http://www.sklogwiki.org/SklogWiki/api.php?action=feedcontributions&user=Sergius&feedformat=atomSklogWiki - User contributions [en]2024-03-29T11:15:38ZUser contributionsMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Chemical_potential&diff=14290Chemical potential2014-08-18T15:39:57Z<p>Sergius: </p>
<hr />
<div>==Classical thermodynamics==<br />
Definition:<br />
<br />
:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math><br />
<br />
where <math>G</math> is the [[Gibbs energy function]], leading to <br />
<br />
:<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><br />
<br />
where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math><br />
is the [[Boltzmann constant]], <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>V</math><br />
is the volume.<br />
<br />
==Statistical mechanics==<br />
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the <br />
number of particles<br />
<br />
:<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><br />
<br />
where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math><br />
identical particles<br />
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math><br />
and <math>Q_N</math> is the <br />
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]<br />
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math><br />
<br />
==Kirkwood charging formula==<br />
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><br />
<br />
:<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><br />
<br />
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]].<br />
==See also==<br />
*[[Ideal gas: Chemical potential]]<br />
*[[Widom test-particle method]]<br />
*[[Overlapping distribution method]]<br />
<br />
==References==<br />
<references/><br />
'''Related reading'''<br />
*[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)]<br />
*[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)]<br />
*[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)]<br />
<br />
<br />
[[category:classical thermodynamics]]<br />
[[category:statistical mechanics]]</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Lennard-Jones_model&diff=13871Lennard-Jones model2013-10-17T00:44:12Z<p>Sergius: </p>
<hr />
<div>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]] <br />
<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)] &sect; 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>.<br />
The Lennard-Jones [[models |model]] consists of two 'parts'; a steep repulsive term, and<br />
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,<br />
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 <br />
most faithful representation of the potential energy surface, but rather its use is widespread due to its computational expediency.<br />
For example, the repulsive term is maybe better described with the [[exp-6 potential]].<br />
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]].<br />
<br />
== Functional form == <br />
The Lennard-Jones potential is given by<br />
<br />
:<math> \Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math><br />
<br />
where<br />
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math><br />
* <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles or ''sites''<br />
* <math> \sigma </math> is the value of <math>r</math> at which <math> \Phi_{12}(r)=0</math><br />
* <math> \epsilon </math> is the well depth (energy)<br />
In reduced units: <br />
* 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>)<br />
* 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]]<br />
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><br />
[[Image:Lennard-Jones.png|500px]]<br />
<br />
==Special points==<br />
* <math> \Phi_{12}(\sigma) = 0 </math><br />
* Minimum value of <math> \Phi_{12}(r) </math> at <math> r = r_{min} </math>; <br />
: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math><br />
<br />
==Critical point==<br />
The location of the [[Critical points |critical point]] is <br />
<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><br />
:<math>T_c^* = 1.326 \pm 0.002</math><br />
at a reduced density of<br />
:<math>\rho_c^* = 0.316 \pm 0.002</math>.<br />
