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Chemical potential

2014-08-18T15:39:57Z

Sergius:

==Classical thermodynamics==
Definition:

:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math>

where <math>G</math> is the [[Gibbs energy function]], leading to

:<math>\frac{\mu}{k_B T}=\frac{G}{N k_B T}=\frac{A}{N k_B T}+\frac{p V}{N k_B T}</math>

where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math>
is the [[Boltzmann constant]], <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>V</math>
is the volume.

==Statistical mechanics==
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the
number of particles

:<math>\mu= \left. \frac{\partial A}{\partial N}\right\vert_{T,V}=\frac{\partial (-k_B T \ln Z_N)}{\partial N} = - k_B T \left[ \frac{3}{2} \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N} \right]</math>

where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
identical particles
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math>
and <math>Q_N</math> is the
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math>

==Kirkwood charging formula==
The Kirkwood charging formula is given by <ref>[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]</ref>

:<math>\beta \mu_{\rm ex} = \rho \int_0^1 d\lambda \int \frac{\partial \beta \Phi_{12} (r,\lambda)}{\partial \lambda} {\rm g}(r,\lambda) dr</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>{\rm g}(r)</math> is the [[Pair distribution function | pair correlation function]].
==See also==
*[[Ideal gas: Chemical potential]]
*[[Widom test-particle method]]
*[[Overlapping distribution method]]

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.17844 G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics '''63''' pp. 737-742 (1995)]
*[http://dx.doi.org/10.1007/s10955-005-8067-x T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics '''122''' pp. 1237-1260 (2006)]
*[http://dx.doi.org/10.1063/1.4758757 Federico G. Pazzona, Pierfranco Demontis, and Giuseppe B. Suffritti "Chemical potential evaluation in NVT lattice-gas simulations", Journal of Chemical Physics '''137''' 154106 (2012)]

[[category:classical thermodynamics]]
[[category:statistical mechanics]]

Lennard-Jones model

2013-10-17T00:44:12Z

Sergius:

The '''Lennard-Jones''' [[intermolecular pair potential]] is a special case of the [[Mie potential]] and takes its name from [[ Sir John Edward Lennard-Jones KBE, FRS | Sir John Edward Lennard-Jones]]
<ref>[http://dx.doi.org/10.1098/rspa.1924.0081 John Edward Lennard-Jones "On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character '''106''' pp. 441-462 (1924)] § 8 (ii)</ref> <ref>[http://dx.doi.org/10.1098/rspa.1924.0082 John Edward Lennard-Jones "On the Determination of Molecular Fields. II. From the Equation of State of a Gas", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character '''106''' pp. 463-477 (1924)] Eq. 2.05</ref>.
The Lennard-Jones [[models |model]] consists of two 'parts'; a steep repulsive term, and
smoother attractive term, representing the London dispersion forces <ref>[http://dx.doi.org/10.1007/BF01421741 F. London "Zur Theorie und Systematik der Molekularkräfte", Zeitschrift für Physik A Hadrons and Nuclei '''63''' pp. 245-279 (1930)]</ref>. Apart from being an important model in itself,
the Lennard-Jones potential frequently forms one of 'building blocks' of many [[force fields]]. It is worth mentioning that the 12-6 Lennard-Jones model is not the
most faithful representation of the potential energy surface, but rather its use is widespread due to its computational expediency.
For example, the repulsive term is maybe better described with the [[exp-6 potential]].
One of the first [[Computer simulation techniques |computer simulations]] using the Lennard-Jones model was undertaken by Wood and Parker in 1957 <ref>[http://dx.doi.org/10.1063/1.1743822 W. W. Wood and F. R. Parker "Monte Carlo Equation of State of Molecules Interacting with the Lennard‐Jones Potential. I. A Supercritical Isotherm at about Twice the Critical Temperature", Journal of Chemical Physics '''27''' pp. 720- (1957)]</ref> in a study of liquid [[argon]].

