HYBRID PARTICLE-FINITE ELEMENT ELASTODYNAMICS SIMULATIONS OF NEMATIC LIQUID CRYSTAL ELASTOMERS

Transcription

1 HYBRID PARTICLE-FINITE ELEMENT ELASTODYNAMICS SIMULATIONS OF NEMATIC LIQUID CRYSTAL ELASTOMERS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Badel L. Mbanga May 2012

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3 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 ~~~ Dr. Eugene C. Gartland, Members, Doctoral Dissertation Committee Dr.~~ ~- ~ /\ - (:1. 7 ~ ;/ ~--' -!.-;G:...-_( _~~_~ l- ~ Dr~ LianjkhY Ch;6n '--/ -J Dr. Timothy Moerland Accepted by Chair. Department of Chemical Physics Denn, College of Arts and Science ii ii

4 TABLE OF CONTENTS LIST OF FIGURES vi Acknowledgements xi Dedication xii 1 INTRODUCTION Liquid Crystals Liquid Crystal Elastomers Isotropic-Nematic transition in Nematic LCE Photoexcitation in NLCE Freedericksz transition in NLCE Flexoelectric effect in NLCE Strains, strain energy, rubber elasticity Strain tensor Strain energy Rubber elasticity Theory of NLCE De Gennes phenomenological theory Neo-classical theory iii

5 2 FINITE ELEMENT ELASTODYNAMICS SIMULATIONS OF LCE Introduction Forces calculations Algorithm Dissipation Simulations of Rubbery materials Simulations of LCE Isotropic-Nematic phase transition Semisoft Elasticity POLYDOMAIN-MONODOMAIN TRANSITION IN NEMATIC ELASTOMERS Introduction Initial configuration Simulation and results Discussion MODELING THE STRIPE INSTABILITY IN NEMATIC ELASTOMERS Introduction Simulations and results Stripes width Threshold for stripe instability Rate of strain Discussion iv

6 5 MODELING DEVICES Polarization tuner Actuators Peristaltic pumps Self-propelled earthworm CONCLUSION Future work Role of the Frank-Oseen elastic energy Volumetric change (swelling) of LCE BIBLIOGRAPHY v

7 LIST OF FIGURES 1 Some liquid crystal phases Cyanobiphenyl, a commercially available liquid crystal molecule Hexaazatriphenylene liquid crystal (hat), a discotic liquid crystal forming molecule cyano-resorcinol, an example of a bent-core liquid crystal molecule Nematic and columnar phases of discotic liquid crystals Distortions of nematic liquid crystals Liquid crystal elastomers interaction with external stimuli. Image courtesy Dr. B. Ratna Main chain and side chain polymers Isotropic nematic transition in nematic elastomers Photoexcitation in nematic elastomers. The conformation change of the Azo dyes drives a shape change of the polymer matrix Freederickzs transition in liquid crystals Relative positions of two material points in the reference (dr) and target (dr) spaces Soft elastic deformation. The director rotates to accommodate the strain Soft vs. semisoft response in a nematic elastomer (a) and (b) show a tetrahedron in the unstrained and deformed states respectively 30 vi

8 16 Beam of rubber twisting Potential (black dots) and kinetic (red dots) energy of the elastic beam experiencing torsional deformation. In the absence of dissipation, the energy is conserved (green dots) 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 Cartoon of the semisoft response in a NLCE 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 Initial configurations with different thermomechanical histories 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 Texture and director configuration in a region near the center of the strip in the polydomain state of I-PNE Texture and director configuration in a region near the center of the strip in the monodomain state of I-PNE vii

