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| The '''9-3 Lennard-Jones potential''' is related to the [[Lennard-Jones model| Lennard-Jones potential]]. | | The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model|standard Lennard-Jones potential]]. |
| It has the following form: | | |
| | It takes the form: |
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| : <math> | | : <math> |
| \Phi_{12}(r) = \frac{ 3 \sqrt{3}}{ 2} \epsilon \left[ \left( \frac{\sigma}{r} \right)^9 -
| | V(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> |
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| where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>.
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| The minimum value of <math> \Phi(r) </math> is obtained at <math> r = r_{min} </math>, with
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| * <math> \Phi \left( r_{min} \right) = - \epsilon </math>,
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| * <math> \frac{ r_{min} }{\sigma} = 3^{1/6} </math>
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| == Applications ==
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| It is commonly used to model the interaction between the particles
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| of a fluid with a flat structureless solid wall or ''vice versa'' (Ref. 1).
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| == Interaction between a solid and a fluid molecule ==
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| Let us consider the space divided in two regions:
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| * <math> x < 0 </math>: this region is occupied by a ''diffuse'' solid with density <math> \rho_s </math> composed of 12-6 [[Lennard-Jones model|Lennard-Jones]] atoms
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| with parameters <math> \sigma_s </math> and <math> \epsilon_a </math>
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|
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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>.
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| Such an interaction can be computed using cylindrical coordinates.
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|
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| The interaction will be:
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|
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| :<math>
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| \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}
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| \left[ \sigma^{12} \frac{ r} {(r^2 + z^2)^{6}}
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| - \sigma^6 \frac{r}{(r^2 + z^2 )^{3} }\right] .
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| </math>
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|
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| :<math>
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| \Phi_{W} \left( x \right) = 8 \pi \epsilon_{sf} \rho_{s} \int_{-\infty}^{-x} {\textrm d z}
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| \left[ \frac{ \sigma^{12}} { 10 (r^2 + z^2)^5}
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| - \frac{\sigma^6 }{ 4 (r^2 + z^2 )^{2} }\right]^{r=0}_{r=\infty} .
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| </math>
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|
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| : <math>
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| \Phi_{W} \left( x \right) = 8 \pi \epsilon_{sf} \rho_{s} \int_{-\infty}^{-x} {\textrm d z}
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| \left[ \frac{ \sigma^{12}} { 10 z^{10} }
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| - \frac{\sigma^6 }{ 4 z^4 } \right];
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| </math>
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|
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|
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| : <math>
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| \Phi_{W} \left( x \right) = 8 \pi \epsilon_{sf} \rho_s
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| \left[ - \frac{ \sigma^{12}} { 90 z^{9} }
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| + \frac{\sigma^6 }{ 12 z^3 } \right]_{z=-\infty}^{z=-x};
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| </math>
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|
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| : <math>
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| \Phi_{W} \left( x \right) = \frac{4 \pi \epsilon_{sf} \rho_s \sigma^3}{3}
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| \left[ \frac{ \sigma^{9}} { 15 x^{9} }
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| - \frac{\sigma^3 }{ 2 x^3 } \right]
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| </math>
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| ==References==
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| #[http://dx.doi.org/10.1063/1.435080 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)]
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| [[category:models]]
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