== Functional form ==
The Lennard-Jones potential is given by

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

where
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles or ''sites''
* <math> \sigma </math> is the value of <math>r</math> at which <math> \Phi_{12}(r)=0</math>
* <math> \epsilon </math> is the well depth (energy)
In reduced units:
* Density: <math> \rho^* := \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^* := k_B T/\epsilon </math>, where <math> T </math> is the absolute [[temperature]] and <math> k_B </math> is the [[Boltzmann constant]]
The following is a plot of the Lennard-Jones model for the Rowley, Nicholson and Parsonage parameter set <ref>[http://dx.doi.org/10.1016/0021-9991(75)90042-X L. A. Rowley, D. Nicholson and N. G. Parsonage "Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon", Journal of Computational Physics '''17''' pp. 401-414 (1975)]</ref> (<math>\epsilon/k_B = </math> 119.8 K and <math>\sigma=</math> 0.3405 nm). See [[argon]] for other parameter sets.<br>
[[Image:Lennard-Jones.png|500px]]

==Special points==
* <math> \Phi_{12}(\sigma) = 0 </math>
* Minimum value of <math> \Phi_{12}(r) </math> at <math> r = r_{min} </math>;
: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math>

==Critical point==
The location of the [[Critical points |critical point]] is
<ref>[http://dx.doi.org/10.1063/1.477099 J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics '''109''' pp. 4885-4893 (1998)]</ref>
:<math>T_c^* = 1.326 \pm 0.002</math>
at a reduced density of
:<math>\rho_c^* = 0.316 \pm 0.002</math>.

Vliegenthart and Lekkerkerker
<ref>[http://dx.doi.org/10.1063/1.481106 G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics '''112''' pp. 5364-5369 (2000)]</ref>
<ref>[http://dx.doi.org/10.1063/1.3496468 L. A. Bulavin and V. L. Kulinskii "Generalized principle of corresponding states and the scale invariant mean-field approach", Journal of Chemical Physics ''''133''' 134101 (2010)]</ref>
have suggested that the critical point is related to the [[second virial coefficient]] via the expression

:<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math>

==Triple point==
The location of the [[triple point]] as found by Mastny and de Pablo <ref name="Mastny"> [http://dx.doi.org/10.1063/1.2753149 Ethan A. Mastny and Juan J. de Pablo "Melting line of the Lennard-Jones system, infinite size, and full potential", Journal of Chemical Physics '''127''' 104504 (2007)]</ref> is
:<math>T_{tp}^* = 0.694</math>

:<math>\rho_{tp}^* = 0.84</math> (liquid);

:<math>\rho_{tp}^* = 0.96</math> (solid).

==Radial distribution function==
The following plot is of a typical [[radial distribution function]] for the monatomic Lennard-Jones liquid<ref>[http://dx.doi.org/10.1063/1.1700653 John G. Kirkwood, Victor A. Lewinson, and Berni J. Alder "Radial Distribution Functions and the Equation of State of Fluids Composed of Molecules Interacting According to the Lennard-Jones Potential", Journal of Chemical Physics '''20''' pp. 929- (1952)]</ref> (here with <math>\sigma=3.73</math>Å and <math>\epsilon=0.294</math> kcal/mol at a [[temperature]] of 111.06K):
[[Image:LJ_rdf.png|center|450px|Typical radial distribution function for the monatomic Lennard-Jones liquid.]]

==Helmholtz energy function==
An expression for the [[Helmholtz energy function]] of the [[Building up a face centered cubic lattice | face centred cubic]] solid has been given by van der Hoef <ref>[http://dx.doi.org/10.1063/1.1314342 Martin A. van der Hoef "Free energy of the Lennard-Jones solid", Journal of Chemical Physics '''113''' pp. 8142-8148 (2000)]</ref>, applicable within the density range <math>0.94 \le \rho^* \le 1.20</math> and the temperature range <math>0.1 \le T^* \le 2.0</math>. For the liquid state see the work of Johnson, Zollweg and Gubbins <ref>[http://dx.doi.org/10.1080/00268979300100411 J. Karl Johnson, John A. Zollweg and Keith E. Gubbins "The Lennard-Jones equation of state revisited", Molecular Physics '''78''' pp. 591-618 (1993)]</ref>.