9 25 Texture and director configuration in a region near the center of the strip in the polydomain state of N-PNE Texture and director configuration in a region near the center of the strip in the monodomain state of N-PNE 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) 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) Experiment: stretching a nematic elastomer film at an angle of 90 o to the director results in a microstructure consisting stripes of alternating director orientation. Image courtesy H. Finkelmann Simulation: stretching a nematic elastomer film at an angle of 90 o to the director. Initially a monodomain, the director field evolves to form a striped microstructure Engineering stress (circles) and director rotation (squares) vs applied strain, for the system shown in Figure 30. Onset of director rotation and the stressstrain plateau both occur at the same strain Engineering stress (circles) and director rotation (squares) vs applied strain, applied at an angle of 60 o from the nematic director Simulation: stretching a nematic elastomer film at an angle of 60 o to the director. Initially a monodomain, the director field rotates smoothly without sharp gradients in orientation viii

10 34 Dependence of the stress-strain response on strain rate Simulation: A nematic elastomer disk is stretched radially. The director field smoothly transforms from a homogeneous monodomain to a radial configuration Cartoon of the polarization modulation by the elastomer strip Simulation of the photo-deformation of a beam of nematic liquid crystal elastomer Simulation of the photobending of a strip of polydomain nematic liquid crystal elastomer Simulation of a soft peristalsis tube made of nematic liquid crystal elastomer Simulation of a nematic elastomer robotic earthworm moving on a rugged surface shaped like a size wave, with height z = A sin(kx) ix

11 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

12 To my family xi

13 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 partially 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 depending 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

14 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 dimensions 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

15 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 characterized 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 another. 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 Figure 5(a). The direction of average orientation of the molecules is called the director n(r). n

16 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 discotic liquid crystal (b) Columnar of a discotic crystal phase liquid 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 Q ij = S(n i n j 1 3 δ ij) is a suitable quantity for characterizing the transition. n is the local nematic director. S = 1 2 < 3v in i 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 1 polynomial, S = 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

17 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 5 expansion in Q ij E = 1 2 a(1 T c T )Q ijq ij bq ijq jk Q ki c[q ijq ij ] 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 2 K 11( n) K 22( n n) K 33( n n) 2 (2) Here K 11, K 22, and K 33 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.

18 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].

19 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. Conversely, 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

20 8 LCE can exist in the main chain or side chain configurations. The former has the mesogens 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 polymer 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 reported the first liquid crystal elastomer formed with nematogens attached to the backbone

21 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] 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, elongating 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 example, 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. Experiments 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

22 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 monodomain 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 residual 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

23 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 crystal 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 besides 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 dyes drives 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

24 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 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: F frank = 1 2 K( n)2 1 2 K(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 F frank = 1 2 K(1 d )2. The contribution of the electrostatic energy can be expressed in the form: F electric = 1 2 ϵ o ϵe 2 d (4) where ϵ is the dielectric anisotropy, E is the applied electric field, and ϵ o is the dielectric

25 13 permittivity of free space. The electric energy will overcome the elastic energy when the field reaches the critical value: E c = π K (5) d ϵ o ϵ K This corresponds to a voltage V c = π which is independent of the sample thickness, ϵ o ϵ 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 E (a) With the field OFF (b) With the field ON 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: E c = µ ϵ o ϵ (6)

26 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] 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. Indeed, 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]: P f = e 1 n( n) + e 3 n( 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 nematic 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 (e 3 50nC/m) to be three orders of magnitude larger than those obtained from estimation [30, 31] and measurements with calamitic liquid crystals such as

27 15 5CB [32, 33]. Simulations by Dhakal and Selinger [34] have found similar results. Subsequent experiments have also approached the converse flexoelectric effect, which is characterized 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 periodic 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 deformations 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 nematic liquid crystal. Observe that both experiments yielded bend flexoelectric coefficients comparable to that of the low molecular weight bent core nematic. These experiments reinforce 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 dissertation 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

28 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 λ ij = R i r j 16 referred 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 dr 2 dr 2 = (λ ik dr k )(λ il dr l ) (δ ik dr k )(δ il dr l ) = (λ ik λ il δ ik δ il )dr k dr l = (λ ik λ il δ kl )dr k dr l 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: λ ij = R i r j = r i r j + u i r j = δ ij + u i r j

