A Landau de Gennes theory of liquid crystal elastomers

In this article, we study minimization of the Landau-de Gennes energy for liquid crystal elastomer.The total energy, is of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal. Our model consider two sources of anisotropy represented by the traceless nematic order tensor $Q$ (rigid units), and the positive definite step-length tensor $L$ (network). This work is motivated by the study of cytoskeletal networks which can be regarded as consisting of rigid rod units crosslinked into a polymeric-type network. Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential that the deformation maps $\varphi$ be invertible. We require sufficient regularity of the fields $(\varphi, Q)$ of the problem, and that the deformation map satisfies the Ciarlet-Necas condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. The approach of including the Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also discussed. The regularity requirements lead us to consider powers of the gradient of the order tensor $Q$ higher than quadratic in the energy. We assume polyconvexity of the stored energy function with respect to the effective deformation tensor and apply methods from isotropic nonlinear elasticity. We formulate a necessary and sufficient condition to guarantee this invertibility property in terms of the growth to infinity of the bulk liquid crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular limit of $-\frac{1}{3}$. $L$ becomes singular as the minimum eigenvalue of $Q$ reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe of nematics (see Ball 2010).

1. Introduction. We investigate existence of minimizers of the energy of a liquid crystal elastomer that in the reference configuration occupies a domain Ω ⊂ R 3 . We assume that the total energy consists of the Lagrangian contribution of the anisotropic elastomer and the Landau-de Gennes liquid crystal energy in the Eulerian frame. The mixed Lagrangian-Eulerian setting of the problem requires injectivity of the deformation map. For this, we require the admissible deformation maps to have sufficent regularity and to satisfy the Ciarlet-Necas condition as well. This work aims at investigation the coupling of elastic energy and nematic order, and it applies to, both, thermotropic and lyotropic systems.
Liquid crystal elastomers are anisotropic nonlinear elastic materials, with the source of anisotropy stemming from the presence of elongated rigid monomer units, that are either inserted in the back-bone as part of the polymer main-chain or are present as side groups. They are elastic solids that may also present fluid and mixed regimes [18], [20], [21], [27], [43]. In nematic fluids, molecules tend to align themselves along preferential directions but do not present ordering of centers of mass.
The nature of the connections between polymer chains and rigid monomer units plays a main role in determining the behavior of liquid crystal elastomers [43]. From a different perspective, studies of actin and cytoskeletal networks ( [11], [28], [42]) show lyotropic rigid rod systems crosslinked into networks by elastomer chains or linkers that present qualitative properties of liquid crystal elastomoers. A main feature of these networks is the average number of connections between rods and the location of these connections in the rod. These motivates us to consider two limiting types of lyotropic systems, the first one corresponding to a nematic liquid with rods weakly coupled into the network. The second case is that of a solid where rod rotation occurs uniquely as a result of elastic deformation, such as an elastomer made of material fibers. Consequeantly, we postulate Eulerian and Lagrangian Landaude Gennes liquid crystal energies, respectively, for these systems. Many physical systems are found having intermediate properties between these two limiting cases, and it, then, may be appropriate to postulate the energy as a weighted sum of the Eulerian and Lagrangian energies, scaled according to a macroscopic parameter representing the density of crosslinks. The analogous property holds for the elastic energy of the system that we discuss in this article.
The study of a lyotropic elastomer with Lagrangian liquid crystal energy was carried out in previous work where we also analyzed liquid crystal phase transitions triggered by change in rod density [11]. In this work, we assume that the anisotropic behavior of the rigid units is represented by a liquid-like, Eulerian liquid crystal energy. Sufficient regularity of the deformation map is required to guarantee its invertibility, in order to pass from the current configuration of the nematic liquid crystal to the reference configuration of the elastic solid.
