9-3 Lennard-Jones potential: Difference between revisions

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: <math>
: <math>
V(r) = \frac{ 3 \sqrt{3}}{ 2} \epsilon \left[ \left( \frac{\sigma}{r} \right)^9 -  
\Phi(r) = \frac{ 3 \sqrt{3}}{ 2} \epsilon \left[ \left( \frac{\sigma}{r} \right)^9 -  
\left( \frac{ \sigma }{r} \right)^3 \right].
\left( \frac{ \sigma }{r} \right)^3 \right].
</math>
</math>


The minimum value of <math> V(r) </math> is obtained at <math> r = r_{min} </math>, with
where <math>\Phi(r)</math> is the [[intermolecular pair potential]].
The minimum value of <math> \Phi(r) </math> is obtained at <math> r = r_{min} </math>, with


* <math> V \left( r_{min} \right) = - \epsilon </math>,
* <math> \Phi \left( r_{min} \right) = - \epsilon </math>,


* <math> \frac{ r_{min} }{\sigma} = 3^{1/6} </math>
* <math> \frac{ r_{min} }{\sigma} = 3^{1/6} </math>
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Our aim is to compute the total interaction between this solid and a molecule located at a position <math> x_f > 0 </math>.
Our aim is to compute the total interaction between this solid and a molecule located at a position <math> x_f > 0 </math>.
Such an interaction can be computed using cylindrical coordinates ( I GUESS SO, at least).
Such an interaction can be computed using cylindrical coordinates.


The interaction will be:
The interaction will be:


:<math>
:<math>
  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}   
  \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}}
\left[ \sigma^{12} \frac{ r} {(r^2 + z^2)^{6}}
- \sigma^6 \frac{r}{(r^2 + z^2 )^{3} }\right] .
- \sigma^6 \frac{r}{(r^2 + z^2 )^{3} }\right] .
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:<math>
:<math>
  V_{W} \left( x \right) = 8 \pi  \epsilon_{sf} \rho_{s}  \int_{-\infty}^{-x} {\textrm d z}  
  \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}
\left[  \frac{ \sigma^{12}} { 10 (r^2 + z^2)^5}
- \frac{\sigma^6 }{ 4 (r^2 + z^2 )^{2} }\right]^{r=0}_{r=\infty} .
- \frac{\sigma^6 }{ 4 (r^2 + z^2 )^{2} }\right]^{r=0}_{r=\infty} .
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: <math>
: <math>
  V_{W} \left( x \right) = 8 \pi  \epsilon_{sf} \rho_{s}  \int_{-\infty}^{-x} {\textrm d z}  
  \Phi_{W} \left( x \right) = 8 \pi  \epsilon_{sf} \rho_{s}  \int_{-\infty}^{-x} {\textrm d z}  
\left[  \frac{ \sigma^{12}} { 10 z^{10} }
\left[  \frac{ \sigma^{12}} { 10 z^{10} }
- \frac{\sigma^6 }{ 4 z^4  } \right];
- \frac{\sigma^6 }{ 4 z^4  } \right];
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: <math>
: <math>
  V_{W} \left( x \right) = 8 \pi  \epsilon_{sf} \rho_s
  \Phi_{W} \left( x \right) = 8 \pi  \epsilon_{sf} \rho_s
\left[  - \frac{ \sigma^{12}} { 90 z^{9} }
\left[  - \frac{ \sigma^{12}} { 90 z^{9} }
+ \frac{\sigma^6 }{ 12 z^3  } \right]_{z=-\infty}^{z=-x};
+ \frac{\sigma^6 }{ 12 z^3  } \right]_{z=-\infty}^{z=-x};
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: <math>
: <math>
  V_{W} \left( x \right) = \frac{4 \pi  \epsilon_{sf} \rho_s \sigma^3}{3}  
  \Phi_{W} \left( x \right) = \frac{4 \pi  \epsilon_{sf} \rho_s \sigma^3}{3}  
\left[  \frac{ \sigma^{9}} { 15  x^{9} }
\left[  \frac{ \sigma^{9}} { 15  x^{9} }
- \frac{\sigma^3 }{ 2 x^3  } \right]
- \frac{\sigma^3 }{ 2 x^3  } \right]

Revision as of 14:05, 21 June 2007

Functional form

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

It takes the form:

where is the intermolecular pair potential. The minimum value of is obtained at , with

  • ,

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:

  • : this region is occupied by a diffuse solid with density composed of 12-6 Lennard-Jones atoms

with parameters and

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

The interaction will be: