MOLECULAR FIELD AND LANDAU THEORIES FOR BIAXIAL SYSTEMS

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MOLECULAR FIELD AND LANDAU THEORIES FOR BIAXIAL
SYSTEMS Ken Thomas School of Electronics and
Computer Science University of Southampton
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Collaborators

• Geoffrey Luckhurst
• Tim Sluckin

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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

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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
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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

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Case Study Landau-de Gennes
Order parameter Quadratic and cubic invariants
Expansion
Hence transition first order
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MAIER-SAUPE THEORY ALSO PREDICTS FIRST ORDER
TRANSITION
Familiar Grandjean-Maier-Saupe graphical
construction Shows hysteresis and first-order
transition
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Symmetry and Order Parameter Manifold
well-hidden in Grandjean-Maier-Saupe theory Must
be there even though well-hidden!
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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

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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)
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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.
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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
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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 !
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Strategy of Katriel et al (2)
Minimise TS term subject to given OP

1. f(?) a function of auxiliary parameter ?
2. Partition function Z(?)
3. OP a function of ?

F is now a function of ? and OP
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Strategy of Katriel et al (3)

1. Invert eq. ()
2. F was a function of OP and ?
3. 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
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RESULT
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• 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?