There are two main quantities that describe the anisotropy associated with a liquid crystal elastomer: the traceless order tensor Q describing the nematic order of rigid rod units, and the positive definite step-length tensor L that encodes the shape of the network. L is spherical for isotropic polymers and spheroidal for uniaxial nematic elastomers, in which case, it has eigenvalues l and l ⊥ (double).
The quantity r := l l ⊥ − 1 measures the degree of anisotropy of the network, with positive values corresponding to prolate shape and negative ones to oblate. In the prolate geometry, the eigenvector n associated with l is the director of the theory, giving the average direction of alignment of the rods and also the direction of shape A LANDAU-DE GENNES THEORY OF LIQUID CRYSTAL ELASTOMERS 3 elongation of the network. It is natural to assume that L and Q share the same eigenvectors. In this article, we set Q = L − 1 3 (trL)I so as to obtain a traceless Q, from a given positive definite tensor L ( [43], page 49). The free energy may also carry information on the anisotropy L 0 imprinted in the network at crosslinking the original polymer melt. In this work, we choose L 0 = I.
The standard Landau-de Gennes free energy density consists of the sum of scalar quadratic terms of ∇Q and the bulk scalar function f (Q), usually a polynomial of the trace of powers of Q, describing the phase transition between the isotropic and the nematic phases of the liquid crystal [29], [38]. In this work, we take a departure from these forms but still keeping consistentcy with the original Landau-de Gennes theory. We first observe that the polynomial growth is not physically realistic, since it is expected that the energy should grow unboundedly near limiting alignment configurations. In the uniaxial case, these correspond to perfect alignment, characterized by scalar order parameter s = 1, and configurations where the rods are confined to the plane perpendicular to the director n [12], [23]. In terms of the order tensor, the limiting configuration is represented by the minimum eigenvalue taking the value − 1 3 . Bounds on eigenvalues of Q are not part of the original theory, but are needed for it to be compatible with the Maier-Saupe theory from statistical physics [5]. This turns out to be as well an essential element of our analysis. Also, we take the gradient of Q with respect to space variables in the deformed configuration, in which case the coupling with the deformation gradient F emerges naturally. It turns out that powers of gradient of Q, higher than quadratic, in the energy, are required for compactness.
The elastic energy density proposed by Blandon, Terentjev and Warner is given by the trace form [43]: This is the analog of the Neo-Hookean energy of isotropic elasticity and is also derived from Gaussian statistical mechanics. Let us call G = L − 1 2 F L 1 2 0 the effective deformation tensor of the network; of course, in general, G is not itself a gradient. Then W BTW = µ(|G| 2 − 1). Motivated by the theory of existence of minimizers of isotropic nonlinear elasticity ( [3]), in his PhD, thesis [32], Luo generalizes W BTW to polyconvex stored energy density functionsŵ(G(x)); that is, w(G) = Ψ(G, adj G, det G) is a convex function of (G, adj G, det G). This approach was used later in [11]) to model phase transitions in rod networks. In order to recover the limiting deformation gradient F * from the minimizing sequences {G k } k≥1 , it is necessary that the minimizing sequences {L k } yield a nonsingular limit. This is achieved by requiring the blowup of f (Q) as the minimum eigenvalue of Q tends to − 1 3 , that is, f (Q) → ∞ as det(Q + 1 3 I) = det L → 0. We point out that this lends another significance to the minimum eigenvalue limit.
In a related work [14], Calderer and Luo carried out a mixed finite element analysis of the elastomer trace energy coupled with that of the Ericksen model of uniaxial nematic liquid crystals. This was later used in numerical simulations of domain formation in two-dimensional extensional deformations. From a different persepective, a study of uniaxial elastomers with variable length director was carried out in [13], for a restricted set of deformation maps.