29 The aforementioned square of the change in separation between neighbouring material points can thus be expressed as 17 dr 2 dr 2 = (λ ik λ il δ kl )dr k dr l = [ (δ ik + u i )(δ il + u ] i ) δ kl dr k dr l r k r l = ( u l r k + u k r l = 2ε kl dr k dr l + u i r k u i r l )dr k dr l The quantity ε kl = 1 2 (λ ikλ il δ kl ) 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 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

30 18 quadratic function of the strain tensor. U = 1 2 C ijklε ij ε kl Here ε ij is the Green-Lagrange strain tensor, C ijkl represents the elastic constants associated with the material. The stiffness tensor C ijkl is a function of the shear modulus and Poisson ratio of the material. We note that since the Green-Lagrange strain ε ij is defined in the material or body frame, we must also define C ijkl 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 = R θ F. The stretching tensor is F = λ T λ, and the rotation is R θ = λf 1. A deformation such that F = δ ij is a mere rotation of the body and costs no elastic energy. In its eigenframe, the stretching tensor is expressed as F = λ T λ = λ λ 2 0 (8) 0 0 λ 3

31 19 such that C = F 2 = λ T λ = λ λ λ 2 3 ; ε = λ λ λ (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 (λ 1, λ 2, λ 3 ), its invariants are : I 1 (C) = λ λ λ 2 3 I 2 (C) = λ 2 1λ λ 2 1λ λ 2 2λ 2 3 J(C) = λ 1 λ 2 λ 3 The neo-hookean strain energy density for an incompressible material is expressed as : U neo H = µ 2 (I 1 3) + C o (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. U MR = 1 2 C m1(ī1 3) C m2(ī2 3) where Ī1 = I 1 J 2/3 and Ī2 = I 2 J 4/3. The coefficients C m1 and C m2 are related to the shear and bulk moduli of the materials.

32 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 L c = 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 3 P (R) = ( 2πb 2 N 2 )3/2 exp[ 3R 2 /2bL c ] The entropic free energy of the polymer chain in this configuration is thus F = k B T ln(z(r)) = F o + k B T (3R 2 /2bL c ) = F o + k B T (3R 2 /2R 2 o) For N x chains in a volume element, one obtains F = F o + µ(3r 2 /2R 2 o) (10) where µ = N x k B T is the shear modulus.

33 Theory of NLCE 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 applications. 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: ω ij = 1 2 ( iu j j u i ). Here u i are the components of the displacement vector u. This rotation can further be decoupled into a rotation around the director (ω = 1 2 n iϵ ijk ω jk ), and two rotations around axes perpendicular to it (ω = n j ω ij ), 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. E 1 = 1 2 D 1 [(Ω ω) n] 2 = 1 2 D 1 Ω i Ωi

34 22 E 2 = 1 2 D 2 n ε (Ω ω) n = 1 2 D 2 Ω i ε jk n j δ ik where δ ik = δ ik n i n k E 3 = 1 2 C ijklε ij ε kl As can be seen, the first expression (E 1 ) is the cost for relative rotations between the director and the polymer matrix. The second expression (E 2 ) couples the strain tensor to the relative rotations of the polymer network and the director field. Finally, E 3 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 2 K 11( n) K 22(n n) K 33(n n) D 1Ω i Ω i D 2Ω i ε jk n j δik C ijklε ij ε kl 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 D 1 and D 2 and microscopic and macroscopic 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.

35 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 interactions of NLCE with external stimuli requires a more strict, preferably microscopic approach. The most successful model for describing NLCE is the neo-classical theory introduced 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 quantity is the step length tensor l ij, which is usually isotropic in ordinary polymers. The chain anisotropy r is usually more pronounced in main chain NLCE than in side chain materials. l 0 0 l ij = 0 l l 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 (11)

