Surface energy as thermodynamic potential Consider free energy of sharp interface $\Gamma$
  $$ \int_\Gamma \sigma\;\mathrm{d}S $$
or also free energy of diffuse interface of characteristic width $\epsilon$ given by Cahn-Hilliard/Allen-Cahn functional of phase-field $c\in [0,1]$
  $$ \int_\Omega \left[ 12\frac{\sigma}{\epsilon}c^2(1-c)^2 + \frac{3}{4}\sigma\epsilon|\nabla c|^2 \right]\;\mathrm{d}V .$$


*

*Is there some reasoning to conclusively deduce if these are Gibbs energies or Helmholtz energies?

*Or would such a conclusion be equivalent to definition of circumstances under which surface tension/energy $\sigma$ is measured?

*If latter holds what are these conditions for either Gibbs or Helmholtz energy and which are of practical significance and being typically measured?

 A: As Mark Rovetta says, the Gibbs energy is for constant temperature and pressure, whereas the Helmholtz energy is for constant temperature and volume. Typically, when dealing with fluids you're in a constant pressure situation, so technically the relevant free energy is usually going to be the Gibbs energy.
However, I doubt it makes any practical difference. The reason is that a change in the Helmholtz energy is $\Delta A = \Delta U-T \Delta S$, whereas a change in the Gibbs energy is $\Delta G = \Delta U-T \Delta S + p \Delta V$, so they are only numerically different if there is a volume change involved. A change in the configuration of a fluid surface might make a tiny volume change because of a change in the orientation of the molecules at the fluid interface, but the number of molecules involved is tiny compared to the bulk of the fluid, and I would expect this volume change to be un-measurably tiny compared to the change in $U$ due to surface tension effects.
The upshot of this is that you can almost certainly assume $\Delta V=0$ for all practical purposes, which means that $\Delta A = \Delta G$ and the fluid interface behaviour won't change depending on whether the pressure or the volume is held constant.
A: Gibbs is for constant temperature and pressure, Helmholtz for constant temperature and volume. 
A: I provide answer of my colleague in this post as it seems interesting and contributive.
Let's describe equilibrium thermodynamics of surface by Gibbs equation
  $$ T\;\mathrm{d}S^i = \mathrm{d}U^i - \sigma\;\mathrm{d}A^i ,$$
where 


*

*$T$ is temperature,

*$S^i$ is entropy fo surface,

*$U^i$ is internal energy of surface and

*$A^i$ is surface area.


Exploiting one-homogenity of entropy $S^i(\lambda U^i, \lambda A^i) = \lambda S^i(U^i, A^i)$ one gets Euler equation
  $$ T^iS^i = U^i - \sigma A^i .$$
For surface density of entropy $s^i=S^i/A^i$ and energy $u^i=U^i/A^i$ one have
  $$ T^i s^i = u^i - \sigma ,$$
from which you read that $\sigma$ is surface density of Helmholtz potential.