<br />
Vliegenthart and Lekkerkerker<br />
<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><br />
<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><br />
have suggested that the critical point is related to the [[second virial coefficient]] via the expression <br />
<br />
:<math>B_2 \vert_{T=T_c}= -\pi \sigma^3</math><br />
<br />
==Triple point==<br />
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<br />
:<math>T_{tp}^* = 0.694</math><br />
<br />
:<math>\rho_{tp}^* = 0.84</math> (liquid); <br />
<br />
:<math>\rho_{tp}^* = 0.96</math> (solid).<br />
<br />
==Radial distribution function==<br />
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>&Aring; and <math>\epsilon=0.294</math> kcal/mol at a [[temperature]] of 111.06K):<br />
[[Image:LJ_rdf.png|center|450px|Typical radial distribution function for the monatomic Lennard-Jones liquid.]]<br />
<br />
==Helmholtz energy function==<br />
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>.<br />
<br />
==Equation of state==<br />
:''Main article: [[Lennard-Jones equation of state]]''<br />
<br />
==Virial coefficients==<br />
:''Main article: [[Lennard-Jones model: virial coefficients]]''<br />
<br />
==Phase diagram==<br />
:''Main article: [[Phase diagram of the Lennard-Jones model]]''<br />
<br />
==Zeno line==<br />
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]].<br />
<br />
==Widom line==<br />
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]].<br />
<br />
==Perturbation theory==<br />
The Lennard-Jones model is also used in [[Perturbation theory |perturbation theories]], for example see: [[Weeks-Chandler-Andersen perturbation theory]].<br />
== Approximations in simulation: truncation and shifting ==<br />
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.<br />
<br />
== n-m Lennard-Jones potential ==<br />
It is relatively common to encounter potential functions given by:<br />
: <math> \Phi_{12}(r) = c_{n,m} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m <br />
\right].<br />
</math><br />
with <math> n </math> and <math> m </math> being positive integers and <math> n > m </math>.<br />
<math> c_{n,m} </math> is chosen such that the minimum value of <math> \Phi_{12}(r) </math> being <math> \Phi_{min} = - \epsilon </math>.<br />
Such forms are usually referred to as '''n-m Lennard-Jones Potential'''.<br />
For example, the [[9-3 Lennard-Jones potential |9-3 Lennard-Jones interaction potential]] is often used to model the interaction between<br />
a continuous solid wall and the atoms/molecules of a liquid.<br />
On the '9-3 Lennard-Jones potential' page a justification of this use is presented. Another example is the [[n-6 Lennard-Jones potential]],<br />
where <math>m</math> is fixed at 6, and <math>n</math> is free to adopt a range of integer values.<br />
The potentials form part of the larger class of potentials known as the [[Mie potential]].<br />
<br><br />
Examples:<br />
*[[8-6 Lennard-Jones potential]]<br />
*[[9-3 Lennard-Jones potential]]<br />
*[[9-6 Lennard-Jones potential]]<br />
*[[10-4-3 Lennard-Jones potential]]<br />
*[[200-100 Lennard-Jones potential]]<br />
*[[n-6 Lennard-Jones potential]]<br />
<br />
==Mixtures==<br />
*[[Binary Lennard-Jones mixtures]]<br />
*[[Multicomponent Lennard-Jones mixtures]]<br />
<br />
==Related models==<br />
*[[Kihara potential]]<br />
*[[Lennard-Jones model in 1-dimension]] (rods)<br />
*[[Lennard-Jones disks | Lennard-Jones model in 2-dimensions]] (disks)<br />
*[[Lennard-Jones model in 4-dimensions]] <br />
*[[Lennard-Jones sticks]]<br />
*[[Mie potential]]<br />
*[[Soft-core Lennard-Jones model]]<br />
*[[Soft sphere potential]]<br />
*[[Stockmayer potential]]<br />
<br />
==References==<br />
<references /><br />
[[Category:Models]]</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Chemical_potential&diff=13844Chemical potential2013-08-23T10:14:27Z<p>Sergius: See the discussion page</p>
<hr />
<div>==Classical thermodynamics==<br />
Definition:<br />
<br />
:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math><br />
<br />
where <math>G</math> is the [[Gibbs energy function]], leading to <br />
<br />
:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math><br />
<br />
where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math><br />
is the [[Boltzmann constant]], <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>V</math><br />
is the volume.<br />
<br />
==Statistical mechanics==<br />
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the <br />
number of particles<br />
<br />
:<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><br />
<br />
where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math><br />
identical particles<br />
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math><br />
and <math>Q_N</math> is the <br />
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]<br />
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math><br />
<br />
==Kirkwood charging formula==<br />
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><br />
<br />
:<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><br />
<br />
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]].<br />
==See also==<br />
*[[Ideal gas: Chemical potential]]<br />
*[[Widom test-particle method]]<br />
*[[Overlapping distribution method]]<br />
<br />
==References==<br />
<references/><br />
'''Related reading'''<br />
*[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)]<br />
*[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)]<br />
*[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)]<br />
<br />
<br />
[[category:classical thermodynamics]]<br />
[[category:statistical mechanics]]</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Chemical_potential&diff=13843Talk:Chemical potential2013-08-23T10:05:49Z<p>Sergius: </p>
<hr />
<div>In this formula in the article <br />
:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math><br />
<br />
<math>\mu</math> <br />
has wrong dimensionality.<br />
Truly dimensionality of <math>\mu</math> is dimensionality of energy.--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 11:49, 23 August 2013 (CEST)<br />
<br />
Just in case there is old version of formula <br />
:<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><br />
<br />
with different dimensionality of 1st and 2nd items.--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 12:05, 23 August 2013 (CEST)</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Chemical_potential&diff=13842Talk:Chemical potential2013-08-23T09:49:42Z<p>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 ..."</p>
<hr />
<div>In this formula in the article <br />
:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math><br />
<br />
<math>\mu</math> <br />
has wrong dimensionality.<br />
Truly dimensionality of <math>\mu</math> is dimensionality of energy.--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 11:49, 23 August 2013 (CEST)</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Stockmayer_potential&diff=13311Stockmayer potential2013-02-16T00:10:56Z<p>Sergius: /* References */</p>
<hr />
<div>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>):<br />
<br />
:<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><br />
<br />
where:<br />
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math><br />
* <math> \Phi(r) </math> is the [[intermolecular pair potential]] between two particles at a distance <math>r</math><br />
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math> <br />
* <math> \epsilon </math> represents the well depth (energy)<br />
* <math> \epsilon_0 </math> is the permittivity of the vacuum<br />
* <math>\mu</math> is the dipole moment<br />
* <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.<br />
* <math>\phi</math> is the azimuth angle between the two dipole moments<br />
If one defines a reduced dipole moment, <math>\mu^*</math>, such that: <br />
<br />
:<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math><br />
<br />
one can rewrite the expression as <br />
:<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><br />
<br />
For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.<br />
==Critical properties==<br />
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>:<br />