==Equation of state==
:''Main article: [[Lennard-Jones equation of state]]''

==Virial coefficients==
:''Main article: [[Lennard-Jones model: virial coefficients]]''

==Phase diagram==
:''Main article: [[Phase diagram of the Lennard-Jones model]]''

==Zeno line==
It has been shown that the Lennard-Jones model has a straight [[Zeno line]] <ref>[http://dx.doi.org/10.1021/jp802999z E. M. Apfelbaum, V. S. Vorob’ev and G. A. Martynov "Regarding the Theory of the Zeno Line", Journal of Physical Chemistry A '''112''' pp. 6042-6044 (2008)]</ref> on the [[Phase diagrams: Density-temperature plane |density-temperature plane]].

==Widom line==
It has been shown that the Lennard-Jones model has a [[Widom line]] <ref>[http://dx.doi.org/10.1021/jp2039898 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and E. N. Tsiok "Widom Line for the Liquid–Gas Transition in Lennard-Jones System", Journal of Physical Chemistry B Article ASAP (2011)]</ref> on the [[Phase diagrams: Pressure-temperature plane | pressure-temperature plane]].

==Perturbation theory==
The Lennard-Jones model is also used in [[Perturbation theory |perturbation theories]], for example see: [[Weeks-Chandler-Andersen perturbation theory]].
== Approximations in simulation: truncation and shifting ==
The Lennard-Jones model is often used with a cutoff radius of <math>2.5 \sigma</math>, beyond which <math> \Phi_{12}(r)</math> is set to zero. Setting the well depth <math> \epsilon </math> to be 1 in the potential on arrives at <math> \Phi_{12}(r)\simeq -0.0163</math>, i.e. at this distance the potential is at less than 2% of the well depth. For an analysis of the effect of this cutoff on the melting line see the work of Mastny and de Pablo <ref name="Mastny"> </ref> and of Ahmed and Sadus <ref>[http://dx.doi.org/10.1063/1.3481102 Alauddin Ahmed and Richard J. Sadus "Effect of potential truncations and shifts on the solid-liquid phase coexistence of Lennard-Jones fluids", Journal of Chemical Physics '''133''' 124515 (2010)]</ref>. See Panagiotopoulos for critical parameters <ref>[http://dx.doi.org/10.1007/BF01458815 A. Z. Panagiotopoulos "Molecular simulation of phase coexistence: Finite-size effects and determination of critical parameters for two- and three-dimensional Lennard-Jones fluids", International Journal of Thermophysics '''15''' pp. 1057-1072 (1994)]</ref>. It has recently been suggested that a truncated and shifted force cutoff of <math>1.5 \sigma</math> can be used under certain conditions <ref>[http://dx.doi.org/10.1063/1.3558787 Søren Toxvaerd and Jeppe C. Dyre "Communication: Shifted forces in molecular dynamics", Journal of Chemical Physics '''134''' 081102 (2011)]</ref>. In order to avoid any discontinuity, a piecewise continuous version, known as the [[modified Lennard-Jones model]], was developed.

== n-m Lennard-Jones potential ==
It is relatively common to encounter potential functions given by:
: <math> \Phi_{12}(r) = c_{n,m} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m
\right].
</math>
with <math> n </math> and <math> m </math> being positive integers and <math> n > m </math>.
<math> c_{n,m} </math> is chosen such that the minimum value of <math> \Phi_{12}(r) </math> being <math> \Phi_{min} = - \epsilon </math>.
Such forms are usually referred to as '''n-m Lennard-Jones Potential'''.
For example, the [[9-3 Lennard-Jones potential |9-3 Lennard-Jones interaction potential]] is often used to model the interaction between
a continuous solid wall and the atoms/molecules of a liquid.
On the '9-3 Lennard-Jones potential' page a justification of this use is presented. Another example is the [[n-6 Lennard-Jones potential]],
where <math>m</math> is fixed at 6, and <math>n</math> is free to adopt a range of integer values.
The potentials form part of the larger class of potentials known as the [[Mie potential]].
<br>
Examples:
*[[8-6 Lennard-Jones potential]]
*[[9-3 Lennard-Jones potential]]
*[[9-6 Lennard-Jones potential]]
*[[10-4-3 Lennard-Jones potential]]
*[[200-100 Lennard-Jones potential]]
*[[n-6 Lennard-Jones potential]]

