9-3 Lennard-Jones potential: Difference between revisions
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== Functional form == | == Functional form == | ||
The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model| | The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model| Lennard-Jones potential]]. | ||
It has the following form: | |||
It | |||
: <math> | : <math> | ||
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where <math>\Phi(r)</math> is the [[intermolecular pair potential]]. | 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 | The minimum value of <math> \Phi(r) </math> is obtained at <math> r = r_{min} </math>, with | ||
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math> | |||
* <math> \Phi \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> | ||
== 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. | 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 | * <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> | with parameters <math> \sigma_s </math> and <math> \epsilon_a </math> | ||
Revision as of 15:06, 17 July 2008
Functional form
The 9-3 Lennard-Jones potential is related to the Lennard-Jones potential. It has the following form:
![{\displaystyle \Phi (r)={\frac {3{\sqrt {3}}}{2}}\epsilon \left[\left({\frac {\sigma }{r}}\right)^{9}-\left({\frac {\sigma }{r}}\right)^{3}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/525d6d00a02cb123ca1b57010c8c5c7be3c1807b)
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(r)} is the intermolecular pair potential. The minimum value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(r) } is obtained at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = r_{min} } , with
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|}
- ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ r_{min} }{\sigma} = 3^{1/6} }
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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x < 0 } : this region is occupied by a diffuse solid with density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_s } composed of 12-6 Lennard-Jones atoms
with parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_s } and
Our aim is to compute the total interaction between this solid and a molecule located at a position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_f > 0 } . Such an interaction can be computed using cylindrical coordinates.
The interaction will be:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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] . }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} . }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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]; }
![{\displaystyle \Phi _{W}\left(x\right)=8\pi \epsilon _{sf}\rho _{s}\left[-{\frac {\sigma ^{12}}{90z^{9}}}+{\frac {\sigma ^{6}}{12z^{3}}}\right]_{z=-\infty }^{z=-x};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bfe145b725f03951d6be57ea8b3d1df20f69f76)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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] }