Editing Surface tension

The '''surface tension''',
<math> \gamma </math>, is a measure of the [[work]] required to create an [[interface]] between
two bulk phases.

== Thermodynamics == 
In the [[Canonical ensemble]] the surface tension is formally given as:

:<math> \gamma = \frac{ \partial A (N,V,T, {\mathcal A} )}{\partial  {\mathcal A} } </math>;

where

*<math>A</math> is the [[Helmholtz energy function]]
* <math> N </math> is the number of particles
*<math> V </math> is the volume
*<math> T </math> is the [[temperature]]
*<math> {\mathcal A} </math> is the surface area

==Computer Simulation==
Different techniques may be used to compute this quantity, such as the traditional [[stress | stress tensor]] route. More recently, several methods have been proposed which avoid the some times difficult calculation of the stress tensor, e.g., the [[test area method]] and [[wandering interface method]]. A review can be found in the paper by Gloor ''et al.'' (Ref. 1).

==Liquid-Vapour Interfaces of one component systems ==
=== Binder procedure===
Here, only an outline 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, <math> A(N;V,T)</math>. The calculation is usually carried out using [[Monte Carlo]] simulation using [[boundary conditions |periodic boundary conditions]] 
If liquid-vapour equilibrium occurs, a plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, 
as a function of <math> N </math> shows a loop.
Using basic thermodynamic procedures ([[Maxwell's equal area construction]]) it is possible
to compute the densities of the two phases; <math> \rho_v, \rho_l </math> at liquid-vapour equilibrium.
Considering the thermodynamic limit for densities <math> \rho </math>  with  <math> \rho_v < \rho < \rho_l </math> the 
[[Helmholtz energy function]] will be:

:<math> A(N) = - p_{eq} V + \mu_{eq} N + \gamma {\mathcal A}(N)  </math>

where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilibrium situation.
From the previous equation one can write

:<math> \Omega (N) \equiv A(N)  - \mu_{eq} N = - p_{eq} V + \gamma {\mathcal A}(N)  </math>.

For appropriate values of <math> N </math> one can estimate the value of the surface area, <math> {\mathcal A} </math> (See MacDowell ''et al.'', Ref. 3), and compute <math> \gamma </math> directly as:

:<math> \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)} </math>

where <math> N_l </math> and <math> N_v </math> are given by: <math> N_l = V \cdot \rho_l </math> and <math> N_v = V \cdot \rho_v </math>

=== Explicit interfaces ===
In these methods one performs a direct simulation of the two-phase system. [[boundary conditions |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.
Taking into account the [[canonical ensemble]] definition (see above), one  computes the change in the [[Helmholtz energy function]] when a small (differential)
change of the surface area is performed at constant <math> V, T, </math> and <math> N </math>.
The explicit equations can be written in terms of the diagonal components of the [[pressure]] tensor of the system.
Mechanical arguments can also be invoked to arrive at equivalent conclusions (see Ref 1 for a detailed discussion of these issues).

=== System-size analysis ===

The [[Finite size effects |system-size dependence]] of the results for <math> \gamma </math> have to be taken into account in order to  obtain  accurate results for [[Models |model systems]].
Spurious effects that occur due to small system sizes can appear in the ''explicit interface'' methods. (See P.Orea ''et al.'' Ref. 4).

== Mixtures ==

Different ensembles can be used to compute the surface tension between two phases in the case of mixtures (See for example see Y. Zhang ''et al.'' Ref 5).
The simulation techniques are  essentially the same as those for  one-component systems, but different ensembles can be more adequate.
For instance, for binary mixtures (with components 1 and 2), the [[isothermal-isobaric ensemble]], <math> N_1,N_2,p.T </math> is a ''more natural'' ensemble to compute
<math> \gamma </math> using explicit interface techniques (See Ref. 6 as an example). 
In the case of the Binder technique
the analysis can be carried out by fixing the total number of particles: <math> N \equiv N_1 + N_2 </math>, the [[pressure]], <math> p </math>, and the [[temperature]] <math> T </math>. Then
one will have to compute the variation of the adequate thermodynamic potential as a function of the composition, e.g. <math> x_1 = N_1/N </math>.

==See also==
*[[Line tension]]
*[[Boundary tension]]
*[[Droplets]]

==References==
#[http://dx.doi.org/10.1063/1.2038827     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)]
#[http://dx.doi.org/10.1103/PhysRevA.25.1699 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)]
#[http://dx.doi.org/10.1063/1.2218845  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)]
#[http://dx.doi.org/10.1063/1.2018640  Pedro Orea, Jorge López-Lemus, and  José Alejandre,  "Oscillatory surface tension due to finite-size effects", Journal of Chemical Physics '''123'''  114702 (6 pages) (2005)]
#[http://dx.doi.org/10.1063/1.469927  Yuhong Zhang, Scott E. Feller, Bernard R. Brooks, and Richard W. Pastor, "Computer simulation of liquid/liquid interfaces. I. Theory and application to octane/water", Journal of Chemical Physics, '''103''', pp. 10252-10266 (1995)]
#[http://dx.doi.org/10.1063/1.2751153  E. de Miguel, N. G. Almarza, and G. Jackson, "Surface tension of the Widom-Rowlinson model", Journal of Chemical Physics, '''127''', 034707  (10 pages)  (2007)]
#[http://dx.doi.org/10.1063/1.1747248     John G. Kirkwood and Frank P. Buff "The Statistical Mechanical Theory of Surface Tension", Journal of Chemical Physics '''17''' pp. 338-343 (1949)]
#[http://dx.doi.org/10.1016/0021-9991(76)90078-4 Charles H. Bennett  "Efficient estimation of free energy differences from Monte Carlo data", Journal of Computational Physics  '''22''' pp. 245-268 (1976)]
#[http://dx.doi.org/10.1063/1.432627 J. Miyazaki,  J. A. Barker and G. M. Pound "A new Monte Carlo method for calculating surface tension", Journal of Chemical Physics '''64'''  pp. 3364-3369 (1976)]
[[category: statistical mechanics]]