# Hard core Yukawa potential

The hard core Yukawa potential [1] has two forms, the attractive Yukawa potential:

${\displaystyle \Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\infty &;&r<\sigma \\-\left({\frac {\epsilon \sigma }{r}}\right)\exp \left[-\kappa \left({\frac {r}{\sigma }}-1\right)\right]&;&r\geq \sigma \end{array}}\right.}$

and the repulsive form

${\displaystyle \Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\infty &;&r<\sigma \\\left({\frac {\epsilon \sigma }{r}}\right)\exp \left[-\kappa \left({\frac {r}{\sigma }}-1\right)\right]&;&r\geq \sigma \end{array}}\right.}$

where ${\displaystyle \Phi \left(r\right)}$ is the intermolecular pair potential, ${\displaystyle r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|}$ is the distance between site 1 and site 2, ${\displaystyle \sigma }$ is the hard diameter, ${\displaystyle \epsilon }$ is the energy well depth (${\displaystyle \epsilon >0}$), and ${\displaystyle \kappa }$ is a parameter that controls the interaction range (${\displaystyle \kappa >0}$).

The repulsive form has been used to study charge-stabilised colloid-colloid interactions.

## Critical point

For the attractive form of the potential, from a study of the law of corresponding states, one has (Eq. 3 in [2])

${\displaystyle P_{c}=0.0228+0.0742T_{c}}$

and (Eq. 4)

${\displaystyle \rho _{c}=0.2534+0.071{\frac {1}{T_{c}}}}$

The repulsive form of the potential has no critical point.

## Triple points

The triple points for this model have been studied by Azhar and co-workers [3].

## Virial coefficients

For the attractive form of the potential the virial coefficients have been calculated by Naresh and Singh [4].