# Stockmayer potential

The **Stockmayer potential** consists of the Lennard-Jones model with an embedded point dipole. Thus the Stockmayer potential becomes (Eq. 1 ^{[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[edit]

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

## Bridge function[edit]

A bridge function for use in integral equations has been calculated by Puibasset and Belloni ^{[3]}.

## References[edit]

- ↑ W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics
**9**pp. 398-402 (1941) - ↑ M.E. Van Leeuwen "Deviation from corresponding-states behaviour for polar fluids", Molecular Physics
**82**pp. 383-392 (1994) - ↑ Joël Puibasset and Luc Belloni "Bridge function for the dipolar fluid from simulation", Journal of Chemical Physics
**136**154503 (2012)

**Related reading**

- 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) - M. E. van Leeuwen "Derivation of Stockmayer potential parameters for polar fluids", Fluid Phase Equilibria
**99**pp. 1-18 (1994) - Osvaldo H. Scalise "On the phase equilibrium Stockmayer fluids", Fluid Phase Equilibria
**253**pp. 171–175 (2007) - Reinhard Hentschke, Jörg Bartke, and Florian Pesth "Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid", Physical Review E
**75**011506 (2007) - 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)