:<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math><br />
:<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math><br />
:<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math><br />
==Bridge function==<br />
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>.<br />
==References==<br />
<references/><br />
'''Related reading'''<br />
*[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)] <br />
*[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)] <br />
*[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)] <br />
*[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)]<br />
{{numeric}}<br />
[[category: models]]</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Stockmayer_potential&diff=13310Stockmayer potential2013-02-15T23:48:14Z<p>Sergius: </p>
<hr />
<div>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>):<br />
<br />
:<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><br />
<br />
where:<br />
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math><br />
* <math> \Phi(r) </math> is the [[intermolecular pair potential]] between two particles at a distance <math>r</math><br />
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math> <br />
* <math> \epsilon </math> represents the well depth (energy)<br />
* <math> \epsilon_0 </math> is the permittivity of the vacuum<br />
* <math>\mu</math> is the dipole moment<br />
* <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.<br />
* <math>\phi</math> is the azimuth angle between the two dipole moments<br />
If one defines a reduced dipole moment, <math>\mu^*</math>, such that: <br />
<br />
:<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math><br />
<br />
one can rewrite the expression as <br />
:<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><br />
<br />
For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.<br />
==Critical properties==<br />
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>:<br />
:<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math><br />
:<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math><br />
:<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math><br />
==Bridge function==<br />
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>.<br />
==References==<br />
<references/><br />
'''Related reading'''<br />
*[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)] <br />
*[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)] <br />
*[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)]<br />
{{numeric}}<br />
[[category: models]]</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Stockmayer_potential&diff=13309Stockmayer potential2013-02-15T23:34:39Z<p>Sergius: /* References */</p>
<hr />
<div>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>):<br />
<br />
:<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><br />
<br />
where:<br />
* <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math><br />
* <math> \Phi(r) </math> is the [[intermolecular pair potential]] between two particles at a distance <math>r</math><br />
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math> <br />
* <math> \epsilon </math> represents the well depth (energy)<br />
* <math> \epsilon_0 </math> is the permittivity of the vacuum<br />
* <math>\mu</math> is the dipole moment<br />
* <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.<br />
* <math>\phi</math> is the azimuth angle between the two dipole moments<br />
If one defines a reduced dipole moment, <math>\mu^*</math>, such that: <br />
<br />
:<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math><br />
<br />
one can rewrite the expression as <br />
:<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><br />
<br />
For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.<br />
==Critical properties==<br />
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>:<br />
:<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math><br />
:<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math><br />
:<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math><br />
==Bridge function==<br />
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>.<br />
