Surface tension and interface free energy

Extension: interface free energy and surface tension

In systems with phase coexistence, an interface separating two stable phases costs free energy proportional to its area. The proportionality constant (possibly direction-dependent) is the surface tension.

A canonical setting is the Ising model below its critical temperature, where distinct infinite-volume phases exist (see phase transitions and infinite-volume Gibbs measures ).

Definition via boundary conditions (Ising example)

Let $\Lambda_L$ be a large box with linear size $L$, and impose boundary conditions that force an interface with normal direction $\mathbf{n}$ (for example, “$+$” on one side of a plane and “$-$” on the other). Let $Z_L^{+-}(\mathbf{n})$ be the finite-volume partition function with such mixed boundary conditions, and let $Z_L^{++}$ be the partition function with uniform “$+$” boundary conditions.

If $A_L(\mathbf{n})$ is the cross-sectional area of the imposed interface, the surface tension in direction $\mathbf{n}$ is defined (when the limit exists) by

Interpretation: the ratio of partition functions isolates the excess free energy due to creating the interface, normalized by area.

Key consequences and context

• Coexistence and order: A positive $\tau(\mathbf{n})$ is tied to coexistence of distinct phases and nonzero spontaneous magnetization (an order parameter ).
• Geometry: Directional dependence of $\tau(\mathbf{n})$ leads to equilibrium crystal shapes via Wulff-type constructions (a geometric “dual” of interface costs).
• Large deviations of profiles: Interface free energies govern probabilities of atypical magnetization profiles and droplet formation; this connects to large deviations and to the thermodynamic potential viewpoint via pressure (log-partition density) .

Prerequisites / cross-links

• lattice Hamiltonians , finite-volume Gibbs measures , DLR equation
• ferromagnetic Ising model
• thermodynamic stability (interface costs as excess free energies)