SklogWiki - User contributions [en]

Lennard-Jones model

2007-02-28T13:51:20Z

161.111.27.40: /* References */

The Lennard-Jones potential is given by

:<math> V(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math>

where:

* <math> V(r) </math> : potential energy of interaction between two particles at a distance r;

* <math> \sigma </math> : diameter (length);

* <math> \epsilon </math> : well depth (energy)

Reduced units:

* Density, <math> \rho^* \equiv \rho \sigma^3 </math>, where <math> \rho = N/V </math> (number of particles <math> N </math> divided by the volume <math> V </math>.)

* Temperature; <math> T^* \equiv k_B T/\epsilon </math>, where <math> T </math> is the absolute temperature and <math> k_B </math> is the [[Boltzmann constant]]

==References==

#[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)]
[[Category:Models]]

Mean spherical approximation

2007-02-28T13:39:35Z

161.111.27.40:

The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by

:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>

The '''Blum and Høye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by

:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>

and

:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math>

where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>

where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones]] potential.
By introducing the definition (Eq. 10 Ref. 4)

:<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math>

one can arrive at (Eq. 11 in Ref. 4)

:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>

The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.

==References==
#[http://dx.doi.org/10.1103/PhysRev.144.251 J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review '''144''' pp. 251 - 258 (1966)]
#[http://dx.doi.org/10.1007/BF01011750 L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal of Statistical Physics, '''19''' pp. 317-324 (1978)]
#[http://dx.doi.org/10.1007/BF01013935 Lesser Blum "Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure" Journal of Statistical Physics, '''22''' pp. 661-672 (1980)]
#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]

[[Category:Integral equations]]

Mean spherical approximation

2007-02-28T13:38:44Z

161.111.27.40: /* References */

The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by

:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>

The '''Blum and Hoye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by

:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>

and

:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math>

where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>

where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones]] potential.
By introducing the definition (Eq. 10 Ref. 4)

:<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math>

one can arrive at (Eq. 11 in Ref. 4)

:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>

The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.

==References==
#[http://dx.doi.org/10.1103/PhysRev.144.251 J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review '''144''' pp. 251 - 258 (1966)]
#[http://dx.doi.org/10.1007/BF01011750 L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal of Statistical Physics, '''19''' pp. 317-324 (1978)]
#[http://dx.doi.org/10.1007/BF01013935 Lesser Blum "Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure" Journal of Statistical Physics, '''22''' pp. 661-672 (1980)]
#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]

[[Category:Integral equations]]

Mean spherical approximation

2007-02-28T13:36:05Z

161.111.27.40: /* References */

The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by

:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>

The '''Blum and Hoye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by

:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>

and

:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math>

where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>

where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones]] potential.
By introducing the definition (Eq. 10 Ref. 4)

:<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math>

one can arrive at (Eq. 11 in Ref. 4)

:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>

The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.

==References==
#[http://dx.doi.org/10.1103/PhysRev.144.251 J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review '''144''' pp. 251 - 258 (1966)]
#[http://dx.doi.org/10.1007/BF01011750 L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal of Statistical Physics, '''19''' pp. 317-324 (1978)]
#[http://dx.doi.org/
#[JSP_1980_22_0661_nolotengoSpringer]
#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]

[[Category:Integral equations]]

Mean spherical approximation

2007-02-28T13:29:25Z

161.111.27.40: /* References */

The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by

:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>

The '''Blum and Hoye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by

:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math>

and

:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math>

where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as

:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>

where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones]] potential.
By introducing the definition (Eq. 10 Ref. 4)

:<math>\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)</math>

one can arrive at (Eq. 11 in Ref. 4)

:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>

The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>.

==References==
#[http://dx.doi.org/10.1103/PhysRev.144.251 J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review '''144''' pp. 251 - 258 (1966)]
#[http://dx.doi.org/
#[http://dx.doi.org/

#[JSP_1978_19_0317_nolotengoSpringer]
#[JSP_1980_22_0661_nolotengoSpringer]
#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]

[[Category:Integral equations]]

Main Page

2007-02-27T12:44:38Z

161.111.27.40:

SklogWiki is a Wiki for the thermodynamics, statistical mechanics, and computer simulation of materials, community.
*[[History]]
*[[Classical thermodynamics]]
*[[Statistical mechanics]]
*[[Non-equilibrium thermodynamics]]
*[[Ideal gas]]
*[[Models]]
*[[Force fields]]
*[[Equations of state]]
*[[Integral equations]]
*[[Perturbation theory]]
*[[Ergodic hypothesis]]
*[[Path integral formulation]]
*[[Monte Carlo]]
*[[Molecular dynamics]]
*[[Ewald sum]]
*[[Materials modeling and computer simulation codes]]
*[[Virial coefficients of real substances]]
*[[Virial coefficients of model systems]]
*[[Phase diagrams]]
*[[Liquid crystals]]
*[[Colloids]]
*[[Gels]]
*[[Polymers]]
*[[Water]]
*[[Glasses]]
*[[Proteins]]
*[[Mathematics]]
*[[Researchers and research groups]]
*[[Conferences]]

Boltzmann constant

2007-02-20T15:12:27Z

161.111.27.40: New page: The '''Boltzmann constant''' (<math>k</math> or <math>k_B</math>) is the physical constant relating temperature to energy. It is named after the Austrian physicist [[Ludwig Ed...