36 24 the step length tensor depends on the nematic director as: l ij = l δ ij + (l l )n i n j (12) The sample in the reference state exists in a monodomain conformation with a director n o and step length tensor lij. o An arbitrarily chosen polymer chain has a fully extended contour length L c and end-to-end distance R o. The mean square displacement is Ri o Rj o = 1 3 L clij o. 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 (R o ) = Det [ l o ij ] exp[ 3 2L c R o i (l o ij) 1 R o j] (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 R i = λ ij ( o R j ) where λ ij is the macroscopic deformation gradient tensor. The current step length tensor also differs from the reference l ij l o ij and the probability distribution of the end-to-end length of the chain is P (R) = Det [l ij ] exp( 3 2L c R i l 1 ij R j) (14) Using this, one easily obtains the elastic energy as : F = 1 2 N xk B T Tr[l o ij λ T ij l 1 ij λ ij ] (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

37 25 above, with the substitution λ ij = δ ij + ε ij, yields the coefficients of de Gennes relative rotations between nematogens and the polymer network Soft and semisoft elasticity D 1 = N x k B T (l l ) 2 l l D 2 = N x k B T l2 l2 l l (16) 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 elastomers, 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 director, 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

38 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 λ ij = l 1/2 ij R θ (l o ij) 1/2, where R θ is an arbitrary rotation by an angle θ, one obtains a zero energy cost from the configuration with step length tensor lij o to that with step length tensor l ij. Although this is easily thought of for a single polymer strand, it is clearly an idealization, 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 elasticity [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 accommodate the strains.

39 Figure 14: Soft vs. semisoft response in a nematic elastomer. 27

40 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 macroscopic 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

41 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.

42 30 u(x, y, z) = a 1 + a 2 x + a 3 y + a 4 z v(x, y, z) = b 1 + b 2 x + b 3 y + b 4 z w(x, y, z) = c 1 + c 2 x + c 3 y + c 4 z In the unstrained states (Figure 15(a)), the vertices of the tetrahedron are at coordinates (x 1, y 1, z 1 ), (x 2, y 2, z 2 ), (x 3, y 3, z 3 ), and (x 4, y 4, z 4 ). 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

43 31 u 1 u 2 u 3 u 4 = 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 a 1 a 2 a 3 a 4 v 1 v 2 v 3 v 4 = 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 b 1 b 2 b 3 b 4 w 1 w 2 w 3 w 4 = 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 c 1 c 2 c 3 c 4 Here u, v, and w are displacements in the x, y, and z directions respectively. The coefficients of the mapping functions a i, b i, and c i 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 deformation gradient tensor, strain tensor, the total elastic energy, and the forces on its vertices as follows:

44 32 λ ij = 1 + a 2 a 3 a 4 b b 3 b 4 c 2 c c 4 ε 11 = u x [ ( u ) 2 + x = a (a2 2 + b c 2 2) ( ) 2 v + x ( ) ] 2 w x ε 22 = v y [ ( u ) 2 + y = b (a2 3 + b c 2 3) ( ) 2 v + y ( ) ] 2 w y ε 33 = w z [ ( u ) 2 + z = c (a2 4 + b c 2 4) ( ) 2 v + z ( ) ] 2 w z The off-diagonal components of the strain tensor are found as well. ε 12 = 1 2 ( u y + v ) + 1 x 2 ( u u x y + v x = 1 2 (a 3 + b 2 ) (a 2a 3 + b 2 b 3 + c 2 c 3 ) v y + w x ) w y

45 33 ε 13 = 1 2 ( u z + w ) + 1 x 2 ( u u x z + v x = 1 2 (a 4 + c 2 ) (a 2a 4 + b 2 b 4 + c 2 c 4 ) v z + w x ) w z ε 23 = 1 2 ( v z + w ) + 1 y 2 ( u u y z + v v y z + w y = 1 2 (b 4 + c 3 ) (a 3a 4 + b 3 b 4 + c 3 c 4 ) ) w z Rewriting the strain energy U = 1 2 C ijklε ij ε kl in terms of the strain tensor, one obtains: U = 1 2 C xxxx(ε 2 11+ε 2 22+ε 2 33)+2C xxyy (ε 11 ε 22 +ε 22 ε 33 +ε 33 ε 11 )+4C xyxy (ε 2 12+ε 2 23+ε 2 31) (17) The elastic constants C ijkl are obtained as C xxxx = λ + 2µ C xxyy = λ C xyxy = 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)

