Università di Pavia Dipartimento di Matematica F. Casorati 1/23 http://www-dimat.unipv.it I seminari del giovedì Transizione di fase in cristalli liquidi II: Ricostruzione d ordine in celle nematiche frustrate Fulvio Bisi mailto:bisi@dimat.unipv.it
Historical introduction Material frustration is a common feature in defects and microconfinements in liquid crystals still offering unsolved problems. 2/23
Historical introduction Material frustration is a common feature in defects and microconfinements in liquid crystals still offering unsolved problems. 2/23 In nematic liquid crystals the boundary surfaces produce frustration...
Historical introduction Material frustration is a common feature in defects and microconfinements in liquid crystals still offering unsolved problems. 2/23 In nematic liquid crystals the boundary surfaces produce frustration... indirectly, dictating the presence of defects in the bulk...
Historical introduction Material frustration is a common feature in defects and microconfinements in liquid crystals still offering unsolved problems. 2/23 In nematic liquid crystals the boundary surfaces produce frustration... indirectly, dictating the presence of defects in the bulk... or directly, causing extreme confinements of the material.
Order reconstruction For defects,first shown by Schopohl and Sluckin (Defect core structure in nematic liquid crystals, Phys. Rev. Lett. 59 (1987), 2582-2584). 3/23 Within the core of a disclination two uniaxial states with orthogonal directors changed one into the other through a transformation not involving any director rotation bridged through biaxial configurations.
Order reconstruction For defects,first shown by Schopohl and Sluckin (Defect core structure in nematic liquid crystals, Phys. Rev. Lett. 59 (1987), 2582-2584). 3/23 Within the core of a disclination two uniaxial states with orthogonal directors changed one into the other through a transformation not involving any director rotation bridged through biaxial configurations. eigenvalue exchange: in the tensorial description of the nematic order, two uniaxial states with orthogonal directors can be connected with no eigenframe rotation by only changing the eigenvector that possesses the dominant eigenvalue.
Order reconstruction In a hybrid cell: Palffy-Muhoray, Gartland and Kelly (Palffy-Muhoray et al., A new configurational transition in inhomogeneous nematics. Liq. Cryst. 16 (1994), 713-718). 4/23 Two orthogonal uniaxial directors prescribed on the plates (one planar and the other homeotropic) connected either through a director bend or through an order reconstruction employing biaxial states.
Order reconstruction In a hybrid cell: Palffy-Muhoray, Gartland and Kelly (Palffy-Muhoray et al., A new configurational transition in inhomogeneous nematics. Liq. Cryst. 16 (1994), 713-718). 4/23 Two orthogonal uniaxial directors prescribed on the plates (one planar and the other homeotropic) connected either through a director bend or through an order reconstruction employing biaxial states. An interesting result of it is that, when the cell thickness is sufficiently small, only the order reconstruction exists as an equilibrium configuration for the free energy of the cell.
Order reconstruction In a hybrid cell: Palffy-Muhoray, Gartland and Kelly (Palffy-Muhoray et al., A new configurational transition in inhomogeneous nematics. Liq. Cryst. 16 (1994), 713-718). 4/23 Two orthogonal uniaxial directors prescribed on the plates (one planar and the other homeotropic) connected either through a director bend or through an order reconstruction employing biaxial states. An interesting result of it is that, when the cell thickness is sufficiently small, only the order reconstruction exists as an equilibrium configuration for the free energy of the cell. Assumption of strong anchoring on both bounding plates.
Order reconstruction In a hybrid cell: Palffy-Muhoray, Gartland and Kelly (Palffy-Muhoray et al., A new configurational transition in inhomogeneous nematics. Liq. Cryst. 16 (1994), 713-718). 4/23 Two orthogonal uniaxial directors prescribed on the plates (one planar and the other homeotropic) connected either through a director bend or through an order reconstruction employing biaxial states. An interesting result of it is that, when the cell thickness is sufficiently small, only the order reconstruction exists as an equilibrium configuration for the free energy of the cell. Assumption of strong anchoring on both bounding plates. Further investigations: similar lack of solution for weak anchorings.
Order reconstruction In a hybrid cell: Palffy-Muhoray, Gartland and Kelly (Palffy-Muhoray et al., A new configurational transition in inhomogeneous nematics. Liq. Cryst. 16 (1994), 713-718). 4/23 Two orthogonal uniaxial directors prescribed on the plates (one planar and the other homeotropic) connected either through a director bend or through an order reconstruction employing biaxial states. An interesting result of it is that, when the cell thickness is sufficiently small, only the order reconstruction exists as an equilibrium configuration for the free energy of the cell. Assumption of strong anchoring on both bounding plates. Further investigations: similar lack of solution for weak anchorings. Easy axes invariably at right angles.
