Minimizers of the Landau-de Gennes energy around a spherical colloid particle Lia Bronsard McMaster University Results obtained with: S. Alama, X. Lamy Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 1 / 1
Nematic Liquid Crystals Fluid of rod-like particles, partially ordered: translation but rotational symmetry is broken. Nematic phase: νηµα , thread/fil, particles prefer to order parallel to their neighbors Director n ( x ), | n ( x ) | = 1 indicates local axis of preference: gives on average the direction of alignment. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 2 / 1
Oseen–Frank energy A variational model for equilibrium configurations of liquid crystals. Equilibria n : Ω ⊂ R 3 → S 2 minimize elastic energy, � E ( n ) = e ( n , ∇ n ) dx Ω e ( n , ∇ n ) = K 1 ( ∇ · n ) 2 + K 2 [ n · ( ∇ × n )] 2 + K 3 [ n × ( ∇ × n )] 2 Simple case: one-constant approximation K 1 = K 2 = K 3 = 1, E ( n ) = 1 � the S 2 harmonic map energy. |∇ n | 2 dx , 2 Ω n is not oriented, − n ∼ n gives same physical state. n : Ω → R P 2 . = ⇒ Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 3 / 1
Harmonic Maps to S 2 (or R P 2 ) Real-valued minimizers f : Ω → R of the Dirichlet energy Ω |∇ f | 2 dx are harmonic functions, ∆ f = 0. E ( f ) = 1 � 2 ◮ Linear elliptic PDE; solutions are smooth, bounded singularities removable. When u : Ω → M , M a smooth manifold, minimizers solve a nonlinear elliptic system of PDE. For M = S k or R P k , − ∆ n = |∇ n | 2 n Regularity theory for S 2 or R P 2 -valued harmonic maps: ◮ Schoen-Uhlenbeck (1982): S 2 -valued minimizers are H¨ older continuous except for a discrete set of points. ◮ Brezis-Coron-Lieb (1986): singularities have degree ± 1, n ≃ Rx | x | , R orthogonal. (“hedgehog”, “antihedgehog”) ◮ Hardt-Kinderlehrer-Lin (1986): for Oseen-Frank, min are real analytic except for a closed set Z , H 1 ( Z ) = 0. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 4 / 1
Applications of colloidal suspensions in nematic liquid crystals: photonics, biomedical sensors, ... I. Musevic, M. Skarabot and M. Ravnik, Phil Trans Roy Soc A, 2013 Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 5 / 1
The spherical colloid Consider a nematic in R 3 surrounding a spherical particle B r 0 (0). n ( x ) ' e z , | x | ! 1 Ω = R 3 \ B r 0 (0), exterior domain. ∂ Ω = ∂B r 0 As | x | → ∞ , tend to vertical director, n ( x ) → ± e z On ∂ B r 0 , homeotropic (normal) anchoring: ◮ Strong (Dirichlet) with x n = e r = | x | , ◮ Weak anchoring, via surface ∂ B r 0 | n − e r | 2 dS W � energy, n ( x ) ' x 2 | x | x 2 ∂B r 0 Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 6 / 1
Size matters Physicists observe that the character of the minimizers should depend on particle radius r 0 and anchoring strength W . (a) (b) (c) Kleman & Lavrentovich, Phil. Mag. 2006 . (a) For large r 0 , a “dipolar” configuration, with a detached (antihedghog) defect; (b) For small r 0 with large W , a “quadripolar” minimizer, with no point singularity but a “Saturn ring” disclination; (c) For small r 0 and low W , no singular structure at all. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 7 / 1