==Mixtures==
*[[Binary Lennard-Jones mixtures]]
*[[Multicomponent Lennard-Jones mixtures]]

==Related models==
*[[Kihara potential]]
*[[Lennard-Jones model in 1-dimension]] (rods)
*[[Lennard-Jones disks | Lennard-Jones model in 2-dimensions]] (disks)
*[[Lennard-Jones model in 4-dimensions]]
*[[Lennard-Jones sticks]]
*[[Mie potential]]
*[[Soft-core Lennard-Jones model]]
*[[Soft sphere potential]]
*[[Stockmayer potential]]

==References==
<references />
[[Category:Models]]

Chemical potential

2013-08-23T10:14:27Z

Sergius: See the discussion page

==Classical thermodynamics==
Definition:

:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math>

where <math>G</math> is the [[Gibbs energy function]], leading to

:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math>

where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math>
is the [[Boltzmann constant]], <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>V</math>
is the volume.

==Statistical mechanics==
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the
number of particles

:<math>\mu= \left. \frac{\partial A}{\partial N}\right\vert_{T,V}=\frac{\partial (-k_B T \ln Z_N)}{\partial N} = - k_B T \left[ \frac{3}{2} \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N} \right]</math>

where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
identical particles
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math>
and <math>Q_N</math> is the
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math>

==Kirkwood charging formula==
The Kirkwood charging formula is given by <ref>[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]</ref>

:<math>\beta \mu_{\rm ex} = \rho \int_0^1 d\lambda \int \frac{\partial \beta \Phi_{12} (r,\lambda)}{\partial \lambda} {\rm g}(r,\lambda) dr</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>{\rm g}(r)</math> is the [[Pair distribution function | pair correlation function]].
==See also==
*[[Ideal gas: Chemical potential]]
*[[Widom test-particle method]]
*[[Overlapping distribution method]]

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.17844 G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics '''63''' pp. 737-742 (1995)]
*[http://dx.doi.org/10.1007/s10955-005-8067-x T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics '''122''' pp. 1237-1260 (2006)]
*[http://dx.doi.org/10.1063/1.4758757 Federico G. Pazzona, Pierfranco Demontis, and Giuseppe B. Suffritti "Chemical potential evaluation in NVT lattice-gas simulations", Journal of Chemical Physics '''137''' 154106 (2012)]

[[category:classical thermodynamics]]
[[category:statistical mechanics]]

Talk:Chemical potential

2013-08-23T10:05:49Z

Sergius:

In this formula in the article
:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math>

<math>\mu</math>
has wrong dimensionality.
Truly dimensionality of <math>\mu</math> is dimensionality of energy.--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 11:49, 23 August 2013 (CEST)

Just in case there is old version of formula
:<math>\mu= \left. \frac{\partial A}{\partial N}\right\vert_{T,V}=\frac{\partial (-k_B T \ln Z_N)}{\partial N} = -\frac{3}{2} k_BT \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N}</math>

with different dimensionality of 1st and 2nd items.--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 12:05, 23 August 2013 (CEST)

Talk:Chemical potential

2013-08-23T09:49:42Z

Sergius: Created page with "In this formula in the article :<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math> <math>\mu</math> has wrong dimensionality. Truly dimensionality of <math>\mu</math> is ..."

Stockmayer potential

2013-02-16T00:10:56Z

Sergius: /* References */

The '''Stockmayer potential''' consists of the [[Lennard-Jones model]] with an embedded point [[Dipole moment |dipole]]. Thus the Stockmayer potential becomes (Eq. 1 <ref>[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398-402 (1941)]</ref>):

:<math> \Phi_{12}(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\mu_1 \mu_2}{4\pi \epsilon_0 r^3} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) </math>

where:
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math> \Phi(r) </math> is the [[intermolecular pair potential]] between two particles at a distance <math>r</math>
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math>
* <math> \epsilon </math> represents the well depth (energy)
* <math> \epsilon_0 </math> is the permittivity of the vacuum
* <math>\mu</math> is the dipole moment
* <math>\theta_1</math> and <math>\theta_2 </math> are the angles associated with the inclination of the two dipole axes with respect to the intermolecular axis.
* <math>\phi</math> is the azimuth angle between the two dipole moments
If one defines a reduced dipole moment, <math>\mu^*</math>, such that:

