITE ELEMENT ELASTODYNAMICS SIMULATIONSOF NEMATIC LIQUID CRYSTAL ELASTOMERS

A dissertation submitted toKent State University in partial

fulfillment of the requirements for thedegree of Doctor of Philosophy

by

Badel L. Mbanga

May 2012

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Dissertation written by

Badel L. Mbanga

B.S., University of Buca, 200 I

M.A., Dalarna University, 2003

Ph.D .• Kent State University, 2012

Approved by

, Chair, Doctoral Dissertation Committee

, Members, Doctoral Dissertation Committee ----------~~~-------------Dr. Eugene C. Gartland

Dr.~~

Accepted by

~- ~ /\ - (:1.7 ~ ;/ ~--' -!.-;G:....-_(... .... _~~_~-+-__ -+ ____ l-__ ~ _____ • Chair. Department of Chemical Physics Dr~ LianjkhY Ch;6n

'--/ -J

------------------------- • Denn, College of Arts and Science Dr. Timothy Moerland

ii

ii

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Liquid Crystal Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Isotropic-Nematic transition in Nematic LCE . . . . . . . . . . . . . 9

1.2.2 Photoexcitation in NLCE . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Freedericksz transition in NLCE . . . . . . . . . . . . . . . . . . . . 12

1.2.4 Flexoelectric effect in NLCE . . . . . . . . . . . . . . . . . . . . . 14

1.3 Strains, strain energy, rubber elasticity . . . . . . . . . . . . . . . . . . . 15

1.3.1 Strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.3 Rubber elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4 Theory of NLCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.1 De Gennes phenomenological theory . . . . . . . . . . . . . . . . . 21

1.4.2 Neo-classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . 23

iii

2 FINITE ELEMENT ELASTODYNAMICS SIMULATIONS OF LCE . . . . . 28

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.1 Forces calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Simulations of Rubbery materials . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Simulations of LCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.1 Isotropic-Nematic phase transition . . . . . . . . . . . . . . . . . . . 42

2.5.2 Semisoft Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 POLYDOMAIN-MONODOMAIN TRANSITION IN NEMATIC ELASTOMERS 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Initial configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 MODELING THE STRIPE INSTABILITY IN NEMATIC ELASTOMERS . . 62

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Simulations and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Stripes width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.2 Threshold for stripe instability . . . . . . . . . . . . . . . . . . . . . 70

4.2.3 Rate of strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

iv

5 MODELING DEVICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1 Polarization tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Peristaltic pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4 Self-propelled earthworm . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.1.1 Role of the Frank-Oseen elastic energy . . . . . . . . . . . . . . . 83

6.1.2 Volumetric change (swelling) of LCE . . . . . . . . . . . . . . . . 85

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

v

LIST OF FIGURES

1 Some liquid crystal phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 5-Cyanobiphenyl, a commercially available liquid crystal molecule . . . . . . 2

3 Hexaazatriphenylene liquid crystal (hat), a discotic liquid crystal forming

molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 4-cyano-resorcinol, an example of a bent-core liquid crystal molecule . . . . 3

5 Nematic and columnar phases of discotic liquid crystals. . . . . . . . . . . . 4

6 Distortions of nematic liquid crystals . . . . . . . . . . . . . . . . . . . . . . 6

7 Liquid crystal elastomers interaction with external stimuli. Image courtesy

Dr. B. Ratna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8 Main chain and side chain polymers . . . . . . . . . . . . . . . . . . . . . . 8

9 Isotropic nematic transition in nematic elastomers . . . . . . . . . . . . . . . 10

10 Photoexcitation in nematic elastomers. The conformation change of the Azo

dyes drives a shape change of the polymer matrix. . . . . . . . . . . . . . . . 11

11 Freederickzs transition in liquid crystals. . . . . . . . . . . . . . . . . . . . . 13

12 Relative positions of two material points in the reference (dr) and target (dR)

spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

13 Soft elastic deformation. The director rotates to accommodate the strain. . . . 25

14 Soft vs. semisoft response in a nematic elastomer. . . . . . . . . . . . . . . 27

15 (a) and (b) show a tetrahedron in the unstrained and deformed states respectively 30

vi

16 Beam of rubber twisting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

17 Potential (black dots) and kinetic (red dots) energy of the elastic beam expe-

riencing torsional deformation. In the absence of dissipation, the energy is

conserved (green dots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

18 Simulation of the isotropic-nematic transition in a liquid crystal elastomer.

The sample experiences a macroscopic shape change as it is cooled down to

the nematic phase and heated back into the isotropic phase. The sample is

clamped at the top end. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

19 Cartoon of the semisoft response in a NLCE. . . . . . . . . . . . . . . . . . 44

20 A cartoon of a polydomain nematic elastomer shows no correlation between

the orientation of the director in neighboring domains. The blue arrows show

the orientation of the nematic director in the domains. . . . . . . . . . . . . 46

21 Initial configurations with different thermomechanical histories. . . . . . . . 53

22 Simulation studies of a I-PNE. (a) and (b) show the entire strip of nematic

elastomer in the polydomain and monodomain configuration respectively. In

the unstrained state, the sample strongly scatters light, whereas when strained

the sample is sandwiched between crossed polarizers and aligned with either

the polarizer or analyzer, total extinction of the incident light will occur. . . . 54

23 Texture and director configuration in a region near the center of the strip in

the polydomain state of I-PNE. . . . . . . . . . . . . . . . . . . . . . . . . 55

24 Texture and director configuration in a region near the center of the strip in

the monodomain state of I-PNE. . . . . . . . . . . . . . . . . . . . . . . . . 56

vii

25 Texture and director configuration in a region near the center of the strip in

the polydomain state of N-PNE. . . . . . . . . . . . . . . . . . . . . . . . . 57

26 Texture and director configuration in a region near the center of the strip in

the monodomain state of N-PNE. . . . . . . . . . . . . . . . . . . . . . . . 58

27 Engineering stress σ ,(black curve) and Global order parmeter, S (red curve)

vs strain for a nematic elastomer crosslinked in the isotropic phase ( I-PNE) . 59

28 Engineering stress σ ,(black curve) and Global order parmeter, S (red curve)

vs strain for a nematic elastomer crosslinked in the nematic phase ( N-PNE) . 60

29 Experiment: stretching a nematic elastomer film at an angle of 90o to the

director results in a microstructure consisting stripes of alternating director

orientation. Image courtesy H. Finkelmann . . . . . . . . . . . . . . . . . . 63

30 Simulation: stretching a nematic elastomer film at an angle of 90o to the

director. Initially a monodomain, the director field evolves to form a striped

microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

31 Engineering stress (circles) and director rotation (squares) vs applied strain,

for the system shown in Figure 30. Onset of director rotation and the stress-

strain plateau both occur at the same strain. . . . . . . . . . . . . . . . . . . 70

32 Engineering stress (circles) and director rotation (squares) vs applied strain,

applied at an angle of 60o from the nematic director. . . . . . . . . . . . . . 71

33 Simulation: stretching a nematic elastomer film at an angle of 60o to the

director. Initially a monodomain, the director field rotates smoothly without

sharp gradients in orientation. . . . . . . . . . . . . . . . . . . . . . . . . . 72

viii

34 Dependence of the stress-strain response on strain rate. . . . . . . . . . . . . 73

35 Simulation: A nematic elastomer disk is stretched radially. The director field

smoothly transforms from a homogeneous monodomain to a radial configu-

ration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

36 Cartoon of the polarization modulation by the elastomer strip. . . . . . . . . 76

37 Simulation of the photo-deformation of a beam of nematic liquid crystal elas-

tomer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

38 Simulation of the photobending of a strip of polydomain nematic liquid crys-

tal elastomer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

39 Simulation of a soft peristalsis tube made of nematic liquid crystal elastomer. 79

40 Simulation of a nematic elastomer robotic earthworm moving on a rugged

surface shaped like a size wave, with height z = A sin(kx). . . . . . . . . . . 81

ix

Acknowledgements

I thank Professors Jonathan Selinger, Mark Warner, Eugene Terentjev, T.C. Lubensky,

E. C. Gartland, and Antonio Desimone for very fruitful discussions. Dr. Fangfu Ye and

Ms. Vianney Gimenez-Pinto have greatly contributed to the work that is reported in this

dissertation. Finally, I am profoundly grateful to my advisor Professor Robin L. B. Selinger

for her guidance, support and mentoring.

x

To my family

xi

CHAPTER 1

INTRODUCTION

1.1 Liquid Crystals

In this chapter we review fundamental background information from the literature which

serves as a basis for the work that follows. Since their discovery at the end of the nineteen

century [1, 2], liquid crystals have generated a widespread interest from physical scientists

involved in areas such as soft condensed matter, cosmology [3, 4] and biological physics due

to their remarkable properties and potential applications. The liquid crystalline state is a par-

tially ordered state of matter that can be observed in compound materials with anisotropic

molecules or aggregates. This phase can itself be subdivided into several mesophases de-

pending on the type of broken symmetry. A simplistic view of the accepted nomenclature of

the liquid crystalline mesophases is shown in Figure 1.

Besides the anisotropy usually observed in their physical properties (electrical, magnetic,

optical), liquid-crystal-forming molecules share a common characteristic, which is the shape

anisotropy of the molecules. In particular, elongated molecules (rod-like) are known to form

liquid crystalline phases. A common example is 5CB, shown in Figure 2. One can also obtain

a liquid crystal phase from discotic (disk-like) molecules [5] or bent-core (also referred to as

banana) molecules. Figures 3 and 4 show examples of discotic and bent-core liquid crystals

forming molecules, respectively.

The transition between two mesophases can be induced either by a temperature change

1

2

Figure 1: Some liquid crystal phases

Figure 2: 5-Cyanobiphenyl, a commercially available liquid crystal molecule

or by a concentration change if the material is in the presence of a solvent. Materials that

fall in the former category are known as thermotropic liquid crystals, and those in the latter

are labelled lyotropic liquid crystals. There exist materials that respond to both temperature

and concentration variations. Those are known as amphotropic liquid crystals. In the works

reported here, any reference to a liquid crystal will implicitly assume a thermotropic material

unless otherwise stated.

The smectic phase is characterized by a spontaneously broken translational symmetry

along one direction as shown in Figure 1. It is best thought of as being a fluid in two dimen-

sions and a solid in the third. X-ray scattering experiments suggest a layered-like structure in

smectic materials. There exist several mesophases of smectic materials. The most studied are

3

Figure 3: Hexaazatriphenylene liquid crystal (hat), a discotic liquid crystal forming molecule.

Figure 4: 4-cyano-resorcinol, an example of a bent-core liquid crystal molecule

the smectic A and the smectic C phase. They differ in the molecular orientation relative to

the layer normal. In the smectic A phase, the molecules are on average parallel to the normal

to the layers, whereas in the smectic C phase, they are tilted with respect to the layer normal,

as illustrated by the cartoon in Figure 1.

The nematic phase is the most well studied, as it is used in most applications. It is char-

acterized by its long range orientational order and absence of translational order. For rod-like

nematogens, this means that the long axes of the molecules tend to align parallel to one an-

other. A snapshot of the centers of masses of the molecules will show no correlation, just

as in the isotropic phase. On the other hand, discotic liquid crystals will have the axes of

the disk-like molecules more or less aligned with one another in the nematic phase as in Fig-

ure 5(a). The direction of average orientation of the molecules is called the director n(r). n

4

is invariant under inversion symmetry, meaning that it is a headless vector, or defined such

that n ≡ −n . Under polarizing optical microscopy, a sample of liquid crystal in the nematic

phase exhibits a birefringence conferred to it by the anisotropy of its molecules.

(a) Nematic phase of a dis-cotic liquid crystal

(b) Columnar phaseof a discotic liquidcrystal

Figure 5: Nematic and columnar phases of discotic liquid crystals.

The transition between the isotropic and the nematic phases can be described with the

aid of an order parameter. This quantity is carefully chosen to be nonzero in the nematic

phase and zero in the isotropic phase. Moreover, it obeys the symmetry of both phases. In

the particular case of a uniaxial nematic phase, the nematic order parameter tensor Qij =

S(ninj −1

3δij) is a suitable quantity for characterizing the transition. n is the local nematic

director. S =1

2< 3vini − 1 > is the scalar order parameter order parameter. Here v is

along the long molecular axis, and one notes that S is just the average of the second Legendre

polynomial, S =

⟨1

2(3 cos2 θ − 1)

⟩, where θ = n · v. It is immediate from what precedes

that S = 0 when the nematogens are randomly oriented in space as is the case in the isotropic

5

phase, whereas S = 1 when they are perfectly aligned with one another. The Landau-de

Gennes theory for the Isotropic to nematic phase transition is built around a power series

expansion in Qij

E =1

2a(1− Tc

T)QijQij +

1

3bQijQjkQki +

1

4c[QijQij]

2 (1)

While global transformations, that is, translations and uniform rotations of the material

should cost no elastic energy, gradients in the orientation of the nematic director are penalized.

Figure 6 shows an illustration of the three types of deformations of the director field that are

penalized, namely the splay, twist, and bend deformations. Those spatial variations in the

orientation of the nematic director are penalized by the Frank-Oseen free energy:

F =1

2K11(∇ · n)2 + 1

2K22(n · ∇ × n)2 +

1

2K33(n×∇× n)2 (2)

Here K11, K22, and K33 are the elastic constants corresponding to the splay, twist and

bend deformations respectively. Note that additional terms associated with boundaries are

ignored in the equation above.

What precedes was just a brief survey of certain liquid crystals properties that are pertinent

to this dissertation. References [5–9] provide excellent reviews about the physics of liquid

crystals.

6

Figure 6: Distortions of nematic liquid crystals

1.2 Liquid Crystal Elastomers

Here we summarize findings from several research groups that have helped contribute to

a better understanding of the behavior and properties of liquid crystal elastomers. Emphasis

is put mainly on the information that is most relevant to the substance of the work reported in

this dissertation.

Liquid Crystal Elastomers (LCE) are materials that exhibit some of the elastic properties

of rubber along with the orientational order properties of liquid crystals. They are composed

of liquid crystal mesogens covalently bonded to a weakly cross-linked polymer backbone

[10,11]. Similar to low molecular weight liquid crystals, LCE respond to external stimuli such

as a temperature change, applied electric or magnetic fields, or mechanical stress [10, 12].

7

These materials display strong coupling between orientational order of the mesogens and

mechanical deformation of the polymer network. For instance in a nematic LCE, any change

in the magnitude of the nematic order parameter can induce shape change, e.g. the isotropic

nematic phase transition induces strains of up to several hundred percent in a strip of nematic

elastomer [12]. Thus LCE have been proposed for use as artificial muscles or soft actuators.

The cartoon in Figure 7 illustrates the stimulus-response of liquid crystal elastomers. Con-

versely, applied strain can also drive changes in orientational order, producing the fascinating

phenomenon of semisoft elasticity [13].

Figure 7: Liquid crystal elastomers interaction with external stimuli. Image courtesy Dr. B.Ratna

8

LCE can exist in the main chain or side chain configurations. The former has the meso-

gens linked together within the polymer backbone as in Figure 8(a), whereas in the latter,

the liquid crystal molecules are pendant and attached to the polymer backbone by flexible

spacers as shown in Figure 8(b). Note that discotic liquid crystal elastomers have also been

reported [14], with properties similar to LCE made with rod-like nematogens.

(a) main chain

(b) side chain

Figure 8: Main chain and side chain polymers

The first few attempts to blend polymers and liquid crystals date as far back as the

1960’s [10]. The effort then was mostly directed towards obtaining polymer networks with

a certain amount of frozen-in long range order. The approach was thus to crosslink poly-

mer networks in the presence of a liquid crystalline solvent in the nematic phase. Several

groups successfully obtained such materials, but it was only in 1981 that Finkelmann re-

ported the first liquid crystal elastomer formed with nematogens attached to the backbone

9

with crosslinkers [15]. Such materials had been predicted theoretically by De Gennes less

than a decade earlier in a seminal paper [16]. The Finkelmann experiment reported liquid

crystal elastomers in the smectic, cholesteric, and nematic phases. These were side-chain

LCE with a polysiloxane polymer backbone. Other approaches have been used for making

the polymer backbones; in particular, acrylate polymer backbones have been reported [?].

Polysiloxane however remains the backbone of choice due to its high anisometry, which has

been observed to display the most dramatic shape change in experiments [17] .

1.2.1 Isotropic-Nematic transition in Nematic LCE

Nematic elastomer materials constitute the main focus of this dissertation. A spontaneous

shape change as depicted in the cartoon of Figure 9 is usually observed in nematic elastomers

undergoing the isotropic to nematic phase transition. The mechanism is a simple one whereby

the polymer network is distorted and tends to depart from the average spherical shape, elon-

gating along the direction of the spontaneous director that arises upon cooling. In other words,

the radius of gyration tensor of the polymer chains acquire the anisotropy of the liquid crystal.

Conventional liquid crystals undergo a nematic to isotropic phase transition characterized by

a sharp discontinuity in the nematic order parameter and in the materials properties. For ex-

ample, a plot of the birefringence as a function of temperature will show a sharp change at the

transition. In the Ehrenfest classification of phase transitions, this is a first order transition.

