9-3 Lennard-Jones potential: Difference between revisions
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with paremeters <math> \sigma_s </math> and <math> \epsilon_a </math> | with paremeters <math> \sigma_s </math> and <math> \epsilon_a </math> | ||
Our aim is to compute the total interaction between this solid and a molecule located at a position <math> | Our aim is to compute the total interaction between this solid and a molecule located at a position <math> x_f > 0 </math>. | ||
Such an interaction can be computed using cylindrical coordinates ( I GUESS SO, at least). | |||
The interaction will be: | |||
:<math> | |||
V_{W} \left( x \right) = 4 \epsilon_{sf} \rho_{s} \int_{0}^{2\pi} d \phi \int_{-\infty}^{x} d z \int_{0}^{\infty} \textrm{d r} | |||
\left[ \sigma^{12} (r^2 + z^2)^{-6} | |||
- \sigma^6 (r^2 + z^2 )^{-3} \right] r . | |||
</math> | |||
[TO BE CONTINUED] | [TO BE CONTINUED] | ||
Revision as of 14:06, 23 March 2007
[EN CONSTRUCCION]
Functional form
The 9-3 Lennard-Jones potential is related to the standard Lennard-Jones potential.
It takes the form:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V(r)={\frac {3{\sqrt {3}}}{2}}\epsilon \left[\left({\frac {\sigma }{r}}\right)^{9}-\left({\frac {\sigma }{r}}\right)^{3}\right].}
The minimum value of is obtained at , with
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V\left(r_{min}\right)=-\epsilon } ,
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {r_{min}}{\sigma }}=3^{1/6}}
Applications
It is commonly used to model the interaction between the particles of a fluid with a flat structureless solid wall.
Interaction between a solid and a fluid molecule
Let us consider the space divided in two regions:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x<0} : this region is occupied by a diffuse solid with density composed of 12-6 Lennard-Jones atoms
with paremeters 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 \epsilon_a }
Our aim is to compute the total interaction between this solid and a molecule located at a position 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 x_f > 0 } . Such an interaction can be computed using cylindrical coordinates ( I GUESS SO, at least).
The interaction 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 V_{W} \left( x \right) = 4 \epsilon_{sf} \rho_{s} \int_{0}^{2\pi} d \phi \int_{-\infty}^{x} d z \int_{0}^{\infty} \textrm{d r} \left[ \sigma^{12} (r^2 + z^2)^{-6} - \sigma^6 (r^2 + z^2 )^{-3} \right] r . }
[TO BE CONTINUED]