Stockmayer potential

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

\[ \Phi_{12}(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\mu_1 \mu_2}{4\pi \epsilon_0 r^3} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right) \]

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 \( \Phi(r)=0\)
• \( \epsilon \) represents the well depth (energy)
• \( \epsilon_0 \) 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.
• \(\phi\) is the azimuth angle between the two dipole moments

If one defines a reduced dipole moment, \(\mu^*\), such that:

\[\mu^* := \sqrt{\frac{\mu^2}{4\pi\epsilon_0\epsilon \sigma^3}}\]

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 \(0 \leq \mu^* \leq 2.45\) : \[T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)\] \[\rho_c^* = 0.3009 - 0.00785\mu^{*2} - 0.00198\mu^{*4}\] \[P_c^* = 0.127 + 0.0023\mu^{*2}\]

Bridge function

A bridge function for use in integral equations has been calculated by Puibasset and Belloni .

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)