Surprisingly, nematic elastomers show a totally different behavior near the transition. Exper-

iments have reported a transition that is neither first order nor second order, albeit smooth. It

is more like a smooth crossover between the two phases. Selinger et al. [18] have shown that

there are several mechanisms that could account for that peculiar behavior. First, they showed

10

that the underlying heterogeinity of elastomers, that is, the quenched disorder of which the

crosslink points are the sources, results in regions of different isotropic-nematic transition

temperatures. This means that the smooth transition observed is due to the fact that domains

are continuously undergoing the transition, much like an avalanche. The strain also varies like

other material properties during the transition. The other possible cause for the broadening

of the transition proposed by Selinger et al. is the non-uniform distribution of local stresses

as a result of crosslinking. We make a similar observation in our studies of the polydomain to

monodomain transition in nematic elastomers in Chapter 3. De Gennes proposed that mon-

odomain samples produced by means of the Finkelmann’s two-step crosslinking method are

paranematic when heated up above the isotropic nematic transition. That is, they have a resid-

ual anisotropy acquired during the second stage of crosslinking, and hence no real isotropic

nematic transition could be expected.

Figure 9: Isotropic nematic transition in nematic elastomers

11

1.2.2 Photoexcitation in NLCE

Experiments on LCE have demonstrated that these materials show a coupling between

optical and mechanical energy [19, 20] . A beam of light incident on a sample of liquid crys-

tal elastomer was reported to induce mechanical deformations comparable to those attained

at the N-I transition. This is a direct modulation of the degree of nematic order in the material

by light, followed by a response of the polymer matrix due to the coupling between order

and strain in LCE. Warner and Terentjev [10] have coined this phenomenon “stress-optical

coupling.” In order to observe stress-optical coupling, there must be another component be-

sides the liquid crystal molecules and the polymer network that responds to illumination,

as depicted in Figure 10. This is usually achieved by embedding a low concentration of

light-sensitive molecules (Azo dyes) in the blend [21, 22].

Figure 10: Photoexcitation in nematic elastomers. The conformation change of the Azo dyesdrives a shape change of the polymer matrix.

These molecules can exist in either the Trans (elongated) or the Cis (kinked) configuration

and undergo a conformational change from Trans to Cis when illuminated at the appropriate

12

wavelength. The Cis configuration is usually a metastable state and hence the molecules tend

to relax from Cis to Trans when the stimulus is removed.

1.2.3 Freedericksz transition in NLCE

A sample of nematic liquid crystal confined between two parallel substrates can be aligned

by an external electric or magnetic field. In a typical experiment with an electric field, the

director at the surface of the substrates is anchored and has a preferred orientation dictated

by an alignment layer. An electric field is applied perpendicular to the substrates, and the

director tends to align parallel (resp. perpendicular) to the field if the liquid crystal molecules

have a positive (resp. negative) dielectric anisotropy. Figure 11 shows a typical aligned

sample before and after application of a field.

In the one-elastic-constant approximation, one can rewrite the Frank-Oseen elastic energy

density as:

Ffrank =1

2K(∇n)2

≈ 1

2K(

π

d)2 (3)

Here K is the elastic constant and n = n(r) is the nematic director; d is the cell gap. The

total elastic energy in the cell is thus Ffrank =1

2K(

1

d)2 .

The contribution of the electrostatic energy can be expressed in the form:

Felectric = −1

2ϵo∆ϵE2d (4)

where ∆ϵ is the dielectric anisotropy, E is the applied electric field, and ϵo is the dielectric

13

permittivity of free space.

The electric energy will overcome the elastic energy when the field reaches the critical

value:

Ec =π

d

√K

ϵo∆ϵ(5)

This corresponds to a voltage Vc = π

√K

ϵo∆ϵwhich is independent of the sample thickness,

but just depends on the material parameters. The onset of re-orientation of the molecules

occurs at a critical voltage. This phenomenon is known as the Freedericksz transition and

is used in liquid crystal display applications to modulate light as it passes through the liquid

crystal cell. Similar experiments carried out on nematic elastomers have failed to show such

(a) With the eld OFF (b) With the eld ON

E

Figure 11: Freederickzs transition in liquid crystals.

a dramatic characteristic behavior unless the applied field was very large.

Using a similar argument to the aforementioned for low molecular weight liquid crystals,

and taking into account that the elastic energy scale is of the order of the shear modulus

µ, and that the contribution of the Frank-Oseen elastic energy is negligibly small, one can

approximate the electric field required to switch the orientation of the nematogens as:

Ec =

õ

ϵo∆ϵ(6)

14

From this it follows that nematic elastomers respond to a critical field and not to a critical

voltage [23–26] . This is intuitive when one considers that contrary to low molecular weight

liquid crystal where the anchoring occurs at the surfaces, in nematic elastomers, the director

is anchored throughout the bulk of the sample, hence offering more resistance to macroscopic

deformation. Note however that even small director reorientations have been shown to drive

macroscopic shape changes in unconfined samples [27].

1.2.4 Flexoelectric effect in NLCE

The flexoelectric effect is the induction of a spontaneous polarization as a response to a

mechanical deformation. Although very much pronounced in polar dielectric materials, it

does not require the material’s molecules to have a nonzero permanent dipole moment. In-

deed, the flexoelectric effect can be observed in liquid crystals of apolar molecules subjected

to splay or bend deformations [28]. The induced polarization is of the form [28]:

Pf = e1n(∇ · n) + e3n(∇× n) (7)

A direct method of measuring the flexoelectric coefficients was recently introduced by

Harden et al. [29], and allowed them to measure the flexoelectric response of bent-core ne-

matic liquid crystals. Bent-core nematic liquid crystals are special because they have a strong

permanent dipole moment conferred to them by their shape anisotropy. They found bend

flexoelectric coefficients (e3 ≈ 50nC/m) to be three orders of magnitude larger than those

obtained from estimation [30, 31] and measurements with calamitic liquid crystals such as

15

5CB [32, 33]. Simulations by Dhakal and Selinger [34] have found similar results. Subse-

quent experiments have also approached the converse flexoelectric effect, which is charac-

terized by a shape change under the application of a current. Note that the changes in shape

observed here are orders of magnitude larger than in piezoelectric devices. This motivated

experiments on bent-core nematic elastomers by several groups. Harden et al. applied a peri-

odic mechanical deformation to a thin film of bent-core nematic liquid crystal using a small

speaker-driven motor. The induced electric current they obtained with small to moderate de-

formations was in the nanoampere range. Chambers et al. [35] also obtained similar results

using a calamitic nematic liquid crystal elastomer film swollen in a solution of bent-core ne-

matic liquid crystal. Observe that both experiments yielded bend flexoelectric coefficients

comparable to that of the low molecular weight bent core nematic. These experiments rein-

force the view that nematic liquid crystal elastomers are excellent candidates for engineering

device applications that strive to convert electrical to mechanical energy, and vice-versa.

1.3 Strains, strain energy, rubber elasticity

Before delving into the details of the method we will introduce for modelling nematic

elastomers, it is useful to quickly review some concepts that are used throughout this disser-

tation.

1.3.1 Strain tensor

Elasticity is a material property observed when the forces causing deformation remain

below a certain threshold, thus allowing the material to return to its (relaxed) state before

deformation upon removal of the forces. We will refer to the space in which the material exists

16

prior to deformation as the reference space, and that in which it is found when deformed as the

target space. Consider a material point initially at a position r in the reference space as shown

in Figure 12. After a deformation of the material, the position of the material point in the

target space can be found by a mapping R(r). In order to express how neighbouring material

points are displaced with respect to one another, we define a quantity λij =∂Ri

∂rjreferred to

as the deformation gradient tensor. It can be shown that the difference in the square of the

Euclidean distances between two such neighbouring points due to the deformation is

dR2 − dr2 = (λikdrk)(λildrl)− (δikdrk)(δildrl)

= (λikλil − δikδil)drkdrl

= (λikλil − δkl)drkdrl

One can express the position of a material point in the target space as the sum of its position

in the reference space and a displacement field: R(r) = r + u(r). Note that a uniform

displacement field simply corresponds to moving the whole body, and should cost no elastic

energy. In terms of the displacement field, one can express the deformation gradient tensor

as:

λij =∂Ri

∂rj

=∂ri∂rj

+∂ui

∂rj

= δij +∂ui

∂rj

17

The aforementioned square of the change in separation between neighbouring material points

can thus be expressed as

dR2 − dr2 = (λikλil − δkl)drkdrl

=

[(δik +

∂ui

∂rk)(δil +

∂ui

∂rl)− δkl

]drkdrl

= (∂ul

∂rk+

∂uk

∂rl+

∂ui

∂rk

∂ui

∂rl)drkdrl

= 2εkldrkdrl

The quantity εkl =1

2(λikλil − δkl) is the Green-Lagrange strain tensor which is invariant

under rotations in the target frame.

Figure 12: Relative positions of two material points in the reference (dr) and target (dR)spaces.

1.3.2 Strain energy

The energy cost of deforming an elastic material can be described in several forms, all

using strain tensors and elastic moduli as main ingredients. Perhaps the most familiar form

of strain energy is that proposed by Hooke, which states that the strain energy is simply a

18

quadratic function of the strain tensor.

U =1

2Cijklεijεkl

Here εij is the Green-Lagrange strain tensor, Cijkl represents the elastic constants asso-

ciated with the material. The stiffness tensor Cijkl is a function of the shear modulus and

Poisson ratio of the material.

We note that since the Green-Lagrange strain εij is defined in the material or body frame,

we must also define Cijkl in the body frame. Thus if the sample undergoes a rotation, these

quantities conveniently rotate with the body rather than being defined in the lab frame.

Other forms of strain energies have been proposed to describe deformations of rubbery

materials, in particular the Neo-Hookean strain and the Mooney-Rivlin strain energies. These

are mainly constructed by creating a series expansion in terms of the invariants of the strain

tensor. Polar decomposition allows one to write the deformation gradient tensor as the product

of a positive-semidefinite stretching (F ) and a unitary rotation (R) tensors such that λ =

RθF . The stretching tensor is F =√λTλ, and the rotation is Rθ = λF−1. A deformation

such that F = δij is a mere rotation of the body and costs no elastic energy. In its eigenframe,

the stretching tensor is expressed as

F =√λTλ =

λ1 0 0

0 λ2 0

0 0 λ3

(8)

19

such that

C = F 2 = λTλ =

λ21 0 0

0 λ22 0

0 0 λ23

; ε =

λ21 − 1 0 0

0 λ22 − 1 0

0 0 λ23 − 1

(9)

are defined in the same eigenframe. Here C is known as the first Cauchy-Green tensor; in

terms of the components of the diagonalized stretching tensor (λ1, λ2, λ3), its invariants are :

I1(C) = λ21 + λ2

2 + λ23

I2(C) = λ21λ

22 + λ2

1λ23 + λ2

2λ23

J(C) = λ1λ2λ3

The neo-Hookean strain energy density for an incompressible material is expressed as :

Uneo−H =µ

2(I1 − 3) + Co(J − 1)2

Note that for an incompressible material, the third invariant (J) is unity, thus making the

second term in the expression above vanish. However, rubbery materials do not really deform

at constant volume, and this is taken into account by the Mooney-Rivlin strain energy density,

which allows for deviations from the volume before deformation.

UMR =1

2Cm1(I1 − 3) +

1

2Cm2(I2 − 3)

where I1 = I1J−2/3 and I2 = I2J

−4/3. The coefficients Cm1 and Cm2 are related to the shear

and bulk moduli of the materials.

20

1.3.3 Rubber elasticity

The statistical theory of rubber elasticity, in the simplest approximation, uses a random

walk model to describe a polymer chain. Given an articulated polymer chain with N freely

jointed segments of length b, the contour length of the chain is Lc = N b. We consider

a configuration in which the end-to-end distance of the chain is R. The number of such

configurations is Z(R) = P (R)Z, where Z =∑

configurations exp(−H/∥BT ) is the partition

function. Here H is just a constant, as energy plays no role in this model. P (R) is the

probability for the chain to have the end-to-end distance R and is expressed as

P (R) = (3

2πb2N2)3/2 exp[−3R2/2bLc]

The entropic free energy of the polymer chain in this configuration is thus

F = −kBT ln(Z(R))

= Fo + kBT (3R2/2bLc)

= Fo + kBT (3R2/2R2

o)

For Nx chains in a volume element, one obtains

F = Fo + µ(3R2/2R2o) (10)

where µ = NxkBT is the shear modulus.

21

1.4 Theory of NLCE

1.4.1 De Gennes phenomenological theory

The first theoretical description of nematic elastomers was proposed by De Gennes, and is

antecedent to the first successful synthesis of a NLCE. Using a completely phenomenological

approach, he postulated that coupling rubber elasticity with nematic order would produce a

new class of soft materials with unprecedented properties and potential for engineering ap-

plications. De Gennes theory suggests that the energy cost of deforming a nematic elastomer

must be due to the relative rotations between the director field and the polymer network.

Considering a rotation of the body about an axis ω, it can be represented as: ωij =

1

2(∇iuj −∇jui). Here ui are the components of the displacement vector u. This rotation can

further be decoupled into a rotation around the director (ω∥ =1

2niϵijkωjk), and two rotations

around axes perpendicular to it (ω⊥ = njωij), where n is the nematic director. Naturally,

only those rotations about axes perpendicular to the director are expected to cost energy. Let

us define an infinitesimal change in the director due to the rotation as δn = Ω × n , such

that n · δn = 0. The relative rotation btween the body and the nematic director can then be

expressed as: δn− ω⊥ = (Ω− ω)× n.

The following three contributions to the energy are allowed by symmetry.

E1 =1

2D1 [(Ω− ω)× n]2

=1

2D1ΩiΩi

22

E2 =1

2D2 n · ε · (Ω− ω)× n

=1

2D2 Ωiεjknjδ

⊥ik

where δ⊥ik = δik − nink

E3 =1

2Cijklεijεkl

As can be seen, the first expression (E1) is the cost for relative rotations between the

director and the polymer matrix. The second expression (E2) couples the strain tensor to the

relative rotations of the polymer network and the director field. Finally, E3 is simply a strain

energy.

A free energy density that completely describes the elasticity of a nematic elastomer can

thus be written as:

F =1

2K11(∇ · n)2 + 1

2K22(n · ∇ × n)2 +

1

2K33(n ×∇× n)2

+1

2D1ΩiΩi +

1

2D2Ωiεjknjδ

⊥ik +

1

2Cijklεijεkl

De Gennes theory, although purely phenomenological, is able to adequately describe the

behavior of nematic elastomers at a macroscopic level. Experiments [36] have successfully

established a relationship between the coefficients D1 and D2 and microscopic and macro-

scopic material constants for side chain nematic LCE. One minor pitfall of this theory is its

inapplicability to main chain materials, as it is prohibitively difficult to decouple the rotations

of the nematogens from that of the network in main chain LCE.

23

1.4.2 Neo-classical theory

Although a macroscopic theory of nematic liquid crystal elastomers seems satisfactory

enough, a better, more comprehensive picture of the intrinsic mechanical properties and in-

teractions of NLCE with external stimuli requires a more strict, preferably microscopic ap-

proach. The most successful model for describing NLCE is the neo-classical theory intro-

duced by Warner and Terentjev [10] . It is an extension of rubber elasticity to anisotropic

polymer chains. In this description, one of the starting considerations is that some anisotropy

is conferred to the polymer chains due to the presence of the nematogens. That is, the shape

of gyration of the polymer chain deviates from a sphere to an ellipsoid. The relevant quantity

used to describe this anisotropy is r =l∥l⊥

, where l∥ is the length in the direction parallel

to the nematic director and l⊥ is the length in the transverse direction. An important quan-

tity is the step length tensor lij , which is usually isotropic in ordinary polymers. The chain

anisotropy r is usually more pronounced in main chain NLCE than in side chain materials.

lij =

l⊥ 0 0

0 l⊥ 0

0 0 l∥

(11)

Similar to low molecular weight liquid crystals, where order on a microscopic scale can

be related to macroscopic measurable quantities such as the dielectric anisotropy, the values

of l∥, l⊥ and hence r can be obtained by making measurements on a larger scale. In particular,

cooling a sample from the isotropic to the nematic phase leads to a spontaneous elongation

along the direction of the nematic director, and contraction perpendicular to it. The ratio of

this resulting extension and contraction provides a good measure for r. It can be shown that

24

the step length tensor depends on the nematic director as:

lij = l⊥δij + (l∥ − l⊥)ninj (12)

The sample in the reference state exists in a monodomain conformation with a director no

and step length tensor loij . An arbitrarily chosen polymer chain has a fully extended contour

length Lc and end-to-end distance Ro. The mean square displacement is⟨Ro

iRoj =

1

3Lcl

oij

⟩.

In light of the the discussion of rubber elasticity from the preceding section, one can write

the end-to-end distance as a Gaussian distribution.

