Nematic-Isotropic phase transition in Liquid crystals: a variational derivation of effective geometric motions

In this work, we study the nematic-isotropic phase transition based on the dynamics of the Landau--De Gennes theory of liquid crystals. At the critical temperature, the Landau--De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.


Introduction
Nematic liquid crystals react to shear stress like a conventional liquid while the molecules are oriented in a crystal-like way. One of the successful continuum theories modeling nematic liquid crystals is the Q-tensor theory, also referred to as Landau-De Gennes theory, which uses a 3 × 3 traceless and symmetric matrix-valued function Q(x) as order parameter to characterize the orientation of molecules near a material point x (cf. [8]). The matrix Q, also called Q-tensor, can be interpreted as the second moment of a number density function f (x, p) dp, (1.1) where f (x, p) corresponds to the number density of liquid crystal molecules which orient along the direction p ∈ S 2 near the material point x (cf. [5]). The configuration space of the Q-tensor is the 5-dimensional linear space By elementary linear algebra, each such Q can be written as for some s, t ∈ R and u, v ∈ S 2 which are perpendicular. In the physics literature, for instance De Gennes-Prost [8], such a representation is called the biaxial nematic configurations, cf. [23]. In case Q has repeated eigenvalues, it is called uniaxial. These Q's form a 3-dimensional manifold in Q, denoted by U := Q ∈ Q Q = s u ⊗ u − 1 3 I 3 for some s ∈ R and u ∈ S 2 , (1.4) with a conical singularity at s = 0. Here the parameter s is called the degree of orientation. To study static configurations of the liquid crystal material in a physical domain Ω, a natural approach 1 is to consider the Ginzburg-Landau type energy (1.5) where Ω ⊂ R d is a bounded domain with smooth boundary, |∇Q| = ijk |∂ k Q ij | 2 , and F (Q) is the bulk energy density Here a, b, c ∈ R + are material and temperature dependent constants, and ε denotes the relative intensity of elastic and bulk energy, which is usually quite small. It can be proved that all critical points of F (Q) are uniaxial (1.4), (cf. [23]), and thus Moreover, F (Q) has two families of stable local minimizers corresponding to the following choices of s = s ± : (1. 8) In this work we shall consider the bistable case when where N := Q ∈ Q | Q = s + u ⊗ u − 1 3 I 3 for some u ∈ S 2 , with s + = 3a c . (1.11) At this point we digress to mention that the Landau-De Gennes model (1.5) is closely related to Ericksen's model, where the energy is e E (s, u) := Ω κ|∇s| 2 + s 2 |∇u| 2 + ψ(s) dx. (1.12) This model was introduced by Ericksen [9] for the purpose of studying line defects. It can be formally obtained by plugging the uniaxial Ansatz (1.4) into (1.5). In contrast to (1.5) which uses Q ∈ Q as order parameter, Ericksen's model uses (s, u) ∈ R×S 2 and is very useful to describe liquid crystal defects. The analysis of this model is very challenging, mainly due to the reason that the geometry of the uniaxial configuration (1.4) corresponds to a double-cone, and the energy (1.12) is highly degenerate when s = 0. The analytical aspects of such a model have been investigated by many authors, for instance, by Lin [18], Hardt-Lin-Poon [20], Bedford [7], Alper-Hardt-Lin [2], and Alper [1].
To model nematic-isotropic phase transitions in the framework of Landau-De Gennes theory, we shall investigate the small-ε limit of the natural gradient flow dynamics of (1.5) with initial data undergoing a sharp transition near a smooth interface. To be more precise, we consider the system (1.14) The system (1.13a) is the L 2 -gradient flow of energy (1.5) on the slow time scale ε. Our main result, Theorem 2.1, states that starting from initial conditions with a reasonable nematic-isotropic phase transition from a nematic region Ω + (0) into an isotropic region Ω − (0), before the occurrence of topological changes, the solution Q ε of (1.13) converges to the isotropic phase Q ≡ 0 in Ω − (t) and to a field Q ∈ N taking values in the nematic phase in Ω + (t), where the interface between Ω + (t) and Ω − (t) moves by mean curvature flow. Furthermore, we show that the limit Q is a harmonic map heat flow from Ω + (t) into the closed manifold N . Finally, if the region Ω + (t) is simply-connected, there exists a director field u such that Q = s + (u ⊗ u − 1 3 I 3 ), u is a harmonic map heat flow from Ω + (t) into S 2 , and satisfies homogenous Neumann boundary conditions on the evolving boundary ∂Ω + (t).
The proof consists of two key steps: (i) an adaptation of the modulated energy inequality in [12] to the vector-valued case to control the leading-order energy contribution, which is of order O(1) and comes from the phase transition across ∂Ω + (t). (ii) A version of Chen-Shatah's wedge-product trick in the sense that (1.13) implies where [·, ·] denotes the commutator. In (i) we basically follow [12] but need to carefully regularize the metric d F on Q induced by the conformal structure F (Q) in order to exploit the fine properties of its derivative ∇ q d F ε . In particular, we will use the crucial commutator relation ∇ q d F ε (Q ε ), Q ε = 0 for a.e. (x, t). This seems to lie beyond the realm of generalized chain rules as in [3], which was employed in the work of Simon and one of the authors in [17]. Regarding (ii), we emphasize that the Neumann boundary condition along the free boundary ∂Ω + (t) can be naturally encoded in the distributional formulation of (1.15) by enlarging the space of test functions. This however, requires uniform L 2 -estimates on the commutators [∂ t Q ε , Q ε ] and [∇Q ε , Q ε ], which are one order of ε better than the a priori estimates suggest. We show that these estimates are guaranteed by our bounds on the modulated energy.

Main results
To state the main result of this work, we assume is a smoothly evolving closed surface in Ω, (2.1) starting from a closed smooth surface I 0 ⊂ Ω. Let Ω + (t) be the domain enclosed by I t , and d(x, I t ) be the signed-distance from x to I t which takes positive values in Ω + (t), and negative values in Moreover, for each T > 0 we shall denote the 'distorted' parabolic cylinder by For δ > 0, the δ-neighborhood of I t is denoted by So there exists a sufficiently small number δ I ∈ (0, 1) such that the nearest point projection P I (·, t) : I t (δ I ) → I t is smooth for any t ∈ [0, T ], and the interface (2.1) stays at least δ I distance away from the boundary of the domain ∂Ω.
To introduce the modulated energy for (1.13), we extend the inner normal vector field n I of I t to a neighborhood of it by where η is a cutoff function satisfying η is even in R and decreases in [0, ∞); Following [16,12], we define the modulated energy by and the convolution is understood in the space Q ≃ R 5 . Moreover, we set a family of mollifiers in the 5-dimensional configuration space (1.2). Here φ is smooth, non-negative, having support in B Q 1 (the unit ball in Q), and isotropic, i.e. for any orthogonal matrix R ∈ O(3) and any q ∈ Q it holds φ(R T qR) = φ(q). The function d F in (2.8b) is the quasi-distance function which was introduced by Sternberg [26] and independently by Tartar-Fonseca [13] for the study of the singular perturbation problem. Some properties of d F are stated in Lemma 3.1 below, and interested readers can find the proof in [26,19]. One can refer to Section 3 for more details of these functions. Throughout, we will assume an L ∞ -bound of Q ε , i.e.
Q ε L ∞ (Ω×(0,T )) ≤ c 0 (2.11) for some fixed constant c 0 . Such an estimate can be obtained by assuming an uniform L ∞ -bound of the initial data Q in ε and then applying maximum principle to (1.13a), see Lemma 3.3 in the sequel. Note that the choice K = 4 in (2.8a) is due to a technical reason, and is used in the proof of Lemma 4.1.
The main result of this work is the following: for some constant c 1 that does not depend on ε, then for some ε k ↓ 0 as k ↑ +∞, where s ± are given by (1.8) and u ∈ H 1 (Ω + T ; S 2 ). (2.14) Moreover, u is a harmonic map heat flow into S 2 with homogenous Neumann boundary conditions in the sense that If Ω + (t) is multi-connected, for instance when Ω + (t) is the region outside I t , then a well-known orientability issue arises and the conclusion (2.14) usually only holds away from defects. See the work of Bedford [7] for more discussions of such issues.
