Subsections

# Interface Layer in Landau Theory

## Euler-Lagrange Equation

We need to minimize the functional

with

This is a Lagrange problem with

On the coexistence line H=0 (Why?). We have:

and Euler-Lagrange equation becomes

 gM''=aM+2dM3 (6)

## Solution of Euler-Lagrange Equation

Multiply (6) by M'. Then

and

Therefore

and

To find C note that at we have

We obtain:

This gives

or
 (7)
and

Solution:
 (8)
This gives x(M). Inverting this function, we obtain density profile M(x)

Width of the interface
 (9)
The length of the interface is proportional to the correlation length!

## Surface Tension

In the bulk the energy per unit volume is

In the interface layer we have free energy per unit volume

Additional free energy per unit volume

Surface tension is surface energy per unit area, i.e.

Change of variables:

or, substituting and M'
 (10)

## Critical Behavior

What happens at ? Coefficient a tends to zero as .

From (9) and (10) we obtain:

As we are closer to the critical point, the interface layer becomes thicker, and the surface tension drops. Since , correlation length diverges, and means that the difference between the phases disappears. The penalty for forming interface becomes lower, and fluctuations grow.

Next: Quiz Up: Spatial Inhomogeneity. Interfaces Previous: Mathematical Digression: Euler-Lagrange Equation

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