==References==<br />
<references/><br />
'''Related reading'''<br />
*[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)] <br />
*[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)] <br />
*[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)]<br />
{{numeric}}<br />
[[category: models]]</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:FORTRAN_code_for_the_Kolafa_and_Nezbeda_equation_of_state&diff=13281Talk:FORTRAN code for the Kolafa and Nezbeda equation of state2013-01-08T11:48:02Z<p>Sergius: </p>
<hr />
<div> + +(-19.58371655*2+rho*( <br />
+ +((-19.58371655)*2+rho*(<br />
--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 12:41, 8 January 2013 (CET)</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=FORTRAN_code_for_the_Kolafa_and_Nezbeda_equation_of_state&diff=13280FORTRAN code for the Kolafa and Nezbeda equation of state2013-01-08T11:45:28Z<p>Sergius: +(-19.58371655* >> +((-19.58371655)*</p>
<hr />
<div>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]]<br />
<br />
c===================================================================<br />
C Package supplying the thermodynamic properties of the<br />
C LENNARD-JONES fluid<br />
c<br />
c J. Kolafa, I. Nezbeda, Fluid Phase Equil. 100 (1994), 1<br />
c<br />
c ALJ(T,rho)...Helmholtz free energy (including the ideal term)<br />
c PLJ(T,rho)...Pressure<br />
c ULJ(T,rho)...Internal energy<br />
c===================================================================<br />
DOUBLE PRECISION FUNCTION ALJ(T,rho)<br />
C Helmholtz free energy (including the ideal term)<br />
c<br />
implicit double precision (a-h,o-z)<br />
data pi /3.141592654d0/<br />
eta = PI/6.*rho * (dC(T))**3<br />
ALJ = (dlog(rho)+betaAHS(eta)<br />
+ +rho*BC(T)/exp(gammaBH(T)*rho**2))*T<br />
+ +DALJ(T,rho)<br />
RETURN<br />
END<br />
C/* Helmholtz free energy (without ideal term) */<br />
DOUBLE PRECISION FUNCTION ALJres(T,rho)<br />
implicit double precision (a-h,o-z)<br />
data pi /3.141592654d0/<br />
eta = PI/6. *rho*(dC(T))**3<br />
ALJres = (betaAHS(eta)<br />
+ +rho*BC(T)/exp(gammaBH(T)*rho**2))*T<br />
+ +DALJ(T,rho)<br />
RETURN<br />
END<br />
C/* pressure */<br />
DOUBLE PRECISION FUNCTION PLJ(T,rho)<br />
implicit double precision (a-h,o-z)<br />
data pi /3.141592654d0/<br />
eta=PI/6. *rho*(dC(T))**3<br />
sum=((2.01546797*2+rho*(<br />
+ (-28.17881636)*3+rho*(<br />
+ 28.28313847*4+rho*<br />
+ (-10.42402873)*5)))<br />
+ +((-19.58371655)*2+rho*(<br />
+ +75.62340289*3+rho*(<br />
+ (-120.70586598)*4+rho*(<br />
+ +93.92740328*5+rho*<br />
+ (-27.37737354)*6))))/dsqrt(T)<br />
+ + ((29.34470520*2+rho*(<br />
+ (-112.35356937)*3+rho*(<br />
+ +170.64908980*4+rho*(<br />
+ (-123.06669187)*5+rho*<br />
+ 34.42288969*6))))+<br />
+ ((-13.37031968)*2+rho*(<br />
+ 65.38059570*3+rho*(<br />
+ (-115.09233113)*4+rho*(<br />
+ 88.91973082*5+rho*<br />
+ (-25.62099890)*6))))/T)/T)*rho**2<br />
PLJ = ((zHS(eta)<br />
+ + BC(T)/exp(gammaBH(T)*rho**2)<br />
+ *rho*(1-2*gammaBH(T)*rho**2))*T<br />
+ +sum )*rho<br />
RETURN<br />
END<br />
C/* internal energy */<br />
DOUBLE PRECISION FUNCTION ULJ( T, rho)<br />
implicit double precision (a-h,o-z)<br />
data pi /3.141592654d0/<br />
dBHdT=dCdT(T)<br />
dB2BHdT=BCdT(T)<br />
d=dC(T)<br />
eta=PI/6. *rho*d**3<br />
sum= ((2.01546797+rho*(<br />
+ (-28.17881636)+rho*(<br />
+ +28.28313847+rho*<br />
+ (-10.42402873))))<br />
+ + (-19.58371655*1.5+rho*(<br />
+ 75.62340289*1.5+rho*(<br />
+ (-120.70586598)*1.5+rho*(<br />
+ 93.92740328*1.5+rho*<br />
+ (-27.37737354)*1.5))))/dsqrt(T)<br />
+ + ((29.34470520*2+rho*(<br />
+ -112.35356937*2+rho*(<br />
+ 170.64908980*2+rho*(<br />
+ -123.06669187*2+rho*<br />
+ 34.42288969*2)))) +<br />
+ (-13.37031968*3+rho*(<br />
+ 65.38059570*3+rho*(<br />
+ -115.09233113*3+rho*(<br />
+ 88.91973082*3+rho*<br />
+ (-25.62099890)*3))))/T)/T) *rho*rho<br />
ULJ = 3*(zHS(eta)-1)*dBHdT/d<br />
+ +rho*dB2BHdT/exp(gammaBH(T)*rho**2) +sum<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION zHS(eta)<br />
implicit double precision (a-h,o-z)<br />
zHS = (1+eta*(1+eta*(1-eta/1.5*(1+eta)))) / (1-eta)**3<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION betaAHS( eta )<br />