46 Forces calculations The force on the i th node is then obtained as derivatives of U with respect to the node displacement. F x,i = C xxxx ε 11 M 2i (1 + a 2 ) C xxxx ε 22 M 3i a 3 C xxxx ε 33 M 4i a 4 C xxyy ε 11 M 3i a 3 C xxyy ε 22 M 2i (1 + a 2 ) C xxyy ε 22 M 4i a 4 C xxyy ε 33 M 3i a 3 C xxyy ε 33 M 2i (1 + a 2 ) C xxyy ε 11 M 4i a 4 2C xyxy ε 12 M 3i (1 + a 2 ) 2C xyxy ε 12 M 2i a 3 2C xyxy ε 23 M 4i a 3 2C xyxy ε 23 M 3i a 4 2C xyxy ε 31 M 4i (1 + a 2 ) 2C xyxy ε 31 M 2i a 4 F y,i = C xxxx ε 11 M 2i (1 + b 2 ) C xxxx ε 22 M 3i a 3 C xxxx ε 33 M 4i a 4 C xxyy ε 11 M 3i a 3 C xxyy ε 22 M 2i (1 + a 2 ) C xxyy ε 22 M 4i a 4 C xxyy ε 33 M 3i a 3 C xxyy ε 33 M 2i (1 + a 2 ) C xxyy ε 11 M 4i a 4 2C xyxy ε 12 M 3i (1 + a 2 ) 2C xyxy ε 12 M 2i a 3 2C xyxy ε 23 M 4i a 3 2C xyxy ε 23 M 3i a 4 2C xyxy ε 31 M 4i (1 + a 2 ) 2C xyxy ε 31 M 2i a 4 F z,i = C xxxx ε 11 M 2i (1 + a 2 ) C xxxx ε 22 M 3i a 3 C xxxx ε 33 M 4i a 4 C xxyy ε 11 M 3i a 3 C xxyy ε 22 M 2i (1 + a 2 ) C xxyy ε 22 M 4i a 4 C xxyy ε 33 M 3i a 3 C xxyy ε 33 M 2i (1 + a 2 ) C xxyy ε 11 M 4i a 4 (18) 2C xyxy ε 12 M 3i (1 + a 2 ) 2C xyxy ε 12 M 2i a 3 2C xyxy ε 23 M 4i a 3 2C xyxy ε 23 M 3i a 4 2C xyxy ε 31 M 4i (1 + a 2 ) 2C xyxy ε 31 M 2i a 4

47 Here the quantities M 2i, M 3i, and M 4i are the components of the inverse of the shape matrix 35 defined as 1 M = 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3. 1 x 4 y 4 z 4 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 approximation, 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

48 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]: x i (t + t) = x i (t) + v i (t) t a i(t)( t) 2 v i (t + t) = v i (t) + a i(t) + a i (t + t) t 2 This finite element explicit dynamics algorithm closely resembles the familiar molecular 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 modified form of Kelvin dissipation. In its standard form, the Kelvin dissipation force (e.g. between 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 conservation 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

49 their absolute velocity difference. With these modifications, the dissipation force between a pair of neighboring nodes separated by distance d is F 12 = η (v 1 v 2 ) (r 1 r 2 ) ˆr 12, with d 12 η = 10 7 kg.m/sec. The resulting dissipation is isotropic in character and does not depend on the orientation of the director field 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 = 1.5MP a, and Poisson ratio very close to the

50 38 e Figure 17: Potential (black dots) and kinetic (red dots) energy of the elastic beam experiencing torsional deformation. In the absence of dissipation, the energy is conserved (green dots). incompressibility limit ν = 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 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 distribution 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.