Our Model A twisted cell with strong anchoring on two plates. 5/23 Boundary states prescribed to be uniaxial, internal states possibly biaxial. We release the condition that the two easy axes of the plates be orthogonal d e z +d x Figura 1: A cell bounded by two parallel plates. e y e x
Our Model with the constraint (traceless tensor): 3 Q = λ i e i e i, (1) i=1 6/23 λ 1 + λ 2 + λ 3 =, λ i [ 1 3, 2 3 ] Order parameter s for uniaxial states: (two eigenvalues equal): Degree of biaxiality of Q Uniaxial states: β 2 = ; Q = s(n n 1 3 I), s [ 1, 1] (2) 2 β 2 := 1 6 (trq3 ) 2 (trq 2 with β [, 1] (3) ) 3 Maximal biaxiality: β 2 = 1 ( detq = ) since trq 3 = 3detQ.
Our Model We impose uniaxial states at both plates Q = Q = s b ( e z e z 1 3 I ), (4) 7/23 and Q = Q + = s b ( n n 1 3 I ), (5) where n := cos ϕ e z + sin ϕ e y. (6) Two opposite twists: in general, one more strained than the other; they are one the mirror image of the other only for ϕ = π 2.
Our Model The symmetry of the system allows the use of 3 parameters for the q-representation (q := (q 1, q 2, q 3 )): where q i = q i (x) and Q = λ 1 = 2q 1, λ 2 = (q 1 + 2q 1 (q 1 q 2 ) q 3 q 3 (q 1 + q 2 ) q 2 2 + q2 3 ), λ 3 = (q 1 q 2 2 + q2 3 ). (7) trq 2 = 2[3q 2 1 + q2 2 + q2 3 ] trq 3 = 6q 1 [q 2 1 q2 2 q2 3 ]. (8) 8/23
Our Model Admissible q in the cone with axis along q 1 defined by the inequalities 1 3 q 1 1 q2 2 6 + q2 3 q 1 + 1 3. (9) 9/23 Uniaxial-state conical surface (λ 1 = λ 2, when q 1 <, and λ 1 = λ 3, when q 1 > ) (along with q 1 -axis): 9q 2 1 q 2 2 q 2 3 = (1) (Biscari P., G. Capriz, and E.G. Virga: On surface biaxiality. Liq. Cryst. 16 (1994), 479-489). Recast BC s: q 1 ( d) = q 1 (d) = s b 6 q 2 ( d) = s b 2 q 2 (d) = s b 2 cos 2ϕ q 3 ( d) = q 3 (d) = s b 2 sin 2ϕ. (11)
Free Energy: A. Bulk potential Classical Landau-de Gennes theory 1/23 B, C positive constitutive parameters f b := A 2 trq2 B 3 trq3 + C 4, (trq2 ) 2 (12) A = a(t T ) (T : nematic supercooling temperature). In homogeneous phase, with q 3 = : trq 2 = 2 3 s2 = 2(3q 2 1 + q 2 2), f b = 1 3 [As2 2 9 B s 2 3q2 2(s2 12q2 2) + 1 3 Cs4 ] (13) = 1 3 As2 2 27 Bs3 + 1 9 Cs4
Free Energy: A. Bulk potential f [kj/m 3 ] 25 2 11/23 15 1 5 T** T NI 5 1 15 2 25.5.5 1 s Figura 2: The uniaxial bulk potential function f vs. the order parameter s. T<T* T*
Free Energy: A. Bulk potential s := B 4C, 12/23 superheating temperature: T := T + B2 24aC (inflection point) T NI := T + B2 27aC < T (Nematic-Isotropic transition) reduced temperature ϑ := A A = T T T T s ± := s [1 ± 1 θ], with s b = s +
Free Energy: B. Equilibrium equations One-constant approximation 13/23 F[Q] := B (f e + f b )dv f e := L 2 Q 2 (14) δf := df[q ε] dε = ε= B Q ε := Q + εu, { L Q U + ( AQ BQ 2 + C(trQ 2 )Q + λi ) U } dv Euler-Lagrange equation: LQ + AQ BQ 2 + C(trQ 2 )Q + B 3 (trq2 )I =, (15) First integral: L 2 Q 2 + A 2 trq2 B 3 trq3 + C 4 (trq2 ) 2 = H, (16)
Free Energy: C. Non dimensional forms Biaxial coherence length: ξ b := L Bs + = [ 4LC B 2 ( 1 θ + 1) ] 1/2. (17) 14/23 Moreover, x := x ξ b Q = s Q F := 3A B 2 d 4C 2 F[Q] = F 1 1 { ( ξb d ) 2 ( 1 θ + 1) ( ) 2 dq dx + θ 3 trq2 2 3 trq3 + 1 8 (trq2 ) 2 } dx, (18)
Differential equations 15/23 F [q] = 1 2 1 1 { ξ 2 b d 2 ( 1 θ + 1 ) [ 3(q 1 ) 2 + (q 2) 2 + (q 3) 2] + θ 6 (3q2 1 + q 2 2 + q 2 3) + 2q 1 (q 2 1 q 2 2 q 2 3) + 1 4 (3q2 1 + q 2 2 + q 2 3) 2 } dx, (19) ξ 2 b ( 1 θ + 1)q d 2 1 = ϑ 6 q 1 1 3 [q2 2 + q2 3 3q2 1 ] + 1 2 (3q2 1 + q2 2 + q2 3 )q 1 ξb 2 ( 1 θ + 1)q d 2 2 = [ ϑ 6 2q 1 + 1 2 (3q2 1 + q2 2 + q2 3 )]q 2 ξb 2 ( 1 θ + 1)q d 2 3 = [ ϑ 6 2q 1 + 1 2 (3q2 1 + q2 2 + q2 3 )]q 3. Polar plots for (q 2, q 3 ). Bifurcation/Unfolding diagrams r := q 2 2 () + q2 3 () (2)
Polar Plot: ϕ = π 2 16/23 d/ξ b P + P Figura 3: Polar plot of q 2, q 3 components of order-tensor equilibrium solutions. ϕ = π 2, reduced temperature θ = 8, several values of dimensionless cell half-width d/ξ b ranging between 2.475 and 8 in the sense indicated by the arrows.
