9-3 Lennard-Jones potential

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[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 (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(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 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(r) } is obtained at 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 r = r_{min} } , 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 V \left( r_{min} \right) = - \epsilon } ,
  • 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 \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 (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 < 0 } : this region is occupied by a diffuse solid with density 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_s } composed of 12-6 Lennard-Jones atoms

with paremeters 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 \sigma_s } 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{W}\left(x\right)=4\epsilon _{sf}\rho _{s}\int _{0}^{2\pi }d\phi \int _{-\infty }^{-x}dz\int _{0}^{\infty }{\textrm {dr}}\left[\sigma ^{12}{\frac {r}{(r^{2}+z^{2})^{6}}}-\sigma ^{6}{\frac {r}{(r^{2}+z^{2})^{3}}}\right].}
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) = 8 \pi \epsilon_{sf} \rho_{s} \int_{-\infty}^{-x} {\textrm d z} \left[ \frac{ \sigma^{12}} { 10 (r^2 + z^2)^5} - \frac{\sigma^6 }{ 4 (r^2 + z^2 )^{2} }\right]^{r=0}_{r=\infty} . }
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) = 8 \pi \epsilon_{sf} \rho_{s} \int_{-\infty}^{-x} {\textrm d z} \left[ \frac{ \sigma^{12}} { 10 z^{10} } - \frac{\sigma^6 }{ 4 z^4 } \right]; }


Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{W}\left(x\right)=8\pi \epsilon _{sf}\rho _{s}\int _{-\infty }^{-x}{{\textrm {d}}z}\left[{\frac {\sigma ^{12}}{10z^{10}}}-{\frac {\sigma ^{6}}{4z^{4}}}\right];}


[TO BE CONTINUED]

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