Static continuum theory of nematic liquid crystals.

Transcription

1 Static continuum theory of nematic liquid crystals. Dmitry Golovaty The University of Akron June 11, Some images are from Google Earth and bisenyuk/liquidcrystals Dmitry Golovaty (UA) June 11, / 32

2 Nematic Liquid Crystals Figure: Logs in the Spirit Lake, Mt. St. Helens. Dmitry Golovaty (UA) June 11, / 32

3 Nematic Liquid Crystals Figure: Logs in the Spirit Lake, Mt. St. Helens. Dmitry Golovaty (UA) June 11, / 32

4 Director-Based Theory Suppose that a nematic occupies a domain Ω R 3 and n : Ω S 2. The director field n(x) represents local orientation of nematic molecules near x Ω. To formulate a continuum variational theory, need a functional space and an energy functional that take into account Elastic distortions of the director field n in Ω Interactions of the nematic with the walls of the container, i.e. the boundary or anchoring conditions satisfied by the director field n on Ω. Note: Additional effects (magnetic field, etc.) can be taken into account beyond the scope of this talk. Dmitry Golovaty (UA) June 11, / 32

5 Oseen-Frank Model Oseen-Frank elastic energy density (Frank, 1958): f OF (n, n) := K 1 2 ( div n)2 + K 2 2 ( curl n n)2 + K 3 curl n n K 2 + K 4 (tr ( n) 2 ( div n) 2) 2 Splay Twist Bend Saddle Splay Dmitry Golovaty (UA) June 11, / 32

6 Anchoring Conditions Controlled, e.g., by mechanical treatment or use of surfactants. Two possible types of boundary conditions: Strong anchoring: Homeotropic (Dirichlet) Planar Tilted Weak anchoring via a surface energy density term, e.g.: f s OF (n, ν) = γ(n ν)2 or f s OF (n, ν) = γ ((n ν) 2 cos 2 α where ν is an outward unit normal to Ω. The first expression is a Rapini-Papoular surface energy density. Dmitry Golovaty (UA) June 11, / 32 ) 2

7 Variational problem (Strong anchoring): Minimize F OF [n] := Ω { K1 2 ( div n)2 + K 2 2 ( curl n n)2 + K 3 curl n n K 2 + K 4 (tr ( n) 2 ( div n) 2)} 2 in H 1 ( Ω, S 2) subject to the appropriate boundary data, i.e., n Ω = ν for the homeotropic anchoring. (Hardt, Kinderlehrer and Lin, 1986) For the positive K 1, K 2, K 3 global minimizers of F OF exist among all maps in H 1 ( Ω, S 2) subject to Lipschitz, S 2 -valued Dirichlet boundary data. Any minimizer is smooth except for a closed set of Hausdorff dimension strictly less than 1. Dmitry Golovaty (UA) June 11, / 32

8 Facts: When K 1 = K 2 = K 3 = K and K 4 = 0, the Oseen-Frank energy reduces to the Dirichlet integral F OF [n] = K n 2. The saddle-splay term is a null Lagrangian, i.e., its integral over Ω depends only on the boundary data this term reduces to a constant for Dirichlet boundary conditions on n. Any configuration with a line singularity (observed experimentally), e.g., n(x) = (x/ x 2 + y 2, y/ x 2 + y 2, 0) has an infinite energy. Ω Dmitry Golovaty (UA) June 11, / 32

9 Inability to model line defects with a finite energy within OF theory can be addressed by Moving to higher-dimensional order parameters (Ericksen s theory for nematics with variable degree of orientations, Landau-de Gennes theory) Making suitable modifications to the energy functional and the class of admissible maps (Ball and Bedford, 2014) e.g., replacing n 2 with n p, where 1 < p < 2 and/or allowing n to jump across surfaces by assuming that n SBV ( Ω, S 2). The second modification addresses another shortcoming of director-based theories, the issue of orientability. Note: Any version of a continuum theory based on a single vector field only works for uniaxial nematics and does not allow to model biaxiality. Dmitry Golovaty (UA) June 11, / 32

10 Nematics with variable degree of orientation To allow for line defects, Ericksen (1991) proposed to supplement n with a scalar field s : Ω ( 1 2, 1) to describe the degree of local orientational order. Simplified version of the energy functional: { } F E [s, n] := K s s 2 + K n s 2 n 2 + W (s, T ) Here and Ω min s ( 1,1) W (s, T ) = W (s 0(T ), T ) = 0 2 lim W (s, T ) = lim W (s, T ) =. s 1/2 s 1 Dmitry Golovaty (UA) June 11, / 32

11 The model allows for a phase transition between nematic and isotropic states: { s0 0, T < T s(t 0 ) = c, (nematic state = order), 0, T > T c, (isotropic state = disorder), where T c R is a critical temperature. If K s = K n = K, set u = sn then F E [u] = K u 2 + W ( u, T ) - Ginzburg-Landau model. Ω Note: As formulated, Ericksen s model does not resolve orientability issue and it cannot be used to model biaxiality. Dmitry Golovaty (UA) June 11, / 32

12 Orientability Experimental fact: Probability of finding the head of a molecule pointing in a given directions is equal to the probability of finding the tail of a molecule pointing in the same direction. Consequence: n(x) = n 0 and n(x) = n 0 for some n 0 S 2 correspond to the same nematic state at x. The tensor field n n (possibly, translated and/or dilated) is, however, invariant under inversion n n. Conclusion: Local orientation of nematic molecules is described by a line field with values in RP 2 and not a vector field with values in S 2. The classical OF theory will give incorrect predictions when a minimizing line field is not orientable (Ball and Zarnescu, 2010). Dmitry Golovaty (UA) June 11, / 32

13 A B a 3b Stadium configuration (Ball and Zarnescu, 2010) re 3: A situation in which the energy minimizer is non-orientable Nematic disclination s ( (0, 1, 0) (0, 1, 0) 1 3 Q(x) Id), x M 3 = s ( n δ(x) n δ(x) 1 3 Id), x M 1 \ M 4, x 2 δ s ( m δ(x) m δ(x) 1 3 Id), x M 2 \ M 5, x 2 δ ( ) Dmitry def Golovaty x 2 δ(ua) x 1 June 11, / 32

14 Q-tensor theory Let Ω R 3 and ρ(n, x) be a pdf of molecular orientations at x Ω, where n S 2. Since head and tail are equiprobable = ρ( n, x) = ρ(n, x) and the first moment of ρ vanishes. Nontrivial information about LC configuration at x is given by the second moment M(x) = (n n)ρ(n, x)dn S 2 Note: M T (x) = M(x) and tr M(x) = 1 for all x Ω. LC is isotropic at x if ρ(n, x) 1 4π = M(x) = M iso = 1 3 I. Q tensor: Q(x) = M(x) M iso so that Q vanishes in the isotropic state. Dmitry Golovaty (UA) June 11, / 32

15 Nematic Q-tensor Q M 3 3 sym is a traceless tensor eigenvalues satisfy λ 1 + λ 2 + λ 3 = 0 with a mutually orthonormal eigenframe {e 1, e 2, e 3 }. Uniaxial nematic: repeated nonzero eigenvalues λ 1 = λ 2 Q = S ( n n 1 3 I), where S := 3λ 3 2 is the uniaxial nematic order parameter and n S 2 is the nematic director. Biaxial nematic: no repeated eigenvalues Q = S 1 ( e1 e I) + S 3 ( e3 e I), where S 1 := 2λ 1 + λ 3 and S 3 = λ 1 + 2λ 3 are biaxial order parameters. Isotropic: all eigenvalues are equal zero Q = 0. By construction, λ i [ 1 3, 2 3], where i = 1, 2, 3. Dmitry Golovaty (UA) June 11, / 32

16 Landau-de Gennes Model Bulk elastic energy density: f e (Q, Q) := L 1 2 Q 2 + L 2 2 Q ik,jq ij,k + L 3 2 Q ij,jq ik,k + L 4 2 Q lkq ij,k Q ij,l Bulk Landau-de Gennes energy density: f LdG (Q) := a tr ( Q 2) + 2b 3 tr ( Q 3) + c 2 ( tr ( Q 2 )) 2 Here a(t ) is temperature-dependent, c > 0, and f LdG 0 by adding an appropriate constant. Function of eigenvalues of Q only. Depending on T, minimum is either isotropic or nematic w/specific s. Surface energy density (Either strong or weak anchoring): f s (Q) := f (Q, ν) on the boundary of the container and ν S 2 is a normal to the surface of the liquid crystal. Dmitry Golovaty (UA) June 11, / 32

17 Remarks: When L 4 0, the energy F LdG [Q] = Ω f e (Q, Q) + f LdG (Q) is unbounded from below. The existence of the global minimizer can be established, subject to constraints on L i, i = 1,..., 4 if the potential term is modified according to the Ericksen s idea (Ball and Majumdar (2009)). When L 4 = 0, the functional is coercive subject to constraints on L i, i = 1, 2, 3 (Gartland and Davis (1998), Longa et al (1987)). However, in this case, two of the three elastic constants in a Oseen-Frank reduction (when Q = s ( n n 1 3 I ) with a constant s and n = 1) must be equal. Dmitry Golovaty (UA) June 11, / 32

18 Nematic Film Z ^z Y X h Ω ν Figure: Geometry of the problem. Here Ω R 2 and h > 0 is small. Dmitry Golovaty (UA) June 11, / 32

19 Nematic energy functional: E[Q] := {f e (Q, Q) + f LdG (Q)} dv + Ω [0,h] Ω {0,h} f s (Q, ẑ) da Uniaxial data on the lateral boundary of the film: Q Ω [0,h] = g H 1/2 ( Ω; A). Admissible class: C g h := { Q H 1 (Ω [0, h]; A) : Q Ω [0,h] = g }, where A is the set of three-by-three symmetric traceless matrices. Dmitry Golovaty (UA) June 11, / 32

20 Osipov-Hess surface energy Bare surface energy (Osipov-Hess): f s (Q, ν) := c 1 (Qν ν) + c 2 Q Q + c 3 (Qν ν) 2 + c 4 Qν 2 where c i, i = 1,..., 4 are constants. Observe that: Q Q = 2 Qν 2 (Qν ν) 2 + Q 2 Q 2, where Q 2 M 2 2 sym is a nonzero square block of (I ν ν) Q (I ν ν). The traceless condition for Q: where x := Qν R 3. tr Q 2 + x ν = 0 Dmitry Golovaty (UA) June 11, / 32

21 In terms of x and Q 2 : f s (Q, ν) = c 1 (x ν) + c 2 Q 2 Q 2 + (c 3 c 2 ) (x ν) 2 + (2c 2 + c 4 ) x 2 This expression has a family of surface-energy-minimizing tensors that is 1 parameterized by at least one free eigenvalue 2 normal to the surface of the liquid crystal is an eigenvector as long as c 2 = 0, α = c 3 + c 4 > 0, and γ = c 4 > 0. Then the surface energy has the form where β = c 1 2(c 3 +c 4 ). f s (Q, ν) = α [(Qν ν) β] 2 + γ (I ν ν) Qν 2 Dmitry Golovaty (UA) June 11, / 32

22 Nondimensionalization Let L 4 = 0 and where D := diam(ω). Set x = x D, ỹ = y D, z = z h, F ɛ = 2 L 1 h E, ξ = L 1 2D 2, ɛ = h D, δ = K 2 = L 2 L 1, K 3 = L 3 L 1 2ξ c A = a c, B = b c α = α ξ, γ = γ ξ Dmitry Golovaty (UA) June 11, / 32

23 Nondimensional energy F ɛ [Q] = Ω [0,1] ( f e ( Q) + 1 ) δ 2 f LdG (Q) dv + 1 f s (Q, ẑ) da, ɛ Ω {0,1} where ] f e ( Q) := [ xy Q 2 + K 2 Q ik,j Q ij,k + K 3 Q ij,j Q ik,k + 2 ɛ [K 2Q i3,j Q ij,3 + K 3 Q ij,j Q i3,3 ] + 1 ] [ Q ɛ 2 z 2 + (K 2 + K 3 )Qi3,3 2, f LdG (Q) = 2A tr ( Q 2) B tr ( Q 3) + ( tr ( Q 2)) 2, f s (Q, ẑ) = α [(Qν ν) β] 2 + γ (I ν ν) Qν 2. Dmitry Golovaty (UA) June 11, / 32

24 Assumptions Suppose for simplicity that K 2 = K 3 = 0 then for every Q C g 1 F ɛ [Q] = and set Ω [0,1] { Q x 2 + Q y ɛ 2 Q z ( δ 2 2A tr ( Q 2) B tr ( Q 3) + ( tr ( Q 2)) )} 2 dv + 1 ( α [(Qẑ ẑ) β] 2 + γ (I ẑ ẑ) Qẑ 2) da, ɛ Ω {0,1} f s (Q, ẑ) =: f (0) s (Q, ẑ) + ɛf (1) s (Q, ẑ) this allows for different asymptotic regimes for α and γ. Dmitry Golovaty (UA) June 11, / 32

