3.G. Landau Theory

 

1st order transition

Continuous transition

1st derivatives of G

Discontinuous at transition point

Continuous

Order parameter

Discontinuous across coexistent curve

(except at the critical point)

Continuous

Symmetry

May or may not be broken

Aways broken

 

The Ginzburg-Landau theory is a mean field theory that describes an order-disorder transition.  Thus, the low temperature, or ordered, phase is characterized by a nonzero order parameter η, which vanishes in the high temperature, or disordered phase.  In a continuous transition, there is also an accompanying change of symmetries with the ordered phase having the lower symmetries.  The molar free energy φ is assumed to be analytic (i.e., a Taylor series expansion exists) so that near the transition point where η is small, we can write

ϕ( T,Y,η ) ϕ 0 ( T,Y )+ α 2 ( T,Y ) η 2 + α 3 ( T,Y ) η 3 + α 4 ( T,Y ) η 4             (3.63a)

where terms O( η 5 )  are neglected and we have set α 1 =0  so that the disordered phase η=0  is always a local minimum.  In the presence of an external force conjugate to η , (3.63a) is generalized to  

ψ( T,Y,f )=ϕ( T,Y,η )fη

ϕ 0 ( T,Y )+ α 2 ( T,Y ) η 2 + α 3 ( T,Y ) η 3 + α 4 ( T,Y ) η 4 fη             (3.63)

 

3.G.1.          Continuous Phase Transitions

3.G.2.          First Order Transitions