\section{Flux relations vol. 2}

The goal of this section is to derive relations between the Soret coefficient and the phenomenological coefficients. For a binary system, the independent flux-force relations may be written as

\begin{equation}
    \begin{split}
        \flux{q}{V}' &= \Lqq' \nabla \pfrac{1}{T} + \Lqm' \nabla \pfrac{\mu_1}{T}\\
        \flux{1}{V} &= \Lmq' \nabla \pfrac{1}{T} + \Lmm' \nabla \pfrac{\mu_1}{T}
    \end{split}
\end{equation}

where $\flux{q}{V}'$ is the \textit{total} heat flux, and $\flux{1}{V}$ is the molar flux of species $1$ in the CoV FoR. For the remainder of this section, the superscript $V$ indicating the FoR will be omitted to simplify notation. $\nabla_T$ indicates that the gradient is taken at constant temperature.

The system under investigation is a closed container in contact with a heat bath on one side, and a heat sink on the other. The system is allowed to reach a steady state, where the mass flux vanishes. The pressure is uniform, such that no convection is taking place. Further, it is assumed that all transport coefficients are independent of the fluxes and gradients, depending only on the state point of the system (linear response).

Long story short:
\begin{equation}
    \begin{split}
        \Vec{J} &= \Mat{L} \Vec{X}\\
        \Vec{X} &= \Mat{R} \Vec{J} 
    \end{split}
\end{equation}
in the steady state for the two-bulb system, $\flux{1}{} = 0$, such that 
\begin{equation}
    \begin{split}
        \nabla \pfrac{\mu_1}{T} &= R_{\mu q} \flux{q}{}'\\
        &= \frac{- L_{\mu q}}{\det\lrp\Mat{L}\rrp} \lrp - k \nabla T \rrp
    \end{split}
    \label{eq:gradient_flux_relation}
\end{equation}

\todo[inline]{Relasjonene nedenfor er for målbar varmefluks, de er ikke oppdattert for denne seksjonen!}

\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 the final equality defines the quantity $\der{\mu_1}{x_1}_{T,p}$. Alternatively, the chemical potential may be expressed as $\mu_1 = \mu_1(T, \rho, \vn)$, giving the relation
\begin{equation}
    \begin{split}
        \nabla \mu_1 &= \ppder{\mu_1}{T}_{\rho,\vx} \nabla T + \ppder{\mu_1}{\rho}_{T, \vx}\nabla \rho + \sum_i \ppder{\mu_1}{x_i}_{T,\rho}\nabla x_i\\
        \left(\nabla_T \mu_1\right)_{\nabla p = 0} &= \left[\ppder{\mu_1}{x_1}_{T,\rho} - \ppder{\mu_1}{x_2}_{T,\rho} \right]\nabla x_1 + \ppder{\mu_1}{\rho}_{T, \vx} \nabla_T \rho\\
        &= \left[\ppder{\mu_1}{x_1}_{T,\rho} - \ppder{\mu_1}{x_2}_{T,\rho} \right]\nabla x_1 + \ppder{\mu_1}{\rho}_{T, \vx} \left[\ppder{\rho}{x_1}_{T,p} -  \ppder{\rho}{x_2}_{T,p}\right]\nabla x_1\\
        &\equiv \left[\der{\mu_1}{x_1}_{T,\rho} + \ppder{\mu_1}{\rho}_{T,\vx} \der{\rho}{x_1}_{T,p}\right] \nabla x_1
    \end{split}
\end{equation}
where the final equality defines $\der{\mu_1}{x_1}_{T,\rho}$ and $\der{\rho}{x_1}_{T,p}$, and it has been used that

\begin{equation}
    \begin{split}
        \nabla \rho &= \ppder{\rho}{T}_{p,\vx} \nabla T + \ppder{\rho}{p}_{T,\vx}\nabla p + \ppder{\rho}{x_1}_{T,p}\nabla x_1 +  \ppder{\rho}{x_2}_{T,p}\nabla x_2 \\
        \left(\nabla_T \rho\right)_{\nabla p = 0} &= \left[\ppder{\rho}{x_1}_{T,p} -  \ppder{\rho}{x_2}_{T,p}\right] \nabla x_1.
    \end{split}
\end{equation}

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

\begin{equation}
    \begin{split}
          \left(\flux{1}{}\right)_{\nabla T = 0} = - \frac{\Lmm}{T} \nabla_T \mu_1 &= - D_{12} \nabla x_1\\
        \frac{\Lmm}{T} \der{\mu_1}{x_1}_{T,p} \nabla x_1 &= D_{12} \nabla x_1\\
        \Lmm &= \frac{T D_{12}}{\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(1 - x_1) S_{T,1}\\
        &= T^2 D_{12} x_1(1 - x_1) S_{T,1}.
    \end{split}
\end{equation}

Finally, the coefficient $\Lqq$ can be related to the conductivity, $k$, 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) = - k \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 Eq. \ref{eq:Lqq_k_relate_1} yields
\begin{equation}
    \begin{split}
        -\frac{\Lqq}{T^2} \nabla T + \frac{\Lqm^2}{\Lmm T^2} \nabla T &= -k \nabla T\\
        \Lqq &= T^2 k + \frac{\Lmq^2}{\Lmm}\\
        \Lqq &= T^2 k + \left(T \der{\mu_1}{x_1}_{T,p} x_1 (1 - x_1) S_{T,1}\right)^2 \Lmm\\
        \Lqq &= T^2 \left(k + \left(x_1 (1 - x_1) S_{T,1}\right)^2 \der{\mu_1}{x_1}_{T,p} T D_{12}\right)
    \end{split}
\end{equation}