  Transport of liquid mixtures through porous membranes is central to processes
such as desalination, chemical separations and energy harvesting, with
ultrathin membranes made from novel 2D nanomaterials showing exceptional
promise. Here we derive, for the first time, general equations for the solution
and solute fluxes through a circular pore in an ultrathin planar membrane
induced by a solute concentration gradient. We show that the equations
accurately capture the fluid fluxes measured in finite-element numerical
simulations for weak solute-membrane interactions. We also derive scaling laws
for these fluxes as a function of the pore size and the strength and range of
solute-membrane interactions. These scaling relationships differ markedly from
those for concentration-gradient-driven flow through a long cylindrical pore or
for flow induced by a pressure gradient or electric field through a pore in an
ultrathin membrane. These results have broad implications for transport of
liquid mixtures through membranes with a thickness on the order of the
characteristic pore size.