P (Ro) = Det[loij]exp[

−3

2Lc

Roi (l

oij)

−1Roj ] (13)

If the sample undergoes a deformation from its reference state to the current state, the

theory assumes that the deformation is affine, i.e. the separation between two crosslinks

is proportional to the shape change of the whole sample. In other words, one can write

the new end-to-end distance as Ri = λij(oRj) where λij is the macroscopic deformation

gradient tensor. The current step length tensor also differs from the reference lij = loij and the

probability distribution of the end-to-end length of the chain is

P (R) = Det [lij] exp(−3

2Lc

Ril−1ij Rj) (14)

Using this, one easily obtains the elastic energy as :

F =1

2NxkBT Tr[loij λ

Tij l

−1ij λij] (15)

Recalling that the step length tensor is a function of the nematic director, one easily sees

that strains are coupled to the director configuration. In the small strain regime, the equation

25

above, with the substitution λij = δij + εij , yields the coefficients of de Gennes’ relative

rotations between nematogens and the polymer network.

D1 = NxkBT(l∥ − l⊥)

2

l∥l⊥

D2 = NxkBTl2∥ − l2⊥

l∥l⊥(16)

1.4.2.1 Soft and semisoft elasticity

Figure 13: Soft elastic deformation. The director rotates to accommodate the strain.

The trace formula above has been successfully used in various studies of nematic elas-

tomers, yielding results in remarkable agreement with experiments. Of particular interest to

the studies reported in this dissertation is the prediction from this theory that certain modes

of deformation can be achieved at almost no energy cost. These energy-free deformations are

those in which a rotation of the system’s internal degree of freedom, i.e. the nematic direc-

tor, occurs in order to balance the cost of the elastic distortion. As its roots, soft elasticity

assumes an isotropic reference state which, when cooled into the nematic phase, results in

the elongation of the sample in the direction of the spontaneously formed director. As there

are infinitely many equiprobable possibilities for the orientation of the nematic director at the

26

transition, one can expect that finding a path that maps two or more such states is equivalent

to achieving a deformation at no energy cost.

It is easily proved that such modes of deformation exist, at least mathematically [37, 38].

As an illustration, choosing a deformation of the form

λij = l1/2ij Rθ (l

oij)

1/2,

where Rθ is an arbitrary rotation by an angle θ, one obtains a zero energy cost from the

configuration with step length tensor loij to that with step length tensor lij .

Although this is easily thought of for a single polymer strand, it is clearly an idealiza-

tion, as the polymer chains differ slightly in their composition; that is, a deformation may be

soft for one strand but not for the other. In other words, the onset of director rotation might

occur at different thresholds for different polymer strands, hence the term semisoft elastic-

ity [39–41]. A modified model exists [42] that accounts for the compositional fluctuations in

the sample. This phenomenon is experimentally observed when a thin sheet of monodomain

nematic elastomer is subjected to a uniaxial stretch. The stress-strain curve of such an elastic

deformation shows a very peculiar behavior that depends on the relative orientation of the

nematic director with respect to that of the imposed strain. For strains applied parallel to the

director, the stress-strain curve is similar to that of a pure elastic material, whereas strains

applied perpendicular to the director display two elastic regimes separated by a plateau. The

plateau corresponds to the soft region, i.e. that in which the nematic director rotates to ac-

commodate the strains.

27

Figure 14: Soft vs. semisoft response in a nematic elastomer.

CHAPTER 2

FINITE ELEMENT ELASTODYNAMICS SIMULATIONS OF LCE

2.1 Introduction

The dynamics of shape evolution in nematic liquid crystal elastomers is modeled using a

three dimensional finite element elastodynamics approach. This model predicts the macro-

scopic mechanical response to an external stimulus such as a change in nematic order, e.g.

by heating or cooling through the isotropic-nematic transition or, in azo-doped materials, by

exposure to light. The mechanics of nematic LCEs are thus controlled by intrinsic coupling

between nematic order and mechanical strain. Theoretical models of this coupling can be

solved analytically, e.g. if the goal is to predict the mechanical response of a representative

volume element, as described at length in [10] and references therein. The goal of the present

work, however, is more ambitious: we wish to model entire devices containing LCE actuators

and simulate their behavior on laboratory length and time scales, including dynamics as well

as static response, in three dimensions. Geometries and boundary conditions of interest are

not simple enough to allow for analytical solutions, so we turn to finite element simulation

methods to simulate the elastodynamics. Finite element methods have been used previously

to model 2-d statics of LCEs but have not, to our knowledge, previously been used to model

dynamics in 3-d [43].

Instead of using a preconfigured software package, we have developed our own finite

element simulation code based on a Hamiltonian approach. Our algorithm is based on a

28

29

marvelously simple approach to finite element elastodynamics proposed by Broughton et

al. [44, 45]. However Broughtons algorithm relies on the approximation that both strain and

rotation are small, and their finite element Hamiltonian is not invariant under rotation. As a

result, dynamics calculated from such a Hamiltonian conserve energy poorly, particularly in

the case of finite rotations. As a solution to this difficulty, we replace the linear strain tensor in

Broughtons approach with the Green-Lagrange strain tensor, a measure of deformation that is

invariant under sample rotation, containing both linear and nonlinear terms. This substitution

renders our Hamiltonian rotationally invariant, and thus our algorithm is not limited to the

small-rotation limit. The resulting dynamics shows remarkable numerical stability, and total

energy and momentum are both conserved to high precision.

In the spirit of the finite element method, we model a sample of nematic elastomer by

discretizing its volume into a mesh of tetrahedral elements. Consider a 3-d tetrahedral element

composed of an elastic material. Any arbitrary deformation u, v, or w of the element can be

described via a standard set of linear mapping functions that find the displacement of a given

interior point of an element by interpolating the displacements of its vertices. Note that using

such an affine mapping results in a uniform strain inside the element is uniform. The choice

of tetrahedral elements in 3-dimensions is not arbitrary, as simplexes or elements with n + 1

vertices in n dimension make this linear mapping very convenient.

30

u(x, y, z) = a1 + a2x+ a3y + a4z

v(x, y, z) = b1 + b2x+ b3y + b4z

w(x, y, z) = c1 + c2x+ c3y + c4z

In the unstrained states (Figure 15(a)), the vertices of the tetrahedron are at coordinates

(x1, y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4). Any displacement of the four vertices

can be written as a function of mapping coefficients as shown below:

(a) Unstrained (b) Deformed

Figure 15: (a) and (b) show a tetrahedron in the unstrained and deformed states respectively

31

u1

u2

u3

u4

=

1 x1 y1 z1

1 x2 y2 z2

1 x3 y3 z3

1 x4 y4 z4

a1

a2

a3

a4

v1

v2

v3

v4

=

1 x1 y1 z1

1 x2 y2 z2

1 x3 y3 z3

1 x4 y4 z4

b1

b2

b3

b4

w1

w2

w3

w4

=

1 x1 y1 z1

1 x2 y2 z2

1 x3 y3 z3

1 x4 y4 z4

c1

c2

c3

c4

Here u, v, and w are displacements in the x, y, and z directions respectively. The coeffi-

cients of the mapping functions ai, bi, and ci are known as the shape factors and are calculated

every time step. They are calculated by simply inverting the matrix above. The inverse matrix

however is obtained once at the start of the simulation, that is, it needs not be updated as the

reference state remains the same throughout the simulation. When the tetrahedral element

experiences a deformation, with displacement fields u, v, and w, we can calculate the defor-

mation gradient tensor, strain tensor, the total elastic energy, and the forces on its vertices as

follows:

32

λij =

1 + a2 a3 a4

b2 1 + b3 b4

c2 c3 1 + c4

ε11 =∂u

∂x+

1

2

[(∂u

∂x

)2

+

(∂v

∂x

)2

+

(∂w

∂x

)2]

= a2 +1

2(a22 + b22 + c22)

ε22 =∂v

∂y+

1

2

[(∂u

∂y

)2

+

(∂v

∂y

)2

+

(∂w

∂y

)2]

= b3 +1

2(a23 + b23 + c23)

ε33 =∂w

∂z+

1

2

[(∂u

∂z

)2

+

(∂v

∂z

)2

+

(∂w

∂z

)2]

= c4 +1

2(a24 + b24 + c24)

The off-diagonal components of the strain tensor are found as well.

ε12 =1

2

(∂u

∂y+

∂v

∂x

)+

1

2

(∂u

∂x

∂u

∂y+

∂v

∂x

∂v

∂y+

∂w

∂x

∂w

∂y

)=

1

2(a3 + b2) +

1

2(a2a3 + b2b3 + c2c3)

33

ε13 =1

2

(∂u

∂z+

∂w

∂x

)+

1

2

(∂u

∂x

∂u

∂z+

∂v

∂x

∂v

∂z+

∂w

∂x

∂w

∂z

)=

1

2(a4 + c2) +

1

2(a2a4 + b2b4 + c2c4)

ε23 =1

2

(∂v

∂z+

∂w

∂y

)+

1

2

(∂u

∂y

∂u

∂z+

∂v

∂y

∂v

∂z+

∂w

∂y

∂w

∂z

)=

1

2(b4 + c3) +

1

2(a3a4 + b3b4 + c3c4)

Rewriting the strain energy U =1

2Cijklεijεkl in terms of the strain tensor, one obtains:

U =1

2Cxxxx(ε

211+ε222+ε233)+2Cxxyy(ε11ε22+ε22ε33+ε33ε11)+4Cxyxy(ε

212+ε223+ε231) (17)

The elastic constants Cijkl are obtained as

Cxxxx = λ+ 2µ

Cxxyy = λ

Cxyxy = µ

where λ and µ are the are the bulk and shear moduli respectively. These depend on the

Young’s modulus(E) and Poisson’s ratio(ν) of the material as shown below.

λ =E ν

(1− 2ν)(1 + ν)

µ =E ν

2(1 + ν)

34

2.1.1 Forces calculations

The force on the ith node is then obtained as derivatives of U with respect to the node

displacement.

Fx,i = −Cxxxxε11M2i(1 + a2)− Cxxxxε22M3ia3 − Cxxxxε33M4ia4

−Cxxyyε11M3ia3 − Cxxyyε22M2i(1 + a2)− Cxxyyε22M4ia4

−Cxxyyε33M3ia3 − Cxxyyε33M2i(1 + a2)− Cxxyyε11M4ia4

−2Cxyxyε12M3i(1 + a2)− 2Cxyxyε12M2ia3 − 2Cxyxyε23M4ia3

−2Cxyxyε23M3ia4 − 2Cxyxyε31M4i(1 + a2)− 2Cxyxyε31M2ia4

Fy,i = −Cxxxxε11M2i(1 + b2)− Cxxxxε22M3ia3 − Cxxxxε33M4ia4

−Cxxyyε11M3ia3 − Cxxyyε22M2i(1 + a2)− Cxxyyε22M4ia4

−Cxxyyε33M3ia3 − Cxxyyε33M2i(1 + a2)− Cxxyyε11M4ia4

−2Cxyxyε12M3i(1 + a2)− 2Cxyxyε12M2ia3 − 2Cxyxyε23M4ia3

−2Cxyxyε23M3ia4 − 2Cxyxyε31M4i(1 + a2)− 2Cxyxyε31M2ia4

Fz,i = −Cxxxxε11M2i(1 + a2)− Cxxxxε22M3ia3 − Cxxxxε33M4ia4

−Cxxyyε11M3ia3 − Cxxyyε22M2i(1 + a2)− Cxxyyε22M4ia4

−Cxxyyε33M3ia3 − Cxxyyε33M2i(1 + a2)− Cxxyyε11M4ia4

−2Cxyxyε12M3i(1 + a2)− 2Cxyxyε12M2ia3 − 2Cxyxyε23M4ia3

−2Cxyxyε23M3ia4 − 2Cxyxyε31M4i(1 + a2)− 2Cxyxyε31M2ia4

(18)

35

Here the quantities M2i, M3i, and M4i are the components of the inverse of the shape matrix

defined as

M =

1 x1 y1 z1

1 x2 y2 z2

1 x3 y3 z3

1 x4 y4 z4

−1

.

It is easy to see from the above that for a single tetrahedral element with no external forces

or constraints, all these node forces sum to zero component by component, and the total

torque also must sum to zero. This is a consequence of the conservation of linear and angular

momentum. For a mesh of connected elements, again with no external forces or constraints,

each node receives force contributions from each element of which it is a member. The sum

of forces on any node is in general not zero. Observe that the forces on the vertices should be

derivatives of the total elastic energy. This is important to point out because in the following

chapters, the elastic energy will have other terms added, which will need to be taken into

account when calculating the node forces.

2.2 Algorithm

To construct a Hamiltonian we also need to specify the kinetic energy of the system. Here

we follow the method used by Broughton et al. [44] and apply the “lumped mass approxima-

tion”, which assumes that all the mass is concentrated in the nodes of the finite element mesh.

In our simulation, the initial position and velocity of each node in the material are specified

in the initial state. The subsequent dynamics of the system are calculated explicitly using a

finite time step. After each time step, the potential energy in Equation 17 is calculated for

36

each element as a function of the corner nodes displacements from their initial positions in

the reference state. Forces on each node are calculated as a derivative of the total potential

energy in all adjacent volume elements with respect to the nodes position. The node positions

and velocities are then updated using the velocity Verlet method [46]:

xi(t+∆t) = xi(t) + vi(t)∆t+1

2ai(t)(∆t)2

vi(t+∆t) = vi(t) +ai(t) + ai(t+∆t)

2∆t

This finite element explicit dynamics algorithm closely resembles the familiar molecu-

lar dynamics method and is almost as easy to code. However, here we are moving nodes

rather than atoms, and instead of an interatomic potential we are using the continuum elastic

potential energy, expressed as a function of node displacements.

2.3 Dissipation

To add internal damping associated with velocity gradients in the sample, we use a mod-

ified form of Kelvin dissipation. In its standard form, the Kelvin dissipation force (e.g. be-

tween two particles, or between two nodes in a finite element mesh) is proportional to the

velocity difference between them (see e.g. [47]). This form conserves linear momentum but

violates conservation of angular momentum; internal dissipation forces could create torque,

which is of course unphysical. We modified the Kelvin dissipation form to provide for con-

servation of angular momentum, that is, dissipation forces between any pair of nodes must

act along the line of sight between them, so they create no torque [48]. We also scale the

dissipation force so it depends on the effective strain rate between two nodes rather than

37

their absolute velocity difference. With these modifications, the dissipation force between a

pair of neighboring nodes separated by distance d is F12 = −η(v1 − v2) · (r1 − r2)

d12r12, with

η = 10−7 kg.m/sec. The resulting dissipation is isotropic in character and does not depend

on the orientation of the director field.

2.4 Simulations of Rubbery materials

Figure 16: Beam of rubber twisting.

Equipped with the tools described in the previous sections, we venture into a simple test

case which consists in simulating an isotropic piece of rubber of size 10mm× 5mm× 1mm

that is subjected to a torsional deformation; see Figure 16 . The material parameters used

for this simulation are a Young’s modulus E = 1.5MPa, and Poisson ratio very close to the

38

e

Figure 17: Potential (black dots) and kinetic (red dots) energy of the elastic beam experi-encing torsional deformation. In the absence of dissipation, the energy is conserved (greendots).

incompressibility limit ν = 0.499. Note that a Poisson ratio of ν = 0.5 is not numerically

achievable as that would result in an infinite bulk modulus. As shown in Figure 17, in the

absence of dissipative forces, energy is conserved up to a part in 108.

It is worth mentioning here that prior to performing all the simulations reported from this

point on, several numerical experiments were performed for benchmarking. In particular, the

discretization of the volume into a tetrahedral mesh was handled in such a way that the distri-

bution of element sizes was close to uniform. The appropriate time step for each simulation

was chosen to be smaller than any characteristic time scale of the system. All benchmarks

showed that for a Poisson ratio ν = 0.499, the sample deforms with volume fluctuations are

less than a part in a thousand.

39

2.5 Simulations of LCE

To apply the method described in the previous section to nematic elastomers, there must

be extra terms in the potential energy to account for the nematic interaction and the strain-

order coupling. In the simplest approximation we consider a linear coupling between the

strain tensor and the nematic order as proposed by de Gennes [16]. One can thus rewrite the

potential energy as:

U = Ustrain − αεijQij (19)

The nematic order parameter tensor Qij is defined in the material frame and transforms like

the strain tensor; that is, rotations of the whole sample in the laboratory frame leave these

quantities unchanged. While one normally couples the left strain tensor to the order parameter

tensor in order to have a frame indifferent energy [49] , the coupling above is also valid if

the nematic director nref used to construct Qij lives in the reference frame; that is, nref is

obtained by an inverse transformation from the target to the reference space before coupling

the Green-Lagrange strain tensor εij to Qij . The said inverse transformation is precisely

described by the unitary matrix (rotation) obtained by polar decomposition of the deformation

gradient tensor.

To illustrate this, let us define by nt the local nematic director in the target space. Recall that

the polar decomposition of the deformation gradient tensor introduced earlier reads:

λ = RU = V R

Where U and V are positive definite and R is unitary. From this, it follows that λT = RT V .

First, coupling the left Green tensor to the order parameter reads:

40

(λλT )(nt ⊗ nt) = nt · (λλT )nt

= | λT nt |2

= | RTV nt |2

= | V nt |2

If we write nref = RT nt , then coupling the right Green tensor to the order parameter

tensor in the reference space reads:

(λTλ)(nref ⊗ nref ) = | λnref |2

= | λRT nt |2

= | V nt |2

Clearly, coupling the left Green tensor to the target order tensor is equivalent to coupling the

right Green tensor to the appropriately transformed order tensor described above.

