\subsection{Review of Mie parameters}
\label{sec:mie_params}
With the exception of \ce{H_2O}, all Mie parameters used to represent real fluids in this work are currently available in the ThermoPack database.\cite{thermopack} This section gives an overview regarding what physical properties the parameters have been fitted to, and to what degree the parameters have been successful in reproducing those properties. A summary of the parameters and some reported deviations from data, either obtained from experiment, a reference EoS, or simulations, is given in Table \ref{tab:mie_params}.

With the exception of hydrogen and deuterium heat capacities, and \ce{CO2} and deuterium speed of sound, SAFT-type equations of state appear to quite reliably reproduce the equilibrium behaviour of these fluids. Aasen et al., who have regressed the parameters for hydrogen, deuterium and neon, report significantly improved accuracy when using the Feynman-Hibbs quantum corrected Mie potential.\cite{saft_mixtures} The implementation of the Enskog solutions developed in this work makes it possible to utilise such a potential, but this route was not pursued. However, Aasen et al. also report mixing parameters $k_{ij}$ and $\ell_{ij}$ for mixtures of helium, neon and deuterium. These are used in this work when computing cross-interaction parameters for mixtures of these components.

For computations involving water, the temperature dependent parameters reported by Lobanova et. al. were used.\cite{water_mie_params} These are given as 
\begin{equation}
    \begin{split}
        \sigma &= 3.119 - 4.554\cdot 10^{-4} T - 8.720 \cdot 10^{-8} T^2 + 1.262 \cdot 10^{-9} T^3, \hspace{.5cm} [\text{Å}]\\
        \varepsilon &= 586.8 - 0.3077 T + 1.105 \cdot 10^{-5} T^2, \hspace{.5cm} [k_B]
    \end{split}
    \label{eq:temp_dep_water_params}
\end{equation}
with $(\lambda_r, \lambda_a) = (8, 6)$. These parameters have been regressed to reproduce the vapour-liquid equilibrium properties of water. As shown in Table \ref{tab:mie_params}, the agreement with experimental data is not excellent. However, water is commonly known as a notoriously difficult molecule to model with a simple potential model such as the Mie potential, due to the high degree of hydrogen bonding, and the polar nature of the molecule. It is worth mentioning that when using these parameters in MD simulations they report systematically under-predicting the self diffusion coefficient of water.

An important point to note is that, with the exception of oxygen, the parameters here are fit to equilibrium properties. This carries the implication that quality of the model predictions should be representative of the predictive power of the Enskog solutions for Mie fluids, rather than a result of parameter fitting.

\begin{table}[htb]
    \centering
    \caption{Mie potential parameters used in this work. The rightmost columns show deviations from reference data for some common properties, the vapour pressure $p_vap$, the saturated liquid density $\rho_{liq}^{sat}$, the enthalpy of vaporisation $\Delta_{vap}H$, the critical temperature $T_c$, the speed of sound $u_s$ and the isobaric heat capacity $C_p$,  predicted by either a SAFT-type EoS or MD simulations, as reported by those that have regressed the parameters. The reference data is obtained either from experimental measurements, MD or a reference EoS.}
    \begin{tabular}{c c c c c c c c c c c c}
    \toprule
     & & & & & \multicolumn{6}{c}{Reported deviations [\%]} & \\ \cline{6-11}
    Specie & $\sigma$ [Å] & $\varepsilon$ [$k_B$] & $\lambda_a$ [-] & $\lambda_r$ [-] & $p_{vap}$ & $\rho_{liq}^{sat}$ & $\Delta_{vap} H$ & $T_c$ & $u_s$ & $C_p$ & Ref.\\  
    \midrule
        \ce{Xe}	&	3.9011	 &	227.55	 &	6.0	 &	12.0	 & 3 & 0.5 & 2 & & & & \cite{xe_param} \\
        \ce{O2}$^{(a)}$	&	3.433	 &	113.0	 &	6.0	 &	12.0 & & & & & & & \cite{o2_param}\\ %Chen, Rex and Yuen, W., Oxidation of Metals, 73 (2009),  	 \\ 
        \ce{CO2}	&	3.1916	 &	231.88	 &	5.1646	 &	27.557	 & 0.40 & 1.18 & 3.26 & & 10.70 & 3.49 & \cite{saft_vr_mie}\\ %Lafitte et al. 2013, doi: 10.1063/1.4819786	 \\ 
        \ce{Ne}	&	2.8019	 &	29.875	 &	6.0	 &	9.6977	 & 0.06 & 0.4 & & 1.69 & 1.37 & 5.98 & \cite{saft_mixtures}\\ %Aasen 2019, doi: 10.1063/1.5111364	 \\ 
        \ce{CH4} &	3.7412	 &	153.36	 &	6.0	 &	12.65 &  0.63 & 0.78 & 2.86 & & 1.29 & 4.73 & \cite{saft_vr_mie}\\ %Lafitte et al. 2013, doi: 10.1063/1.4819786
        \ce{D2}	&	3.1538	 &	21.2	 &	6.0	 &	8.0	 & 2.15 & 5.24 & & 1.25 & 9.34 & 45.15 & \cite{saft_mixtures}\\ %Aasen 2019, doi: 10.1063/1.5111364	 \\ 
        \ce{He}$^{(b)}$	&	3.353	 &	4.44	 &	6.0	 &	14.84 & & & & & & & \cite{he_param}\\ %Herdes 2015, doi: 10.1016/j.fluid.2015.07.014	 \\ 
        \ce{H2}	&	3.2574	 &	17.931	 &	6.0	 &	8.0	 & 3.84 & 9.01 & & 3.38 & 3.35 & 93.32 & \cite{saft_mixtures}\\ %Aasen 2019, doi: 10.1063/1.5111364	 \\ 
        \ce{Kr}$^{(c)}$	&	3.64	 &	166.66	 &	6.0	 &	12.0	 & & & & & & & \cite{kr_param}\\ %Irma et. al (1999) ISSN : 1029-0435/0892-7022	 \\ 
        \ce{Ar}	&	3.404	 &	117.84	 &	6.0	 &	12.085	 & 0.21 & 0.66 & 3.09 & 2.19 & 2.13 & 1.77& \cite{ar_param} \\ %Unpublished (as of 08/2018) parameters from G. Jackon's group	 \\ 
        \ce{N2}$^{(d)}$	&	3.656	 &	98.94	 &	6.0	 &	12.26	 & & & & & & & \cite{n2_param}\\ %Ronald A. Aziz, J. Chem. Phys. 99, 4518 (1993), DOI : https://doi.org/10.1063/1.466051	 \\ 
        \ce{H2O} & (e) & (e) & 6 & 8 & 4 & 1 & 15.9 & 3.09 & - & 34.6 & \cite{water_mie_params} \\
    \bottomrule
    \multicolumn{12}{l}{$^{(a)}$Fitted to diffusion coefficient for use in metallurgical modelling.}\\
    \multicolumn{12}{l}{$^{(b)}$Values obtained from the correlation proposed by Meija et al.\cite{he_param_2}} \\
    \multicolumn{12}{l}{$^{(c)}$It is not reported how the parameters are obtained.} \\
    \multicolumn{12}{l}{$^{(d)}$Deviations in the order of 2-5 \% measured against interfacial tensions.} \\
    \multicolumn{12}{l}{$^{(e)}$Temperature dependent parameters given in Equation \eqref{eq:temp_dep_water_params}.} \\
    \end{tabular}
    \label{tab:mie_params}
\end{table}