Let us now comment on the deeper mechanical significance of G (also denoted F eff ) in connection with the special class of spontaneous deformation as brought up by Agostiniani, DeSimone and Teresi [1], [22]. Spontaneous deformations are those that do not cost any elastic energy. In uniaxial nematics, they correspond to volume preserving uniaxial extensions along the eigenvector n, the latter being defined in the current configuration of the elastomer. (In the context of biaxial nematics, volume preserving biaxial extensions are also spontaneous deformations represented by F s = √ L s , where, in its (unit) eigenvector representation, L s := am ⊗ m + r ⊗ r) + cn ⊗ n, a, b, c > 0, abc = 1). Spontaneous strains have crystallographic significance in that they represent variants in domain patterns present in low energy microstructure. The chevron domains observed by Sanchez and Finkelman in liquid crystal elastomers realize the microstructure in these materials. [31], [40] and [44].
We note that F eff does not represent the deformation of the elastomer with respect to the identity but with respect to spontaneous deformations √ L s . Consequently, the Trieste group also pointed out that F eff is the only deformation tensor that contributes to the elastic energy of the elastomer, this energy being isotropic, since it has the same form regardless the current value of n. These authors proposed to include as well an anisotropic component in the energy depending on F T F (2.8). We observe the regularizing role of the latter part of the energy. Without loss of generality, in this work, we neglect the latter term and address the more challenging problem of the energy depending only on G (F eff ).
In addition to the trace models of liquid crystal elastomer energy studied by Terentjev and Warner ([43] and references therein), generalizations of these earlier forms have been proposed and studied by several authors ( [2] and [26]; [1], [15], [16] and [22]). In these references, the authors propose energies based on powers of the earlier trace form, including Ogden type energies, and study their extensions to account for semisoft elasticity. Also, the analysis of equilibrium states presented in [1] applies to elastomer energy density functions that are not quasiconvex, these being appropriate to model crystal-like phase transitions.
The types of boundary conditions that we consider include prescribing the deformation map on the boundary or part of it. In such cases, we also prescribe components of the order tensor Q there. In an alternate approach, instead of prescribing boundary conditions on Q, we include a surface free energy of the Rapini-Papoular type. The latter also extends to those parts of the boundary with no prescribed Dirichlet conditions on the deformation map. In such a case, the energy integral is formulated in terms of the pull-back order tensorQ(x), x ∈ Ω.
This article is organized as follows. Section 2 is devoted to modeling, with special emphasis on analyzing the type of coupling between the Landau-de Gennes model of nematic liquid crystal and the nonlinear elasticity of the anisotropic network. In Section 3, we study the admissible set of fields corresponding to finite energy, focusing on classes of deformation maps and boundary conditions for which the invertibility property of the maps holds. Section 4 addresses minimization of the energy. The conclusions are described in Section 5.
2. The Landau-de Gennes liquid crystal elastomer. Equilibrium configurations of nematic liquid crystal elastomers are characterized by the deformation gradient F together with the symmetric tensors L and Q, describing the shape of the material and the nematic order, respectively. Within the point of view of the mean-field theory, the state of alignment of a nematic liquid crystal is given by a probability distribution function ρ in the unit sphere. The order tensor is defined as the second order moment of ρ: From this definition, it follows that Q is a symmetric, traceless, 3 × 3 matrix with bounded eigenvalues ( [25] and [5]) For biaxial nematic, Q admits the representation where r and s correspond to the biaxial order parameters and e 1 and e 2 are unit eigenvectors. If r = 0, then (2.3) yields the uniaxial order tensor Q = s(e 2 ⊗ e 2 − 1 3 I), where s ∈ (− 1 2 , 1) corresponds to the uniaxial nematic order parameter and n = e 2 is the unit director field of the theory.
In a rigorous study of the Landau-De Gennes model [34], Majumdar observes that the definition of Q given by (2.1) is that of the Maier-Saupe mean-field theory [33]. In the original theory by Landau and de Gennes [19], [29], Q has a phenomenological role as a dielectric or magnetic susceptibility tensor, and its eigenvalues do not satisfy any inequality constraints. It is the latter that bring compatibility to these theories.