We have developed two distinct approaches to simulating nematic elastomers. In the first

and the simplest approximation, we can control the state of the nematic order and observe the

mechanical response of the sample. In particular, by increasing (resp. reducing) the amount of

order in the sample, one can mimic the cooling (resp. heating) of the material, hence allowing

to study the shape change of nematic LCE undergoing a Nematic-Isotropic phase transition.

In chapters 3 and 4, we present a more advanced approach in which we simultaneously model

shape change and microstructural evolution. This approach allows us to track the dynamics

of the nematic director field and predict mechanical response. The main assumption used is

that the nematic director is always in quasi-static equilibrium with the slowly evolving strain

41

field. This is justified by the observed discrepancy between the time scales of the network and

director relaxations [50]. This permits one to study the nucleation and evolution of stripes

and the polydomain to monodomain transition. It is appropriate to highlight here that this

approach that considers isotropic rubber elasticity and a coupling between strains and order

tensors is not at odds with others such as the neoclassical which is widely used. One can

show that one form can be recovered from the other [51]. To such end, we start by rewriting

the de Gennes form of the energy in the following form.

U =1

2µ[Tr(λλT )− αQ · (λλT )]

=1

2µ[Tr(λλT )− αλλT · S(n⊗ n− 1

3I)]

=1

2µ[Tr(λλT )− α S n · (λλT )n− 1

3Tr(λλT )]

=1

2µ[Tr(λλT )− α S | λT n |2 −1

3Tr(λλT )]

=1

2µ[(1 +

αS

3)Tr(λλT )− α S | λT n |2]

Similarly, assuming an isotropic reference state, i.e. lo = I , we write neoclassical energy

as:

U =1

2µTr(l−1λλT )

=1

2µTr

[((1

l∥− 1

l⊥

)n⊗ n+

1

l⊥

)λλT

]=1

2µTr

[(1

l∥− 1

l⊥

)(λλT )(n⊗ n) +

1

l⊥λλT

]=1

2µ

[(1

l∥− 1

l⊥

)| λT n |2 + 1

l⊥Tr(λλT )

]Equating the two expressions above, the phenomenological constant α that couples the

strain tensor to the nematic order parameter tensor is be expressed in terms of the components

42

of the step length tensor as: αS = le

(1

l⊥− 1

l∥

). Here one defines le =

3

l∥ + 2l⊥.

2.5.1 Isotropic-Nematic phase transition

We present here a simulation of a thin strip of nematic elastomer undergoing the transition

from the isotropic to the nematic phase. The film has dimensions 1.5 mm × 0.5 mm with a

thickness of 100 µm. The shear and bulk moduli are µ = 5.7×105Pa and Br = 2.8×107Pa

, respectively, comparable to that of an isotropic rubber. In this simulation, the strain-order

coupling parameter is α = µ. The sample is initially in the high temperature isotropic phase,

and we use the first approach described above to evolve the simulation. The strength of the

nematic scalar order parameter is increased, and as a response, the sample elongates in one

direction and contracts in the other two. In a transition from the isotropic to nematic phase

in NLCE, the spontaneous nematic director can point in any direction, and this will normally

lead to an uncontrolled shape change of the sample. In order to break the symmetry, one end

of the sample of the sample is constrained not to move. This study provides a qualitative

agreement with the large shape change observed in similar experiments [12].

2.5.2 Semisoft Elasticity

Following the description of the semisoft elastic response presented in the preceding chap-

ter, we can model this phenomenon. If we consider an element with the director initially

oriented at an angle ϕ with respect to, say, the z-direction, then a uniaxial strain applied in

the in y-direction has the effect of rotating the director to a new angle θ. The strain energy

for the deformation as described in the neoclassical theory then becomes a function of θ.

43

Figure 18: Simulation of the isotropic-nematic transition in a liquid crystal elastomer. Thesample experiences a macroscopic shape change as it is cooled down to the nematic phaseand heated back into the isotropic phase. The sample is clamped at the top end.

U = µ1

4(1

r− 1)(2a2(a3 cos(2θ)((r − 1) cos(2ϕ) + r + 1) + (r − 1) sin(2ϕ)((b3 + 1) cos(2θ)

−b2 sin(2θ))− (r − 1) sin(2θ) cos(2ϕ)− (r + 1) sin(2θ)) + 2(r − 1) sin(2ϕ)

((a3(b3 + 1)− b2) sin(2θ) + (a3b2 + b3 + 1) cos(2θ))

+(r − 1) cos(2ϕ)((a23 + b22 − b3(b3 + 2)− 2) sin(2θ) + 2(a3 − b2(b3 + 1)) cos(2θ))

+a22 sin(2θ)(−((r − 1) cos(2ϕ) + r + 1)) + a23r sin(2θ) + 2a3r cos(2θ)

+a23 sin(2θ) + 2a3 cos(2θ)− b22r sin(2θ) + b23r sin(2θ)

+2b3r sin(2θ) + 2b2r cos(2θ) + 2b2b3r cos(2θ)− b22 sin(2θ)

+b23 sin(2θ) + 2b3 sin(2θ) + 2b2 cos(2θ) + 2b2b3 cos(2θ)

−2c22 sin(2θ) + 2c23 sin(2θ) + 4c2c3 cos(2θ))

44

Figure 19: Cartoon of the semisoft response in a NLCE.

Solving for θ is straightforward and gives the orientation of the director for a given strain.

In particular, for the most commonly studied experiment in which the strain is imposed per-

pendicular to the director orientation, one uses ϕ =π

2. This yields a rotation of the director

by an amount

θ =1

2tan−1

(2((a2 + 1)a3 + b2(b3 + 1)r + c2c3)

(a2 + 1)2 − a23 + b22r − (b3 + 1)2r + c22 − c23

)

This suggests that one can always know the state of the nematic director given the strain.

However, as appealing as this may appear, it will not be our method of choice in the subse-

quent chapters. This is simply because here the strain is treated as an external global variable

that couples to the local nematic director. In order to study microstructure formation and evo-

lution, it is important to allow the strains to vary locally in response to changes in the director

orientation and vice-versa.

CHAPTER 3

POLYDOMAIN-MONODOMAIN TRANSITION IN NEMATIC ELASTOMERS

3.1 Introduction

Liquid crystal elastomers, when crosslinked, usually exist in the polydomain configura-

tion, consisting of a large array of randomly oriented, micron sized domains, see Figure 20.

The domains may exhibit a local average degree of order, but globally there is no long range

order. A sample of such material will strongly scatter light that is incident upon it due to the

size of the domains.

Monodomain liquid crystal elastomers can be obtained by means of the Finkelmann two-

step crosslinking method, which consists in a weak crosslink followed by another crosslink

under the influence of an aligning field, e.g. an electric or magnetic field, or external strain.

Deformation of an initially polydomain nematic elastomer film induces a transition to

the monodomain configuration. We model the resulting microstructural evolution and stress-

strain response using a novel finite element elastodynamics simulation approach. We explore

how the thermomechanical history of the sample, e.g. its crosslink density and phase at time

of network formation, affects the width of the poly-monodomain transition and the associated

stress-strain behavior. We find that when the sample is cross-linked in the isotropic phase,

the material shows a semi-soft response with a well-defined plateau in the stress-strain curve.

By contrast, when the sample is cross-linked in the nematic phase, the resulting strong local

disorder broadens the transition, and the plateau is much less pronounced. These simulation

45

46

Figure 20: A cartoon of a polydomain nematic elastomer shows no correlation between theorientation of the director in neighboring domains. The blue arrows show the orientation ofthe nematic director in the domains.

results yield qualitative agreement with recent experimental observations. We also study the

rate-dependent material response under uniaxial extension. This simulation approach allows

us to explore the fundamental physics governing dynamic mechanical response of nematic

elastomers and also provides a potentially useful computational tool for engineering device

applications.

When a polydomain LCE thin film is stretched uniaxially, its orientational domains align,

producing long-range orientational order. This poly-to-monodomain (P-M) transition has

been well-characterized [52–54]. Some polydomain materials exhibit semi-soft mechanical

response, while others do not, with no universal behavior. It was not clear what aspect of

composition or processing determines the nature of a sample’s mechanical response.

47

Recent work by Uruyama and coworkers [55] shed some light on this mystery by study-

ing the P-M transition in samples with the same chemical composition but different thermo-

mechanical history. They prepared polydomain samples in two different ways. Nematic-

crosslinked polydomain nematic elastomers, or N-PNE, were prepared by cross-linking in

the nematic phase with no aligning field. N-PNE samples display short-range orientational

order with approximately micron-sized domains. Isotropic-crosslinked polydomain nematic

elastomers, or I-PNE, were prepared by cross-linking in the isotropic phase with no aligning

field and then cooling through the I-N transition. I-PNE samples display stronger disorder,

with orientational domains too small to observe via polarization microscopy.

Uruyama et al found that under uniaxial extension, I-PNE samples show a sharp P-M tran-

sition with a clear semi-soft mechanical response, while N-PNE materials show a broadened

P-M transition and no pronounced plateau in the stress-strain curve. When the applied strain

was relaxed, both types of samples recovered their initial shape, but N-PNE also recovered

the same initial polydomain texture, indicating a strong local memory effect.

Discrete lattice models in both 2-d [56,57] and 3-d [58] have provided key insights into the

role of local heterogeneity in both monodomain and polydomain LCE. Yu et al. explored how

heterogeneity affects long-range correlations in director orientation in polydomain LCE in the

absence of applied strain. Uchida [56] modeled the P-M transition in LCE films cross-linked

in the isotropic state, demonstrating that local heterogeneity broadens the transition region. J.

Selinger and Ratna [58] showed that the Isotropic-Nematic (I-N) transition in monodomain

LCE may be broadened by heterogeneity in either random fields or bonds. In these lattice-

based approaches, strain is treated as a global variable playing the role of an external applied

48

field; thus the details of sample shape evolution are not predicted.

Desimone et al. [43] used 2-d finite element elastostatics methods to model the mechanical

response of a thin monodomain LCE film under uniaxial strain. They modeled the sample

as a 2-d homogenized composite of domains with different orientations. This innovative

approach successfully reproduced both the soft mechanical response and the sample’s overall

shape evolution, but without explicitly modeling the resulting microstructure or the associated

rate-dependence.

In this chapter, we describe 3-d finite element elastodynamics simulation studies of the

P-M transition in both N-PNE and I-PNE polydomain samples, with spatial resolution of the

nematic director field down to the micron scale. This model allows us to simulate mechanical

response, shape change, microstructural evolution, and strain-rate effects. We use the model

to investigate in detail how a sample’s mechanical response depends on its thermo-mechanical

history. We also quantify the degree of global orientational order induced via applied strain.

To study the mechanical response of LCE thin films, we carry out computer simulations

using the 3-d finite element elastodynamics approach described in the previous chapter. A

sample of arbitrary shape is discretized into a nonuniform mesh of volume elements using

the Netgen algorithm [59]. Each element represents an approximately micron-sized nematic

domain. The deformations of an element are expressed in terms of shape functions that

interpolate the position of every point within the element from the position of its vertices [60].

The shape function used in this case is an affine one, and for simplicity, we use elements of

simplectic form, here tetrahedra in 3-d; this implies that the strain is uniform within each

element, though it may vary from one element to the next. We define a free energy density

49

for each element that comprises three parts:

U = Ustrain + Ustrain−order + Umemory (20)

The first term is the well-known neo-Hookean strain energy:

Ustrain =1

2µTr(λkiλkj) +K(Det[λij]− 1)2 (21)

where λij is the deformation gradient tensor . Volume conservation is maintained by a large

bulk modulus K. The second term Ustrain−order describes the coupling between mechanical

strain and orientational order:

Ustrain−order = −αεij(Qij −Qoij) (22)

Here we use the rotationally invariant Green-Lagrange strain tensor εij = 12(λikλkj −

δij), defined locally for each element. Local orientational order is characterized by Qij, the

symmetric and traceless uniaxial order parameter tensor. Qoij is the local order parameter

tensor present at the time of crosslinking. The parameter α is proportional to the density

of crosslinks in the sample. Both Qoij and α depend on the details of sample preparation as

discussed below.

The third term Umemory describes the crosslink memory effect:

Umemory =1

2β(Qij −Qo

ij)2 (23)

This term biases the nematic director to remain parallel to its orientation at the time of

crosslinking, thus playing the role of a local field whose orientation is defined by Qoij . The

50

strength of the field is assumed to be uniform throughout the sample and is defined by the

parameter β. We expect that samples crosslinked in the nematic phase have ’stronger’ mem-

ory of their initial state, and thus larger β, than those crosslinked in the isotropic phase. The

uniaxial order parameter tensor Qij has three independent degrees of freedom. We assume

no biaxiality in the system and hold the nematic scalar order parameter S constant within the

element; we further assume that S does not vary spatially across the sample. Thus, the only

degrees of freedom of Qij that are allowed to vary are the polar and azimuthal angles that

define the orientation of the local nematic director.

When the sample is subjected to an external uniaxial strain, the forces on the vertices

of each element are calculated as derivative of the free energy density F = − U . The

contribution to the force from the elastic potential energy is :

Fx,i = K (2 (b4c3 − (b3 + 1) (c4 + 1))M2iZ + 2 (a3 (c4 + 1)− a4c3)M3iZ

+2 (a4 (b3 + 1)− a3b4)M4iZ)

+µ ((a2 + 1)M2i + b2M3i + c2M4i)

Fy,i = K (2 (b2 (c4 + 1)− b4c2)M2iZ + 2 (a4c2 − (a2 + 1) (c4 + 1))M3iZ

+2 ((a2 + 1) b4 − a4b2)M4iZ)

+µ (a3M2i + (b3 + 1)M3i + c3M4i)

Fz,i = K (2 ((b3 + 1) c2 − b2c3)M2iZ + 2 ((a2 + 1) c3 − a3c2)M3iZ

+2 (a3b2 − a2 (b3 + 1)− b3 − 1)M4iZ) +

µ (a4M2i + b4M3i + (c4 + 1)M4i)

51

where

Z = (c2 (a4 (b3 + 1)− a3b4) + c3 ((a2 + 1) b4 − a4b2)

− (c4 + 1) (−a3b2 + a2 (b3 + 1) + b3 + 1) + 1)

The momenta of the elements are computed using the lumped mass approximation whereby

the mass of each element is equally distributed to its vertices, also refered to here as nodes

[45, 61]. The simulation evolves in two steps. First, holding Qij fixed, the nodes forces are

calculated and integrated forward in time via the velocity Verlet algorithm. The new strain

tensor can then be defined from the current and old positions of the nodes. The next step

consists in holding the nodes positions fixed and relaxing the nematic order parameter tensor

in order to minimize the free energy density. Since the local nematic director is defined as

n = (sin θ cosϕ, sin θ sinϕ, cos θ) with respect to the laboratory frame, relaxing the nematic

order tensor amounts to minimizing the free energy density with respect to its degrees of free-

dom, namely the azimuthal and polar angles of the local director ϕ and θ respectively. Note

that the only terms that depend on ϕ and θ are Ustrain−order and Umemory. The minimization

is performed using Powell’s method, which is a form of conjugate gradient method that does

not require the derivative of the function to be computed.

minimizeθ,ϕ

U(θ, ϕ)

This algorithm allows us to simultaneously model microstructural evolution and macro-

scopic shape change of the sample.

52

3.2 Initial configuration

The thermomechanical history of the sample as stated above plays a very important role in

the sample’s mechanical response. Here we will attempt to study the response of two samples,

one crosslinked deep in the nematic phase, and the other in the isotropic phase. It is worth

describing how we obtain the initial configuration prior to the imposition of the strain. With

the sample discretized in a tetrahedral mesh, we assign a nematic director to each element.

To mimic an isotropic crosslinked nematic elastomer, we start with a random distribution

of the orientations of the nematic directors (Figure 21(a)). Urayama et al [55] have shown

that the nematic elastomers with nematic genesis show the schlieren texture observable in

low molecular weight liquid crystals under polarizing microscopy. In our studies, a nematic

crosslinked initial state (Figure 21(b)) is achieved by running a simulation that takes the

sample into the nematic phase. This is done by using the Lebwohl-Lasher model of nematic

liquid crystals. It is simply equivalent to a lattice approximation of the Maier-Saupe theory.

It uses a Hamiltonian of the form

H = J∑<i,j>

[1− (ni · nj)2]

where the summation is carried over neighbouring pairs (i, j). J is the field coupling strength.

Note that the nematic directors are defined at the centroids of the elements, and, although the

mesh is not regular, the distance between the centroids of neighbouring elements is approx-

imately constant throughout the mesh. The Monte Carlo Metropolis algorithm is used to

anneal the sample, and one obtains different sizes of correlation length by quenching. Note

that this simulation involves only the nematic directors, and not the elastic degrees of freedom

53

of the sample.

(a) Isotropic genesis (IPNE) (b) Nematic genesis (NPNE)

Figure 21: Initial configurations with different thermomechanical histories.

