An Overview of the Oseen-Frank Elastic Model plus Some Symmetry Aspects of the Straley Mean-Field Model for Biaxial Nematic Liquid Crystals

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1 An Overview of the Oseen-Frank Elastic Model plus Some Symmetry Aspects of the Straley Mean-Field Model for Biaxial Nematic Liquid Crystals Chuck Gartland Department of Mathematical Sciences Kent State University, Kent, Ohio, USA Thanks: John Toland (Isaac Newton Institute) John Ball, David Chillingworth, Mikhail Osipov, Peter Palffy-Muhoray, Mark Warner (Programme) David Chillingworth, Peter Palffy-Muhoray (Workshop) Support: NSF DMS-2597 Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide Background macroscopic continuum model, equilibrium states, orientational properties variational, phenomenological nematic LC mesophase: orientational order, no positional order flavors of the model: uniaxial nematic ( nematic ), chiral nematic ( cholesteric ) assumes uniform degree of order, constant temperature, constant mass density director: n = n(x), n(x) =, x average orientation of distinguished molecular axis distinguished eigenvector of uniaxial Q Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 2

2 Free Energy Functional: min F[n], N = admissible (regularity, BCs, n = ) n N F[n] = W e (n, n) dv, W e = distortional elastic energy density W e penalizes distortions from ground state (e.g., parallel steric, Van der Waals) additional terms: magnetic and/or electric fields surface anchoring potential flexoelectric polarization Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 3 Modeling Assumptions continuum: n frame indifference: molecular length scale W e (Qn,Q nq T ) = W e (n, n), Q SO(3) material symmetry (nematic only):, Q O(3) evenness: n n W e ( n, n) = W e (n, n) quadratic in n non-negative: W e (n, n) (W e = on ground state... usually ) Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 4

3 Nematic Model 2W nem = K (div n) 2 + K 2 (n curl n) 2 + K 3 n curl n 2 + (K 2 + K 4 ) [ tr ( ( n) 2) (div n) 2] K i = K i (T), K,K 2,K 3, K 2 + K 4 2K Ground State: uniform field n = const div n =, curl n = W nem = Surface Term / Null Lagrangian: tr ( ( n) 2) (div n) 2 = div [ curl n n (div n)n ] often zero or an additive constant... Equal Elastic Constants: K = K 2 = K 3 = K, K 4 = W nem = K 2 n 2 = K 2 3 ( ni ) 2 x j i,j= Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 5 Cholesteric Model 2W chol = K (div n) 2 + K 2 (n curl n + q ) 2 + K 3 n curl n 2 ground state: Beltrami field K,K 2,K 3, ±q, 2π q = cholesteric pitch n = cos q z e x + sinq z e y div n =, curl n = q n W chol = Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 6

4 Interpretation Splay (K ) Twist (K 2 ) Bend (K 3 ) n = e r n = cos qz e x + sinqz e y n = e θ K Saddle Splay...? Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 7 Dimensions and Scales Dimensions: [F] = energy, [W e ] = energy volume, [K i] = energy length = force, [q ] = Typical Values: 5CB (26 C) length K = J/m, K 2 = J/m, K 3 = J/m K i pn, K K 3, K 2 2 K,3 Intrinsic Length Scales: cholesteric pitch boundary extrapolation length magnetic/electric coherence lengths generally not of molecular order (vs Landau-de Gennes, mesoscopic )... Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 8

5 Electric Field Typical Situation: capacitor, constant voltage [ε] = ε E = U, U = electrostatic potential D = ε E + P = electric displacement / flux P = ε χ(n)e = induced polarization ( ) D = ε(n)e, ε(n) = ε I + χ(n) = dielectric tensor ε ε ε = ε [ ε I + ε a n n ], ε a := ε ε ε l,m,n Electrostatics: Gauss s Law div D = div ( ε(n) U ) = 3 i,j= ( U ) ε ij = x i x j Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 9 Perturbative Effect: div D = U + ε a ε div ( ( U n)n ) =, ε a ε = ε ε ε = O(). Free Energy: F[n,U] = [ We (n, n) W elec (n, U) ] W elec = 2 D E = 2 ε(n) U U = 2 ε [ ε U 2 + ε a ( U n) 2] δ U F = div D = div ( ε(n) U ) = Equilibrium: min n max U F[n,U] min F[n,U(n)], subject to div( ε(n) U ) = n Influence: ε a > n E vs ε a < n E Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide

6 Magnetic Field similar, but... Free Energy: F[n] = [ We (n, n) W mag (n) ], W mag = 2 B H B = µ (M + H), M = χ(n)h B H = µ ( + χ )H 2 + µ χ a (H n) 2, χ a := χ χ Typical Form Taken: χ a + χ = χ χ + χ = O( 6 ) H const F[n] = [W e (n, n) 2 µ χ a (H n) 2] Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide Equilibrium: intrinsic length scale: min F[n], H = uniform field n K = magnetic coherence length H µ χ a Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 2

7 Flexoelectricity distortion-induced polarization... Meyer (969) free-energy density: P f (n) = e (div n)n + e 3 n curl n e,e 3 = flexoelectric coefficients (Lagerwall sign convention) W total = P f E electrostatics: D = ε(n)e + P f div ( ε(n) U ) = divp f (n) Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 3 Surface Anchoring Potential weak anchoring vs strong anchoring... F = W vol + various forms... e.g., Rapini-Papoular (969) enters natural BCs... dimensions: [W surf ] = energy area W surf, W surf = W surf (n,ν,...) W surf = ± 2 W (n ν) 2 typical values: W = 6 to 4 J/m 2 intrinsic length scale: K W = surface extrapolation length Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 4

8 Equilibrium Equations various forms... Ex: director components, no E-field F[n] = W vol (n, n) dv + W surf (n) ds, = Γ Γ, Γ strong Γ [ Wvol δf[n ](u) = n u + W ] vol n u W surf + Γ n u weak form #: δf[n ](u) =, u {n u = in,u = on Γ } =: T n weak form #2: δf[n ](v) = λ n v + µ n v, v {v = on Γ } Γ Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 5 strong form: ( Wvol ) div + W vol n n = λ n, in ( Wvol ) ν + W surf n n = µ n, on Γ Stability: δ 2 F[n ](u) λ u 2 µ u 2, u T n Γ more complicated with coupled E-field... Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 6

9 Orientation Angles deal with n = pointwise constraint... e.g., n = sin θ cos φe + sin θ sin φe 2 + cos θ e 3 θ = θ(x), φ = φ(x), x F[n] = F[θ,φ] = W(θ, θ,φ, φ)... Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 7 -D Example System: chiral nematic film electric field + negative dielectric anisotropy strong homeotropic anchoring z d E x L L Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 8

10 z Influences: Phases: BCs n = e z q n = helical twist (P = 2π/q = pitch ) ε a < n E z z 3 2 x x x Homeotropic Translation Independent Cholesteric (TIC) Cholesteric Finger Type (CF) Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 9 -D Analysis Assumption: n = n(z), U = U(z) (includes Homeotropic and TIC, not CF) Free Energy Density: representation, θ = θ(z), φ = φ(z) 2W = ( K sin 2 θ + K 3 cos 2 θ ) θ 2 z + ( K 2 sin 2 θ + K 3 cos 2 θ ) sin 2 θ φ 2 z 2K 2 q sin 2 θ φ z + K 2 q 2 ε ( ε sin 2 θ + ε cos 2 θ ) U 2 z Equilibrium Equations: θ(z) =, U(z) = V z/d Homeotropic d [( K sin 2 θ + K 3 cos 2 θ ) ] { θ z = sinθ cos θ (K K 3 )θz 2 dz + [ (2K 2 K 3 ) sin 2 θ + K 3 cos 2 θ ] φ 2 z 2K 2 q φ z + ε ε a U 2 z d { sin 2 θ [( K 2 sin 2 θ + K 3 cos 2 θ ) ] } φ z K 2 q = dz d [( ε sin 2 θ + ε cos 2 θ ) ] U z = dz } Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 2

11 Phase Diagram Phases and Coexistence Regions 2nd Order Transition st Order Transition Metastability Limit Metastability Limit.5.45 Phases and Coexistence Regions 2nd Order Transition st Order Transition Metastability Limit Metastability Limit.5 TIC.4 V V.35 HOMEOTROPIC d / P d / P Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide V d / P V =.3 V =.37 V = θ m θ m θ m d / P d / P d / P Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 22

12 Notes and Issues Effective for modeling many problems at device or experiment scale Consistent with Ericksen-Leslie hydrostatics, provided... Admits point defects (n = e r spherical F < )... but does not admit line defects (n = e r cylindrical F = ) Parity issues: globally OK: F[ n] = F[n] locally?... workarounds (n n n ) Ball and Zarnescu (2) n(x) = pointwise constraint a nuisance. Math analysis: Hardt-Kinderlehrer-Lin (986) References (books): Virga (994), Stewart (24) Symmetry, Bifurcation, and Order Parameters, Newton Institute, Cambridge Slide 23