51 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 strainorder 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 = U strain αε ij Q ij (19) The nematic order parameter tensor Q ij 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 ˆn ref used to construct Q ij lives in the reference frame; that is, ˆn ref is obtained by an inverse transformation from the target to the reference space before coupling the Green-Lagrange strain tensor ε ij to Q ij. 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 ˆn t 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 λ T = R T V. First, coupling the left Green tensor to the order parameter reads:

52 40 (λλ T )(ˆn t ˆn t ) = ˆn t (λλ T )ˆn t = λ T ˆn t 2 = R T V ˆn t 2 = V ˆn t 2 If we write ˆn ref = R T ˆn t, then coupling the right Green tensor to the order parameter tensor in the reference space reads: (λ T λ)(ˆn ref ˆn ref ) = λˆn ref 2 = λr T ˆn t 2 = V ˆn t 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

53 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 2 µ[tr(λλt ) αq (λλ T )] = 1 2 µ[tr(λλt ) α{λλ T S(ˆn ˆn 1 3 I)}] = 1 2 µ[tr(λλt ) α S {ˆn (λλ T )ˆn 1 3 Tr(λλT )}] = 1 2 µ[tr(λλt ) α S { λ T ˆn Tr(λλT )}] = 1 2 µ[(1 + αs 3 ) Tr(λλT ) α S { λ T ˆn 2 }] Similarly, assuming an isotropic reference state, i.e. l o = I, we write neoclassical energy as: U = 1 2 µ Tr(l 1 λλ T ) = 1 [(( 1 2 µ Tr 1 ) l = 1 2 µ Tr [( 1 l 1 l = 1 [( 1 2 µ 1 ) l l l ˆn ˆn + 1 ) ] λλ T l ) (λλ T )(ˆn ˆn) + 1 l λλ T λ T ˆn l Tr(λλ T ) 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 ] ]

54 ( 1 of the step length tensor as: αs = l e 1 ). Here one defines l e = l l 3 l + 2l 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 = P a and B r = P a, 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] Semisoft Elasticity Following the description of the semisoft elastic response presented in the preceding chapter, 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.

55 43 Figure 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. U = µ 1 4 (1 r 1)(2a 2(a 3 cos(2θ)((r 1) cos(2ϕ) + r + 1) + (r 1) sin(2ϕ)((b 3 + 1) cos(2θ) b 2 sin(2θ)) (r 1) sin(2θ) cos(2ϕ) (r + 1) sin(2θ)) + 2(r 1) sin(2ϕ) ((a 3 (b 3 + 1) b 2 ) sin(2θ) + (a 3 b 2 + b 3 + 1) cos(2θ)) +(r 1) cos(2ϕ)((a b 2 2 b 3 (b 3 + 2) 2) sin(2θ) + 2(a 3 b 2 (b 3 + 1)) cos(2θ)) +a 2 2 sin(2θ)( ((r 1) cos(2ϕ) + r + 1)) + a 2 3r sin(2θ) + 2a 3 r cos(2θ) +a 2 3 sin(2θ) + 2a 3 cos(2θ) b 2 2r sin(2θ) + b 2 3r sin(2θ) +2b 3 r sin(2θ) + 2b 2 r cos(2θ) + 2b 2 b 3 r cos(2θ) b 2 2 sin(2θ) +b 2 3 sin(2θ) + 2b 3 sin(2θ) + 2b 2 cos(2θ) + 2b 2 b 3 cos(2θ) 2c 2 2 sin(2θ) + 2c 2 3 sin(2θ) + 4c 2 c 3 cos(2θ))

56 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 perpendicular to the director orientation, one uses ϕ = π. This yields a rotation of the director 2 by an amount θ = 1 ( ) 2((a 2 + 1)a 3 + b 2 (b 3 + 1)r + c 2 c 3 ) 2 tan 1 (a 2 + 1) 2 a b 2 2r (b 3 + 1) 2 r + c 2 2 c 2 3 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 subsequent 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 evolution, it is important to allow the strains to vary locally in response to changes in the director orientation and vice-versa.

57 CHAPTER 3 POLYDOMAIN-MONODOMAIN TRANSITION IN NEMATIC ELASTOMERS 3.1 Introduction Liquid crystal elastomers, when crosslinked, usually exist in the polydomain configuration, 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 twostep 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 stressstrain 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

58 46 Figure 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. 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.