3.3 Simulation and results

We present here simulation results obtained for a uniaxial stretching of a polydomain

film of dimensions 1.5 mm × 0.5 mm with a thickness of 100 µm, with shear modulus

µ = 5.7 × 105Pa, bulk modulus Br = 2.8 × 107Pa. The strain-order and memory cou-

pling parameters for are α = µ and β = 0.3µ, respectively. In these simulations, we used

γ = 10−7J . The strip of nematic elastomer was discretized in a mesh of approximately

80,000 tetrahedral elements. The initial director orientation of the directors is either random

(isotropic crosslinked), or composed of uncorrelated domains of size about an order of mag-

nitude larger than the element size (nematic crosslinked); that is, there is no global order in

the system. The sample is clamped at its end, i.e. the components of the displacement and

velocity perpendicular to the direction of imposed strain are always zero for the nodes in these

regions. All other nodes of the finite element mesh are allowed to move in any direction. The

54

(a) ∆λ/λ = 0

(b) ∆λ/λ = 0.5

Figure 22: Simulation studies of a I-PNE. (a) and (b) show the entire strip of nematic elas-tomer in the polydomain and monodomain configuration respectively. In the unstrained state,the sample strongly scatters light, whereas when strained the sample is sandwiched betweencrossed polarizers and aligned with either the polarizer or analyzer, total extinction of theincident light will occur.

clamped regions are moved at a constant speed of 1 mm/sec.

As the experiments on this materials are usually carried out in a quasi static manner due

to their slow stress relaxation [62], it was of paramount importance in these simulations to

investigate the strain rate dependance of the transition. High rates of strain resulted in out of

equilibrium situations, often missing entirely the dynamics of the polydomain-monodomain

transition when the latter occured within a narrow range of strains. A discussion of the strain

rate dependance of the elastic response in this materials can be found in the next chapter. This

was easily alleviated by using smaller time steps, with the drawback that the computation

time was greatly increased. Our programs thus had to be optimized for parallel computations

using MPI, and the simulations were executed on a cluster architecture hosted by the Ohio

Supercomputer Center. The typical time step used in the simulations shown here is 0.1 µs.

55

Figure 23: Texture and director configuration in a region near the center of the strip in thepolydomain state of I-PNE.

While the simulation evolves as described in the previous section, we track the time evolution

of the local director in each element, together with each element and the overall mesh shape

change. As demonstrated in [52], we observe that the polydomain to monodomain transition

in nematic elastomers proceeds by rotation of the domains rather than domain growth. During

the deformation, the sample goes from a scattering state as a result of the nonuniformity in

the director orientations to a transparent one.

56

Figure 24: Texture and director configuration in a region near the center of the strip in themonodomain state of I-PNE.

The textures as would be observed under optical polarizing microscopy are shown in

Figures 25 and 26. The local directors in the regions near the clamps are not allowed to

rotate, resulting in the formation of defects in the director orientation. These regions strongly

scatter light, even after the director rotation is complete far from the clamps. The resulting

stress-strain behavior is shown in Fig. 27 for samples with crosslinked in the isotropic phase,

and in Fig. 28 for a sample crosslinked in the nematic phase. The stress here is the engineering

57

Figure 25: Texture and director configuration in a region near the center of the strip in thepolydomain state of N-PNE.

stress which is obtained by dividing the average of the normal forces applied to the sample

by the area of the cross-sectional where these forces are applied. The same plots also show

the dependence of the global order parameter S of the sample on the imposed strain. S is

simply the average of the second Legendre polynomial (S = ⟨P2(cos θ)⟩) over the whole

mesh, where θ is the angle between the local director and the direction of the applied strain.

For samples crosslinked in the nematic phase, the stress-strain curve displays a linear

58

Figure 26: Texture and director configuration in a region near the center of the strip in themonodomain state of N-PNE.

regime followed by a plateau in the range of strains that correspond to the rotation of the

domains towards the direction of the applied stress. Upon completion of the directors rotation,

the linear regime is recovered. The height of the plateau, and hence the amount of work

required to achieve the transition is higher for samples crosslinked deep in the nematic phase.

Samples crosslinked in the isotropic phase on the other hand display a much more pronounced

plateau, with an ideally soft behavior. We thus anticipate the polydomain to monodomain

transition to be a reversible process for samples crosslinked in the nematic phase and not

59

for those crosslinked in the isotropic phase. The slope of the stress-strain curve in the linear

regime (after the rotation of the domains is completed) however does not reveal any detail on

the history of the sample.

DL/L

Figure 27: Engineering stress σ ,(black curve) and Global order parmeter, S (red curve) vsstrain for a nematic elastomer crosslinked in the isotropic phase ( I-PNE) .

It can be seen from the range of strains over which the transition occurs that samples

crosslinked in the isotropic phase offer less resistance to the rotation of the domains than their

counterpart crosslinked in the nematic phase. One can thus think of LCE crosslinked in the

isotropic phase as having a weak anchoring of the director to the polymer matrix, and those

crosslinked in the nematic phase as having a strong anchoring. It is worth noting however

that the amount of order in the obtained monodomain state is roughly the same regardless of

the whether the sample was crosslinked in the isotropic or in the nematic phase.

60

DL/L

Figure 28: Engineering stress σ ,(black curve) and Global order parmeter, S (red curve) vsstrain for a nematic elastomer crosslinked in the nematic phase ( N-PNE) .

3.4 Discussion

This chapter presented simulations of the polydomain to monodomain transition in liq-

uid crystal nematic elastomers samples with different crosslinking histories. Studying these

materials at the continuum level, we used a model that makes no assumption on the detailed

chemical structure, and hence strive to extract universal characteristics of the response of liq-

uid crystal elastomers subjected to external stimuli. The mechanical response of polydomain

nematic elastomers has been investigated in the presence of an external mechanical stimulus,

taking into account the details of the sample preparation. We found that the thermomechan-

ical history of the sample plays a crucial role in determining the dynamics of the transition,

and the simulation results are in a good qualitative agreement with recent experiments on this

61

class of material [55].

CHAPTER 4

MODELING THE STRIPE INSTABILITY IN NEMATIC ELASTOMERS

4.1 Introduction

In the nematic phase, due to strong coupling between mechanical strain and orientational

order, nematic liquid crystal elastomers display strain-induced instabilities [10] associated

with formation and evolution of orientational domains. In a classic experiment, Kundler and

Finkelmann [63] measured the mechanical response of a monodomain nematic LCE thin film

stretched along an axis perpendicular to the nematic director. They observed a semisoft elastic

response with a pronounced plateau in the stress-strain curve arising at a threshold stress.

Accompanying this instability they observed the formation of striped orientational domains

with alternating sense of director rotation, and a stripe width of 15 µm. They repeated the

experiment with samples cut at different orientations to the director axis, and found that the

instability was absent when the angle between the initial director and the stretch axis was less

than 70o [63] ; in this geometry, instead of forming stripes, the director rotates smoothly as a

single domain.

This peculiar behavior reminiscent of a martensitic transformation, promises a bright fu-

ture to nematic liquid crystal elastomers for engineering applications based on soft actua-

tors [64]. An interesting application that was proposed was building an acoustic wave polar-

izer by modulating the internal degree of freedom, namely the nematic director.

Using the aforementioned 3-d finite element elastodynamics simulation, we investigate

62

63

Figure 29: Experiment: stretching a nematic elastomer film at an angle of 90o to the direc-tor results in a microstructure consisting stripes of alternating director orientation. Imagecourtesy H. Finkelmann

the onset of stripe formation in a monodomain film stretched along an axis not parallel to the

nematic director.

In an earlier work, DeSimone et al. [43] carried out numerical simulation studies of the

stripe instability using a two-dimensional finite element elastostatic method. Each area el-

ement in the system was considered as a composite of domains with different orientations.

This simulation model was the first to reproduce successfully the soft elastic response of

nematic elastomers, but did not attempt to resolve the resulting microstructural evolution.

Uchida carried out more detailed studies of director evolution in nematic elastomers using a

two-dimensional lattice model where macroscopic strain is treated as a global variable analo-

gous to an external field, but did not attempt to describe the non-uniform strain and resulting

shape evolution of the sample.

Here we explore this elastic instability in more detail by simultaneously modeling the

sample’s mechanical response, shape evolution, and the associated microstructural evolution

as a function of strain. We use a Hamiltonian-based 3-d finite element elastodynamics model

with terms that explicitly couple strain and nematic order. By resolving the finite element

64

mesh down to the micron scale, we resolve the formation of orientational domains, and be-

cause the model is dynamic rather than static in character, we can examine the effects of strain

rate. We use the simulation to explore the dependence of mechanical response on deformation

geometry.

4.2 Simulations and results

We model this instability in a thin film of nematic elastomer which has been cross-linked

in the nematic phase [65]. Using public domain meshing software [59] we discretize the

volume of the sample into approximately 78, 000 tetrahedral elements. For each volume

element we assign a local variable n that defines the nematic director, and Qij =12S(3ninj −

δij), which is the associated symmetric and traceless nematic order tensor. The initial state is

taken to be a monodomain with n = no in every element; this configuration is defined as the

system’s stress-free reference state.

There are many approaches to finite element simulation of the dynamics of elastic media

[66]; we make use of an elegant Hamiltonian approach developed by Broughton et al. [44,45],

generalizing it to three dimensions and the case of large rotations. We write the Hamiltonian

of an isotropic elastic solid as:

Helastic =∑p

Vp1

2Cijklε

pijε

pkl +

∑i

1

2miv

2i . (24)

Here the first term represents elastic strain energy, with p summing over volume elements. Vp

is the volume of element p in the reference state. For an isotropic material the components of

the elastic stiffness tensor Cijkl are determined from only two material parameters, namely the

65

shear and bulk moduli [67]. As an approximation, Broughton et al developed this formulation

using the linear strain tensor, but we instead use the rotationally invariant Green-Lagrange

strain tensor εij = 12(ui,j + uj,i + uk,iuk,j), where u is the displacement field. We note that

using the linearized strain tensor would make the Hamiltonian unphysical, as rotation of the

sample would appear to cost energy. The second term represents kinetic energy in the lumped

mass approximation [45] whereby the mass of each element is equally distributed among its

vertices, which are the nodes of the mesh. Here i sums over all nodes, mi is the effective

mass and vi the velocity of node i.

To account for the additional energy cost associated with the presence of a director field,

we add to the potential energy,

Hnematic =∑p

Vp

[−αεpij(Q

pij −Qrp

ij ) + β(Qpij −Qrp

ij )2]. (25)

The first term describes coupling between the strain and order parameter tensors using a form

proposed by DeGennes [16]. Here Qprij defines the nematic order in the element’s reference

state. The prefactor α controls the strength of this coupling, and DeGennes [16] argued that

it is of the same order of magnitude as the shear modulus µ. Variables Qij , Qprij , and εij

are all defined in the body frame, i.e. they are invariant under rotations in the target frame.

See [68] for the relation between Qij in the body and lab frames. The second term describes

“cross-link memory,” that is, the tendency of the nematic director to prefer its orientation at

crosslinking. Thus there is an energy cost to rotate the director away from its reference state,

with coupling strength β.

66

The strain tensor εij within each tetrahedral element is calculated in two steps. We calcu-

late the displacement u of each node from the reference state, then perform a linear interpola-

tion of the displacement field within the volume element in the reference state. The resulting

interpolation coefficients represent the derivatives ui,j needed to calculate the components of

the strain tensor. Details were described in chapter 2 and can be found in any introductory text

on finite element methods, e.g. [60]. At this level of approximation, the strain is piecewise

constant within each volume element. The effective force on each node is calculated as the

negative derivative of the potential energy with respect to node displacement.

To evolve the system forward in time, we assume the director is in quasistatic equilibrium

with the strain; that is, the time scale for director relaxation is much faster than that for strain

evolution as observed by Urayama [69]. The first part of each step is elastodynamics: holding

Qij in each element constant, the equations of motion f = ma for all node positions and

velocities are integrated forward in time using the Velocity Verlet algorithm [46], with a time

step of 10−8 sec. In the second part of each step, we relax the nematic director in each element

to instantaneously minimize the element’s potential energy. Because the director relaxes from

a higher energy state to a lower energy state without picking up conjugate momentum, this

is a source of anisotropic dissipation. Thus in our model, as in real nematic elastomers,

strains that rotate the director cause more energy dissipation than those applied parallel to the

director [70].

To add internal damping associated with velocity gradients in the sample, we use a mod-

ified form of Kelvin dissipation. In its standard form, the Kelvin dissipation force (e.g. be-

tween two particles, or between two nodes in a finite element mesh) is proportional to the

67

velocity difference between them (see e.g. [47].) This form conserves linear momentum but

violates conservation of angular momentum; internal dissipation forces could create torque,

which is of course unphysical. We modified the Kelvin dissipation form to provide for con-

servation of angular momentum, that is, dissipation forces between any pair of nodes must

act along the line of sight between them, so they create no torque [48]. We also scale the

dissipation force so it depends on the effective strain rate between two nodes rather than their

absolute velocity difference. With these modifications, the dissipation force between a pair of

neighboring nodes separated by distance d is F12 = −η(v1 − v2) · (r1 − r2)

d12r12. The result-

ing dissipation is isotropic in character and does not depend on the orientation of the director

field.

We simulate uniaxial stretching in an initially monodomain nematic elastomer film of

size 1.5 mm × 0.5 mm with a thickness of 50 µm, with shear modulus µ = 5.7 × 105Pa,

bulk modulus Br = 2.8 × 107Pa, and parameters α = µ, β = 0.3µ, and ζ = 10−7kg.m/s.

We first consider the case where the director is initially oriented along the y axis, transverse

to the direction of applied strain. The sample is clamped on two sides and the clamped

regions are constrained to move apart laterally at a constant speed of 1 mm/sec. The resulting

microstructural evolution is shown in Fig.30. Here color represents Jones matrix imaging

of the director field as viewed through crossed polarizers parallel to the x and y directions;

blue corresponds to a director parallel to the polarizer or analyzer, and red corresponds to a

director at a 45o angle to either. While the simulated sample is three-dimensional, the film’s

microstructure does not vary significantly through the thickness and can thus be visualized in

2-d.

68

Figure 30: Simulation: stretching a nematic elastomer film at an angle of 90o to the director.Initially a monodomain, the director field evolves to form a striped microstructure.

4.2.1 Stripes width

At a strain of 8.5%, the director field in the sample becomes unstable and orientational

domains form, nucleating first from the free edges of the film. Heterogeneity in the finite ele-

ment mesh serves to break the symmetry and nucleate the instability. By 9% strain, the whole

film is occupied by striped orientational domains with alternating sense of director rotation.

69

The stripes are not uniform in width, being slightly larger near the free edges. Near the center

of the sample, each individual stripe has a width of about 25 µm, which is of the same or-

der of magnitude as that observed in experiment [63]. This value is in reasonable agreement

with the theoretical estimate by Warner and Terentjev [10] who predicted a stripe width of

h ∼√ξL/

√1− 1/λ3

1; where ξ is the nematic penetration length , L is the sample width,

and λ1 is the strain threshold of the instability. Indeed , using ξ = 50nm, L = 1.5× 10−3m,

and λ1 = 1.09, one finds h ≈ 18.14µm. The stripes coarsen as the elongation increases.

Eventually this microstructure evolves into a more disordered state with stripes at multiple

orientations. By reaching 35% strain, the stripes have vanished and the film is again in a mon-

odomain state with the director oriented with the direction of strain. Only the regions near

the clamped edges do not fully realign, in agreement with experimental observations [63]

and with the simulation studies of DeSimone [43]. We will explore the dependence of stripe

width on aspect ratio and other parameters in future work.

The resulting stress-strain response is semi-soft [68] in character, as shown in Figure 31.

The initial elastic response is linear, followed by an extended plateau running from about

8.5% to over 30% strain, after which there is a second linear regime. We also measure the

average director rotation⟨sin2(ϕ)

⟩and observe that the thresholds for both the stress-strain

plateau and the rotation of the nematic director occur at the same strain. This finding demon-

strates, in agreement with theory [10,68], that the reorientation of the system’s internal degree

of freedom–namely the nematic director–reduces the energy cost of the deformation.

70

σ(

N/

mm

2)

0

0. 02

0. 04

0. 06

0. 08

0. 1

0. 12

0

0.2

0.4

0.6

0.8

1

DL/L

0 0 .1 0. 2 0 .3 0. 4 0 .5

S

S

S

Figure 31: Engineering stress (circles) and director rotation (squares) vs applied strain, forthe system shown in Figure 30. Onset of director rotation and the stress-strain plateau bothoccur at the same strain.

4.2.2 Threshold for stripe instability

We also performed simulations for monodomain nematic elastomer films with the initial

director orientation at different angles to the pulling direction. In Figure 32 we plot the film’s

stress-strain response when strain is applied at an angle of 60o from the nematic director,

which shows no plateau, and likewise director rotation shows no threshold behavior. As

shown in Figure 33, the director rotates to align with the strain direction without forming

stripes. We performed additional simulations with the director at angles of 70o and 80o to

the pulling direction and again found no stripe formation and no plateau in the stress-strain

response.

