9-3 Lennard-Jones potential

[EN CONSTRUCCION]

Functional form

The 9-3 Lennard-Jones potential is related to the standard Lennard-Jones potential.

It takes the form:

$V(r)={\frac {3{\sqrt {3}}}{2}}\epsilon \left[\left({\frac {\sigma }{r}}\right)^{9}-\left({\frac {\sigma }{r}}\right)^{3}\right].$

The minimum value of $V(r)$ is obtained at $r=r_{min}$, with

• $V\left(r_{min}\right)=-\epsilon$,

• ${\frac {r_{min}}{\sigma }}=3^{1/6}$

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:

• $x<0$: this region is occupied by a diffuse solid with density $\rho _{s}$ composed of 12-6 Lennard-Jones atoms

with paremeters $\sigma _{s}$ and $\epsilon _{a}$

Our aim is to compute the total interaction between this solid and a molecule located at a position Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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]