Surface tension: Difference between revisions
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<math> A(N;V,T) </math> | <math> A(N;V,T) </math> | ||
The calculation is usually carried out using [[Monte Carlo]] simulation | The calculation is usually carried out using [[Monte Carlo]] simulation using [[periodic boundary conditions]] | ||
If liquid-vapour equilibrium occurs, the plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, | If liquid-vapour equilibrium occurs, the plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, | ||
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Using basic thermodynamic procedures (Maxwell 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>. | to compute the densities of the two phases; <math> \rho_v, \rho_l </math>. | ||
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>. | *<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 equilbrium situation. | where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilbrium situation. | ||
From the previous equation we can write | |||
* <math> \Phi(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 in the references), | |||
and compute <math> \gamma </math> directly as: | |||
* <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} </math> | |||
=== Explicit interfaces === | === Explicit interfaces === | ||
Revision as of 11:38, 1 August 2007
The surface tension, , is a measure of the work required to create a surface.
Thermodynamics
In the Canonical ensemble the surface tension is formally given as:
- ;
where
- is the number of particles
- is the volume
- is the temperature
- is the surface area
- is the Helmholtz energy function
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 a sketchy picture of the procedure is presented, more details can be found in Reference xxx
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, the plot of the chemical potential, 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 \mu \equiv (\partial A/\partial N)_{V,T} } , 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 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 } .
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 equilbrium situation. From the previous equation we 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 \Phi(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 in the references), 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{ \Phi(N) + p_{eq} V } { {\mathcal A}(N) } = \frac{ \Phi(N) - \frac{1}{2}(\Phi(N_l)+\Phi(N_v)) }{\mathcal A} }
Explicit interfaces
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)