Stockmayer potential

The Stockmayer potential consists of the Lennard-Jones model with an embedded point dipole. Thus the Stockmayer potential becomes (Eq. 1 ):

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

where:

• $r:=|{\mathbf{r}}_1-{\mathbf{r}}_2|$
• $\mathrm{\Phi }(r)$ is the intermolecular pair potential between two particles at a distance $r$
• $\sigma$ is the diameter (length), i.e. the value of $r$ at $\mathrm{\Phi }(r)=0$
• $ϵ$ represents the well depth (energy)
• ${ϵ}_0$ is the permittivity of the vacuum
• $\mu$ is the dipole moment
• ${\theta }_1$ and ${\theta }_2$ are the angles associated with 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 a reduced dipole moment, ${\mu }^{*}$, such that:

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

one can rewrite the expression as

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

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

Critical properties

In the range $0\le {\mu }^{*}\le 2.45$ :

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

$\underset{c}{\overset{*}{\rho }}=0.3009-0.00785{\mu }^{*2}-0.00198{\mu }^{*4}$

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

References

Related reading

• M. E. van Leeuwen "Derivation of Stockmayer potential parameters for polar fluids", Fluid Phase Equilibria 99 pp. 1-18 (1994)
• Reinhard Hentschke, Jörg Bartke, and Florian Pesth "Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid", Physical Review E 75 011506 (2007)