9-3 Lennard-Jones potential: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
m (Slight tidy + added a reference.)
 
(6 intermediate revisions by 3 users not shown)
Line 1: Line 1:
[EN CONSTRUCCION]
The '''9-3 Lennard-Jones potential''' is related to the [[Lennard-Jones model| Lennard-Jones potential]].
== Functional form ==
It has the following form:
The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model|standard Lennard-Jones potential]].
 
It takes the form:


: <math>
: <math>
V(r) = \frac{ 3 \sqrt{3}}{ 2} \epsilon \left[ \left( \frac{\sigma}{r} \right)^9 -  
\Phi_{12}(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_{12}(r)</math> is the [[intermolecular pair potential]] and <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>.
 
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>
== 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 or ''vice versa'' (Ref. 1).
 
== 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 paremeters <math> \sigma_s </math> and <math> \epsilon_a </math>
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>.
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] .
Line 39: Line 31:


:<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} .
Line 45: Line 37:


: <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];
Line 52: Line 44:


: <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
\left[  \frac{ \sigma^{12}} { 10 z^{10} }
\left[  - \frac{ \sigma^{12}} { 90 z^{9} }
- \frac{\sigma^6 }{ 4 z^4 } \right];
+ \frac{\sigma^6 }{ 12 z^3 } \right]_{z=-\infty}^{z=-x};
</math>
</math>


 
: <math>
 
\Phi_{W} \left( x \right) = \frac{4 \pi  \epsilon_{sf} \rho_s \sigma^3}{3}
[TO BE CONTINUED]
\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]]

Latest revision as of 14:44, 24 July 2008

The 9-3 Lennard-Jones potential is related to the Lennard-Jones potential. It has the following form:

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

  • ,

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:

  • : 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:


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)