25 Limiting problem Let { F 0 [Q] := Here and 2 { } Ω xy Q f δ 2 LdG (Q) + f s (1) (Q, ẑ) da if Q Hg 1, + otherwise. H 1 g := { Q H 1 (Ω; D) : Q Ω = g } { D := Q A : Q argmin Q A f (0) s } (Q), for some boundary data g : Ω D. Dmitry Golovaty (UA) June 11, / 32

26 Theorem (G, Montero, Sternberg (2015)) Fix g : Ω D such that Hg 1 is nonempty. Then Γ-lim ɛ F ɛ = F 0 weakly in C g 1. Furthermore, if a sequence {Q ɛ} ɛ>0 C g 1 satisfies a uniform energy bound F ɛ [Q ɛ ] < C 0 then there is a subsequence weakly convergent in C g 1 to a map in Hg 1. Proof. Idea: can use a trivial recovery sequence. Indeed, if Q ɛ Q C g 1 \H1 g then lim ɛ 0 F ɛ [Q ɛ ] = + = F 0 [Q] and when Q ɛ Q H 1 g then F ɛ [Q ɛ ] = F 0 [Q ɛ ] = F 0 [Q] for all ɛ. Dmitry Golovaty (UA) June 11, / 32

27 Asymtotic regimes - Regime I Let f (0) s = α [(Qẑ ẑ) β] 2 + γ (I ẑ ẑ) Qẑ 2 and f (1) s 0 two types of D-valued uniaxial Dirichlet data on Ω: Q = 3β ( n n 1 3 I), where n ẑ is any S 1 -valued field on Ω. Q = 3β 2 (ẑ ẑ 1 3 I). Note: In the first case the boundary data can have any degree, while in the second case the Dirichlet condition is completely rigid as Q is equal to a constant. Dmitry Golovaty (UA) June 11, / 32

28 Can represent Q Hg 1 as Q = p 1 β 2 p 2 0 p 2 p 1 β β. Then F 0 [Q] = F 0 [p] := Ω {2 p 2 + 1δ } 2 W ( p ) dv, where p = (p 1, p 2 ) and W (t) = 4t 4 + Ct 2 + D, with C = 6β 2 4Bβ + 4A and D R. Dmitry Golovaty (UA) June 11, / 32

29 If Q Ω [0,1] = 3 2 β ( ẑ ẑ 1 3 I ), admissible functions satisfy the boundary condition The minimizer of F 0 [p] = Ω p Ω = 0. {2 p 2 + 1δ } 2 W ( p ) dv then has a constant angular component scalar minimization problem for p := p and 1 If C 0 then the minimizer p 0. 2 If C < 0 then the minimizer p solves the problem p + 1 δ 2 W (p) = 0 in Ω, p = 0 on Ω. Dmitry Golovaty (UA) June 11, / 32

30 Now suppose Q Ω [0,1] = 3β (n n 13 ) I, where n : Ω S 1. We have p = 3β (n ), n 1n 2, on Ω where p = 3β 2. If p is smooth and nonvanishing, it has a well-defined winding number d Z. We set the degree of g to be equal to d/2. Then p 0 must vanish somewhere within a vortex core structure of a characteristic size of δ in Ω. Dmitry Golovaty (UA) June 11, / 32

31 . P(, r) Topologically nontrivial boundary data will cause the director to escape from the xy-plane to the z-direction. The requirement that Q 0 takes values in D forces the escape to happen through biaxial states that are heavily penalized by the Landau-de Gennes energy. Figure: Geometry of the target space. Degree of biaxiality: ( trq ξ(q) 2 3 ) 2 ( β2 4p 2 β 2) 2 := 1 6 = 1 27 (trq 2 3 ) (4p 2 + 3β 2 ) 3 where ξ(q) = 0 implies that Q is uniaxial. Dmitry Golovaty (UA) June 11, / 32

32 Asymtotic regimes - Regime II Let α = ɛa for some a > 0 and γ = O(1). Then and f (0) s (Q, ẑ) = γ (I ẑ ẑ) Qẑ 2 f (1) s (Qẑ) = a [(Qẑ ẑ) β] 2 and the target set D consists of traceless symmetric tensors having ẑ as one of its eigenvectors. The limiting functional for Q H 1 g is F 0 [Q] = 2 { Q 2 + 1δ } Ω 2 f LdG (Q) + a [(Qẑ ẑ) β] 2 da (cf. Bauman-Park-Phillips when δ 0 and a = 0). Dmitry Golovaty (UA) June 11, / 32