Surface tension

The surface tension, $\gamma$, is a measure of the work required to create a surface.

Thermodynamics

In the Canonical ensemble the surface tension is formally given as:

$\gamma ={\frac {\partial A(N,V,T,{\mathcal {A}})}{\partial {\mathcal {A}}}}$;

where

• $N$ is the number of particles
• $V$ is the volume
• $T$ is the temperature
• ${\mathcal {A}}$ is the surface area
• $A$ is the Helmholtz energy function

Computer Simulation

A review on different techniques to compute surface (interface) tension can be found in the paper by Gloor et al.

Liquid-Vapour Interfaces of one component systems

Binder procedure

Here, only a sketchy picture of the procedure is presented, more details can be found in Reference xxx

For given conditions of volume and temperature, the Helmholtz energy function is computed as a function of the number of molecules:

$A(N;V,T)$

The calculation is usually carried out using Monte Carlo simulation

If liquid-vapour equilibrium occurs, the plot of the chemical potential, $\mu \equiv (\partial A/\partial N)_{V,T}$, as a function of $N$ shows a loop.

Using basic thermodynamic procedures (Maxwell construction) it is possible to compute the densities of the two phases; $\rho _{v},\rho _{l}$.

Considering the thermodynamic limit for densities $\rho$ with $\rho _{v}<\rho <\rho _{l}$ the Helmholtz energy function will be:

$A(N)=-p_{eq}V+\mu _{eq}N+\gamma {\mathcal {A}}(N)$.

where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilbrium situation.

References

1. G. J. Gloor, G. Jackson, F. J. Blas, and E. de Miguel "Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials", Journal of Chemical Physics 123 134703 (2005)
2. K. Binder "Monte Carlo calculation of the surface tension for two- and three-dimensional lattice-gas models", Physical Review A 25 pp. 1699 - 1709 (1982)