Following the property of freely joined rods, we assume that L and Q have common eigenvectors and propose the constitutive relation where a 0 = trL > 0 is constant. The linear constitutive equation (2.4) is analogous to those proposed by Terentjev and Warner [43] and Fried and Sellers [26] stating that, given a symmetric and traceless tensor Q and a constant β > 0, there is a one α-parameter family of step-length tensors L with trL = β, and such that L = β(αQ + 1 3 I). The form (2.4) corresponds to taking α = 1 and β = a 0 . Letting l 1 , l 2 , l 3 denote the eigenvalues of L, it reduces to the uniaxial nematic with director n and order parameter a multiple of s in the case that l 2 = l 1 := l ⊥ , l 3 := l . The mechanical response of the elastomer along n is distinguished from that along any of the transverse directions.
From the constitutive assumption (2.4) it follows that where λ min (Q) stands for the minimal eigenvalue of a symmetric tensor Q. This shows another consequence of requiring λ min (Q) > − 1 3 : to gurantee the invertibility of L, and so, to be able to recover the gradient of deformation from the effective deformation tensor G.
We assume that, in the reference configuration, a liquid crystal elastomer occupies a bounded domain Ω ⊂ R 3 with smooth boundary ∂Ω. We denote the deformation map of the elastomer and its gradient as where δ 0 > 0 is a given constant. This last constraint on the determinant expresses the maximum degree of compressibility allowed to the material. The trace-form free energy density W BTW = µtr(L 0 F T L −1 F − 1 3 I) expresses the coupling between the step length tensor L and the deformation gradient F and also encodes the anisotropy L 0 of the reference configuration. Usually, L 0 is a constant positive definite symmetric tensor and, by rescaling, in what follows, we assume L 0 = I. Define G = F eff = L − 1 2 F . In general, scalar functions of the invariants of the tensors F F T and F T eff F eff are admissible. Let us examine how vectors transform in each of these cases as illustrated in figure (2). In [22], the authors propose an elastomer energy of the form 8) with F eff representing the deformation tensor with respect to spontaneous deformations. The α-component of this energy was also analyzed in previous work [11], [32], and in [13] and [14] for the Neo-Hookean trace form of the energy. As indicated in the introduction, these terms correspond to two limiting liquid crystal elastomer behaviors, perhaps ideal, the first corresponding to the case that the material is made of pure fibers and the second to a standard liquid crystal elastomer but showing interaction between alignment and deformation.
We focus on the cases of nontrivial coupling between Q and F and propose a Landau-de Gennes elastomer energy of the following form: where, for certain given ϕ and Q, In what follows, we denote M 3 the space of three-dimensional tensors and M 3 Then, for Q ∈ Q, it follows that |Q| ≤ 2 √ 3 and the tensor L defined by (2.4) is invertible.
As for the density functionsŴ , L and f in (2.9), we make the following assumptions motivated by the analogous ones in isotropic nonlinear elasticity [3].
Polyconvexity and coerciveness ofŴ : There exists a convex function Ψ : (2.14) Also, there exist constants α > 0, p > 3 such that Convexity and growth of L: The Landau-de Gennes energy function L(∇ y Q, Q) is convex in ∇ y Q. Moreover, there exists a constant κ > 0 such that where r is a constant satisfying Blow-up of f : The bulk free energy density f : Q → R + is continuous and satisfies Notation. For r as in (2.17), we define We point out that q > max{ p p−2 , 3p p+3 } > 1.

Remarks.
1. If L is given by (2.4), then 0 < det L ≤ (a 0 ) 3 when Q ∈ Q. So, in the case that det F ≥ δ 0 , we have that det G ≥ δ 0 (a 0 ) − 3 2 . Therefore, no condition on the growth near zero-determinant has to be imposed onŴ (G).
