Static continuum theory of nematic liquid crystals.

Static continuum theory of nematic liquid crystals. Dmitry Golovaty The University of Akron June 11, 2015 1 Some images are from Google Earth and http://www.personal.kent.edu/ bisenyuk/liquidcrystals Dmitry Golovaty (UA) June 11, 2015 1 / 32

Nematic Liquid Crystals Figure: Logs in the Spirit Lake, Mt. St. Helens. Dmitry Golovaty (UA) June 11, 2015 2 / 32

Nematic Liquid Crystals Figure: Logs in the Spirit Lake, Mt. St. Helens. Dmitry Golovaty (UA) June 11, 2015 3 / 32

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, 2015 4 / 32

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 2 2 + K 2 + K 4 (tr ( n) 2 ( div n) 2) 2 Splay Twist Bend Saddle Splay Dmitry Golovaty (UA) June 11, 2015 5 / 32

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, 2015 6 / 32 ) 2

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 2 2 + 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, 2015 7 / 32

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, 2015 8 / 32

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, 2015 9 / 32

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, 2015 10 / 32

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, 2015 11 / 32

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, 2015 12 / 32

000 00000 11111 000000 111111 0000000 1111111 0000000 1111111 000000 111111 00000 11111 111 00000 11111 000000 111111 0000000 1111111 0000000 1111111 000000 111111 00000 11111 000 111 A B 00000 11111 000000 111111 0000000 1111111 0000000 1111111 000000 111111 00000 11111 000 111 00000 11111 000000 111111 0000000 1111111 0000000 1111111 000000 111111 00000 11111 000 111 3a 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, 2015 13 / 32

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, 2015 14 / 32

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 1 1 3 I) + S 3 ( e3 e 3 1 3 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, 2015 15 / 32

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, 2015 16 / 32

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, 2015 17 / 32

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, 2015 18 / 32

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, 2015 19 / 32

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, 2015 20 / 32

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, 2015 21 / 32

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, 2015 22 / 32

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) + 4 3 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, 2015 23 / 32

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 2 + 1 ɛ 2 Q z 2 + 1 ( δ 2 2A tr ( Q 2) + 4 3 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, 2015 24 / 32

Limiting problem Let { F 0 [Q] := Here and 2 { } Ω xy Q 2 + 1 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, 2015 25 / 32

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, 2015 26 / 32

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, 2015 27 / 32

Can represent Q Hg 1 as Q = p 1 β 2 p 2 0 p 2 p 1 β 2 0 0 0 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, 2015 28 / 32

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, 2015 29 / 32

Now suppose Q Ω [0,1] = 3β (n n 13 ) I, where n : Ω S 1. We have p = 3β (n 21 12 ), 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, 2015 30 / 32

. 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, 2015 31 / 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, 2015 32 / 32