Stockmayer potential

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|}
• $\Phi (r)$ is the intermolecular pair potential between two particles at a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma } is the diameter (length), i.e. the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(r)=0}
• $\epsilon$ represents the well depth (energy)
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_0 } is the permittivity of the vacuum
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the dipole moment
• $\theta _{1}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_2 } are the angles associated with the inclination of the two dipole axes with respect to the intermolecular axis.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is the azimuth angle between the two dipole moments

If one defines a reduced dipole moment, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu^*} , such that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq \mu^* \leq 2.45} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c^* = 1.313 + 0.2999\mu^{*2} -0.2837 \ln(\mu^{*2} +1)}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

• 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)