:<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math>

one can rewrite the expression as
:<math> \Phi(r, \theta_1, \theta_2, \phi) = \epsilon \left\{4\left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \mu^{*2} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) \left(\frac{\sigma}{r} \right)^{3} \right\}</math>

For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.
==Critical properties==
In the range <math>0 \leq \mu^* \leq 2.45</math> <ref>[http://dx.doi.org/10.1080/00268979400100294 M.E. Van Leeuwen "Deviation from corresponding-states behaviour for polar fluids", Molecular Physics '''82''' pp. 383-392 (1994)]</ref>:
:<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math>
:<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math>
:<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math>
==Bridge function==
A [[bridge function]] for use in [[integral equations]] has been calculated by Puibasset and Belloni <ref>[http://dx.doi.org/10.1063/1.4703899 Joël Puibasset and Luc Belloni "Bridge function for the dipolar fluid from simulation", Journal of Chemical Physics '''136''' 154503 (2012)]</ref>.
==References==
<references/>
'''Related reading'''
*[http://www.nrcresearchpress.com/doi/abs/10.1139/v77-418 Frank M. Mourits, Frans H. A. Rummens "A critical evaluation of Lennard–Jones and Stockmayer potential parameters and of some correlation methods", Canadian Journal of Chemistry '''55''' pp. 3007-3020 (1977)]
*[http://dx.doi.org/10.1016/0378-3812(94)80018-9 M. E. van Leeuwen "Derivation of Stockmayer potential parameters for polar fluids", Fluid Phase Equilibria '''99''' pp. 1-18 (1994)]
*[http://dx.doi.org/10.1016/j.fluid.2007.02.009 Osvaldo H. Scalise "On the phase equilibrium Stockmayer fluids", Fluid Phase Equilibria '''253''' pp. 171–175 (2007)]
*[http://dx.doi.org/10.1103/PhysRevE.75.011506 Reinhard Hentschke, Jörg Bartke, and Florian Pesth "Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid", Physical Review E '''75''' 011506 (2007)]
{{numeric}}
[[category: models]]

Stockmayer potential

2013-02-15T23:48:14Z

Sergius:

The '''Stockmayer potential''' consists of the [[Lennard-Jones model]] with an embedded point [[Dipole moment |dipole]]. Thus the Stockmayer potential becomes (Eq. 1 <ref>[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398-402 (1941)]</ref>):

:<math> \Phi_{12}(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\mu_1 \mu_2}{4\pi \epsilon_0 r^3} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) </math>

where:
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math> \Phi(r) </math> is the [[intermolecular pair potential]] between two particles at a distance <math>r</math>
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math>
* <math> \epsilon </math> represents the well depth (energy)
* <math> \epsilon_0 </math> is the permittivity of the vacuum
* <math>\mu</math> is the dipole moment
* <math>\theta_1</math> and <math>\theta_2 </math> are the angles associated with the inclination of the two dipole axes with respect to the intermolecular axis.
* <math>\phi</math> is the azimuth angle between the two dipole moments
If one defines a reduced dipole moment, <math>\mu^*</math>, such that:

:<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math>

one can rewrite the expression as
:<math> \Phi(r, \theta_1, \theta_2, \phi) = \epsilon \left\{4\left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \mu^{*2} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) \left(\frac{\sigma}{r} \right)^{3} \right\}</math>

For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.
==Critical properties==
In the range <math>0 \leq \mu^* \leq 2.45</math> <ref>[http://dx.doi.org/10.1080/00268979400100294 M.E. Van Leeuwen "Deviation from corresponding-states behaviour for polar fluids", Molecular Physics '''82''' pp. 383-392 (1994)]</ref>:
:<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math>
:<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math>
:<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math>
==Bridge function==
A [[bridge function]] for use in [[integral equations]] has been calculated by Puibasset and Belloni <ref>[http://dx.doi.org/10.1063/1.4703899 Joël Puibasset and Luc Belloni "Bridge function for the dipolar fluid from simulation", Journal of Chemical Physics '''136''' 154503 (2012)]</ref>.
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1016/0378-3812(94)80018-9 M. E. van Leeuwen "Derivation of Stockmayer potential parameters for polar fluids", Fluid Phase Equilibria '''99''' pp. 1-18 (1994)]
*[http://dx.doi.org/10.1016/j.fluid.2007.02.009 Osvaldo H. Scalise "On the phase equilibrium Stockmayer fluids", Fluid Phase Equilibria '''253''' pp. 171–175 (2007)]
*[http://dx.doi.org/10.1103/PhysRevE.75.011506 Reinhard Hentschke, Jörg Bartke, and Florian Pesth "Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid", Physical Review E '''75''' 011506 (2007)]
{{numeric}}
[[category: models]]

Stockmayer potential

2013-02-15T23:34:39Z

Sergius: /* References */

The '''Stockmayer potential''' consists of the [[Lennard-Jones model]] with an embedded point [[Dipole moment |dipole]]. Thus the Stockmayer potential becomes (Eq. 1 <ref>[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398-402 (1941)]</ref>):

:<math> \Phi_{12}(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\mu_1 \mu_2}{4\pi \epsilon_0 r^3} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) </math>

where:
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>
* <math> \Phi(r) </math> is the [[intermolecular pair potential]] between two particles at a distance <math>r</math>
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math>
* <math> \epsilon </math> represents the well depth (energy)
* <math> \epsilon_0 </math> is the permittivity of the vacuum
* <math>\mu</math> is the dipole moment
* <math>\theta_1</math> and <math>\theta_2 </math> are the angles associated with the inclination of the two dipole axes with respect to the intermolecular axis.
* <math>\phi</math> is the azimuth angle between the two dipole moments
If one defines a reduced dipole moment, <math>\mu^*</math>, such that:

:<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math>

one can rewrite the expression as
:<math> \Phi(r, \theta_1, \theta_2, \phi) = \epsilon \left\{4\left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \mu^{*2} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) \left(\frac{\sigma}{r} \right)^{3} \right\}</math>

For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.
==Critical properties==
In the range <math>0 \leq \mu^* \leq 2.45</math> <ref>[http://dx.doi.org/10.1080/00268979400100294 M. E. Van Leeuwe "Deviation from corresponding-states behaviour for polar fluids", Molecular Physics '''82''' pp. 383-392 (1994)]</ref>:
:<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math>
:<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math>
:<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math>
==Bridge function==
A [[bridge function]] for use in [[integral equations]] has been calculated by Puibasset and Belloni <ref>[http://dx.doi.org/10.1063/1.4703899 Joël Puibasset and Luc Belloni "Bridge function for the dipolar fluid from simulation", Journal of Chemical Physics '''136''' 154503 (2012)]</ref>.
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1016/0378-3812(94)80018-9 M. E. van Leeuwen "Derivation of Stockmayer potential parameters for polar fluids", Fluid Phase Equilibria '''99''' pp. 1-18 (1994)]
*[http://dx.doi.org/10.1016/j.fluid.2007.02.009 Osvaldo H. Scalise "On the phase equilibrium Stockmayer fluids", Fluid Phase Equilibria '''253''' pp. 171–175 (2007)]
*[http://dx.doi.org/10.1103/PhysRevE.75.011506 Reinhard Hentschke, Jörg Bartke, and Florian Pesth "Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid", Physical Review E '''75''' 011506 (2007)]
{{numeric}}
[[category: models]]

Talk:FORTRAN code for the Kolafa and Nezbeda equation of state

2013-01-08T11:48:02Z

Sergius:

+ +(-19.58371655*2+rho*(
+ +((-19.58371655)*2+rho*(
--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 12:41, 8 January 2013 (CET)

