Imagine a long, thin tube with one end sealed and the other end open. It is filled with a volatile solvent with a dissolved non-volatile solute. As the solvent evaporates at the liquid surface, the solvent level falls and the concentration of solution there will increase. This sets up a concentration gradient, leading to diffusion according to Fick's Laws. In this case, convection generated from the concentration gradient can be ignored as the solution is considered to be dilute.
I have found a paper one this topic. In equation 6, the solute flux $J$ at position $z$ and time $t$ is stated, though for a slightly different system than the one I described.
$$J\left(z,t\right)=-v\left(t\right)\phi\left(z,t\right)-\frac{D\left(\phi\right)}{RT}\phi\frac{\partial\mu}{\partial z}$$
where $D$ is the diffusivity, which is a function of concentration $\phi$. $\mu$ represents chemical potential of the solvent and $v$ is the flow velocity of the medium at $z$.
There are two main parts where I am confused.
One, the system that is modelled in the paper is different to the one I am interested in. In the paper, one end of the tube is open for fresh solution to flow in to keep the volume constant, whereas I am interested in the case where the tube is sealed on one end, causing the solvent level to slowly fall. In that case, how should the above equation be modified?
Two, I do not understand what role $\mu$ is playing in this equation and whether it is applicable to the system I described above.
I apologise in advance if there are already answers to my question elsewhere as I am new to the field of transport phenomena and thermodynamics.