Theorem 2.1 solves a special case of the Keller-Rubinstein-Sternberg problem [25] using the energy method. A similar result has been established previously by Fei et al. [10,11] using matched asymptotic expansions and spectral gap estimates. Our approach has the superiority that it allows more flexible initial data, as indicated by Proposition 2.3 below. The general case of the Keller-Rubinstein-Sternberg problem is fairly sophisticated and remains open. We refer the interested readers to a recent work of Lin-Wang [21] for the well-posedness of the limiting system. On the other hand, the static problem has been investigated by Lin et al. [19]. It is worthy to mention that recently Golovaty et al. [15,14] studied a model problem based on highly disparate elastic constants. Most Recently, Lin-Wang [22] studied isotropic-nematic transitions based on an anisotropic Ericksen's model. Now we turn to the construction of initial data Q in ε satisfying (2.12). Let I 0 ⊂ Ω be a smooth closed surface and let I 0 (δ 0 ) be a neighborhood in which the signed distance function d(x, I 0 ) is smooth. Let ζ(z) be a cut-off function such that ζ(z) = 0 for |z| ≥ 1, and ζ(z) = 1 for |z| ≤ 1/2. (2.17) Then we define where S(z) is given by the optimal profile Moreover, there exists a constant c 1 > 0 which only depends on I 0 and u in H 1 (Ω) such that Q in ε is well-prepared in the sense of (2.12).
The rest of this work will be organized as follows. In Section 3 we discuss some properties of the quasi-distance function (2.10) and use them to construct the well-prepared initial data (2.20) and thus prove Proposition 2.3. In Section 4 we establish a relative-entropy type inequality for the parabolic system (1.13). Based on the various estimates given by such an inequality, in Section 5 we study the limit ε ↓ 0 of (1.13) and give the proof of Theorem 2.1.

Preliminaries
We start with a lemma about the quasi-distance function (2.10), which was originally due to [26,13].
where c F is the 1-d energy of the minimal connection between N and 0: By elementary linear algebra, any Q ∈ Q can be expressed by are the eigenvalues ordered increasingly. Furthermore using the identities 3 j=1 λ j (Q) = 0 and I 3 = 3 j=1 P j (Q), we may write The next lemma gives a precise form of d F for uniaxial Q-tensors.
where f (s) is given by (1.7).
Proof. The argument here is similar to that in [24]. Let γ be any curve connecting N to Q. When expressed in the form of eigenframe , we claim that n i are constant in t. Actually using the identity we calculate This implies that the global minimum is achieved by a path γ(t) with constant eigenframe. So by (3.4) we may write It is clear that this energy is minimized when θ ≡ 0, which corresponds to the uniaxial solution r(t) ≡ 0. In view of (1.7) One can verify that the minimum of the right hand side can be achieved by a monotone function s(t), and thus (3.6) follows from a change of variable.
At this point we would like to remark that for the general Keller-Rubinstein-Sternberg problem it is very hard to obtain a precise form of d F like (3.6) (cf. [26, Part 2, Lemmas 5 and 7]).
Before giving the proof of Proposition 2.3, we digress here and discuss the convolution in (2.8b). The space Q (1.2) can be equipped with the norm |q| := tr(q T q), and one can easily verify that defined below form an orthonormal basis: So this establishes an isometry Q ≃ R 5 and thus the convolution operation in (2.8b) can be interpreted as an integration in R 5 . Concerning the choice of φ in (2.9), one can simply choose g ∈ C ∞ c (R) and set φ(q) := g(tr(q 2 )), which is obviously isotropic in q.
