Coronal radiative losses

The Stockmayer potential is a mathematical model for representing the interactions between pairs of atoms or molecules. It consists of the Lennard-Jones potential with an embedded point dipole. Thus the Stockmayer potential becomes:

${\Phi }_{12}(r,{\theta }_1,{\theta }_2,\varphi )=4ϵ[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6]-\frac{{\mu }_1{\mu }_2}{4\pi {ϵ}_0{r}^3}(2\cos {\theta }_1\cos {\theta }_2-\sin {\theta }_1\sin {\theta }_2\cos \varphi )$

where:

• $r:=|{𝐫}_1-{𝐫}_2|$
• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance r;
• $\sigma$ is the diameter (length), i.e. the value of $r$ at $\Phi (r)=0$ ;
• $ϵ$ : well depth (energy)
• ${ϵ}_0$ is the vacuum permittivity
• $\mu$ is the dipole moment
• ${\theta }_1,{\theta }_2$ is the inclination of the two dipole axes with respect to the intermolecular axis.
• $\varphi$ is the azimuth angle between the two dipole moments

If one defines the reduced dipole moment, ${\mu }^{\ast }$

${\mu }^{\ast }:=\sqrt{\frac{{\mu }^2}{4\pi {ϵ}_0ϵ{\sigma }^3}}$

one can rewrite the expression as

$\Phi (r,{\theta }_1,{\theta }_2,\varphi )=ϵ\{4[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6]-{\mu }^{\ast 2}(2\cos {\theta }_1\cos {\theta }_2-\sin {\theta }_1\sin {\theta }_2\cos \varphi ){\left(\frac{\sigma }{r}\right)}^3\}$

For this reason the potential is sometimes known as the Stockmayer 12-6-3 potential.

Critical properties

In the range $0\le {\mu }^{\ast }\le 2.45$ (Ref. 1)

$T_c^{\ast }=1.313+0.2999{\mu }^{\ast 2}-0.2837\ln ({\mu }^{\ast 2}+1)$

${\rho }_c^{\ast }=0.3009-0.00785{\mu }^{\ast 2}-0.00198{\mu }^{\ast 4}$

$P_c^{\ast }=0.127+0.0023{\mu }^{\ast 2}$

References

1. M. E. Van Leeuwe "Deviation from corresponding-states behaviour for polar fluids", Molecular Physics 82 pp. 383-392 (1994)
2. Reinhard Hentschke, Jörg Bartke, and Florian Pesth "Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid", Physical Review E 75 011506 (2007)

Attribution