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| The '''surface tension''', | | The '''surface tension''', |
| <math> \gamma </math>, is a measure of the [[work]] required to create an [[interface]] between | | <math> \gamma </math>, is a measure of the work required to create a surface. |
| two bulk phases.
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| == Thermodynamics == | | == Thermodynamics == |
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| where | | where |
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| *<math>A</math> is the [[Helmholtz energy function]]
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| * <math> N </math> is the number of particles | | * <math> N </math> is the number of particles |
| *<math> V </math> is the volume | | *<math> V </math> is the volume |
| *<math> T </math> is the [[temperature]] | | *<math> T </math> is the [[temperature]] |
| *<math> {\mathcal A} </math> is the surface area | | *<math> {\mathcal A} </math> is the surface area |
| | *<math>A</math> is the [[Helmholtz energy function]] |
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| ==Computer Simulation== | | ==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 name="Gloor">[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)]</ref>.
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| | 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). |
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| ==Liquid-Vapour Interfaces of one component systems == | | ==Liquid-Vapour Interfaces of one component systems == |
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| === Binder procedure=== | | === Binder procedure=== |
| Here, only an outline of the procedure is presented, more details can be found in <ref>[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)]</ref>. | | |
| 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]] | | Here, only a sketchy picture of the procedure is presented, more details can be found in Reference 2. |
| If liquid-vapour equilibrium occurs, a plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, | | |
| | For given conditions of volume and temperature, the [[Helmholtz energy function]] is computed as a function of the number of molecules: |
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| | <math> A(N;V,T) </math> |
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| | The calculation is usually carried out using [[Monte Carlo]] simulation using [[periodic boundary conditions]] |
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| | If liquid-vapour equilibrium occurs, the 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. | | 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. | | Using basic thermodynamic procedures (Maxwell construction) it is possible |
| | to compute the densities of the two phases; <math> \rho_v, \rho_l </math>. |
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| Considering the thermodynamic limit for densities <math> \rho </math> with <math> \rho_v < \rho < \rho_l </math> the | | Considering the thermodynamic limit for densities <math> \rho </math> with <math> \rho_v < \rho < \rho_l </math> the |
| [[Helmholtz energy function]] will be: | | [[Helmholtz energy function]] will be: |
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| :<math> A(N) = - p_{eq} V + \mu_{eq} N + \gamma {\mathcal A}(N) </math>
| | *<math> A(N) = - p_{eq} V + \mu_{eq} N + \gamma {\mathcal A}(N) </math>. |
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| where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilibrium situation. | | where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilbrium situation. |
| From the previous equation one can write | | From the previous equation we can write |
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| :<math> \Omega (N) \equiv A(N) - \mu_{eq} N = - p_{eq} V + \gamma {\mathcal A}(N) </math>.
| | * <math> \Phi(N) \equiv A(N) - \mu_{eq} N = - p_{eq} V + \gamma {\mathcal A}(N) </math>. |
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| 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>[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)]</ref>), and compute <math> \gamma </math> directly as: | | For appropriate values of <math> N </math> one can estimate the value of the surface area, <math> {\mathcal A} </math> (See MacDowell in the references), |
| | and compute <math> \gamma </math> directly as: |
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| :<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{ \Phi(N) + p_{eq} V } { {\mathcal A}(N) } = \frac{ \Phi(N) - \frac{1}{2}(\Phi(N_l)+\Phi(N_v)) }{{\mathcal A}(N)} </math> |
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| 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 === |
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| === Explicit interfaces ===
| | In these methods one perform 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]] definition (see above), one computes the change in the [[Helmholtz energy function]] when a small (differential)
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| change of the surface area is performed at constant <math> V, T, </math> and <math> N </math>.
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| The explicit equations can be written in terms of the diagonal components of the [[pressure]] tensor of the system.
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| Mechanical arguments can also be invoked to arrive at equivalent conclusions (see Ref <ref name="Gloor"></ref> for a detailed discussion of these issues).
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| === System-size analysis ===
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| 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]].
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| Spurious effects that occur due to small system sizes can appear in the ''explicit interface'' methods. (See P.Orea ''et al.'' <ref>[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)]</ref>).
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| == Mixtures == | | == 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>[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)]</ref>).
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| The simulation techniques are essentially the same as those for one-component systems, but different ensembles can be more adequate.
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| 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
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| <math> \gamma </math> using explicit interface techniques (See Ref. <ref>[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)]</ref> as an example).
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| In the case of the Binder technique
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| 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
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| 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>.
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| ==See also==
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| *[[Line tension]]
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| *[[Boundary tension]]
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| *[[Droplets]]
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| ==References== | | ==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)] |
| ;Related reading
| | #[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/10.1063/1.1747248 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/#[http:// |
| *[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)]
| | L.G. MacDowell, V.K .Shen, and J.R. Errington |
| *[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)]
| | "Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems", JOURNAL OF CHEMICAL PHYSICS 125 (3): Art. No. 034705 JUL 21 2006 ] |
| *[https://doi.org/10.1016/j.molliq.2016.11.103 A. Maslechko, K. Glavatskiy, V.L. Kulinskii "Surface tension of molecular liquids: Lattice gas approach", Journal of Molecular Liquids '''235''' pp. 119-125 (2017)]
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| *[https://doi.org/10.1016/j.molliq.2016.12.062 Stephan Werth, Martin Horsch and Hans Hasse "Molecular simulation of the surface tension of 33 multi-site models for real fluids", Journal of Molecular Liquids '''235''' pp. 126-134 (2017)]
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| *[https://doi.org/10.1063/1.5008473 T. Dreher, C. Lemarchand, L. Soulard, E. Bourasseau, P. Malfreyt, and N. Pineau "Calculation of a solid/liquid surface tension: A methodological study", Journal of Chemical Physics '''148''' 034702 (2018)]
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| ==External links==
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| *[http://dx.doi.org/10.4249/scholarpedia.9218 Charles Pfister "Interface free energy", Scholarpedia, 5(2):9218 (2010)]
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| [[category: statistical mechanics]] | | [[category: statistical mechanics]] |