2. We observe that det L can become arbitrarily small. This corresponds to the polymer adopting a needle or plate shape. 3. The growth condition (2.18) has been applied in [5] in the context of studying of minimization of the Landau-de Gennes energy. Its restriction to the uniaxial case was first proposed by Ericksen [23] and used in analysis of defects in liquid crystal flow [7,8,9,12]. In order to achieve a better understanding of the gradient part of the of the Landaude Gennes energy (2.16), let us give a brief review of the standard liquid crystal theory [38]. The total energy is of the form where a(T ) = α(T − T * ), T > 0 denotes the absolute temperature, α, T * , b, c, L i = L i (T ) are constants, and In the special case of a single constant, the energy (2.20) reduces to L|∇Q| 2 . For this energy, existence of global minimizer was discussed in [36], with further studies of regularity, characterization of uniaxial and biaxial states, and structure of defect sets presented in [35].

Admissible classes of fields.
In what follows,we let p > 3 and r > max{3, p p−3 } be as in (2.17). We consider the admissible classes of (ϕ, Q) for the energy E(ϕ, Q), from subclasses of functions ϕ ∈ W 1,p (Ω, R 3 ) and Q ∈ W 1,r (ϕ(Ω), M 3 ). Throughout the paper, given a measurable set B ⊂ R 3 and a measurable function ω on B, we define ω ∈ W 1,s (B), provided there exist an open set O containing B and a functionω ∈ W 1,s (O) such thatω = ω on B.
We assume ϕ ∈ W 1,p (Ω, R 3 ) and that the condition (2.7) above is satisfied. To avoid material inter-penetration, we also require that ϕ be injective. The latter issue is addressed in forthcoming lemmas.
In order to define the admissible set A of the variational problem, we first need to introduce the functional S. In particular, it will help us identify types of boundary conditions of the displacement field that are compatible with the injectivity conditions previously discussed.
Let S : W 1,p (Ω, R 3 ) → R + be a given functional such that S(tϕ) = |t|S(ϕ) for all t ∈ R and ϕ. Assume that S is continuous under the weak convergence of W 1,p (Ω, R 3 ), and that if ϕ is constant and S(ϕ) = 0 then ϕ = 0. Then, by the Sobolev-Rellich-Kondrachov compact embedding of W 1,p (Ω) → L p (Ω), one easily has the following Poincaré-type inequality: there exists a constant C such that For such a functional S and given a constant β > 0, let In many applications, the functional S can be chosen as one of the following: where Q is defined by expression (2.13).
Remarks. 1. Examples of sets B(Ω) include the following ones, associated with standard types of boundary conditions: 2. The set B 3 (Ω) includes B 1 (Ω) and the case that partial Dirichlet boundary conditions are prescribed. B 3 can be considered a subset of D S,β with S = S 3 defined by (3.11). This is mainly due to the assumption p > 3 and the compact embedding W 1,p (Ω) → C(Ω).
4. Deformation maps corresponding to B 2 include anti-plane shear deformation.
Let us now discuss appropriate boundary conditions to impose on Q, either in the form of strong anchoring (Dirichelt) or by modifying the total energy by adding a surface energy penalty of the Rapini-Papoular form [39]. For this, we first assume that ϕ satisfies Dirichlet boundary conditions on Γ ⊆ ∂Ω (Remarks 1 and 3, page 10), and require one of the following on Q: 1. Dirichlet boundary conditions: (3.20) 2. We include a surface energy contribution in the total energy. This term, of the Rapini-Papoular form is whereQ| Γ is the L q −trace ofQ ∈ W 1,q (Ω, Q), α ≥ 0 and h ≥ 0 denotes a continuous function satisfying with A, B ∈ M 3×3 and trA = 0 = trB. This energy form includes relevant expressions of the liquid crystal theory, such as with Q 0 ∈ Q prescribed. In the case that ϕ does not satisfy Dirichlet boundary conditions on any part of ∂Ω, we still allow for the modification of the energy as in (3.21), with Γ ⊆ ∂Ω. We point out that the new energy integral is taken on the boundary of the reference domain and it involves the pull-back tensorQ. Additional regularity is required to pose that energy integral in the current domain; this issue will not be addressed in the current work.
