Latest revision |
Your text |
Line 1: |
Line 1: |
| The '''9-3 Lennard-Jones potential''' is related to the [[Lennard-Jones model| Lennard-Jones potential]]. | | [EN CONSTRUCCION] |
| It has the following form: | | == Functional form == |
| | The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model|standard Lennard-Jones potential]]. |
| | |
| | It takes the form: |
|
| |
|
| : <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> |
|
| |
|
| where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>.
| | The minimum value of <math> V(r) </math> is obtained at <math> r = r_{min} </math>, with |
| The minimum value of <math> \Phi(r) </math> is obtained at <math> r = r_{min} </math>, with | | |
| * <math> \Phi \left( r_{min} \right) = - \epsilon </math>, | | * <math> V \left( r_{min} \right) = - \epsilon </math>, |
| | |
| * <math> \frac{ r_{min} }{\sigma} = 3^{1/6} </math> | | * <math> \frac{ r_{min} }{\sigma} = 3^{1/6} </math> |
| | |
| == Applications == | | == Applications == |
| | |
| It is commonly used to model the interaction between the particles | | 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). | | of a fluid with a flat structureless solid wall. |
| | |
| == Interaction between a solid and a fluid molecule == | | == Interaction between a solid and a fluid molecule == |
| Let us consider the space divided in two regions: | | Let us consider the space divided in two regions: |
| * <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
| |
| with parameters <math> \sigma_s </math> and <math> \epsilon_a </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.
| |
|
| |
|
| The interaction will be:
| | * <math> x < 0 </math>: this region is occupied by a ''diffuse'' solid with density <math> \rho_s </math> composed of Lennard-Jones atoms |
| | with paremeters <math> \sigma_s </math> and <math> \epsilon_a </math> |
|
| |
|
| :<math>
| | Our aim is to compute the total interaction between this solid and a molecule located at a position <math> x > 0 </math>. |
| \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] .
| |
| </math> | |
|
| |
|
| :<math>
| |
| \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} .
| |
| </math>
| |
|
| |
|
| : <math>
| | [TO BE CONTINUED] |
| \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];
| |
| </math>
| |
| | |
| | |
| : <math>
| |
| \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};
| |
| </math>
| |
| | |
| : <math>
| |
| \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]
| |
| </math>
| |
| ==References==
| |
| #[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)]
| |
| [[category:models]]
| |