Proof of Proposition 2.3. As a consequence of the choice of the cutoff function ζ satisfying (2.17), we deduce that (2.21) is fulfilled and S ε is smooth. To compute the modulated energy (2.7) of the initial data Q in ε , we write (2.18) by (3.14) It follows from the exponential decay of (2.19) that for some constant C > 0 that only depends on I 0 . So we can write Recalling the form of the bulk energy (1.7) for uniaxial Q-tensors, in view of (1.9), we have and F (Q in ε ) = f (S +Ŝ ε ). Thus the integrand of E ε [Q ε |I](0) can be written as The first line on the right hand side vanishes due to the identity S ′ (z) = 3f (S(z)): On the other hand, since 0 ≤ S ε ≤ s + , by Lemma 3.2, This together with (2.6) and (3.15) implies Adding up (3.19) and (3.21) yields By the exponential decay of (2.19) and (3.15), To treat the last term, we first deduce from the exponential decay of (2.19) that for some C that only depends on I 0 . This together with (2.6) implies Substituting the above estimate into (3.23) and use (2.8a), we arrive at On the other hand, by (2.9) and (3.1), we have where L = L(M, φ, F ). This pointwise estimate implies which together with (3.26) implies (2.12).
The next result is concerned with a maximum modulus estimate of (1.13a) Proof. On the one hand, by (1.13a), |Q ε | 2 fulfills the following identity On the other hand, there exists µ > 0 (sufficiently large) such that |Q| ≥ µ implies then we obtain the desired estimate. Otherwise there holds ∂ t |Q ε | 2 − ∆|Q ε | 2 ≤ 0, and the weak maximum principle implies the maximum must be achieved on the parabolic boundary (∂Ω × (0, T )) ∪ (Ω × {0}), on which |Q ε | is bounded by our assumptions.

The modulated energy inequality
As the gradient flow of (1.5), the system (1.13a) has the following energy dissipation law Due to the concentration of ∇Q ε near the interface I t , this estimate is not sufficient to derive the convergence of Q ε . Following a recent work of Fisher et al. [12] we shall develop in this section a calibrated inequality, which modulates the surface energy. Recall in (2.5) that we extend the normal vector field n I of the interface I t to a neighborhood of it. We also extend the mean curvature vector H I of (2.1) to a neighborhood by H I (x, t) = η(d(x, I t ))H I (P I (x, t), t) = η(d(x, I t ))(div n I )(P I (x, t), t)n I (P I (x, t), t), (4.2) where η ∈ C ∞ c ((−δ I , δ I )) is a cut-off which is identically equal to 1 for s ∈ (−δ I /2, δ I /2), and P I (x, t) = x − ∇d(x, I t )d(x, I t ) is the projection onto I t . The definitions (2.5) and (4.2) of ξ and H I , respectively, imply the following relations: where ∇H I := {∂ j (H I ) i } 1≤i,j≤d is a matrix with i being the row index. Actually in I t (δ I /2) there holds ∂ t d(x, I t ) = −n I · H I (P I (x, t)) and ∇d(x, I t ) = n I (P I (x, t)). So we obtain (4.3) by chain rule. Moreover, and since H I is extended constantly in normal direction, we have (ξ · ∇)H I = 0 for all (x, t) such that |d(x, I t )| < δ I /2. We denote the phase-field analogs of the mean curvature and normal vectors by respectively, where ψ ε is defined by (2.8b). Here and throughout we use the convention that : denotes the contraction in the indices i, j in three-tensors like ∂ k Q i,j , i.e., the scalar product in the state space Q. By chain rule and (2.8b) This motivates the definition of the following projection of ∂ i Q ε onto the span of (4.10) Hence, (4.9) implies The following inequality will be crucial to show the non-negativity of the modulated energy (2.7) and various lower bounds of it. It states that the upper bound for the gradient of the convolution d F ε is as good as if d F was C 1,1/2 and it simply follows from the fact that the modulus |∇d F | is C 1/2 .
Proof. Recall (2.8a), i.e. F ε (q) = F (q) + ε K−1 with K = 4. It follows from (2.9), (3.1) and where in the last step C 0 is a local Lipschitz constant of F (q) for |q| ≤ c 0 . By (2.9) and the assumption that φ is supported in the unit ball of Q, the integral in the last step can be treated as Finally choosing ε 0 sufficiently small leads to (4.12).
We shall apply the above lemma with c 0 being the constant in (3.29).
