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 ''et al.'' | 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> | :<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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Taking into account the [[canonical ensemble]] definition (see above), one computes the change in the [[Helmholtz energy function]] when a small (differential) | 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>. | 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]] | 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 | Mechanical arguments can also be invoked to arrive at equivalent conclusions (see Ref 1 for a detailed discussion of these issues). | ||
=== System-size analysis === | === 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]]. | 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.'' | Spurious effects that occur due to small system sizes can appear in the ''explicit interface'' methods. (See P.Orea ''et al.'' Ref. 4). | ||
== Mixtures == | == Mixtures == | ||
Different ensembles can be used to compute the surface tension between two phases in the case of mixtures (See for example Y. Zhang et al. in the reference list) | |||
==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)] | |||
#[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) ] | |||
[[category: statistical mechanics]] | [[category: statistical mechanics]] |