9-3 Lennard-Jones potential
From SklogWiki
The 9-3 Lennard-Jones potential is related to the Lennard-Jones potential. It has the following form:
![\Phi_{12}(r) = \frac{ 3 \sqrt{3}}{ 2} \epsilon \left[ \left( \frac{\sigma}{r} \right)^9 -
\left( \frac{ \sigma }{r} \right)^3 \right].](/SklogWiki/images/math/3/1/5/3151b21e39de4a52ead5aea34866c93c.png)
where Φ12(r) is the intermolecular pair potential and 
.
The minimum value of Φ(r) is obtained at r = rmin, with
-
,
-
[edit] 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).
[edit] 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 ρs composed of 12-6 Lennard-Jones atoms
with parameters σs and εa
Our aim is to compute the total interaction between this solid and a molecule located at a position xf > 0. Such an interaction can be computed using cylindrical coordinates.
The interaction will be:
![\Phi_{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} \frac{ r} {(r^2 + z^2)^{6}}
- \sigma^6 \frac{r}{(r^2 + z^2 )^{3} }\right] .](/SklogWiki/images/math/e/f/9/ef943c72387a2c92ca90ca54ade4a0c8.png)
![\Phi_{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} .](/SklogWiki/images/math/b/5/7/b57b9bd316249cd53dd0f1feaf3445d3.png)
![\Phi_{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];](/SklogWiki/images/math/d/8/d/d8d391969bb4b62f8249f2798dccd374.png)
![\Phi_{W} \left( x \right) = 8 \pi \epsilon_{sf} \rho_s
\left[ - \frac{ \sigma^{12}} { 90 z^{9} }
+ \frac{\sigma^6 }{ 12 z^3 } \right]_{z=-\infty}^{z=-x};](/SklogWiki/images/math/2/a/7/2a7227f8574aad21401e2f612c4a53b0.png)
![\Phi_{W} \left( x \right) = \frac{4 \pi \epsilon_{sf} \rho_s \sigma^3}{3}
\left[ \frac{ \sigma^{9}} { 15 x^{9} }
- \frac{\sigma^3 }{ 2 x^3 } \right]](/SklogWiki/images/math/b/5/2/b52d1859a3565d5bb4e50eb04bc47678.png)
