9-3 Lennard-Jones potential - Revision history

Carl McBride: Slight tidy + added a reference.

2008-07-24T13:44:24Z

Slight tidy + added a reference.

← Older revision		Revision as of 15:44, 24 July 2008
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	~~== Functional form ==~~		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\| Lennard-Jones potential]].
	It has the following form:		It has the following form:

	: <math>		: <math>
	\~~Phi~~(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>

	where <math>\~~Phi~~(r)</math> is the [[intermolecular pair potential]].		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		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 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:
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	- \frac{\sigma^3 }{ 2 x^3 } \right]		- \frac{\sigma^3 }{ 2 x^3 } \right]
	</math>		</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]]		[[category:models]]

Nice and Tidy: Better defined r

2008-07-17T14:06:38Z

Better defined r

← Older revision		Revision as of 16:06, 17 July 2008
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	== Functional form ==		== Functional form ==
	The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model\|~~standard~~ Lennard-Jones potential]].		The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model\| Lennard-Jones potential]].
			It has the following form:
	It ~~takes~~ the form:

	: <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>

Nice and Tidy at 13:05, 21 June 2007

2007-06-21T13:05:35Z

← Older revision		Revision as of 15:05, 21 June 2007
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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]

Carl McBride at 10:58, 30 March 2007

2007-03-30T10:58:53Z

← Older revision		Revision as of 12:58, 30 March 2007
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	~~[EN CONSTRUCCION]~~
	== Functional form ==		== Functional form ==
	The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model\|standard Lennard-Jones potential]].		The 9-3 Lennard-Jones potential is related to the [[Lennard-Jones model\|standard Lennard-Jones potential]].
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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		* <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>.
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			[[category:models]]
	[~~TO BE CONTINUED~~]

Noe: /* Interaction between a solid and a fluid molecule */

2007-03-23T13:48:09Z