\subsection{Flux relations}

In this section, relationships between various common transport coefficients and the phenomenological coefficients of a binary mixture subjected to a thermal gradient are presented. Additionally, the relationships between various coefficients used to describe thermal diffusion are given. For a binary system, the independent flux-force relations may be written as

\begin{equation}
    \begin{split}
        \flux{q}{} &= \Lqq \nabla \pfrac{1}{T} + \Lqm \left(- \frac{1}{T} \nabla_T \mu_1 \right)\\
        \flux{1}{} &= \Lmq \nabla \pfrac{1}{T} + \Lmm \left(- \frac{1}{T} \nabla_T \mu_1 \right)
    \end{split}
\end{equation}

where $\flux{q}{}$ is the measurable heat flux, and $\flux{1}{}$ is the molar flux of species $1$ in the centre-of-volume (CoV) frame of reference (FoR), $T$ is the absolute temperature, $\mu_1$ is the chemical potential of specie 1, and $L_{ij}$ are the phenomenological coefficients. $\nabla_T$ indicates that the gradient is taken at constant temperature. It is assumed that all transport coefficients are independent of the fluxes and gradients, depending only on the state point of the system i.e. linear response.

First the gradient in chemical potential is related to the mole fraction gradient. Expressing the chemical potential as $\mu_1 = \mu_1(T, p, \vx)$, 
\begin{equation}
    \begin{split}
        \nabla \mu_1 &= \ppder{\mu_1}{T}_{p,\vx} \nabla T + \ppder{\mu_1}{p}_{T,\vx}\nabla p + \sum_i \ppder{\mu_1}{x_i}_{T,p}\nabla x_i\\
        \left(\nabla_T \mu_1\right)_{\nabla p = 0} &= \left[\ppder{\mu_1}{x_1}_{T,p} - \ppder{\mu_1}{x_2}_{T,p} \right]\nabla x_1\\
        & \equiv \der{\mu_1}{x_1}_{T,p}\nabla x_1
    \end{split}
\end{equation}

where $x_i$ is the mole fraction of specie $i$ and $p$ is the pressure. The final equality defines the quantity $\der{\mu_1}{x_1}_{T,p}$. 

Now the phenomenological coefficients can be related to the Soret coefficient ($\ST$), the interdiffusion coefficient ($D_{12}$) and the conductivity ($\lambda$). In the absence of a temperature gradient $\Lmm$ can be related to the diffusion coefficient as

\begin{equation}
    \begin{split}
          \left(\flux{1}{}\right)_{\nabla T = 0} = - \frac{\Lmm}{T} \nabla_T \mu_1 &= - \frac{D_{12}}{x_1 x_2} \nabla x_1\\
        \frac{\Lmm}{T} \der{\mu_1}{x_1}_{T,p} \nabla x_1 &= \frac{D_{12}}{x_1 x_2} \nabla x_1\\
        D_{12} &= \frac{1}{T}x_1 x_2 \Lmm \der{\mu_1}{x_1}_{T,p}.
    \end{split}
\end{equation}

Further, in the state of no mass flux,

\begin{equation}
    \begin{split}
        \Lmq \nabla \pfrac{1}{T} + \Lmm \left(- \frac{1}{T} \nabla_T \mu_1 \right) &= 0\\
        -\frac{\Lmq}{T^2}\nabla T &= \frac{\Lmm}{T} \der{\mu_1}{x_1}_{T,p} \nabla x_1\\
        \Lmq &= - T \Lmm \der{\mu_1}{x_1}_{T,p} \frac{\nabla x_1}{\nabla T}\\
        \Lmq &= T \Lmm \der{\mu_1}{x_1}_{T,p} x_1 x_2 S_{T,1}\\
        S_{T,1} &= \frac{\Lmq}{T \Lmm \der{\mu_1}{x_1}_{T,p} x_1 x_2}.
    \end{split}
\end{equation}
Where the Soret coefficient is defined as 
\begin{equation}
    S_{T,i} \equiv - \pfrac{\nabla x_i}{x_i (1 - x_i) \nabla T}_{\flux{1}{} = 0},
\end{equation}
with $1 - x_1 = x_2$ for a binary system. The conductivity, $\lambda$, is related to the phenomenological coefficients by observing the heat flux in the absence of a mass flux,

\begin{equation}
    \left(\flux{q}{}\right)_{\flux{1}{} = 0} = \Lqq \nabla \pfrac{1}{T} + \Lqm \left(- \frac{1}{T}\nabla_T \mu_1\right) = - \lambda \nabla T
    \label{eq:Lqq_k_relate_1}
\end{equation}
When $\flux{1}{} = 0$, the chemical potential gradient can be related to the temperature gradient as 
\begin{equation}
    \begin{split}
        \flux{1}{} = \Lmq \nabla \pfrac{1}{T} + \Lmm \left(- \frac{1}{T} \nabla_T \mu_1 \right) &= 0\\
        \left(- \frac{1}{T} \nabla_T \mu_1 \right) &= \frac{\Lmq}{\Lmm T^2} \nabla T
    \end{split}
\end{equation}
Inserting this into Equation \eqref{eq:Lqq_k_relate_1} yields
\begin{equation}
    \begin{split}
        -\frac{\Lqq}{T^2} \nabla T + \frac{\Lqm^2}{\Lmm T^2} \nabla T &= -\lambda \nabla T\\
        \lambda &= \frac{\Lqq}{T^2} - \frac{\Lmq^2}{T^2 \Lmm^2}.
    \end{split}
\end{equation}

There are several common ways of quantifying thermal diffusion, one is through the Soret coefficient. Alternatively, one may use the thermal diffusion coefficient $D_T$, considering the mass flux in the absence of a compositional or pressure gradient,
\begin{equation}
    \begin{split}
        \lrp \flux{1}{} \rrp_{\nabla x_1 = \nabla p = 0} &\equiv - \frac{D_T}{x_1 x_2} \nabla \ln T \\
        &= \Lmq \nabla \pfrac{1}{T} \\
        &\implies D_T = \frac{x_1 x_2 \Lmq}{T}.
    \end{split}
\end{equation}

The thermal diffusion ratio, $k_T \equiv \frac{D_T}{D_{12}}$ is also in common use, related to the phenomenological coefficients as
\begin{equation}
    k_T \equiv \frac{D_T}{D_{12}} = \frac{\Lmq}{\Lmm \der{\mu_1}{x_1}_{T,p}} = T x_1 x_2 S_{T,1}.
\end{equation}

Finally, the dimensionless thermal diffusion factor $\alpha_T$ is commonly used, defined as 
\begin{equation}
    \alpha_{T,i} \equiv \pfrac{T \nabla x_1}{x_1 (1 - x_1) \nabla T}_{\flux{1}{} = 0} = T S_{T,1}.
\end{equation}