71

σ(

N/

mm

2)

0

0. 02

0. 04

0. 06

0. 08

0. 1

0. 12

0

0. 2

0. 4

0. 6

0. 8

1

DL/L0 0 .1 0. 2 0 .3 0. 4 0 .5

S

S

Figure 32: Engineering stress (circles) and director rotation (squares) vs applied strain, ap-plied at an angle of 60o from the nematic director.

4.2.3 Rate of strain

Because liquid crystal elastomers have relatively slow stress relaxation [71], mechanical

experiments are often performed using static strains, i.e. the sample is allowed to relax be-

tween successive elongations. However, these static strain experiment may not accurately

reflect the behaviour of these materials when used in applications. It is thus paramount to

study the rate dependence of the strain of the material’s response. We tried varying the

applied strain rate. Figure34 compares the stress-strain response for samples strained at 1

mm/sec and 5 mm/sec. The higher strain rate produces a significant stress overshoot, and

stripe formation occurs at a strain of 15%. This finding suggests that the threshold strain for

the instability depends in a significant way on strain rate.

72

Figure 33: Simulation: stretching a nematic elastomer film at an angle of 60o to the direc-tor. Initially a monodomain, the director field rotates smoothly without sharp gradients inorientation.

4.3 Discussion

The simulations presented here were performed at far higher strain rates, e.g. 50% per

second, than those used in typical experiments [62,63] where the material is allowed to relax

for minutes or hours between strain increments. In future work we plan to apply our model to

examine deformation of nematic elastomers at slower strain rates and as a function of sample

73

(

Figure 34: Dependence of the stress-strain response on strain rate.

geometry. We will also examine the role of initial microstructure and thermomechanical

history in determining mechanical response. Using the same finite element approach, we

can also test the predictions of other proposed constitutive models, and model geometries

of interest for potential applications. Through this approach we hope to bridge the divide

between fundamental theory of these fascinating materials and engineering design of devices.

CHAPTER 5

MODELING DEVICES

The novel electrical, optical and mechanical properties of liquid crystal elastomers allow

one to envision a vast field of possible engineering applications. The method we have intro-

duced for studying the response of these materials to external stimuli also allows us to model

devices, thus making a step towards bridging our understanding of the fundamental micro-

scopic properties of LCE and their applications. In this chapter, a number of examples of

applications of nematic liquid crystal elastomers are presented, ranging from soft photoactu-

ators to robotic earthworms.

5.1 Polarization tuner

The basic principle of operation of a wave plate consists in modifying the state of po-

larization of the light incident upon it by changing the relative phase of it’s ordinary and

extraordinary waves. The accepted taxonomy refers to a wave plate as λn

wave plate if it re-

tards one state of polarization by πn

with respect to the other. Here n is an integer. The most

commonly used are the half wave plate and the quarter wave plate. Observe that the quarter

wave plate will change the polarization state of light from linear to circular, and vice-versa.

To explore mechanical response of nematic elastomer films in a more complex geometry, we

simulated the radial stretching of a circular monodomain film of diameter 1 cm and thick-

ness 100 µm, with the nematic director oriented initially along the y axis, as indicated by

the arrow in Figure 35. Boundary conditions were imposed that clamp the sample around74

75

Figure 35: Simulation: A nematic elastomer disk is stretched radially. The director fieldsmoothly transforms from a homogeneous monodomain to a radial configuration.

its circumference and stretch radially in all directions, pulling the edge outward at constant

speed. Figure 35 shows the film at different stages of its extension, demonstrating that the

director field smoothly changes from a monodomain to a radial configuration, with no stripe

instability.

With a careful choice of the sample’s thickness, this deformed circular sheet of nematic

elastomer could be used as a tunable spatial polarization converter as described in [72].

5.2 Actuators

The photoresponse of nematic elastomers was extensively discussed by Warner and Ter-

entjev [10]. Camacho-Lopez et al. demonstrated that an azo-doped liquid crystal elastomer

beam anchored on one edge bends spontaneously when illuminated on one side by a fast

laser pulse. Our simulations permit us to model the mechanical response of such a system. In

76

Figure 36: Cartoon of the polarization modulation by the elastomer strip.

particular, we consider a nematic elastomer beam in its initial nematic state with its director

aligned horizontally as in Figure 37. Then we switch the top layer of the sample, shown in

red, from nematic to isotropic, by setting the nematic order tensor Qij to zero only within

volume elements on the top of the sample. This mimics the fact that the light is rapidly at-

tenuated as it goes through the material. The resulting surface contraction induces a rapid

bend, pulling the beam into a curved position. The resulting radius of curvature depends on

many variables including the thickness and elastic properties of the beam; the strength of the

nematic-strain coupling; the magnitude of the reduction in the nematic order parameter in the

surface layer; and the thickness of the surface layer. The time response of the beam depends

also on the kinetics of the trans-cis photoisomerization of azobenzene chromophores in the

material.

Ikeda has studied the photoresponse of sheets of nematic elastomers illuminated by UV

light. In his experiments, he was able to control the folding of the sheet by simply rotating the

77

Figure 37: Simulation of the photo-deformation of a beam of nematic liquid crystal elastomer.

polarization of the incident (linearly) polarized light. We have obtained similar results from

our simulations with sheets of polydomain nematic elastomers that are illuminated by a beam

of linearly polarized light. See Figure 38. In these simulations, as in the previous case, only

a thin portion of the film responds to the incident light. The difference between these and the

previous simulations is that the domains on the top layer of the film do not all contract by

the same amount, but by an amount proportional to the angle between the local director and

polarization state of the incident light.

5.3 Peristaltic pumps

Because of their exceptional ability to change shape and mimic the behavior of muscles,

nematic elastomers are a good candidate for the engineering of peristaltic pumps. Peristalsis

is the process by which propagating undulation in a tube or channel induces motion of its

contents. This is for example the mechanism by which food is transported through the human

digestive system. We have modeled two highly idealized conceptual designs for peristaltic

pumps composed of nematic elastomers [61]. The first of these is a tube-shaped structure

78

Figure 38: Simulation of the photobending of a strip of polydomain nematic liquid crystalelastomer.

79

with the nematic director oriented along the tubes long axis. To induce a propagating wave,

we apply a modulation to the strength of the scalar nematic order parameter along the length

of the tube with a selected wavelength and frequency. Such modulation could be created

e.g. by non-uniform heating or by a pattern of laser illumination switched on/off periodically

along the tube to create a moving wave. Such a device might be useful for transport of highly

viscous fluids or slurries. A second configuration is also shown; here we induce a similar

propagating oscillation in a thin film designed to cover a rigid channel and move the fluid

inside. Alternatively a pair of such films might be used on opposite sides of a channel.

Figure 39: Simulation of a soft peristalsis tube made of nematic liquid crystal elastomer.

5.4 Self-propelled earthworm

Earthworms move by a propagating wave of muscle contraction, alternately shortening

and lengthening the body along its length. This motion can be replicated in a nematic elas-

tomer by applying a modulation in the magnitude of the nematic order parameter, much as

80

in the pumps modeled above. To enable motion of the center of mass, an earthworm also

needs friction with a nearby surface. The underside of a real earthworm is decorated with

tiny bristles (setae) that allow it to anchor one section of the body while another section is in

motion. To mimic this behavior we could also imagine placing bristles on the underside of a

nematic elastomer earthworm robot. Alternatively, asymmetric static friction can be created

by suitable surface morphology on the substrate. For instance, an elongated monodomain

nematic elastomer film can be made to crawl by placing it on an asymmetric substrate, e.g.

on brushed velvet. Local contraction and elongation can be induced by moving a heat source

just above the film from one end to the other. In the simulation, shown in Video 1, we ap-

ply perfect static friction so that the simulated earthworm cannot slide backwards. Kinetic

friction is also included with a finite value of the friction coefficient.

81

Figure 40: Simulation of a nematic elastomer robotic earthworm moving on a rugged surfaceshaped like a size wave, with height z = A sin(kx).

CHAPTER 6

CONCLUSION

Coupling between strain and orientational order in nematic liquid crystal elastomers can

be observed through a variety of experiments. When an LCE film is subject to an external

mechanical strain, it breaks down into orientational domains of alternating director orienta-

tion. The resulting orientational domain patterns can be observed either optically or using

x-ray techniques [10]. The director reorientation is shown to compensate the energy cost of

the elastic deformation, and is visible in the stress-strain response of the material in the form

of a plateau. The spontaneous change in shape of a strip of liquid crystal elastomer near the

nematic-isotropic phase transition is also an evidence of this coupling [12]. A liquid crys-

tal elastomer doped with azo dyes will deform upon illumination by light of an appropriate

wavelength as a result of the conformation change of the azo dyes that tend to reorient the

liquid crystalline molecules, and hence distort the network strands [20, 64, 73].

We presented simulations of shape change and microstructural evolution in nematic elas-

tomer films using 3-d finite element elastodynamics. Studying these materials at the contin-

uum level, we derived a simulation model that makes no assumption on the detailed chemical

structure, and hence attempts to probe universal characteristics of the response of liquid crys-

tal elastomers subjected to external stimuli. In this dissertation, the mechanical response of

nematic LCE has been investigated in the presence of an externally applied uniaxial strain,

and the simulations results are in a good qualitative agreement with experiments on this class

82

83

of material [11, 55]. This work aimed at bridging the divide between the fundamental micro-

scopic theory of nematic liquid crystal elastomers and their applications.

Clearly, the results showed here are in no way an exhaustive presentation of the contribu-

tion that can be expected of the methods we have developed. There are several avenues that

remain to be studied using our method. As an example, the interaction of nematic elastomers

with an electric field can be modeled by modifying the free energy density to account for the

interaction between a nematic director and an electric field. Similarly, the flexoelectric effect

and the converse flexoelectric effect in NLCE [35] are readily available for simulation. A

minor variation to the nematic interaction part of our free energy density will also allow one

to study buckling instability in cholesteric liquid crystal elastomers [74].

6.1 Future work

6.1.1 Role of the Frank-Oseen elastic energy

It can be seen from the preceding chapters that spatial variations of the orientation of

the nematic director were not penalized by in the simulation reported. At first glance, this

seem to pose a strong limitation to the strength of this method we employ for modeling LCE.

However, one must expect this contribution to the energy to be negligibly small, i.e. several

orders of magnitude smaller than all others. This expectation is motivated by the fact that

the energy scales involved differ considerably. A characteristic length scale that emerges in

nematic elastomers is the nematic penetration length ξ =√

K/µ ≈ 10−8m, using K ≈

10−11N and µ ≈ 105Pa. It is a rough estimate of how far perturbations of the orientational

order would persist in a competition between Frank elasticity and that of the rubbery network.

The natural question is whether the smallness of this term justifies its omission from most

84

reported studies of the deformation of LCE (Desimone, Warner, Kundler); is it sufficient to

break a symmetry? e.g. can its presence shift the threshold for stripe instability? While such

a contribution is expected to be of very little importance in the mechanical response of these

soft materials [52], we outline here how it will be accounted for in future work. In the one

elastic constant approximation, one can write the energy penalty for spatial variations of the

director orientation as

Unem =K

2

∫dV Qij,kQij,k (26)

Upon discretization, it approximates to

Unem ≈ 1

2γ

N∑p=1

nb∑q=1

h3

(Qp

ij −Qqij

h

)2

(27)

where h is the element length, N is the total number of elements, nb is the number of neigh-

bours of the pth element, and γ is the Frank elastic constant. In the current formulation of

the coupling between order and strains, the director is defined at the centroid of the element,

whereas the strains are obtained from the displacement of the nodes, which in turn are in-

terpolated within the element. In other words, strains are piecewise-linear and the nematic

order tensor Qij is piecewise-constant throughout the mesh. In order for the gradients of the

order parameter tensor to be defined, one would need to have a node-centered definition of

Qij , which will be piecewise-linear, just as the strain tensor. In our two-step dynamics, given

an increment in displacement and hence a strain tensor ε∗ij , the relaxation of the directors will

85

consist in minimizing the energy.

0 =N∑p=1

∂Utotal

∂Qpij

(28)

=N∑p=1

[−αε∗ij + β(Qp

ij −Q0ij) + γ

nb∑q=1

h3

(Qp

ij −Qqij

h2

)]

Which is a system of N equations to be solved simultaneously for the Qp. This effectively

means that our algorithm will be of the form :

1- Initialize the system

2- Apply a deformation, find the local strains ε∗ij .

3- Solve for the local Qij as shown above.

4- Go to 2.

6.1.2 Volumetric change (swelling) of LCE

The stimulus-induced shape change of elastomeric materials opens door to a wide range

of engineering applications. Heretofore, efforts have been mainly devoted to obtaining three-

dimensional structures from the imposition of a two-dimensional pattern on an isotropic ma-

terial [75,76]. The reverse problem has been approached as well by Dias et al. [77] who made

a demonstration of how one finds the growth pattern that generates a certain shape. Here we

intend to consider a different class of materials, and show that additional frustrations occur

which might preclude the attainment of the desired shape, or enhance some saillant features

of the otained shape.

86

A thin strip of elastomer with a nonuniform crosslink density will, upon exposure to an exter-

nal stimulus such as a solvent, undergo nonuniform deformations. Predicting the state of the

deformed material becomes even more arduous when the polymer strands have an anisotropic

shape of gyration as is the case in nematic elastomers. When a nematic elastomer is put in

the presence of a solvent, it undergoes a volumetric change that depends on both the sym-

metry of the solvent, and the state of the elastomer. Experiments [27, 78–80] have shown

that monodomain nematic elastomers, when embedded in an isotropic solvent elongate along

the nematic director, and shrink in the transverse directions. This anisotropic swelling is not

observed in polydomain samples. Other studies have focused on the mechanical and elec-

trooptical properties of nematic elastomers swelled in the presence of a constraint such as a

nematic solvent, or an aligning field [81]. The orientation of the liquid crystal mesogens was

shown to play a role in the selection of bending directions in swollen nematic elastomers [82].

We intend to study the case in which the pattern of the crosslinks defines a metric, and hence

a desired shape upon relaxation. Starting from the phenomenological formalism of the strain-

order coupling presented in this work, we will investigate how the presence of the nematic de-

gree of freedom frustrates the shape selection expected for the case of non-uniformly swelled

isotropic elastomer. Clearly, a first, more accessible question will be that of finding the equi-

librium shape of a uniformly swelled nematic elastomer with a patterned nematic director. We

will also study how morphological transitions may occur by altering the pitch of a swelled

cholesteric liquid crystal elastomer, e.g. by prescribing a swelling pattern corresponding to

an unduloid whose eccentricity is a function of the cholesteric pitch.

