Stockmayer potential

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

$\Phi _{12}(r,\theta _{1},\theta _{2},\phi )=4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]-{\frac {\mu _{1}\mu _{2}}{4\pi \epsilon _{0}r^{3}}}\left(2\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}\cos \phi \right)$

where:

• $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$
• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance r;
• $\sigma$ is the diameter (length), i.e. the value of $r$ at $\Phi (r)=0$ ;
• $\epsilon$ : well depth (energy)
• $\epsilon _{0}$ is the permittivity of the vacuum
• $\mu$ is the dipole moment
• $\theta _{1},\theta _{2}$ is the inclination of the two dipole axes with respect to the intermolecular axis.
• $\phi$ is the azimuth angle between the two dipole moments

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

$\mu ^{*}:={\sqrt {\frac {\mu ^{2}}{4\pi \epsilon _{0}\epsilon \sigma ^{3}}}}$

one can rewrite the expression as

$\Phi (r,\theta _{1},\theta _{2},\phi )=\epsilon \left\{4\left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]-\mu ^{*2}\left(2\cos \theta _{1}\cos \theta _{2}-\sin \theta _{1}\sin \theta _{2}\cos \phi \right)\left({\frac {\sigma }{r}}\right)^{3}\right\}$

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

Critical properties

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

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

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

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

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)