Biaxial Phase Transitions in Nematic Liquid Crystals

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2 Biaxial Phase Transitions in Nematic Liquid Crystals Cortona, 6 September 5 Giovanni De Matteis Department of Mathematics University of Strathclyde ra.gdem@maths.strath.ac.uk

3 Summary General quadrupolar interaction: Straley s potential Mean-field model and Phase diagram for γ = Reference model parameters domain Conjugated phase diagram Extension to a mildly repulsive model Cortona 5 Round Table Discussion on Biaxial Nematics

4 Molecular Biaxiality We consider a homogeneous nematic liquid crystal intrinsically biaxial in the absence of any external field. It is made up of biaxial molecules which we can imagine as being described by a biaxial tensor decomposed into two traceless, irreducible orthogonal components q := m m 3 I b := e e e e m, e, e axes of any molecular polarizability m e e Cortona 5 Round Table Discussion on Biaxial Nematics 3

5 Interaction potential Letting (q, b) and (q, b ) represent two interacting molecules, the most general quadrupolar interaction potential coupling them is V = U {q q + γ(q b + b q ) + λb b } m U > and λ, γ model parameters Ground State λ m x 3 5 e 4 3 e e e 3 γ Cortona 5 Round Table Discussion on Biaxial Nematics 4

6 Model γ = V = U {q q + λb b } { ( V = λ + ) + ( λ) (m m ) + λ [(e e ) + (e e) ]} U 3 Q := q = S(e z e z 3 I) + T (e x e x e y e y ) B := b = S (e z e z 3 I) + T (e x e x e y e y ) pseudo-potential U = U (q Q + λb B) Cortona 5 Round Table Discussion on Biaxial Nematics 5

7 Mean-field equations partition and distribution functions Z (Q, B,, λ) = T exp ( (Q q + λb b)) f = Z exp [(q Q + λb B)] T := S S, := U k B t k B Boltzmann constant and t absolute temperature Q = q = fq B = b = fb T F (Q, B,, λ) = U { Q Q + λ B B ln ( Z (Q, B,, λ) 8π T )}, Cortona 5 Round Table Discussion on Biaxial Nematics 6

8 λ = /3, /, / F L S F L T *. * dominant stable biaxial states (S,,, T ) Q = S(e z e z 3 I) B = T (e x e x e y e y ) Cortona 5 Round Table Discussion on Biaxial Nematics 7

9 λ = /3, /, / S F L S *.5 * unstable uniaxial states with a greater value of free energy (S,, S, ) Q = S(e y e y 3 I) B = S (e y e y 3 I) Q = S(e x e x 3 I) B = S (e x e x 3 I) Cortona 5 Round Table Discussion on Biaxial Nematics 8

10 λ = 5/6, S T. * * dominant stable biaxial states (S,,, T ) Q = S(e z e z 3 I) B = T (e x e x e y e y ) Cortona 5 Round Table Discussion on Biaxial Nematics 9

11 λ = 5/6, S S * * unstable uniaxial states with a greater value of free energy (S,, S, ) Q = S(e y e y 3 I) B = S (e y e y 3 I) Q = S(e x e x 3 I) B = S (e x e x 3 I) Cortona 5 Round Table Discussion on Biaxial Nematics

12 Phase Diagram and Tricritical points ISOTROPIC C.5 /..5. UNIAXIAL C M BIAXIAL λ Durand, Sonnet, Virga (3) De Matteis, Romano,Virga (5) Cortona 5 Round Table Discussion on Biaxial Nematics

13 Reference triangle D λ 3 ( 3, ), B 9 γ ( ),, C A r r : γ + 3λ = C O D B (, ), A (, ) 3 Cortona 5 Round Table Discussion on Biaxial Nematics

14 Model: γ = 3λ q := e e 3 I, b := m m e e {( ) ( ) } 9λ λ V = U q q + b b ( ) 9λ {q V = U q + λ b b } which is equivalent to γ = model by transforming the model parameter λ and as follows ( ) 9λ =, λ = λ 9λ in [ the (γ, ] λ) plane [ 9, [, ], 9, ] 3 [ ] 3, i.e.da OP, CD CP P (, ) Cortona 5 Round Table Discussion on Biaxial Nematics 3

15 Order parameters Q = S (e z e z 3 I) + T (e x e x e y e y ) B = S (e z e z 3 I) + T (e x e x e y e y ) with the following corrispondence on the order parameters S = S S S = S + 3S, T = T T,, T = T + 3T Cortona 5 Round Table Discussion on Biaxial Nematics 4

16 Conjugated phase diagram.5.4 ISOTROPIC /.3 UNIAXIAL. C C. BIAXIAL / λ γ = 3λ 9 λ Cortona 5 Round Table Discussion on Biaxial Nematics 5

17 Reference triangle D λ 3 ( 3, ), B 9 γ ( ),, C A r r : γ + 3λ = C O D B (, ), A (, ) 3 Cortona 5 Round Table Discussion on Biaxial Nematics 6

18 γ = 3λ, λ 9 : a mildly repulsive model ( ) λ {b V µ = U b µq q } µ = 9λ ( ) λ λ, µ = [ λ, ] µ [, ] BD 9 mean-field f µ = exp ( µ (B b µq q )) Z µ Z µ := exp ( µ (B b µq q )) T Cortona 5 Round Table Discussion on Biaxial Nematics 7

19 { F µ (Q, B ) = U (B B µq Q ) ( Zµ log µ 8π Q = q = f µ q F µ Q = B = b = T T f µ b F µ B = Bogoliubov s minimax principle All the phases are characterized at least by two negative eigenvalues in the Hessian matrix of F µ. In this case we adopt a minimax principle to determine the effective phases. min B max Q F µ (Q, B ) )} which is equivalent to finding a solution of the mean-field equations giving F µ a smaller value than do any other solutions Cortona 5 Round Table Discussion on Biaxial Nematics 8

20 Future work Numerical continuation and Bifurcation Analysis for the mildly repulsive model (also with a γ crossing term in the interaction) Landau theory for both attractive and mildly repulsive cases... and finally I would thank my PhD thesis advisor Prof. E. Virga Prof. Gartland, Prof. Durand, Prof. S. Romano and M. Osipov during my PhD research activity in Pavia Cortona 5 Round Table Discussion on Biaxial Nematics 9

21 λ = TP S order parameter.5 TP LP T order parameter LP dominant stable biaxial states (S,,, T ) Q = S(e z e z 3 I) B = T (e x e x e y e y ) Cortona 5 Round Table Discussion on Biaxial Nematics

22 λ =.36.5 S order parameter.5 TP LP S order parameter unstable uniaxial states with a greater value of free energy (S,, S, ) Q = S(e y e y 3 I) B = S (e y e y 3 I) Q = S(e x e x 3 I) B = S (e x e x 3 I) Cortona 5 Round Table Discussion on Biaxial Nematics

23 λ = S order parameter.5 T order parameter dominant stable biaxial states (S,,, T ) Q = S(e z e z 3 I) B = T (e x e x e y e y ) Cortona 5 Round Table Discussion on Biaxial Nematics

24 λ =.9.5 S order parameter.5 S order parameter unstable uniaxial states with a greater value of free energy (S,, S, ) Q = S(e y e y 3 I) B = S (e y e y 3 I) Q = S(e x e x 3 I) B = S (e x e x 3 I) Cortona 5 Round Table Discussion on Biaxial Nematics 3