Problems with Oseen-Frank “Saturn ring”: ◮ Solution should have a 1-D singular set. ◮ Harmonic map or Oseen-Frank minimizers have only isolated point defects. Dipolar, with detached point defect: ◮ This may be observed in a harmonic map model. ◮ But harmonic map/Oseen-Frank has no fixed length scale; cannot distinguish different radii. New approach: embed the harmonic map problem in a larger family of variational problems with a natural length scale. The harmonic maps may be recovered in an appropriate limit. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 8 / 1
Landau–de Gennes Model A relaxation of the harmonic map energy. Introduce space of Q-tensors: Q ( x ) ∈ Q 3 , symmetric, traceless 3 × 3 matrix-valued maps. Q(x) models second moment of the orientational distribution of the rod-like molecules near x. Eigenvectors of Q ( x ) = principal axes of the nematic alignment. Uniaxial Q-tensor: two equal eigenvalues; principal eigenvector defines a director n ∈ S 2 , Q n = s ( n ⊗ n − 1 3 Id ). Q n = Q − n ; these represent R P 2 -valued maps. Biaxial Q-tensor: all eigenvalues are distinct. Strictly speaking, no director; but the principal eigenvector is an approximate director. Isotropic Q-tensor: all eigenvalues are equal, so Q = 0. No preferred direction, the liquid crystal has no alignment or ordering. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 9 / 1
The LdG Energy � ˆ � � L 2 |∇ Q | 2 + f ( Q ) F ˆ L ( Q ) = dx , Ω , Potential f ( Q ) = − a 2tr ( Q 2 ) + b 3tr ( Q 3 ) + c 4(tr ( Q 2 )) 2 − d , a = a ( T NI − T ) , b , c > 0 constant, d chosen so min Q f ( Q ) = 0. f ( Q ) depends only on the eigenvalues of Q . 3 Id ) with n ∈ S 2 (uniaxial) and Q = s ∗ ( n ⊗ n − 1 f ( Q ) = 0 ⇐ ⇒ √ b 2 + 24 ac ) / 4 c > 0 s ∗ := ( b + Euler–Lagrange equations are semilinear, Q 2 − 1 ˆ 3 | Q | 2 I + c | Q | 2 Q � � L ∆ Q = ∇ f ( Q ) = − aQ − b Uniaxial solutions are the exception; in most geometries expect biaxiality rules [Lamy, Contreras–Lamy] Analogy: Ginzburg–Landau model, a relaxation of the S 1 -harmonic map problem: � Ω [ ε 2 2 |∇ u | 2 + ( | u | 2 − 1) 2 ], u : Ω → C Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 10 / 1
The spherical colloid Consider a nematic in R 3 surrounding a spherical particle B r 0 (0). Ω = R 3 \ B r 0 (0), exterior domain. n ( x ) ' e z , Minimize LdG over Q ( x ) ∈ H 1 (Ω; Q 3 ). | x | ! 1 As | x | → ∞ , Q is uniaxial, with vertical ∂ Ω = ∂B r 0 e z ⊗ e z − 1 � � director, Q ( x ) → s ∗ . 3 I On ∂ B r 0 , homeotropic (normal) anchoring: ◮ Strong (Dirichlet) with n = e r = x | x | , e r ⊗ e r − 1 � � Q ( x ) | ∂ B r 0 = Q s := s ∗ 3 I . ◮ Weak anchoring, via surface energy, ˆ ∂ B r 0 | Q ( x ) − Q s | 2 dS W � 2 n ( x ) ' x | x | ˆ L ∂ Q = ⇒ ∂ν = Q s − Q on ∂ B r 0 . ◮ ˆ W x 2 ∂B r 0 Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 11 / 1
