9-3 Lennard-Jones potential

The 9-3 Lennard-Jones potential is related to the Lennard-Jones potential. It has the following form:

${\mathrm{\Phi }}_{12}(r)=\frac{3\sqrt{3}}{2}ϵ[{(\frac{\sigma }{r})}^9-{(\frac{\sigma }{r})}^3].$

where ${\mathrm{\Phi }}_{12}(r)$ is the intermolecular pair potential and $r:=|{\mathbf{r}}_1-{\mathbf{r}}_2|$. The minimum value of $\mathrm{\Phi }(r)$ is obtained at $r=r_{min}$, with

• $\mathrm{\Phi }(r_{min})=-ϵ$,
• $\frac{r_{min}}{\sigma }=3^{1/6}$

Applications

It is commonly used to model the interaction between the particles of a fluid with a flat structureless solid wall or vice versa (Ref. 1).

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 parameters ${\sigma }_s$ and ${ϵ}_a$

Our aim is to compute the total interaction between this solid and a molecule located at a position $x_f>0$. Such an interaction can be computed using cylindrical coordinates.

The interaction will be:

${\mathrm{\Phi }}_W(x)=4{ϵ}_{sf}{\rho }_s\underset{0}{\overset{2\pi }{\int }}d\varphi \underset{-\mathrm{\infty }}{\overset{-x}{\int }}dz\underset{0}{\overset{\mathrm{\infty }}{\int }}\text{dr}[{\sigma }^{12}\frac{r}{(r^2+z^2{)}^6}-{\sigma }^6\frac{r}{(r^2+z^2{)}^3}].$

${\mathrm{\Phi }}_W(x)=8\pi {ϵ}_{sf}{\rho }_s\underset{-\mathrm{\infty }}{\overset{-x}{\int }}\text{d}z{[\frac{{\sigma }^{12}}{10(r^2+z^2{)}^5}-\frac{{\sigma }^6}{4(r^2+z^2{)}^2}]}_{r=\mathrm{\infty }}^{r=0}.$

${\mathrm{\Phi }}_W(x)=8\pi {ϵ}_{sf}{\rho }_s\underset{-\mathrm{\infty }}{\overset{-x}{\int }}\text{d}z[\frac{{\sigma }^{12}}{10z^{10}}-\frac{{\sigma }^6}{4z^4}];$

${\mathrm{\Phi }}_W(x)=8\pi {ϵ}_{sf}{\rho }_s{[-\frac{{\sigma }^{12}}{90z^9}+\frac{{\sigma }^6}{12z^3}]}_{z=-\mathrm{\infty }}^{z=-x};$

${\mathrm{\Phi }}_W(x)=\frac{4\pi {ϵ}_{sf}{\rho }_s{\sigma }^3}{3}[\frac{{\sigma }^9}{15x^9}-\frac{{\sigma }^3}{2x^3}]$

References

1. Farid F. Abraham and Y. Singh "The structure of a hard-sphere fluid in contact with a soft repulsive wall", Journal of Chemical Physics 67 pp. 2384-2385 (1977)