Surface tension: Difference between revisions
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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). | 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 Reference 2. | |||
Here, only | 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>, | |||
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, | |||
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 | |||
Using basic thermodynamic procedures (Maxwell construction) it is possible | |||
to compute the densities of the two phases; <math> \rho_v, \rho_l </math> at liquid-vapour equilibrium. | 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 | 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: | ||
:<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 | where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilibrium situation. | ||
From the previous equation | 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 | 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: | ||
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> | 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> | ||
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=== Explicit interfaces === | === Explicit interfaces === | ||
In these methods one | 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. | ||
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#[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/10.1063/1.2038827 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/10.1103/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/10.1063/1.2218845 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]] | ||
Revision as of 10:46, 2 August 2007
The surface tension, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma } , is a measure of the work required to create a surface.
Thermodynamics
In the Canonical ensemble the surface tension is formally given as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{ \partial A (N,V,T, {\mathcal A} )}{\partial {\mathcal A} } } ;
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is the Helmholtz energy function
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } is the number of particles
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } is the volume
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } is the temperature
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(N;V,T)} . 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, , as a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } shows a loop. Using basic thermodynamic procedures (Maxwell's equal area construction) it is possible to compute the densities of the two phases; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_v, \rho_l } at liquid-vapour equilibrium. Considering the thermodynamic limit for densities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_v < \rho < \rho_l } the Helmholtz energy function will be:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 equilibrium situation. From the previous equation one can write
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega (N) \equiv A(N) - \mu_{eq} N = - p_{eq} V + \gamma {\mathcal A}(N) } .
For appropriate values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } one can estimate the value of the surface area, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A} } (See MacDowell, Ref. 3), and compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma } directly as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_l } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_v } are given by: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_l = V \cdot \rho_l } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_v = V \cdot \rho_v }
Explicit interfaces
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 such a direction.
Mixtures
References
- 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)
- 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)
- 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)