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

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

Computer Simulation

A review of the different techniques that can be used to compute the surface (interface) tension can be found in the paper by Gloor et al. (Ref. 1).

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 2.

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 using periodic boundary conditions

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; $\rho _{v},\rho _{l}$ at liquid-vapour equilibrium.

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. From the previous equation we can write

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

For appropriate values of $N$ one can estimate the value of the surface area, ${\mathcal {A}}$ (See MacDowell in the references), and compute $\gamma$ directly as:

• $\gamma ={\frac {\Omega (N)+p_{eq}V}{{\mathcal {A}}(N)}}={\frac {\Omega (N)-{\frac {1}{2}}(\Omega (N_{l})+\Omega (N_{v}))}{{\mathcal {A}}(N)}}$

where $N_{l}$ and $N_{v}$ are given by: $N_{l}=V\cdot \rho _{l}$ and $N_{v}=V\cdot \rho _{v}$

Explicit interfaces

In these methods one perform a direct simulation of the two-phase system. Periodic boundary conditions are usually employed. Simulation boxes are elongated in one direction, and the interfaces are built (and expected to stay) perpendicular to such a direction.

References

1. Guy J. Gloor, George Jackson, Felipe J. Blas and Enrique 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)
3. L.G. MacDowell, V.K .Shen, and J.R. Errington "Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems", Journal of Chemical Physics 125 034705 (2006)