<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\subsubsection{Evaluation of \mathsec{$\rdf$}}
\label{sec:derivation_eff_density}

Until now, the intermolecular potential has been regarded as some arbitrary, spherical potential. From this point onwards, considerations will be restricted to a Mie potential, given as
\begin{equation}
    \begin{split}
        u_{ij}^{Mie} = C \varepsilon_{ij} \lsp \pfrac{\sigma_{ij}}{r_{ij}}^{\lambda_{r,ij}} - \pfrac{\sigma_{ij}}{r_{ij}}^{\lambda_{a,ij}}\rsp, \hspace{.5cm}C = \frac{\lambda_{r_{ij}}}{\lambda_{r,ij} - \lambda_{a,ij}}\pfrac{\lambda_{r,ij}}{\lambda_{a,ij}}^{\frac{\lambda_{a,ij}}{\lambda_{r,ij} - \lambda_{a,ij}}},
    \end{split}
\end{equation}
where $\varepsilon$ is the potential well depth, $\sigma_{ij}$ is the molecular size parameter and $\lambda_{a,ij}$ and $\lambda_{r,ij}$ are adjustable exponents. The use of $\rdf^{CS}$, introduced in Section \ref{sec:theory_kingas} as
\begin{equation}
    \rdf^{CS}(\rho; \sigma^{HS}) = \frac{1 - \frac{1}{2}\eta}{(1 - \eta)^3}, \hspace{.5cm} \eta = \frac{\pi}{6} \rho \lrp \sigma^{HS} \rrp ^3,
    \label{eq:CS_rdf_2}
\end{equation}
thereby implies an assumption that the rdf. of a Mie fluid at distances in the order of $\sigma$ resemble that of a hard sphere fluid. There is also the underlying assumption that $\rdf$ is independent of particle velocities, and that one may therefore use its equilibrium value even in the non-uniform state. 

The question arising now is what value to use for $\sigma$ when employing the CS-rdf at contact. As mentioned in Section \ref{sec:teori_eos}, the SAFT-VR Mie EoS uses a hard sphere system as a reference state, and utilises the Barker-Henderson diameter to represent the hard-sphere diameter of the reference fluid. Papaioannou et. al. state that the choice of a hard sphere reference is due to the requirement of a highly accurate rdf for the reference fluid.\cite{saft_vr_mie} Still, the physical interpretation of the BH-diameter is dubious. Therefore a different diameter, more closely related to the physics of the collision is chosen here.

For the particle diameter, the idea is to take the average distance of closest approach ($R$) for particles that deflect, $\Bar{\sigma}$. Deflection is here taken to mean interactions where $R &gt; b$, that is, where the interaction has prevented one particle from entering the volume occupied by another.

It can be shown from Equation \eqref{eq:kin_dyn_R_root} that for any Mie potential, $R(\sigma, \sg) = \sigma$, and $R(b, \sg) &gt; b$ $\forall$ $b &lt; \sigma$. Thereby, the average is to be taken over all velocities and for $b$ from 0 to $\sigma$. Formally

\begin{equation}
    \Bar{\sigma}_{ij} = \frac{1}{n_i n_j \sigma_{ij}} \int_0^\infty \int_0^{\infty} \int_0^{\sigma_{ij}} f_i f_j R(g_{ij}, b) \d b \d \vu_i \d \vu_j
    \label{eq:sigma_bar_integral}
\end{equation}
where 
\begin{equation}
    n_i n_j \sigma_{ij} = \int_0^\infty \int_0^{\infty} f_i f_j \int_0^{\sigma}  \d b \d \vu_i \d \vu_j
\end{equation}
is a normalising factor. The integral of Equation \eqref{eq:sigma_bar_integral} could be solved numerically, but this was decided against, largely because the resulting expression for $\Bar{\sigma}_{ij}$ would contain the parameter $\rdf$, leaving a polynomial equation in $\rdf$ with potentially several roots upon inserting the expression into \eqref{eq:CS_rdf_2}. While this would be a soluble problem, the benefit of the potential increase in precision in the evaluation $\Bar{\sigma}$ is highly uncertain. Therefore the integral was simplified, first by assuming that the effect of particles deviating from the average particle velocity would largely cancel and therefore be insignificant. Specifically, the effect of low collision velocities is to increase the value of the integrand of Equation \eqref{eq:sigma_bar_integral}, while high collision velocities reduce it. Therefore, for particles of like type, the integral is simplified by simply using the average particle speed $\Bar{u_i} = \sqrt{\frac{2 k_B T}{m_i}}$ rather than integrating over the velocity distributions. For particles of the same type, this together gives the simplified integral
\begin{equation}
    \Bar{\sigma}_{ii} \approx \frac{1}{\sigma_{i}}\int_0^{\sigma_i} R(\Bar{u}_i, b) \d b.
\end{equation}
For particles of unlike type, under the standard assumption of uncorrelated collisions, the average relative speed at collisions is
\begin{equation}
    \Bar{g}_{ij} = \frac{\Bar{u}_i + \Bar{u}_j}{2}.
\end{equation}
Thereby,
\begin{equation}
    \Bar{\sigma}_{ij} = \frac{1}{\sigma_{ij}}\int_0^{\sigma_{ij}} R(\Bar{g}_{ij}, b) \d b.
    \label{eq:def:sigma_bar_ij}
\end{equation}

This approach was chosen primarily for its simplicity, as this work focuses on behaviour in the low density region. For the results presented on conductivity in mixtures in Section \ref{sec:enskog_valid}, the single component density correction given in Equation \eqref{eq:single_comp_density_corr} was applied, using the $\rdf$ computed from an average $\Bar{\sigma}$, computed as
\begin{equation}
    \Bar{\sigma} = \sum_i \sum_j x_i x_j \Bar{\sigma}_{ij}.
    \label{eq:def_sigma_bar_avg}
\end{equation}

The value of $\rdf$ is thereby computed as 

\begin{equation}
    \rdf_{ij} = \frac{1 - \frac{1}{2}\eta}{(1 - \eta)^3}, \hspace{.5cm} \eta = \frac{\pi}{6} \rho \Bar{\sigma}_{ij}^3
\end{equation}

for pure fluids and for the diffusion correction in binary mixtures, whilst for the conductivity correction

\begin{equation}
    \rdf = \frac{1 - \frac{1}{2}\eta}{(1 - \eta)^3}, \hspace{.5cm} \eta = \frac{\pi}{6} \rho \Bar{\sigma}^3
\end{equation}

is used in combination with the conductivity correction for a pure fluid. More accurate values for $\rdf$ can be obtained if one is interested in employing Enskog theory directly to describe behaviour in the dense gas or low-density supercritical region, but this is somewhat more involved than the solution presented here.\cite{chapman_cowling,kinetic_review}</pre></body></html>