FORTRAN code for the Kolafa and Nezbeda equation of state

2013-01-08T11:45:28Z

Sergius: +(-19.58371655* >> +((-19.58371655)*

The following are the FORTRAN source code functions (without a "main" program) for the [[Lennard-Jones_equation_of_state#Kolafa and Nezbeda equation of state | Kolafa and Nezbeda equation of state]] for the [[Lennard-Jones model]]

c===================================================================
C Package supplying the thermodynamic properties of the
C LENNARD-JONES fluid
c
c J. Kolafa, I. Nezbeda, Fluid Phase Equil. 100 (1994), 1
c
c ALJ(T,rho)...Helmholtz free energy (including the ideal term)
c PLJ(T,rho)...Pressure
c ULJ(T,rho)...Internal energy
c===================================================================
DOUBLE PRECISION FUNCTION ALJ(T,rho)
C Helmholtz free energy (including the ideal term)
c
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
eta = PI/6.*rho * (dC(T))**3
ALJ = (dlog(rho)+betaAHS(eta)
+ +rho*BC(T)/exp(gammaBH(T)*rho**2))*T
+ +DALJ(T,rho)
RETURN
END
C/* Helmholtz free energy (without ideal term) */
DOUBLE PRECISION FUNCTION ALJres(T,rho)
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
eta = PI/6. *rho*(dC(T))**3
ALJres = (betaAHS(eta)
+ +rho*BC(T)/exp(gammaBH(T)*rho**2))*T
+ +DALJ(T,rho)
RETURN
END
C/* pressure */
DOUBLE PRECISION FUNCTION PLJ(T,rho)
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
eta=PI/6. *rho*(dC(T))**3
sum=((2.01546797*2+rho*(
+ (-28.17881636)*3+rho*(
+ 28.28313847*4+rho*
+ (-10.42402873)*5)))
+ +((-19.58371655)*2+rho*(
+ +75.62340289*3+rho*(
+ (-120.70586598)*4+rho*(
+ +93.92740328*5+rho*
+ (-27.37737354)*6))))/dsqrt(T)
+ + ((29.34470520*2+rho*(
+ (-112.35356937)*3+rho*(
+ +170.64908980*4+rho*(
+ (-123.06669187)*5+rho*
+ 34.42288969*6))))+
+ ((-13.37031968)*2+rho*(
+ 65.38059570*3+rho*(
+ (-115.09233113)*4+rho*(
+ 88.91973082*5+rho*
+ (-25.62099890)*6))))/T)/T)*rho**2
PLJ = ((zHS(eta)
+ + BC(T)/exp(gammaBH(T)*rho**2)
+ *rho*(1-2*gammaBH(T)*rho**2))*T
+ +sum )*rho
RETURN
END
C/* internal energy */
DOUBLE PRECISION FUNCTION ULJ( T, rho)
implicit double precision (a-h,o-z)
data pi /3.141592654d0/
dBHdT=dCdT(T)
dB2BHdT=BCdT(T)
d=dC(T)
eta=PI/6. *rho*d**3
sum= ((2.01546797+rho*(
+ (-28.17881636)+rho*(
+ +28.28313847+rho*
+ (-10.42402873))))
+ + (-19.58371655*1.5+rho*(
+ 75.62340289*1.5+rho*(
+ (-120.70586598)*1.5+rho*(
+ 93.92740328*1.5+rho*
+ (-27.37737354)*1.5))))/dsqrt(T)
+ + ((29.34470520*2+rho*(
+ -112.35356937*2+rho*(
+ 170.64908980*2+rho*(
+ -123.06669187*2+rho*
+ 34.42288969*2)))) +
+ (-13.37031968*3+rho*(
+ 65.38059570*3+rho*(
+ -115.09233113*3+rho*(
+ 88.91973082*3+rho*
+ (-25.62099890)*3))))/T)/T) *rho*rho
ULJ = 3*(zHS(eta)-1)*dBHdT/d
+ +rho*dB2BHdT/exp(gammaBH(T)*rho**2) +sum
RETURN
END
DOUBLE PRECISION FUNCTION zHS(eta)
implicit double precision (a-h,o-z)
zHS = (1+eta*(1+eta*(1-eta/1.5*(1+eta)))) / (1-eta)**3
RETURN
END
DOUBLE PRECISION FUNCTION betaAHS( eta )
implicit double precision (a-h,o-z)
betaAHS = dlog(1-eta)/0.6
+ + eta*( (4.0/6*eta-33.0/6)*eta+34.0/6 ) /(1.-eta)**2
RETURN
END
C /* hBH diameter */
DOUBLE PRECISION FUNCTION dLJ(T)
implicit double precision (a-h,o-z)
DOUBLE PRECISION IST
isT=1/dsqrt(T)
dLJ = ((( 0.011117524191338 *isT-0.076383859168060)
+ *isT)*isT+0.000693129033539)/isT+1.080142247540047
+ +0.127841935018828*dlog(isT)
RETURN
END
DOUBLE PRECISION FUNCTION dC(T)
implicit double precision (a-h,o-z)
sT=dsqrt(T)
dC = -0.063920968*dlog(T)+0.011117524/T
+ -0.076383859/sT+1.080142248+0.000693129*sT
RETURN
END
DOUBLE PRECISION FUNCTION dCdT( T)
implicit double precision (a-h,o-z)
sT=dsqrt(T)
dCdT = 0.063920968*T+0.011117524+(-0.5*0.076383859
+ -0.5*0.000693129*T)*sT
RETURN
END
DOUBLE PRECISION FUNCTION BC( T)
implicit double precision (a-h,o-z)
DOUBLE PRECISION isT
isT=1/dsqrt(T)
BC = (((((-0.58544978*isT+0.43102052)*isT
+ +.87361369)*isT-4.13749995)*isT+2.90616279)*isT
+ -7.02181962)/T+0.02459877
RETURN
END
DOUBLE PRECISION FUNCTION BCdT( T)
implicit double precision (a-h,o-z)
DOUBLE PRECISION iST
isT=1/dsqrt(T)
BCdT = ((((-0.58544978*3.5*isT+0.43102052*3)*isT
+ +0.87361369*2.5)*isT-4.13749995*2)*isT
+ +2.90616279*1.5)*isT-7.02181962
RETURN
END
DOUBLE PRECISION FUNCTION gammaBH(X)
implicit double precision (a-h,o-z)
gammaBH=1.92907278
RETURN
END
DOUBLE PRECISION FUNCTION DALJ(T,rho)
implicit double precision (a-h,o-z)
DALJ = ((+2.01546797+rho*(-28.17881636
+ +rho*(+28.28313847+rho*(-10.42402873))))
+ +(-19.58371655+rho*(75.62340289+rho*((-120.70586598)
+ +rho*(93.92740328+rho*(-27.37737354)))))/dsqrt(T)
+ + ( (29.34470520+rho*((-112.35356937)
+ +rho*(+170.64908980+rho*((-123.06669187)
+ +rho*34.42288969))))
+ +(-13.37031968+rho*(65.38059570+
+ rho*((-115.09233113)+rho*(88.91973082
+ +rho* (-25.62099890)))))/T)/T) *rho*rho
RETURN
END