59 47 Recent work by Uruyama and coworkers [55] shed some light on this mystery by studying the P-M transition in samples with the same chemical composition but different thermomechanical history. They prepared polydomain samples in two different ways. Nematiccrosslinked 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 transition 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 latticebased approaches, strain is treated as a global variable playing the role of an external applied

60 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

61 49 for each element that comprises three parts: U = U strain + U strain order + U memory (20) The first term is the well-known neo-hookean strain energy: U strain = 1 2 µ Tr(λ kiλ kj ) + K(Det[λ ij ] 1) 2 (21) where λ ij is the deformation gradient tensor. Volume conservation is maintained by a large bulk modulus K. The second term U strain order describes the coupling between mechanical strain and orientational order: U strain order = αε ij (Q ij Q o ij) (22) Here we use the rotationally invariant Green-Lagrange strain tensor ε ij = 1 2 (λ ikλ kj δ ij ), defined locally for each element. Local orientational order is characterized by Q ij, the symmetric and traceless uniaxial order parameter tensor. Q o ij 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 Q o ij and α depend on the details of sample preparation as discussed below. The third term U memory describes the crosslink memory effect: U memory = 1 2 β(q ij Q o 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 Q o ij. The

62 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 memory of their initial state, and thus larger β, than those crosslinked in the isotropic phase. The uniaxial order parameter tensor Q ij 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 Q ij 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 : F x,i = K (2 (b 4 c 3 (b 3 + 1) (c 4 + 1)) M 2i Z + 2 (a 3 (c 4 + 1) a 4 c 3 ) M 3i Z +2 (a 4 (b 3 + 1) a 3 b 4 ) M 4i Z) +µ ((a 2 + 1) M 2i + b 2 M 3i + c 2 M 4i ) F y,i = K (2 (b 2 (c 4 + 1) b 4 c 2 ) M 2i Z + 2 (a 4 c 2 (a 2 + 1) (c 4 + 1)) M 3i Z +2 ((a 2 + 1) b 4 a 4 b 2 ) M 4i Z) +µ (a 3 M 2i + (b 3 + 1) M 3i + c 3 M 4i ) F z,i = K (2 ((b 3 + 1) c 2 b 2 c 3 ) M 2i Z + 2 ((a 2 + 1) c 3 a 3 c 2 ) M 3i Z +2 (a 3 b 2 a 2 (b 3 + 1) b 3 1) M 4i Z) + µ (a 4 M 2i + b 4 M 3i + (c 4 + 1) M 4i )

63 51 where Z = (c 2 (a 4 (b 3 + 1) a 3 b 4 ) + c 3 ((a 2 + 1) b 4 a 4 b 2 ) (c 4 + 1) ( a 3 b 2 + a 2 (b 3 + 1) + b 3 + 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 Q ij 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 ˆn = (sin θ cos ϕ, sin θ sin ϕ, 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 freedom, 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 U strain order and U memory. 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. minimize θ,ϕ U(θ, ϕ) This algorithm allows us to simultaneously model microstructural evolution and macroscopic shape change of the sample.

64 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 ( ˆn i ˆn j ) 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 approximately 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

65 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 = P a, bulk modulus B r = P a. The strain-order and memory coupling parameters for are α = µ and β = 0.3µ, respectively. In these simulations, we used γ = 10 7 J. 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 magnitude 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

66 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 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. 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.

67 55 Figure 23: Texture and director configuration in a region near the center of the strip in the polydomain 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.

68 56 Figure 24: Texture and director configuration in a region near the center of the strip in the monodomain 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

69 57 Figure 25: Texture and director configuration in a region near the center of the strip in the polydomain 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 = P 2 (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

70 58 Figure 26: Texture and director configuration in a region near the center of the strip in the monodomain 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

71 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) vs strain 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.