The '''Boltzmann constant''' (<math>k</math> or <math>k_B</math>) is the [[physical constant]] relating [[temperature]] to [[energy]].

It is named after the Austrian physicist [[Ludwig Eduard Boltzmann]].
Its experimentally determined value (in [[SI]] units, 2002 [[CODATA]] value) is:

:<math>k_B =1.380 6505(24) \times 10^{-23} </math> [[joule]]/[[kelvin]]

:<math> =8.617 343(15) \times 10^{-5}</math> [[electron-volt]]/kelvin.

Classical thermodynamics

2007-02-20T15:05:59Z

161.111.27.40:

*[[Zeroth law of thermodynamics]]
*[[First law of thermodynamics]]
*[[Second law of thermodynamics]]
*[[Enthalpy]]
*[[Boltzmann constant]]
*[[Helmholtz energy function]]
*[[Gibbs energy function]]
*[[Thermodynamic relations]]
*[[Chemical potential]]
*[[Compressibility]]

SklogWiki:About

2007-02-20T15:03:51Z

161.111.27.40:

SklogWiki is a Wiki for the thermodynamics, statistical mechanics and computer simulation of materials community.
This initiative is partly financed by the [http://www.madrid.org/universidades/ Dirección General de Universidades e Investigación de la Comunidad de Madrid] under Grant
S0505/ESP/0299, program [http://www.icmm.csic.es/mossnoho/ MOSSNOHO].

SklogWiki was started on Thursday the 15th of February 2007. Its name is derived from the famous equation:
:<math>\left.S\right.=k \log W</math>.

Monte Carlo

2007-02-19T17:43:59Z

161.111.27.40:

*[[Methods]]
*[[Lattice Simulations]]
*[[Random numbers]]

Random numbers

2007-02-19T17:43:09Z

161.111.27.40: New page: *Linear congruential generator *Prime modulus multiplicative linear congruential generator *RANDU *RCARRY *RANLUX

*[[Linear congruential generator]]
*[[Prime modulus multiplicative linear congruential generator]]
*[[RANDU]]
*[[RCARRY]]
*[[RANLUX]]

SklogWiki:Site support

2007-02-17T12:49:06Z

161.111.27.40: New page: The best donation you can make is that of your time, please do so.

The best donation you can make is that of your time, please do so.

Models

2007-02-16T16:59:01Z

161.111.27.40:

*[[Hard sphere]]
*[[Hard ellipsoids]]
*[[Hard hexagons]]
*[[Square well]]
*[[Ramp]]
*[[Gaussian overlap]]
*[[Gay-Berne]]
*[[Lennard-Jones]]
*[[Polyamorphic systems]]
*[[Confined systems]]

Models

2007-02-16T16:58:43Z

161.111.27.40:

*[[Hard sphere]]
*[[Hard ellipsoids]]
*[[Hard hexagons
*[[Square well]]
*[[Ramp]]
*[[Gaussian overlap]]
*[[Gay-Berne]]
*[[Lennard-Jones]]
*[[Polyamorphic systems]]
*[[Confined systems]]

Researchers and research groups

2007-02-16T16:55:28Z

161.111.27.40: New page: *[http://www.qft.iqfr.csic.es/ Theoretical Physical Chemistry Group] *[http://www.ucm.es/info/molecsim/ Statistical Thermodynamics of Molecular Fluids] *[http://www1.phys.uu.nl/scm/defaul...

*[http://www.qft.iqfr.csic.es/ Theoretical Physical Chemistry Group]
*[http://www.ucm.es/info/molecsim/ Statistical Thermodynamics of Molecular Fluids]
*[http://www1.phys.uu.nl/scm/default.htm Soft Condensed Matter Group]
*[http://www.uhu.es/filico/home.html PHYSICS OF COMPLEX LIQUIDS GROUP]

Main Page

2007-02-16T16:50:55Z

161.111.27.40:

SklogWiki is a Wiki for the thermodynamics, statistical mechanics, and computer simulation of materials, community.
*[[History]]
*[[Models]]
*[[Equations of state]]
*[[Integral equations]]
*[[Perturbation theory]]
*[[Monte Carlo]]
*[[Molecular dynamics]]
*[[Virial coefficients of real susbstances]]
*[[Water]]
*[[Proteins]]
*[[Research groups]]
*[[Conferences]]