9-3 Lennard-Jones potential

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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].

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

  •  \Phi \left( r_{min} \right) = - \epsilon ,
  •  \frac{ r_{min} }{\sigma} = 3^{1/6}

Applications[edit]

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[edit]

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  \epsilon_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:


 \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] .

 \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} .

 \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];



 \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};

 \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]

References[edit]

  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)