Subsections

Landau Theory

Lifshits Term

Now we divide system into layers, and

with A(x) being the free energy of one layer.

We want to add the dependence on . Lowest order: linear dependence on . But this is not symmetric with respect to . Solution: use

Integrating by parts:
 (2)
If there is bulk phase at , first term is zero. Otherwise we can include it in the boundary conditions. Result:

where W is the cross-section normal to x

General case: substitute by :
 (3)
The term is called Lifshits term, and free energy (3)--Landau Hamiltonian (or Landau-Ginsburg Hamiltonian).

Estimates for Lifshits Term

1.
Sign of g: if g<0, wavy'' structures are favored:

Therefore g>0.

2.
Value of g: the dimensionality of g/d is (Why?). From previous section it is clear, that this of the order of the square of the molecular size a0:

Since at , we obtain . We will see that this is square of the correlation length:

Next: Mathematical Digression: Euler-Lagrange Equation Up: Spatial Inhomogeneity. Interfaces Previous: Self-Consistent Field Theory

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Oct 16 20:58:44 EDT 1997