Editing Surface tension
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The '''surface tension''', | The '''surface tension''', | ||
<math> \gamma </math>, is a measure of the | <math> \gamma </math>, is a measure of the work required to create a surface. | ||
== Thermodynamics == | == Thermodynamics == | ||
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==Computer Simulation== | ==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 == | ==Liquid-Vapour Interfaces of one component systems == | ||
=== Binder procedure=== | === Binder procedure=== | ||
Here, only an outline of the procedure is presented, more details can be found in | 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 [[periodic boundary conditions]] | 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 [[periodic boundary conditions]] | ||
If liquid-vapour equilibrium occurs, a plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, | If liquid-vapour equilibrium occurs, a plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, | ||
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:<math> \Omega (N) \equiv A(N) - \mu_{eq} N = - p_{eq} V + \gamma {\mathcal A}(N) </math>. | :<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 | For appropriate values of <math> N </math> one can estimate the value of the surface area, <math> {\mathcal A} </math> (See MacDowell, 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> | :<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> | ||
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=== Explicit interfaces === | === Explicit interfaces === | ||
In these methods one performs a direct simulation of the two-phase system. [[Periodic boundary conditions]] are usually employed. | In these methods one performs 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 | Simulation boxes are elongated in one direction, and the interfaces are built (and expected to stay) perpendicular to | ||
such a direction. | such a direction. | ||
Taking into account the [[canonical ensemble]] | |||
Using this setup, one can compute the [[surface tension]], <math> \gamma </math>. Taking into account its definition | |||
in the [[canonical ensemble]] (see above), one has to compute the change in the [[Helmholtz energy function]] when a small (diferential) | |||
change of the surface area is performed at constant <math> V, T, </math> and <math> N </math>. | 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 | |||
Mechanical arguments can also be invoked to | 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 get equivalent conclusions (see Ref 1 for a detailed discussion on these issues) | |||
=== System-size analysis === | === System-size analysis === | ||
The | |||
Spurious effects | The system-size dependence of the results of <math> \gamma </math> has to be taken into account to get accurante results for model systems. | ||
Spurious effects due to small system sizes can appear in the ''explicit surface'' methods. (See P.Orea et al. in the references). | |||
== Mixtures == | == Mixtures == | ||
==References== | ==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)] | |||
[[category: statistical mechanics]] | [[category: statistical mechanics]] |