The next two results will be employed in the proof of existence of energy minimizer. The first one establishes properties of weak limits of sequences of effective deformation tensors and their relation with those of the corresponding deformation gradient tensors. The second lemma refers to the preservation of relation (3.4) under weak convergence.
Proof. Clearly, if {G k } is bounded in L p (Ω), then {det G k } and {adj G k } are bounded in L p/3 (Ω) and L p/2 (Ω), respectively. Hence the weak convergence of a further subsequence of both sequences follows as p > 3. By the bounded convergence theorem, our assumption implies that g(L k ) → g(L) strongly in L q (Ω), for all q ≥ 1 and for all continuous functions g. Therefore,  Proof. Without loss of generality, let us assume that lim k→∞ |ϕ k (Ω)| exists. So, for each > 0, there exists a positive integer N such that Let us denote S = ϕ(Ω). The previous inequality implies that ϕ k (Ω) ⊆ S , for all k ≥ N, where S = {y ∈ R 3 : dist(y, S) < }. Hence |ϕ k (Ω)| ≤ |S |, for all k ≥ N.
A classical weak continuity theorem on determinants [3] asserts that det ∇ϕ k det ∇ϕ in L p/3 (Ω) as p > 3. Also, by the compact embedding of W 1,p (Ω) → C(Ω), it follows that ϕ k → ϕ uniformly on Ω. Taking limits as k → ∞ in the above inequality and applying (3.24), we obtain relation (3.4) for ϕ. This concludes the proof of the lemma.
4. Energy minimization. The following theorem establishes the existence of minimizer of the energy E in the admissible class A defined above.
Theorem 4.1. Let the admissible set A be defined as in (3.12). Suppose there exists a pair (ϕ, Q) ∈ A such that E(ϕ, Q) < ∞. Then, there exists at least one pair (ϕ * , Q * ) ∈ A such that Proof. By assumption, there exists a constant K 1 > 0 such that Let (ϕ k , Q k ) ∈ A be a minimizing sequence for E, that is Denote F k , G k , L k ,Q k ,L k the corresponding quantities defined from (ϕ k , Q k ) as above.
Since ϕ ∈ D S,β (Ω) (as defined in (3.8)), applying (3.7) and (3.17) yields where C(M ) is a constant depending on M > 0 that remains bounded for M in a bounded set.
Step 2: Convergence and compactness. Let (ϕ k , Q k ) ∈ A be the minimizing sequence above. By (4.6), we obtain that Hence, there exists a subsequence (labeled also by k) such that det ∇ϕ k det ∇ϕ * in L p/3 (Ω), adj ∇ϕ k adj ∇ϕ * in L p/2 (Ω). (4.10) Since p > 3, the convergence statements in (4.10) follow from the classical weak continuity theorem of null-Lagrangians [3]. We first prove the following result on compensated compactness.
So, the combination of these three steps establishes the weak lower semicontinuity of E on A. Hence, This, in turn, shows that ϕ * (Ω) |∇ y Q * (y)| r dy < ∞.
In the last part of this section, we will address boundary conditions on Q, in cases compatible with the type of regularity assumed so far. Straightforward modifications of the above proof, allow us to establish the following results. Corollary 1. Let p > 3, and σ ≥ 0. Suppose that ϕ 0 ∈ W 1,p (∂Ω, R 3 ), injective, and Q 0 ∈ W 1,2 (∂Ω, Q) are prescribed. Let E and E S be as in (2.9) and (3.23), respectively, and define E total = E + σE S .
This corollary establishes existence of minimizer, even in the case that the Landau-de Gennes energy is quadratic on gradient of Q, as for standard nematic liquid crystals.