Two scaling limits First rescale by the particle radius r 0 ; Ω = R 3 \ B 1 (0), � � ˆ dx + ˆ L |∇ Q | 2 + f ( Q ) W ∂ B 1 | Q s − Q | 2 dA . � � F ( Q )= Ω 2 r 2 2 r 0 0 and non-dimensionalize by dividing by the reference energy a ( T NI ): � L 2 |∇ Q | 2 + f ( Q ) ˜ dx + W ∂ B 1 | Q s − Q | 2 dA . � � � F ( Q ) = Ω 2 W r 2 ˆ ˆ 0 a ( T NI ) L with L = 0 a ( T NI ) , W = . r 2 ˆ L Set Q ∞ = s ∗ ( e z ⊗ e z − 1 3 I ), and H ∞ = Q ∞ + H , with |∇ Q | 2 + | x | − 2 | Q | 2 � H = { Q ∈ H 1 � � loc : dx < ∞} . Ω For fixed parameters L , W , there exists a minimizer in H ∞ , Q ( x ) → Q ∞ uniformly as | x | → ∞ . Open question: at what rate? We consider two limits: ◮ Small particle limit. L → ∞ , with W → w ∈ (0 , ∞ ]. ◮ Large particle limit. L → 0, with Strong (Dirichlet) anchoring. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 12 / 1
Small particle limit ˜ � Ω [ L 2 |∇ Q | 2 + f ( Q ) ] dx + W � ∂ B 1 | Q s − Q | 2 dA . F ( Q )= 2 When L → ∞ , W → w ∈ (0 , ∞ ]: converge to a harmonic (linear) function, ∆ Q w = 0 in Ω = R 3 \ B 1 (0). Explicit solution, Q w ( x ) !! In spherical coordinates ( r , θ, ϕ ), Q w = α ( r )( e r ⊗ e r − I / 3) + β ( r )( e z ⊗ e z − I / 3) , ( r > 1), w 1 w 1 with α ( r ) = s ∗ r 3 , β ( r ) = s ∗ (1 − r ) . 3+ w 1+ w The eigenvalues of Q w may also be calculated explicitly, � − αβ sin 2 ϕ, λ 1 , 2 ( x ) = [ α + β ] [ α + β ] 2 λ 3 ( x ) = − α + β ± < 0 . 6 4 3 At eigenvalue crossing λ 1 = λ 2 , eigenvectors exchange = ⇒ discontinuous director! This occurs along a circle, ( r w , θ, 0), with r w root of: r 3 − 1+ w r 2 − w w 3+ w = 0 . Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 13 / 1
The Saturn Ring √ w = 1 . 732 ≈ 3. w = ∞ w = 3. w = 1. Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 14 / 1
Colloidal cuboids (homeotropic) “Superellipsoid” � x � 2 p � y � 2 p � z � 2 p + + = 1 b b a p = 1 p = 2 p = 10 Aspect ratio: a/b . “Sharpness”: p . a/b = 1 p = 1 1 . 5 2 2 . 5 3 10 Beller, Gharbi & Liu, Soft Matter, 2015, 11, 1078 Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 15 / 1
Large particle limit Now we consider L → 0 , with Dirichlet Q | ∂ B 1 = s ∗ ( e r ⊗ e r − 1 3 I ). Coincides with singular limit as elastic constant L → 0. (Majumdar-Zarnescu; Nguyen-Zarnescu) Minimizer converges to uniaxial Q -tensor, Q ∗ = s ∗ ( n ⊗ n − 1 3 I ), locally uniformly, away from a discrete set of singularities. Director n ( x ) ∈ S 2 is a minimizing harmonic map. No “Saturn ring”, or any other line defects are possible. (Schoen-Uhlenbeck; Hardt-Kinderlehrer-Lin) Solution must have at least one singularity; but generally, neither boundary topology nor energy determine the number of defects. ◮ Hardt-Lin-Poon (1992) There exist axisymmetric harmonic maps in Ω = B 1 (0), with degree-zero Dirichlet BC and arbitrarily many pairs of degree ± 1 defects on the axis. ◮ Hardt-Lin (1986) For any N , ∃ g : ∂ B 1 (0) → S 2 with degree zero such that the minimizing harmonic map has N defects in B 1 (0). Lia Bronsard (McMaster) Spherical Colloid Lyon 2016 16 / 1
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