==Reference==
*[http://dx.doi.org/10.1016/0378-3812(94)80001-4 Jirí Kolafa, Ivo Nezbeda "The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state", Fluid Phase Equilibria '''100''' pp. 1-34 (1994)]
{{Source}}
{{numeric}}

Talk:FORTRAN code for the Kolafa and Nezbeda equation of state

2013-01-08T11:41:44Z

Sergius:

sum=((2.01546797*2+rho*(
+ (-28.17881636)*3+rho*(
+ 28.28313847*4+rho*
+ (-10.42402873)*5)))
+ +((-19.58371655)*2+rho*(
+ +75.62340289*3+rho*(
+ (-120.70586598)*4+rho*(
+ +93.92740328*5+rho*
+ (-27.37737354)*6))))/dsqrt(T)
+ + ((29.34470520*2+rho*(
+ (-112.35356937)*3+rho*(
+ +170.64908980*4+rho*(
+ (-123.06669187)*5+rho*
+ 34.42288969*6))))+
+ ((-13.37031968)*2+rho*(
+ 65.38059570*3+rho*(
+ (-115.09233113)*4+rho*(
+ 88.91973082*5+rho*
+ (-25.62099890)*6))))/T)/T)*rho**2
--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 12:41, 8 January 2013 (CET)

Talk:FORTRAN code for the Kolafa and Nezbeda equation of state

2013-01-08T11:40:16Z

Sergius: +((-19.58371655)* >> +(-19.58371655*