As we shall not integrate the time variable t throughout this section, we shall abbreviate the spatial integration Ω by and sometimes we omit the dx. The following lemma shows that the energy E ε [Q ε |I] defined by (2.7) controls various quantities.
The following result was first proved in [12] in the case of the Allen-Cahn equation, and can be generalized to the vectorial case.
The following lemma, the proof of which will be given at the end of this section, provides the exact computation of the time derivative of the energy E ε [Q ε |I].

Lemma 4.4. Under the assumptions of Theorem 2.1, the following identity holds
where we use the notation In order to prove the proposition, we only need to estimate the terms on the RHS of (4.19).
Proof of Proposition 4.3. According to Lemma 4.4, we only need to estimate the RHS of (4. 19) by E ε [Q ε |I] up to a constant that only depends on I t . We start with (4.19a): it follows from the triangle inequality that The first integral is controlled by (4.13c). Using (4.4), the second integral is controlled by (4.13d).
The third integral can be treated using (4.4) and controlled by (4.13e). The integrals in (4.19b) can be controlled using (4.13b) and (4.13d), recalling The first term in (4.19c) can be controlled using (4.13a), and the second term can be estimated by (4.13d). It remains to estimate (4.20) and (4.21). The last two terms on the RHS of J 1 ε can be bounded using (4.13b), and the first integral can be rewritten using n ε = n ε − ξ + ξ: Note that in the last step we employed ∇H I : n ε ⊗ ξ = (ξ · ∇H I ) · n ε (4. 24) and the fact that (ξ · ∇)H I vanishes in the neighborhood I t ( δ I 2 ) by definition (4.2). So we employ (4.13d) and (4.13b), and (4.11a) Finally applying the Cauchy-Schwarz inequality and then (4.13d) and (4.13b), we obtain As for J 2 ε (4.21), we employ (4.3a) and (4.3b) and yield after applying (4.13d) and (4.13e). So we proved that the RHS of (4.19) is bounded by E ε [Q ε |I] up to a multiplicative constant which only depends on I t .
The following lemma will be used in the proof of Lemma 4.4.
Proof. The LHS of (4.28) can be written as To treat the second term on the RHS of (4.28), we introduce the energy stress tensor T ε In view of (4.8a), we have the identity Testing this identity by H I , integrating by parts and using (4.6), we obtain (4.32) So adding zero leads to Substituting this identity into (4.29) leads to (4.28).
Proof of Lemma 4.4. Using the energy dissipation law (4.1) and adding zero, we compute the time derivative of the energy (2.7) by Due to the symmetry of the Hessian of ψ ε and the boundary conditions (4.6), we have Hence, the first integral on the RHS above can be rewritten as Using (4.28) in Lemma 4.5 to replace the third and fourth integrals on the RHS of the above identity and rewriting the last integral, we arrive at (4.34) First, note that the third to last and second to last integrals combine to J 2 ε . Next, by the property (4.11b) of the orthogonal projection (4.10), we can also find J 1 ε on the right-hand side. Indeed, Using the definition (4.8b) of n ε , we may merge the second, third, and the last integral on the RHS of (4.34) to obtain Now we complete squares for the first four terms on the RHS of (4.35): Reordering terms, we have Using the definition (4.8b) of the normal n ε and the chain rule in form of (4.11a), the terms in (4.35) form the last missing square. Integrating over the domain Ω and substituting into (4.35) we arrive at (4.19).

Convergence to the harmonic map heat flow
This section is devoted to the proof of Theorem 2.1. We start with a lemma about uniform estimates of Q ε .
Lemma 5.1. There exists a universal constant C = C(I 0 ) such that Moreover, for any fixed δ ∈ (0, δ I ), there holds Proof. We first establish a priori estimates of the solutions Q ε which are independent of ε. It follows from (4.18) and the assumption (2.12) that This together with the orthogonal projection (4.10) yields The above two estimates together with (4.13b) implies (5.1). Moreover, (5.2) follows from (5.4) and (4.13e). Now we turn to the time derivative. It follows from (5.4) that Using (4.8a), we expand the integrand in the above estimate and deduce So combining (5.2) with (5.7) leads us to (5.3).
With the above uniform estimates, we can prove the following convergence result.