72 60 DL/L Figure 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). 3.4 Discussion This chapter presented simulations of the polydomain to monodomain transition in liquid 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 liquid 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 thermomechanical 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

73 class of material [55]. 61

74 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 70 o [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 future to nematic liquid crystal elastomers for engineering applications based on soft actuators [64]. An interesting application that was proposed was building an acoustic wave polarizer by modulating the internal degree of freedom, namely the nematic director. Using the aforementioned 3-d finite element elastodynamics simulation, we investigate 62

75 63 Figure 29: Experiment: stretching a nematic elastomer film at an angle of 90 o to the director results in a microstructure consisting stripes of alternating director orientation. Image courtesy 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 element 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 analogous 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

76 64 mesh down to the micron scale, we resolve the formation of orientational domains, and because 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 Q ij = 1S(3n 2 in j δ ij ), which is the associated symmetric and traceless nematic order tensor. The initial state is taken to be a monodomain with n = n o 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: H elastic = p V p 1 2 C ijklε p ij εp kl + i 1 2 m iv 2 i. (24) Here the first term represents elastic strain energy, with p summing over volume elements. V p is the volume of element p in the reference state. For an isotropic material the components of the elastic stiffness tensor C ijkl are determined from only two material parameters, namely the

77 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 ε ij = 1(u 2 i,j + u j,i + u k,i u k,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, m i is the effective mass and v i 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, H nematic = p V p [ αε p ij (Qp ij Qrp ij ) + β(qp ij Qrp ij )2]. (25) The first term describes coupling between the strain and order parameter tensors using a form proposed by DeGennes [16]. Here Q pr ij 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 Q ij, Q p r ij, and ε ij are all defined in the body frame, i.e. they are invariant under rotations in the target frame. See [68] for the relation between Q ij 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.

78 66 The strain tensor ε ij within each tetrahedral element is calculated in two steps. We calculate the displacement u of each node from the reference state, then perform a linear interpolation of the displacement field within the volume element in the reference state. The resulting interpolation coefficients represent the derivatives u i,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 Q ij 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 modified form of Kelvin dissipation. In its standard form, the Kelvin dissipation force (e.g. between two particles, or between two nodes in a finite element mesh) is proportional to the

79 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 conservation 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 F 12 = η (v 1 v 2 ) (r 1 r 2 ) ˆr 12. The result- d 12 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 = P a, bulk modulus B r = P a, and parameters α = µ, β = 0.3µ, and ζ = 10 7 kg.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 45 o 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. 67

80 68 Figure 30: Simulation: stretching a nematic elastomer film at an angle of 90 o to the director. Initially a monodomain, the director field evolves to form a striped microstructure 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 element 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.

81 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 order 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/λ 3 1; where ξ is the nematic penetration length, L is the sample width, and λ 1 is the strain threshold of the instability. Indeed, using ξ = 50nm, L = m, and λ 1 = 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 monodomain 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 sin 2 (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 demonstrates, 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.

82 S 0.8 σ( N/ mm 2 ) S S DL/L Figure 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 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 60 o 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 70 o and 80 o to the pulling direction and again found no stripe formation and no plateau in the stress-strain response.

83 S 0.8 σ( N/ mm 2 ) S DL/L 0 Figure 32: Engineering stress (circles) and director rotation (squares) vs applied strain, applied at an angle of 60 o from the nematic director 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 between 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.

84 72 Figure 33: Simulation: stretching a nematic elastomer film at an angle of 60 o to the director. Initially a monodomain, the director field rotates smoothly without sharp gradients in orientation. 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

85 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.

86 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 introduced 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 microscopic 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 photoactuators to robotic earthworms. 5.1 Polarization tuner The basic principle of operation of a wave plate consists in modifying the state of polarization 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 λ n wave plate if it retards one state of polarization by π n 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 thickness 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 around 74

87 75 Figure 35: Simulation: A nematic elastomer disk is stretched radially. The director field smoothly 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 Terentjev [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

88 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 Q ij to zero only within volume elements on the top of the sample. This mimics the fact that the light is rapidly attenuated 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

89 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

90 Figure 38: Simulation of the photobending of a strip of polydomain nematic liquid crystal elastomer. 78

91 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 elastomer by applying a modulation in the magnitude of the nematic order parameter, much as