1. 17/06/2015Meeting on biaxial liquid crystals April 2005. 1 MOLECULAR FIELD AND LANDAU THEORIES FOR BIAXIAL SYSTEMS Ken Thomas School of Electronics and Computer Science University of Southampton

2. 17/06/2015Meeting on biaxial liquid crystals April 2005. 2 Collaborators Geoffrey Luckhurst Tim Sluckin

3. 17/06/2015Meeting on biaxial liquid crystals April 2005. 3 GOALS OF RESEARCH Construction of consistent Landau theories for systems of biaxial molecules Construction of consistent molecular field theories for these systems Phase diagrams often more easily understood from Landau theories Illuminate relationship between Landau and molecular-field theories Illuminate relationship between molecular parameters and phase diagrams

4. 17/06/2015Meeting on biaxial liquid crystals April 2005. 4 Some previous work (not exhaustive!) Ancient Freiser (1970)General idea… similar strategy to us Alben, McColl and Shih (1972)New uniaxial order parameter Straley (1974)Biaxial volume exclusion Old Govers and Vertogen (1984)Continuum theory Palffy-Muhoray and Hoatson (1990) Mixtures

5. 17/06/2015Meeting on biaxial liquid crystals April 2005. 5 Building a Landau theory (reprise) 1.Isolate order parameters (here tensors) 2.Construct invariants 3.Build Landau expansion from sums of powers of invariants subject to symmetry constraints 4.Minimise with respect to all variables 5.Analyse global minima 6.Bifurcation analysis to determine nature of phase transitions

6. 17/06/2015Meeting on biaxial liquid crystals April 2005. 6 Case Study: Landau-de Gennes Order parameter Quadratic and cubic invariants Expansion Hence transition first order

7. 17/06/2015Meeting on biaxial liquid crystals April 2005. 7 MAIER-SAUPE THEORY ALSO PREDICTS FIRST ORDER TRANSITION Familiar Grandjean-Maier-Saupe graphical construction Shows hysteresis and first-order transition

8. 17/06/2015Meeting on biaxial liquid crystals April 2005. 8 Symmetry and Order Parameter Manifold well-hidden in Grandjean-Maier-Saupe theory Must be there even though well-hidden!

9. 17/06/2015Meeting on biaxial liquid crystals April 2005. 9 OUR PROGRAMME Build Maier-Saupe like theory for biaxial system using simplest building blocks (Straley, Boccara et alia) Find effective free energy by working backwards Expand free energy in terms of order parameter

10. 17/06/2015Meeting on biaxial liquid crystals April 2005. 10 Strategy borrowed from Free energies in the Landau and molecular field approaches J. Katriel, G.F. Kventsel, G.R. Luckhurst and T.J.Sluckin Liquid Crystals 1, 337-355 (1986)

11. 17/06/2015Meeting on biaxial liquid crystals April 2005. 11 Strategy borrowed from Free energies in the Landau and molecular field approaches J. Katriel, G.F. Kventsel, G.R. Luckhurst and T.J.Sluckin Liquid Crystals 1, 337-355 (1986) This paper performed the Landau expansion for the simple Grandjean-Maier-Saupe theory.

12. 17/06/2015Meeting on biaxial liquid crystals April 2005. 12 Strategy borrowed from Free energies in the Landau and molecular field approaches J. Katriel, G.F. Kventsel, G.R. Luckhurst and T.J.Sluckin Liquid Crystals 1, 337-355 (1986) This paper performed the Landau expansion for the simple Grandjean-Maier-Saupe theory. We shall do the same thing for a biaxial system

13. 17/06/2015Meeting on biaxial liquid crystals April 2005. 13 Strategy of Katriel et al (1) (1)Free energy (2)Order parameter (3)Entropy a functional of distribution function f(  ) But F not yet a function of OP !

14. 17/06/2015Meeting on biaxial liquid crystals April 2005. 14 Strategy of Katriel et al (2) (4)f(  ) a function of auxiliary parameter  (5)Partition function Z(  ) (6)OP a function of  Minimise –TS term subject to given OP F is now a function of  and OP

15. 17/06/2015Meeting on biaxial liquid crystals April 2005. 15 Strategy of Katriel et al (3) (7)Invert eq. (*) (8)F was a function of OP and  (9)F now a function of OP Expand (*) in a power series in  Invert power series to required order Expand F in power series in OP

16. 17/06/2015Meeting on biaxial liquid crystals April 2005. 16 RESULT

17. 17/06/2015Meeting on biaxial liquid crystals April 2005. 17 OPEN QUESTIONS Full expansion in all molecular order parameters? Compatibility with other approaches? Nature of phase diagram? More complex molecular structure? Mixtures ? Full tensor expansion?