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| 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>): | | The '''Stockmayer potential''' consists of the [[Lennard-Jones model]] with an embedded point dipole. Thus the Stockmayer potential becomes: |
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| :<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> | | :<math> \Phi(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\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> |
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| where: | | 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 r; |
| * <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> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math> | | * <math> \epsilon </math> : well depth (energy) |
| * <math> \epsilon </math> represents the well depth (energy) | |
| * <math> \epsilon_0 </math> is the permittivity of the vacuum | | * <math> \epsilon_0 </math> is the permittivity of the vacuum |
| * <math>\mu</math> is the dipole moment | | * <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>\theta_1,\theta_2 </math> is 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
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| If one defines a reduced dipole moment, <math>\mu^*</math>, such that: | | If one defines the reduced dipole moment, <math>\mu^*</math> |
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| :<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math> | | :<math>\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}</math> |
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| For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential. | | For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential. |
| ==Critical properties==
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| 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>:
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| :<math>T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)</math>
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| :<math>\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}</math>
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| :<math>P_c^* = 0.127 + 0.0023\mu^{*2}</math>
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| ==Bridge function==
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| 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>.
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| ==References== | | ==References== |
| <references/>
| | #[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)] |
| '''Related reading'''
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| *[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)]
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| *[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)]
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| *[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)]
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| *[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)]
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| *[http://dx.doi.org/10.1063/1.4821455 Jun Wang , Pankaj A. Apte , James R. Morris and Xiao Cheng Zeng "Freezing point and solid-liquid interfacial free energy of Stockmayer dipolar fluids: A molecular dynamics simulation study", Journal of Chemical Physics '''139''' 114705 (2013)]
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| {{numeric}}
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| [[category: models]] | | [[category: models]] |