\subsection{Modelling the Soret effect}
\label{subsec:intro_models}
A variety of different models have been proposed to quantitatively predict the Soret effect. Already in the early 20th century Chapman and Enskog developed solutions to the Boltzmann equations that provided results in agreement with experiment for dilute gases.\cite{chapman_cowling} These solutions are central to this work, and will be referred to as the Enskog solutions. The theory was later extended to the denser gas region, however extending the theory to the liquid phase has been shown to introduce significant difficulties.\cite{liquid_bad_kinetic_1} 

The most common method of employing the Enskog solutions is by evaluating the solutions for mixtures of hard spheres. Approximating a real gas as a hard sphere system greatly simplifies the evaluation of the solutions, and allows one to use tabulated values for a range of otherwise complicated computations.\cite{molecular_theory,chapman_cowling} Additionally, one can find in the literature a variety of correlations that intend to correct the tabulated hard sphere values such that they can be used with higher accuracy for real systems.\cite{properties_liq_and_gas} In fact, in the making of this work, no attempts at explicitly employing a more advanced potential to the Enskog solutions was found. 

It has been suggested that in order to develop a model capable of reproducing the Soret effect in both the gas and liquid phase, one can split the Soret effect in to several contributions. One manner of accomplishing this is splitting the Soret coefficient into an isotopic contribution, $S_T^{iso}$, depending only on molecular masses and moments of inertia, and a chemical contribution, $S_T^{ch}$, depending on molecular interactions.\cite{isotope_effect} The Soret coefficient is then $S_T = S_T^{iso} + S_T^{ch}$, Debuschewitz and Köhler have shown that this manner of splitting the coefficient is likely reasonable. Others have proposed subdividing the Soret coefficient into a reversible and an irreversible contribution,\cite{eastman1926thermodynamics} or a kinetic and a configurational contribution.\cite{kempers01}.

Through the years, a series of different approaches to modelling thermal diffusion in dense fluids have been proposed. The different approaches primarily have their roots in kinetic theory, irreversible thermodynamics or statistical thermodynamics.\cite{soret_intro} Among the more prominent examples are the kinetic based model by Artola et al.\cite{transition_theory_model} the Shukla and Firoozabadi model based on relating the Soret coefficient to the heats of transport,\cite{firoozbadi,firoozabadi2000theoretical} and the Kempers model, based in maximisation of the number of microstates in a system.\cite{kempers89,kempers01} There are other models than those presented here, the models by Artola et al., Kempers, and Firoozabadi and Shukla are used here to exemplify the general frameworks within which most models are developed. Different models developed within the same framework tend to be similar in form, and exhibit similar challenges and advantages.\cite{soret_intro}

The kinetic, or transition theory approach is based in the works by Prigogine and Tichacek.\cite{PRIGOGINE1950,tichacek1956thermal} This approach seeks to describe thermal diffusion as an activated process, and to predict the Soret coefficient from the activation free enthalpies of the components in a mixture. The immediate difficulty within this approach is obtaining these activation energies, with the result that they have been used by some as fitting parameters in the interpretation of experimental data.\cite{transition_fitting} Additionally, the original models within this approach fail to account for the effects of the different molecular masses of the diffusing species, i.e. they fail to capture the isotopic contribution.\cite{soret_review_2022} Artola et al. propose a new derivation within the transition theory framework that incorporates the isotope effect, and obtain the activation energies from MD simulations.\cite{transition_theory_model} They demonstrate that their model is in good quantitative agreement with simulation results for some simple fluids, and also in somewhat qualitative agreement with measurements in the ethanol-water system. The issue still remains, however, of obtaining the relevant activation energies for a given mixture without requiring time-consuming simulations, and they propose that the reason for the poor quantitative agreement with data in the ethanol-water system is due to a failure of the potential model they use to obtain said energies.

Approaching the task of modelling the the Soret effect by use of the heats of transport originated with the model by Denbigh.\cite{first_irrev_model,soret_intro} Much work and improvement within this framework has been made since, resulting in the model proposed by Shukla and Firoozabadi.\cite{firoozbadi,firoozabadi2000theoretical} The challenge of applying this model in practice is similar to the challenges of transition theory based models. Accurate values for the heats of transport are not necessarily trivial to obtain. Additionally the model contains a parameter that is commonly treated as a fitting parameter, thereby somewhat reducing its predictive value. Despite these challenges, it has been shown that the model can be capable of accurately reproducing simulation- and experimental results in dense fluids.\cite{soret_review_2022} 

The statistical thermodynamic approach, here represented by the models proposed by Kempers,\cite{kempers89,kempers01} has its appeal in the lack of dubiously defined or hard-to-obtain parameters. The underlying hypothesis of these models is that the steady state of a system is characterised by a constrained maximum in the canonical partition function, such that by minimising the Helmholtz energy of a system under certain constraints one can relate the Soret coefficient to thermodynamic properties that may be obtained from an equation of state (EoS). The original Kempers model has the issue of being dependent on accurate values for the enthalpy of the reference state, but later Kempers proposed a modification to the model that removes this inconsistency. Using kinetic gas theory to acquire the value of the Soret coefficient in the ideal gas state, the dependency on the reference state is removed. The derivation of this model is covered in more detail in Section \ref{sec:kempers_model}.

It has been shown that the Kempers model is incapable of reliably predicting the Soret coefficient of binary mixtures.\cite{prosjektoppgave,soret_review_2022,ny_kempers} Kempers claims that the model is highly sensitive to the EoS employed, and this has been confirmed in recent work.\cite{prosjektoppgave} An investigation was made into the possibility of predicting whether the model would be capable of predicting the Soret coefficient when coupled with a specific EoS, based on how the given EoS performed in predictions of other thermodynamic quantities. No clear method of achieving this was found, and questions were raised regarding the fundamentals of the statistical thermodynamic approach used by Kempers. Additionally, the question of whether the use of kinetic gas theory should in fact be coupled with an ideal gas reference state was raised.

