Hard spheres

\(\text{FeO}_\text{s}\) provides an implementation of the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state (Boublík, 1970, Mansoori et al., 1971) for hard-sphere mixtures which is often used as reference contribution in SAFT equations of state. The implementation is generalized to allow the description of non-sperical or fused-sphere reference fluids.

The reduced Helmholtz energy density is calculated according to

\[\frac{\beta A}{V}=\frac{6}{\pi}\left(\frac{3\zeta_1\zeta_2}{1-\zeta_3}+\frac{\zeta_2^3}{\zeta_3\left(1-\zeta_3\right)^2}+\left(\frac{\zeta_2^3}{\zeta_3^2}-\zeta_0\right)\ln\left(1-\zeta_3\right)\right)\]

with the packing fractions

\[\zeta_k=\frac{\pi}{6}\sum_\alpha C_{k,\alpha}\rho_\alpha d_\alpha^k,~~~~~~~~k=0\ldots 3\]

The geometry coefficients \(C_{k,\alpha}\) and the segment diameters \(d_\alpha\) depend on the context in which the model is used. The following table shows how the expression can be reused in various models. For details on the fused-sphere chain model, check the repository or the publication.

	Hard spheres	PC-SAFT	Fused-sphere chains
\(d_\alpha\)	\(\sigma_\alpha\)	\(\sigma_\alpha\left(1-0.12e^{\frac{-3\varepsilon_\alpha}{k_\mathrm{B}T}}\right)\)	\(\sigma_\alpha\)
\(C_{0,\alpha}\)	\(1\)	\(m_\alpha\)	\(1\)
\(C_{1,\alpha}\)	\(1\)	\(m_\alpha\)	\(A_\alpha^*\)
\(C_{1,\alpha}\)	\(1\)	\(m_\alpha\)	\(A_\alpha^*\)
\(C_{1,\alpha}\)	\(1\)	\(m_\alpha\)	\(V_\alpha^*\)

Fundamental measure theory

An model for inhomogeneous mixtues of hard spheres is provided by fundamental measure theory (FMT, Rosenfeld, 1989). Different variants have been proposed of which only those that are consistent with the BMCSL equation of state in the homogeneous limit are currently considered in \(\text{FeO}_\text{s}\) (exluding, e.g., the original Rosenfeld and White-Bear II variants).

The Helmholtz energy density is calculated according to

\[\beta f=-n_0\ln\left(1-n_3\right)+\frac{n_{12}}{1-n_3}+\frac{1}{36\pi}n_2n_{22}f_3(n_3)\]

The expressions for \(n_{12}\) and \(n_{22}\) depend on the FMT version.

version	\(n_{12}\)	\(n_{22}\)	references
WhiteBear	\(n_1n_2-\vec n_1\cdot\vec n_2\)	\(n_2^2-3\vec n_2\cdot\vec n_2\)	Roth et al., 2002, Yu and Wu, 2002
KierlikRosinberg	\(n_1n_2\)	\(n_2^2\)	Kierlik and Rosinberg, 1990
AntiSymWhiteBear	\(n_1n_2-\vec n_1\cdot\vec n_2\)	\(n_2^2\left(1-\frac{\vec n_2\cdot\vec n_2}{n_2^2}\right)^3\)	Rosenfeld et al., 1997, Kessler et al., 2021

For small \(n_3\), the value of \(f(n_3)\) numerically diverges. Therefore, it is approximated with a Taylor expansion.

\[\begin{split}f_3=\begin{cases}\frac{n_3+\left(1-n_3\right)^2\ln\left(1-n_3\right)}{n_3^2\left(1-n_3\right)^2}&\text{if }n_3>10^{-5}\\ \frac{3}{2}+\frac{8}{3}n_3+\frac{15}{4}n_3^2+\frac{24}{5}n_3^3+\frac{35}{6}n_3^4&\text{else}\end{cases}\end{split}\]

The weighted densities \(n_k(\mathbf{r})\) are calculated by convolving the density profiles \(\rho_\alpha(\mathbf{r})\) with weight functions \(\omega_k^\alpha(\mathbf{r})\)

\[n_k(\mathbf{r})=\sum_\alpha n_k^\alpha(\mathbf{r})=\sum_\alpha\int\rho_\alpha(\mathbf{r}')\omega_k^\alpha(\mathbf{r}-\mathbf{r}')\mathrm{d}\mathbf{r}'\]

which differ between the different FMT versions.

	WhiteBear/AntiSymWhiteBear	KierlikRosinberg
\(\omega_0^\alpha(\mathbf{r})\)	\(\frac{C_{0,\alpha}}{\pi\sigma_\alpha^2}\,\delta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)	\(C_{0,\alpha}\left(-\frac{1}{8\pi}\,\delta''\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)+\frac{1}{2\pi|\mathbf{r}|}\,\delta'\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\right)\)
\(\omega_1^\alpha(\mathbf{r})\)	\(\frac{C_{1,\alpha}}{2\pi\sigma_\alpha}\,\delta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)	\(\frac{C_{1,\alpha}}{8\pi}\,\delta'\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)
\(\omega_2^\alpha(\mathbf{r})\)	\(C_{2,\alpha}\,\delta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)	\(C_{2,\alpha}\,\delta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)
\(\omega_3^\alpha(\mathbf{r})\)	\(C_{3,\alpha}\,\Theta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)	\(C_{3,\alpha}\,\Theta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)
\(\vec\omega_1^\alpha(\mathbf{r})\)	\(C_{3,\alpha}\frac{\mathbf{r}}{2\pi\sigma_\alpha|\mathbf{r}|}\,\delta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)	-
\(\vec\omega_2^\alpha(\mathbf{r})\)	\(C_{3,\alpha}\frac{\mathbf{r}}{|\mathbf{r}|}\,\delta\!\left(\frac{d_\alpha}{2}-|\mathbf{r}|\right)\)	-