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Appendix 1 : Fortran codes

The following items will also be available on www.e-lc.org

Fortran codes.

program main

call setconfig call forcecalc call pullforce

call initial call dynamics

call closeall

end

93

parameter(ntet=10422) parameter(ntri=999) parameter(np=3193) integer tetlist(ntet,4) integer npick(4),ntetnab(ntet) integer itime, pref, prefa integer npairs,pullme(np), tetnab(ntet,4) integer ione(100000),itwo(100000) integer myrank, numprocs, ierr, tag, myid! integer status(MPI_STATUS_SIZE) real*8 Rot(3,3),centcent(ntet,4) real*8 k1, r, thet1, dir(ntet),ax,bx,cx,dx

c damping parameter(gamma=0.0) parameter(r=1.5) parameter(k1=0.0001) parameter(thet0=0.0)

parameter(d_0 = 1.0e-5)! parameter(thet1=0.0) parameter(dt=1.0E-7) parameter(dt2o2=dt*dt*0.5d0) parameter(dto2=dt*0.5d0) parameter(irealtime=int(2.0/dt))

! parameter(cxxyy=2.79530E+02)! parameter(cxyxy=5.0E+5)! parameter(cxxxx= Cxxyy + 2.0*Cxyxy)! parameter(cxxxx=100.0*Cxyxy)! parameter(cxxyy=1.0*Cxyxy)

parameter(alpha =0.1)! parameter(bet = 1.1*alpha / 4.0) parameter(mu = 1.0)

parameter(rho=1500.0)

character filename1*16 character filename2*16 character filename3*16 character filename4*16 character filename5*16

integer:: istep, nsteps, tetmax(ntet)

character(len=80) filename, frame

real*8 x(np),y(np),z(np), mass(np) real*8 vx(np),vy(np),vz(np) real*8 fx(np),fy(np),fz(np) real*8 fx1(np),fy1(np),fz1(np) real*8 x0(np),y0(np),z0(np) real*8 fxold(np),fyold(np),fzold(np) real*8 st(3,3), xtime

real*8 u(4),v(4),w(4), theta, phi real*8 a(4),b(4),c(4) real*8 eps(3,3) , lambda(3,3), lam(3,3,ntet) real*8 q(3,3),qfac, qloc(ntet),qlocal real*8 q0(3,3), q0field(3,3), qnab(3,3), q0sav(3,3,ntet)

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real*8 fxsum, fysum,fzsum, vtemp(ntet), vol_n(np), mass_n(np) real*8 xm(ntet,4,4),x00(ntet), y00(ntet), z00(ntet) real*8 xmin,xmax,ymin,ymax,zmin,zmax,xlength,ylength,zlength real*8 etot,petot,pe1,pe2,pe3,ketot,pxtot,pytot,pztot real*8 nx(ntet),ny(ntet), nz(ntet), vec(3,3) real*8 nx0(ntet),ny0(ntet), nz0(ntet) real*8 fztot, fytot real*8 xlength0,ylength0,zlength0 character*4 job

95

common x,y,z,vx,vy,vz,x0,y0,z0,u,v,w,a,b,c,fx,fy,fz,fx1,fy1,fz1, & fxold,fyold,fzold,mass,x00,y00,z00,dir,ntetnab,nx,ny,nz,nx0, & ny0,nz0,pi1,lam,lambda,eps,q,qfac,qloc,vtemp,vol_init, & vol_now,vol_n,mass_n,q0sav,q0,q0field, & qnab,rot,xm,fxsum, fysum,fzsum,theta,phi,thet1, & energy,etot,petot,ketot,pxtot,pytot,pztot,xtime,pe, & xmin,xmax,ymin,ymax,zmin,fytot,zmax,vmax,ax,bx,cx,dx, & xlength,ylength,zlength,xlength0,ylength0,zlength0,qlocal, & tetlist,tetnab,centcent,itime,npairs,ione,itwo,pullme, & irun, pref, myrank,numprocs,ierr,tag,myid, & prefa,nb,ibond,filename1,filename2, & filename3,filename4,filename5

96

subroutine setconfig include 'test-inits1.f'! include 'test-inits2.f' integer lda,nn,iflag real*8 aa(4,4),aasave(4,4),bb(4,4) include 'test-inits2.f'

vmax = 1.0e-2

pi1=4*atan(1.0)fytot = 0.0

filename1='theta2.dat'filename2='pic2.xyz'filename3='st-st2.dat'filename4='dir2.dat'filename5='vol2.dat'

open(unit=37,file=filename1,status='unknown')open(unit=28,file=filename2,status='unknown')open(unit=18,file=filename3,status='unknown')open(unit=31,file=filename4,status='unknown')open(unit=117,file=filename5,status='unknown')

lda=4 nn=4 iflag=0

open(unit=41,file='nodes.dat',status='old')

do 2 i=1,np read(41,*)pref,x(i),y(i),z(i) 2 continue close(unit=41)

open(unit=42,file='tetras.dat',status='old') do 4 itet=1,ntet read(42,*)tetlist(itet,1),tetlist(itet,2), & tetlist(itet,3),tetlist(itet,4)

dir(itet) = thet0 4 continue close(unit=42)

call maxmin

xlength0=xmax-xminylength0=ymax-yminzlength0=zmax-zmin

ylength=ylength0

! if(myid ==0) then write(6,*)'in setconfig xmin-max:',xmin,xmax

write(6,*)'in setconfig ymin-max:',ymin,ymaxwrite(6,*)'in setconfig zmin-max:',zmin,zmax

! endif

97

do 608 itet=1,ntet x00(itet)=0.d0 y00(itet)=0.d0 z00(itet)=0.d0 do 609 i=1,4 ip=tetlist(itet,i) x00(itet)=x00(itet)+x(ip) y00(itet)=y00(itet)+y(ip) z00(itet)=z00(itet)+z(ip) 609 continue x00(itet)=x00(itet)/4.d0 y00(itet)=y00(itet)/4.d0 z00(itet)=z00(itet)/4.d0 608 continue

call nbors

do 5 i=1,np pullme(i)=0 if(y(i).eq.ymin)pullme(i)=-1 if(y(i).eq.ymax)pullme(i)=1

5 continue

c generate aa(i,j) matrix and invert it

do 10 itet=1,ntet do 11 i=1,4 ip=tetlist(itet,i) aa(i,1)=1. aa(i,2)=x(ip) aa(i,3)=y(ip) aa(i,4)=z(ip) 11 continue

do 112 i=1,4 do 13 j=1,4 aasave(i,j)=aa(i,j) 13 continue 112 continue

call matinv(aa,lda,nn,iflag)

do 20 i=1,4 do 21 j=1,4 xm(itet,i,j)=aa(i,j) 21 continue 20 continue

10 continue

do 12 i=1,np x0(i)=x(i) y0(i)=y(i) z0(i)=z(i) vx(i)=0.d0 vy(i)=0.d0

98

vz(i)=0.d0 12 continue

c generate a list of nearest neighbor pairs to usec in the damping calculation

npairs=0! if(myid ==0) write(6,*)'finding pairs'

ihaveit=0

if(ihaveit.eq.0)then

do 240 itet=1,ntet

if(mod(itet,500).eq.0)write(6,*)itet

ip1=tetlist(itet,1) ip2=tetlist(itet,2) ip3=tetlist(itet,3) ip4=tetlist(itet,4)

call checkmatch(ip1,ip2) call checkmatch(ip1,ip3) call checkmatch(ip1,ip4) call checkmatch(ip2,ip3) call checkmatch(ip2,ip4) call checkmatch(ip3,ip4) 240 continue

! write(6,*)'we have ',npairs,' unique near nbor node pairs'

open(unit=50,file='ioneitwo.dat',status='unknown')

write(50,*)npairs do 51 ip=1,npairs write(50,*)ip,ione(ip),itwo(ip) 51 continue close(unit=50)

else open(unit=50,file='ioneitwo.dat',status='old') read(50,*)npairs do 151 ip=1,npairs read(50,*)kk,ione(ip),itwo(ip) 151 continue close(unit=50)! write(6,*)'file read successfully'! write(6,*)'we have ',npairs,' unique near nbor node pairs' endif

do 601 i=1,npmass(i)=0.0

99

601 continue vol0=0.d0

vol_init =0.0do 222 itet=1,ntet

call calcvol0(itet,vol0)vol_init = vol_init + vol0

do 602 i=1,4mass(tetlist(itet,i))=mass(tetlist(itet,i)) + 0.25*vol0*rho

602 continue

vtemp(itet)=vol0222 continue

pysum=0.d0 totmass=0.d0

xcom =0.5*(xmax - xmin)ycom =0.5*(ymax - ymin)zcom =0.5*(zmax - zmin)

do 672 i = 1, npvy(i) = vmax*(y(i)-ycom)/ycom

pysum=pysum+vy(i)*mass(i) totmass=totmass+mass(i)672 continue

write(6,*)'totmass=',totmass write(6,*)'pysum=',pysum

write(6,*)'Vol_init',vol_init

vzave=pysum/totmass do 673 i=1,np vy(i)=vy(i)-vzave 673 continue

return end

100

subroutine initialinclude 'test-inits1.f'

include 'test-inits2.f'

strain=100*(ylength-ylength0)/ylength0 write(18,*) strain, -fytot write(6,*)itime,' strain=',strain,' force= ',-fytot

write(28,*)npwrite(28,*)do i=1,npwrite(28,*)'A',x(i),y(i),z(i)

enddo

vol_now=0.d0

do itet=1,ntet call volchange(itet,vol0)

vol_now = vol_now + vol0enddo

del_vol = 100*(vol_now-vol_init)/vol_initwrite(*,*) 'volume change',del_vol,'percent'write(117,*) strain, del_vol

ketot = 0.0do i=1, npketot = ketot + 0.5*mass(i)*(vx(i)**2 + vy(i)**2 + vz(i)**2)enddo

write(*,*) itime,petot, ketot, petot + ketot

write(6,*)itime,' strain=',strain,' force=',-fytot write(6,*)'initial state has potential energy',petot

xtime=0.d0

returnend

101

subroutine nbors include 'test-inits1.f' include 'test-inits2.f'

write(*,*) 'in nbors' open(unit=47, file='tetnab', status='unknown')

open(unit=48, file='ntetnab', status='unknown')

do i=1,ntet do k=1,4 tetnab(i,k)=0 ntetnab(i)=0 centcent(i,k)=0.d0 enddo enddo

do itet=1,ntet-1do jtet=itet+1,ntet

inombre=0 do kk=1,4 ip=tetlist(itet,kk) do mm=1,4 in=tetlist(jtet,mm) if(ip.eq.in)inombre=inombre+1 enddo enddo

if(inombre .eq. 3) then ntetnab(itet)=ntetnab(itet)+1 tetnab(itet,ntetnab(itet))=jtetrr=dsqrt((x00(itet)-x00(jtet))**2 + (y00(itet)-y00(jtet))**2 +

& (z00(itet)-z00(jtet))**2) centcent(itet,ntetnab(itet))=(rr / d_0)**2! write(*,*)centcent(itet,ntetnab(itet)) ntetnab(jtet)=ntetnab(jtet)+1 tetnab(jtet,ntetnab(jtet))=itet centcent(jtet,ntetnab(jtet))=(rr / d_0)**2 endif

enddo

enddowrite(*,*)'smile12'

do itet=1,ntet if(ntetnab(itet).gt.4)then write(6,*)'error in subroutine nbors' write(6,*)'too many neighbors for tetrahedron= ',itet write(6,*)'number of nbors is=',ntetnab(itet) endif enddo

do itet=1,ntet

102

do k=1,ntetnab(itet) write(47,*) itet,k,tetnab(itet,k),centcent(itet,k) enddo

write(48,*) itet,ntetnab(itet) enddo close(unit=47)

close(unit=48)write(*,*) 'out nbors'

return end

103

subroutine dynamicsinclude 'test-inits1.f'

include 'test-inits2.f'

do 999 itime=1,irealtime

call nodeupdate

if (mod(itime,100).eq.0)thencall maxmincall pullforce

call savepic endif

call oldforcescall forcecalccall velocities

999 continue

returnend

104

subroutine nodeupdateinclude 'test-inits1.f'

include 'test-inits2.f'

xtime=xtime+dt

do 410 i=1,np if(pullme(i).eq.0)then x(i)=x(i)+dt*vx(i)+dt2o2*fx(i)/mass(i) y(i)=y(i)+dt*vy(i)+dt2o2*fy(i)/mass(i) z(i)=z(i)+dt*vz(i)+dt2o2*fz(i)/mass(i)

elsey(i)=y(i)+dt*vy(i)

endif410 continue

returnend

105

subroutine velocities ! Velocities update using the Verlet method include 'test-inits1.f' include 'test-inits2.f'

do i=1,np if(pullme(i).eq.0)then vx(i)=vx(i)+dto2*(fx(i)+fxold(i))/mass(i) vy(i)=vy(i)+dto2*(fy(i)+fyold(i))/mass(i) vz(i)=vz(i)+dto2*(fz(i)+fzold(i))/mass(i) endif enddo return end

106

subroutine epstensor(itet) include 'test-inits1.f' include 'test-inits2.f'

do j=1,4 npick(j)=tetlist(itet,j) enddo

do 200 j=1,4 i=npick(j) u(j)=x(i)-x0(i) v(j)=y(i)-y0(i) w(j)=z(i)-z0(i) 200 continue

c now that I have the u,v,w vectors, all I have to do is a matrixc multiply to find the shape function coefficients.

do 210 i=1,4 a(i)=0.d0 b(i)=0.d0 c(i)=0.d0 210 continue do 211 j=1,4 do 212 i=1,4 a(i)=a(i)+xm(itet,i,j)*u(j) b(i)=b(i)+xm(itet,i,j)*v(j) c(i)=c(i)+xm(itet,i,j)*w(j) 212 continue 211 continue

lam(1,1,itet)= 1.0 + a(2) lam(2,2,itet)= 1.0 + b(3) lam(3,3,itet)= 1.0 + c(4) lam(1,2,itet)= a(3) lam(2,1,itet)= b(2) lam(1,3,itet)= a(4) lam(3,1,itet)= c(2) lam(2,3,itet)= b(4) lam(3,2,itet)= c(3)

eps(1,1)=a(2)+0.5d0*(a(2)*a(2)+b(2)*b(2)+c(2)*c(2)) eps(2,2)=b(3)+0.5d0*(a(3)*a(3)+b(3)*b(3)+c(3)*c(3)) eps(3,3)=c(4)+0.5d0*(a(4)*a(4)+b(4)*b(4)+c(4)*c(4)) eps(1,2)=0.5d0*(a(3)+b(2)+a(2)*a(3)+b(2)*b(3)+c(2)*c(3)) eps(2,1)=eps(1,2) eps(1,3)=0.5d0*(a(4)+c(2)+a(2)*a(4)+b(2)*b(4)+c(2)*c(4)) eps(3,1)=eps(1,3) eps(2,3)=0.5d0*(b(4)+c(3)+a(3)*a(4)+b(3)*b(4)+c(3)*c(4)) eps(3,2)=eps(2,3)

return end

107

subroutine forcecalc include 'test-inits1.f' include 'test-inits2.f'

petot=0.d0

do 100 i=1,np fx(i)=0.d0 fy(i)=0.d0 fz(i)=0.d0100 continue

do itet = 1, ntet! call frank(itet) call test(itet)

enddo ! call rayleigh return end

108

subroutine force(itet) include 'test-inits1.f' include 'test-inits2.f'! Force calculation

pe =0.0

call epstensor(itet)

do i = 1,3 do j = 1,3

lambda(i,j) = lam(i,j,itet) enddo enddo

do j=1,4 npick(j)=tetlist(itet,j) enddo

a1 =a(1)b1 =b(1)c1 =c(1)a2 =a(2)b2 =b(2)c2 =c(2)a3 =a(3)b3 =b(3)c3 =c(3)a4 =a(4)b4 =b(4)c4 =c(4)

thet1 =0.5*ATan2(2*cx + (0.5*(2*a4*b4*(1 + (-1 + alpha)*r) + 1(1 + a2)*b2*(1 + (alpha - r)*r) +

1 a3*(1 + b3)*(1 + (alpha - r)*r) + 1 ((-1 - a2)*b2 + a3*(1 + b3))*(1 + r*(-2 + alpha + r)) 1 *Cos(2*thet0) - (1 + a2 + a3*b2 + b3 + a2*b3) 1 *(1 + r*(-2 + alpha + r))*Sin(2*thet0)))/r, 1 ax - bx + (0.25*(-(a2*(2 + a2)) - a3**2 + b2**2 1 + b3*(2 + b3) + 2*b4**2 + 2*a4**2*(-1 + r) 1 + (alpha*(-(a2*(2 + a2)) - a3**2 - 2*a4**2 1 + b2**2 + b3*(2 + b3)) + 2*(-1 + alpha)*b4**2)*r 1 + (a2*(2 + a2) + a3**2 - b2**2 - 2*b3 - b3**2)*r**2 + 1(1 + r*(-2 + alpha + r))*((2 + a2*(2 + a2) - a3**2 1 - b2**2 + b3*(2 + b3))* Cos(2*thet0) + 1 2*((1 + a2)*a3 - b2*(1 + b3))*Sin(2*thet0))))/r)

dir(itet) = thet1

fxsum=0.d0 fysum=0.d0 fzsum=0.d0

do 14 i=1,4 j=npick(i)