Ellipsoids: ϕ = π 2 1 17/23 1 6 4 2 2 x/ξ b 4 6 1 1 1 1 6 1 1 6 4 4 2 2 2 2 x/ξ b 4 4 6 1 1 6 1 1 Figura 4: Order-tensor ellipsoids against position across the cell, in units of the x/ξ b biaxial coherence length ξ b, for 3 basic solutions: opposite twists τ + and τ (top and middle), eigenvalue-exchange χ (bottom).
Polar Plot: ϕ =.45π 18/23 d/ξ b P + 2φ P Figura 5: d/ξ b 8. Upper curves correspond to less twisted solutions τ, approaching uniaxial states on the circumference as d/ξ b increases. Lower solid curves correspond to more twisted solutions τ +, while lower dashed curves are eigenvalue-exchange solutions χ. These merge in the. heavy line as d/ξ b decreases to the critical value d/ξ b = 3.33.
Polar Plot: ϕ = π 19/23 d/ξ b P Figura 6: d/ξ b 2. Solid curves correspond to π-twist solutions, approaching uniaxial states on the circumference as d/ξ b increases. Dashed curves correspond to eigenvalue-exchange solutions. These merge in the heavy. line as d/ξ b decreases to the critical value d/ξ b = 7.72.
Ellipsoids: ϕ = π 2/23 1 1 14 1 5 x/ξ b 5 1 14 1 1 1 1 14 1 1 14 1 1 5 5 x/ξ b x/ξ b 5 5 1 1 14 1 1 14 1 1 Figura 7: Order-tensor ellipsoids vs. position across the cell, in units of ξ b, for 3 basic solutions: uniform uniaxial alignment (top), π-twist (middle), eigenvalue exchange solution (bottom).
Bifurcation diagram (unfolding) 2 1.5 (a) 21/23 r 1.5 (b) (c).5 (b) 1 (a) 1.5 2 2 4 6 8 1 d / ξ b Figura 8: Bifurcation diagram of r vs. the dimensionless half-width d/ξ b of the cell, for several solution paths. Reduced temperature θ = 8, total twist angles: (a) ϕ =, (b) ϕ =.45π, and (c) ϕ = π 2. Curves: solid (metastable), dashed (locally unstable), dotted (spinodal curve).
Lyuksyutov Constraint Lyuksyutov proposed an approximate minimization of the functional in Eq. (14) that has been often employed to explore the fine structure of defect cores (e.g.: S. Kralj, E.G. Virga, and S. Zumer, Phys. Rev. E 6, (1999) 1858). 22/23 Whenever B AC, one can treat the trq 3 term in f b as a small perturbation and first minimize the dominant part of the bulk potential, that is, thus obtaining Then the modified free energy F [Q] = f b = A 2 trq2 + C 4 (trq2 ) 2, B trq 2 = A C. (21) ( L 2 Q 2 B ) 3 trq3 dv can be minimized subject to the constraint (21).
Lyuksyutov Constraint Validity of the constraint (21): related to the growth of the degree of biaxiality within the cell. Since trq 3 = 3detQ, on the equilibrium configurations for F, Eq. (21) is satisfied only if detq is a specific constant. 23/23 det Q 4 3 2 1 8 6 4 2 2 4 6 8 x/ξ b Figura 9: detq through the cell for τ - and χ-solution (dashed and solid line, respectively). θ = 8, ϕ =.45π, d/ξ b = 8.