implicit double precision (a-h,o-z)<br />
betaAHS = dlog(1-eta)/0.6<br />
+ + eta*( (4.0/6*eta-33.0/6)*eta+34.0/6 ) /(1.-eta)**2<br />
RETURN<br />
END<br />
C /* hBH diameter */<br />
DOUBLE PRECISION FUNCTION dLJ(T)<br />
implicit double precision (a-h,o-z)<br />
DOUBLE PRECISION IST<br />
isT=1/dsqrt(T)<br />
dLJ = ((( 0.011117524191338 *isT-0.076383859168060)<br />
+ *isT)*isT+0.000693129033539)/isT+1.080142247540047<br />
+ +0.127841935018828*dlog(isT)<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION dC(T)<br />
implicit double precision (a-h,o-z)<br />
sT=dsqrt(T)<br />
dC = -0.063920968*dlog(T)+0.011117524/T<br />
+ -0.076383859/sT+1.080142248+0.000693129*sT<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION dCdT( T)<br />
implicit double precision (a-h,o-z)<br />
sT=dsqrt(T)<br />
dCdT = 0.063920968*T+0.011117524+(-0.5*0.076383859<br />
+ -0.5*0.000693129*T)*sT<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION BC( T)<br />
implicit double precision (a-h,o-z)<br />
DOUBLE PRECISION isT<br />
isT=1/dsqrt(T)<br />
BC = (((((-0.58544978*isT+0.43102052)*isT<br />
+ +.87361369)*isT-4.13749995)*isT+2.90616279)*isT<br />
+ -7.02181962)/T+0.02459877<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION BCdT( T)<br />
implicit double precision (a-h,o-z)<br />
DOUBLE PRECISION iST<br />
isT=1/dsqrt(T)<br />
BCdT = ((((-0.58544978*3.5*isT+0.43102052*3)*isT<br />
+ +0.87361369*2.5)*isT-4.13749995*2)*isT<br />
+ +2.90616279*1.5)*isT-7.02181962<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION gammaBH(X)<br />
implicit double precision (a-h,o-z)<br />
gammaBH=1.92907278<br />
RETURN<br />
END<br />
DOUBLE PRECISION FUNCTION DALJ(T,rho)<br />
implicit double precision (a-h,o-z)<br />
DALJ = ((+2.01546797+rho*(-28.17881636<br />
+ +rho*(+28.28313847+rho*(-10.42402873))))<br />
+ +(-19.58371655+rho*(75.62340289+rho*((-120.70586598)<br />
+ +rho*(93.92740328+rho*(-27.37737354)))))/dsqrt(T)<br />
+ + ( (29.34470520+rho*((-112.35356937)<br />
+ +rho*(+170.64908980+rho*((-123.06669187)<br />
+ +rho*34.42288969))))<br />
+ +(-13.37031968+rho*(65.38059570+<br />
+ rho*((-115.09233113)+rho*(88.91973082<br />
+ +rho* (-25.62099890)))))/T)/T) *rho*rho<br />
RETURN<br />
END<br />
<br />
<br />
==Reference==<br />
*[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)]<br />
{{Source}}<br />
{{numeric}}</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:FORTRAN_code_for_the_Kolafa_and_Nezbeda_equation_of_state&diff=13279Talk:FORTRAN code for the Kolafa and Nezbeda equation of state2013-01-08T11:41:44Z<p>Sergius: </p>
<hr />
<div> sum=((2.01546797*2+rho*(<br />
+ (-28.17881636)*3+rho*(<br />
+ 28.28313847*4+rho*<br />
+ (-10.42402873)*5)))<br />
+ +((-19.58371655)*2+rho*(<br />
+ +75.62340289*3+rho*(<br />
+ (-120.70586598)*4+rho*(<br />
+ +93.92740328*5+rho*<br />
+ (-27.37737354)*6))))/dsqrt(T)<br />
+ + ((29.34470520*2+rho*(<br />
+ (-112.35356937)*3+rho*(<br />
+ +170.64908980*4+rho*(<br />
+ (-123.06669187)*5+rho*<br />
+ 34.42288969*6))))+<br />
+ ((-13.37031968)*2+rho*(<br />
+ 65.38059570*3+rho*(<br />
+ (-115.09233113)*4+rho*(<br />
+ 88.91973082*5+rho*<br />
+ (-25.62099890)*6))))/T)/T)*rho**2<br />
--[[User:Sergius|Sergius]] ([[User talk:Sergius|talk]]) 12:41, 8 January 2013 (CET)</div>Sergiushttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:FORTRAN_code_for_the_Kolafa_and_Nezbeda_equation_of_state&diff=13278Talk:FORTRAN code for the Kolafa and Nezbeda equation of state2013-01-08T11:40:16Z<p>Sergius: +((-19.58371655)* >> +(-19.58371655*</p>
<hr />
<div>sum=((2.01546797*2+rho*(<br />
+ (-28.17881636)*3+rho*(<br />
+ 28.28313847*4+rho*<br />
+ (-10.42402873)*5)))<br />
+ +((-19.58371655)*2+rho*(<br />
+ +75.62340289*3+rho*(<br />
+ (-120.70586598)*4+rho*(<br />
+ +93.92740328*5+rho*<br />
+ (-27.37737354)*6))))/dsqrt(T)<br />
+ + ((29.34470520*2+rho*(<br />
+ (-112.35356937)*3+rho*(<br />
+ +170.64908980*4+rho*(<br />
+ (-123.06669187)*5+rho*<br />
+ 34.42288969*6))))+<br />
+ ((-13.37031968)*2+rho*(<br />
+ 65.38059570*3+rho*(<br />
+ (-115.09233113)*4+rho*(<br />
+ 88.91973082*5+rho*<br />
+ (-25.62099890)*6))))/T)/T)*rho**2</div>Sergius