109

m2i=xm(itet,2,i) m3i=xm(itet,3,i) m4i=xm(itet,4,i)

fxsum = -0.5*m4i*((2*(-c2 - b3*c2 + b2*c3)*(b3 - a4*c2 1 - a4*b3*c2 + a4*b2*c3 - b4*c3 + c4 + b3*c4 1- a3*(b2 - b4*c2 + b2*c4) + a2*(1 + b3 - b4*c3 + c4 + b3*c4) - 1 k1))/k1 + 2*a4*(1 + (-1 + 1/r)*Cos(thet1)**2) 1 + 2*b4*(-1 + 1/r)*Cos(thet1)*Sin(thet1) 1 + alpha*(2*a4*Cos(thet1)**2 + 2*b4*Cos(thet1)*Sin(thet1))) - 1 0.5*m3i*((2*(-b2 + b4*c2 - b2*c4)*(b3 - a4*c2 - a4*b3*c2 1 + a4*b2*c3 - b4*c3 + c4 + b3*c4 - a3*(b2 - b4*c2 + b2*c4) 1 + a2*(1 + b3 - b4*c3 + c4 + b3*c4) - k1))/ 1 k1 + (1 + (-1 + 1/r)*Cos(thet1)**2)*(2*a3 1 + 2*a3*(-1 + r)*Sin(thet0)**2 + (1 + a2)*(-1 + r) 1 *Sin(2*thet0)) + 2*(-1 + 1/r)*Cos(thet1)*(b2*(-1 + r) 1 *Cos(thet0)*Sin(thet0) + (1 + b3)*(1 1 + (-1 + r)*Sin(thet0)**2))*Sin(thet1) + 1 alpha*(Cos(thet1)**2*(2*a3*Cos(thet0)**2 - 1 (1 + a2)*Sin(2*thet0)) + 2*Cos(thet1) 1 *((1 + b3)*Cos(thet0)**2 - b2*Cos(thet0) 1 *Sin(thet0))*Sin(thet1))) - 1 0.5*m2i*((2*(1 + b3 - b4*c3 + c4 + b3*c4) 1 *(b3 - a4*c2 - a4*b3*c2 + a4*b2*c3 - b4*c3 1 + c4 + b3*c4 - a3*(b2 - b4*c2 + b2*c4) + 1 a2*(1 + b3 - b4*c3 + c4 + b3*c4) - k1))/k1 1 + (1 + (-1 + 1/r)*Cos(thet1)**2)* 1 (2 + 2*a2 + 2*(1 + a2)*(-1 + r)*Cos(thet0)**2 1 + a3*(-1 + r)*Sin(2*thet0)) + 1 (-1 + 1/r)*Cos(thet1)*(b2 + b2*(-1 + r)*Cos(thet0)**2 1 + (1 + b3)*(-1 + r)*Cos(thet0)*Sin(thet0))*Sin(thet1) + 1 (-1 + 1/r)*Cos(thet1)*(b2*(1 + (-1 + r)*Cos(thet0)**2) 1 + (1 + b3)*(-1 + r)*Cos(thet0)*Sin(thet0))*Sin(thet1) + 1 alpha*(Cos(thet1)**2*(2*(1 + a2)*Sin(thet0)**2 1 - a3*Sin(2*thet0)) + 2*Cos(thet1)*(-((1 + b3) 1 *Cos(thet0)*Sin(thet0)) + b2*Sin(thet0)**2)*Sin(thet1)))

fysum = -0.5*m4i*((2*(a3*c2 - c3 - a2*c3)*(b3 - a4*c2 - 1 a4*b3*c2 + a4*b2*c3 - b4*c3 + c4 + b3*c4 - 1 a3*(b2 - b4*c2 + b2*c4) + a2*(1 + b3 - b4*c3 1 + c4 + b3*c4) - k1))/ 1 k1 + 2*a4*(-1 + 1/r)*Cos(thet1)*Sin(thet1) 1 + alpha*(2*a4*Cos(thet1)*Sin(thet1) + 2*b4*Sin(thet1)**2) 1 + 2*b4*(1 + (-1 + 1/r)*Sin(thet1)**2)) - 1 0.5*m3i*((2*(1 - a4*c2 + c4 + a2*(1 + c4))*(b3 - a4*c2 1 - a4*b3*c2 + a4*b2*c3 - b4*c3 + c4 + b3*c4 - a3*(b2 1 - b4*c2 + b2*c4) + a2*(1 + b3 - b4*c3 + c4 + b3*c4) 1 - k1))/k1 + 2*(-1 + 1/r)*Cos(thet1)*((1 + a2)*(-1 + r) 1 *Cos(thet0)*Sin(thet0) + a3*(1 + (-1 + r)*Sin(thet0)**2))* 1 Sin(thet1) + (2 + 2*b3 + 2*(1 + b3)*(-1 + r)*Sin(thet0)**2 1 + b2*(-1 + r)*Sin(2*thet0))*(1 + (-1 + 1/r)*Sin(thet1)**2) + 1 alpha*(2*Cos(thet1)*(a3*Cos(thet0)**2 - (1 + a2) 1 *Cos(thet0)*Sin(thet0))*Sin(thet1) + (2*(1 + b3) 1*Cos(thet0)**2 - b2*Sin(2*thet0))*Sin(thet1)**2)) - 1 0.5*m2i*((2*(a4*c3 - a3*(1 + c4))*(b3 - a4*c2 1 - a4*b3*c2 + a4*b2*c3 - b4*c3 + c4 + b3*c4 - a3*(b2 1 - b4*c2 + b2*c4) + a2*(1 + b3 - b4*c3 + c4 + b3*c4) - k1))/ 1 k1 + (-1 + 1/r)*Cos(thet1)*(1 + a2 + (1 + a2)*(-1 + r) 1 *Cos(thet0)**2 + a3*(-1 + r)*Cos(thet0) 1 *Sin(thet0))*Sin(thet1)+

110

1 (-1 + 1/r)*Cos(thet1)*((1 + a2)*(1 + (-1 + r) 1 *Cos(thet0)**2) + a3*(-1 + r) 1 *Cos(thet0)*Sin(thet0))*Sin(thet1) + 1 (2*b2 + 2*b2*(-1 + r)*Cos(thet0)**2 + (1 + b3) 1 *(-1 + r)*Sin(2*thet0))*(1 + (-1 + 1/r)*Sin(thet1)**2) + 1 alpha*(2*Cos(thet1)*(-(a3*Cos(thet0)*Sin(thet0)) 1 + (1 + a2)*Sin(thet0)**2)*Sin(thet1) 1 + (2*b2*Sin(thet0)**2 - (1 + b3) 1 *Sin(2*thet0))*Sin(thet1)**2))

fzsum = -0.5*(2*(1 + c4) + (2*(1 - a3*b2 + b3 + 1 a2*(1 + b3))*(b3 - a4*c2 - a4*b3*c2 + a4*b2*c3 1 - b4*c3 + c4 + b3*c4 - a3*(b2 - b4*c2 + b2*c4) + 1 a2*(1 + b3 - b4*c3 + c4 + b3*c4) - k1))/k1)*m4i - 0.5*m2i* 1 (c2 + (2*(-a4 - a4*b3 + a3*b4)*(b3 - a4*c2 - a4*b3*c2 1 + a4*b2*c3 - b4*c3 + c4 + b3*c4 - a3*(b2 - b4*c2 + b2*c4) 1 + a2*(1 + b3 - b4*c3 + c4 + b3*c4) - k1))/ 1 k1 + c2*(-1 + r)*Cos(thet0)**2 + c2*(1 + (-1 + r) 1 *Cos(thet0)**2) + 2*c3*(-1 + r)*Cos(thet0)*Sin(thet0)) - 1 0.5*m3i*((2*(a4*b2 - b4 - a2*b4)*(b3 - a4*c2 - a4*b3*c2 1 + a4*b2*c3 - b4*c3 + c4 + b3*c4 - a3*(b2 - b4*c2 + b2*c4) 1 + a2*(1 + b3 - b4*c3 + c4 + b3*c4) - k1))/ 1 k1 + 2*c2*(-1 + r)*Cos(thet0)*Sin(thet0) 1 + 2*c3*(1 + (-1 + r)*Sin(thet0)**2))

! write(*,*) fxsum, fysum, fzsum! call volchange(itet,vol0) fx(j)=fx(j)+fxsum*vtemp(itet) fy(j)=fy(j)+fysum*vtemp(itet) fz(j)=fz(j)+fzsum*vtemp(itet)14 continue

return end

111

subroutine volchange(itet,vol0) ! Calculate the change in volume of an element.

include 'test-inits1.f' include 'test-inits2.f'

do 778 j=1,4 npick(j)=tetlist(itet,j) 778 continue

xa=x(npick(1)) xb=x(npick(2)) xc=x(npick(3)) xd=x(npick(4))

ya=y(npick(1)) yb=y(npick(2)) yc=y(npick(3)) yd=y(npick(4))

za=z(npick(1)) zb=z(npick(2)) zc=z(npick(3)) zd=z(npick(4))

a11=xa-xb a12=ya-yb a13=za-zb

a21=xb-xc a22=yb-yc a23=zb-zc

a31=xc-xd a32=yc-yd a33=zc-zd

vol0=a11*a22*a33+a12*a23*a31+a13*a21*a32 vol0=vol0-a13*a22*a31-a11*a23*a32-a12*a21*a33 vol0=vol0/6.

vol0=abs(vol0)

return end

112

subroutine frank(itet) include 'test-inits1.f' include 'test-inits2.f'

ax = 0.0bx = 0.0cx = 0.0

jtet =ntetnab(itet)

do k=1,jtetdx = centcent(itet,k)ax = ax + (cos(dir(tetnab(itet,k))) / dx )**2bx = bx + ( sin(dir(tetnab(itet,k)))/ dx )**2 cx = cx + 0.5*sin(2.0*dir(tetnab(itet,k))) / ( dx )**2enddo

ax = 0.1*axbx = 0.1*bxcx = 0.1*cx

return end

113

subroutine pullforce include 'test-inits1.f' include 'test-inits2.f'

fytot = 0.d0do 602 i=1, np if(pullme(i).eq.1) thenfytot=fytot + fy(i)endif

602 continue return end

114

subroutine oldforces include 'test-inits1.f'

include 'test-inits2.f'

do 420 i=1,np fxold(i)=fx(i) fyold(i)=fy(i) fzold(i)=fz(i) 420 continue

returnend

115

subroutine savepicinclude 'test-inits1.f'

include 'test-inits2.f'

strain=100*(ylength-ylength0)/ylength0 write(18,*) strain, -fytot write(6,*)itime,' strain=',strain,' force= ',-fytot

write(28,*)npwrite(28,*)do i=1,npwrite(28,*)'A',x(i),y(i),z(i)

enddo

vol_now=0.d0

thetave =0.0

do itet=1,ntet call volchange(itet,vol0)

vol_now = vol_now + vol0 thetave = thetave + sin(dir(itet))**2

write(31,*) itet, dir(itet)

enddowrite(37,*) strain, thetave/real(ntet)write(6,*) 'dir_rot',thetave/real(ntet)

del_vol = 100*(vol_now-vol_init)/vol_initwrite(*,*) 'volume change',del_vol,'percent'write(117,*) strain, del_vol

returnend

116

SUBROUTINE MATINV (A,LDA,N,IFLAG) implicit real*8(a-h,o-z)CC-----------------------------------------------------------------------C MATINV WRITTEN BY CHARLES P. REEVE, STATISTICAL ENGINEERINGC DIVISION, NATIONAL BUREAU OF STANDARDS, GAITHERSBURG,C MARYLAND 20899CC FOR: COMPUTING THE INVERSE OF A GENERAL N BY N MATRIX IN PLACE,C I.E., THE INVERSE OVERWRITES THE ORIGINAL MATRIX. THE STEPS C OF THE ALGORITHM ARE DESCRIBED BELOW AS THEY OCCUR. ROWC INTERCHANGES ARE DONE AS NEEDED IN ORDER TO INCREASE THEC ACCURACY OF THE INVERSE MATRIX. WITHOUT INTERCHANGES THISC ALGORITHM WILL FAIL WHEN ANY OF THE LEADING PRINCIPALC SUBMATRICES ARE SINGULAR OR WHEN THE MATRIX ITSELF ISC SINGULAR. WITH INTERCHANGES THIS ALGORITHM WILL FAIL ONLYC WHEN THE MATRIX ITSELF IS SINGULAR. THE LEADING PRINCIPALCC [A B C]C SUBMATRICES OF THE MATRIX [D E F] ARE [A] AND [A B] .C [G H I] [D E]CC SUBPROGRAMS CALLED: -NONE-CC CURRENT VERSION COMPLETED JANUARY 15, 1987CC REFERENCE: STEWART, G.W., 'INTRODUCTION TO MATRIX COMPUTATIONS',C ACADEMIC PRESS, INC., 1973C-----------------------------------------------------------------------C DEFINITION OF PASSED PARAMETERSCC * A = MATRIX (SIZE NXN) TO BE INVERTED (REAL)CC * LDA = LEADING DIMENSION OF MATRIX A [LDA>=N] (INTEGER)CC * N = NUMBER OF ROWS AND COLUMNS OF MATRIX A (INTEGER)CC IFLAG = ERROR INDICATOR ON OUTPUT (INTEGER) INTERPRETATION: C -2 -> TOO MANY ROW INTERCHANGES NEEDED - INCREASE MXC -1 -> N>LDAC 0 -> NO ERRORS DETECTEDC K -> MATRIX A FOUND TO BE SINGULAR AT THE KTH STEP OFC THE CROUT REDUCTION (1<=K<=N)CC * INDICATES PARAMETERS REQUIRING INPUT VALUES C-----------------------------------------------------------------------C PARAMETER (MX=100) DIMENSION A(LDA,*),IEX(MX,2) IFLAG = 0CC--- CHECK CONSISTENCY OF PASSED PARAMETERSC IF (N.GT.LDA) THEN IFLAG = -1 RETURN ENDIFCC--- COMPUTE A = LU BY THE CROUT REDUCTION WHERE L IS LOWER TRIANGULARC--- AND U IS UNIT UPPER TRIANGULAR (ALGORITHM 3.4, P. 138 OF THEC--- REFERENCE)C

117

NEX = 0 DO 70 K = 1, N DO 20 I = K, N S = A(I,K) DO 10 L = 1, K-1 S = S-A(I,L)*A(L,K) 10 CONTINUE A(I,K) = S 20 CONTINUECC--- INTERCHANGE ROWS IF NECESSARYC Q = 0.0 L = 0 DO 30 I = K, N R = ABS(A(I,K)) IF (R.GT.Q) THEN Q = R L = I ENDIF 30 CONTINUE IF (L.EQ.0) THEN IFLAG = K RETURN ENDIF IF (L.NE.K) THEN NEX = NEX+1 IF (NEX.GT.MX) THEN IFLAG = -2 RETURN ENDIF IEX(NEX,1) = K IEX(NEX,2) = L DO 40 J = 1, N Q = A(K,J) A(K,J) = A(L,J) A(L,J) = Q 40 CONTINUE ENDIFCC--- END ROW INTERCHANGE SECTIONC DO 60 J = K+1, N S = A(K,J) DO 50 L = 1, K-1 S = S-A(K,L)*A(L,J) 50 CONTINUE A(K,J) = S/A(K,K) 60 CONTINUE 70 CONTINUECC--- INVERT THE LOWER TRIANGLE L IN PLACE (SIMILAR TO ALGORITHM 1.5,C--- P. 110 OF THE REFERENCE) C DO 100 K = N, 1, -1 A(K,K) = 1.0/A(K,K) DO 90 I = K-1, 1, -1 S = 0.0 DO 80 J = I+1, K S = S+A(J,I)*A(K,J) 80 CONTINUE A(K,I) = -S/A(I,I)

118

90 CONTINUE 100 CONTINUECC--- INVERT THE UPPER TRIANGLE U IN PLACE (ALGORITHM 1.5, P. 110 OFC--- THE REFERENCE) C DO 130 K = N, 1, -1 DO 120 I = K-1, 1, -1 S = A(I,K) DO 110 J = I+1, K-1 S = S+A(I,J)*A(J,K) 110 CONTINUE A(I,K) = -S 120 CONTINUE 130 CONTINUECC--- COMPUTE INV(A) = INV(U)*INV(L)C DO 160 I = 1, N DO 150 J = 1, N IF (J.GT.I) THEN S = 0.0 L = J ELSE S = A(I,J) L = I+1 ENDIF DO 140 K = L, N S = S+A(I,K)*A(K,J) 140 CONTINUE A(I,J) = S 150 CONTINUE 160 CONTINUECC--- INTERCHANGE COLUMNS OF INV(A) TO REVERSE EFFECT OF ROW C--- INTERCHANGES OF AC DO 180 I = NEX, 1, -1 K = IEX(I,1) L = IEX(I,2) DO 170 J = 1, N Q = A(J,K) A(J,K) = A(J,L) A(J,L) = Q 170 CONTINUE 180 CONTINUE RETURN END

119

SUBROUTINE jacobi(A,V,NEQ,TL)IMPLICIT REAL*8 (A-H,O-Z)DIMENSION A(NEQ,NEQ),V(NEQ,NEQ)

ZERO = 0.0D0SUM = ZEROTOL = DABS(TL)

C---- SET INITIAL EIGENVECTORS -------------DO 200 I=1,NEQDO 190 J=1,NEQIF (TL.GT.ZERO) V(I,J) = ZERO

190 SUM = SUM + DABS(A(I,J))IF (TL.GT.ZERO) V(I,I) = 1.0

200 CONTINUEC---- CHECK FOR TRIVIAL PROBLEM -----------

IF (NEQ.EQ.1) RETURNIF (SUM.LE.ZERO) RETURNSUM = SUM/DFLOAT(NEQ*NEQ)

C-------------------------------------------C---- REDUCE MATRIX TO DIAGONAL ------------C-------------------------------------------400 SSUM = ZERO

AMAX = ZERODO 700 J=2,NEQIH = J - 1DO 700 I=1,IH

C---- CHECK IF A(I,J) IS TO BE REDUCED -----AA = DABS(A(I,J))IF (AA.GT.AMAX) AMAX = AASSUM = SSUM + AAIF (AA.LT.0.1*AMAX) GO TO 700

C---- CALCULATE ROTATION ANGLE ----------AA=ATAN2(2.0*A(I,J),A(I,I)-A(J,J))/2.0SI = DSIN(AA)CO = DCOS(AA)

C---- MODIFY "I" AND "J" COLUMNS --------DO 500 K=1,NEQTT = A(K,I)A(K,I) = CO*TT + SI*A(K,J)A(K,J) = -SI*TT + CO*A(K,J)TT = V(K,I)V(K,I) = CO*TT + SI*V(K,J)

500 V(K,J) = -SI*TT + CO*V(K,J)C---- MODIFY DIAGONAL TERMS -------------

A(I,I) = CO*A(I,I) + SI*A(J,I)A(J,J) =-SI*A(I,J) + CO*A(J,J)A(I,J) = ZERO

C---- MAKE "A" MATRIX SYMMETRICAL -------DO 600 K=1,NEQA(I,K) = A(K,I)A(J,K) = A(K,J)

600 CONTINUEC---- A(I,J) MADE ZERO BY ROTATION ------700 CONTINUEC---- CHECK FOR CONVERGENCE -------------

IF(DABS(SSUM)/SUM .GT.TOL)GO TO 400RETURNEND

120