  In order to model ion transport across protocell membranes in Hadean
hydrothermal vents, we consider both theoretically and experimentally the
planar growth of a precipitate membrane formed at the interface between two
parallel fluid streams in a two-dimensional microfluidic reactor. The growth
rate of the precipitate is found to be proportional to the square root of time,
which is characteristic of diffusive transport. However, the dependence of the
growth rate on the concentrations of hydroxide and metal ions is approximately
linear and quadratic, respectively. We show that such a difference in ionic
transport dynamics arises from the enhanced transport of metal ions across a
thin gel layer present at the surface of the precipitate. The fluctuations in
transverse velocity in this wavy porous gel layer allow an enhanced transport
of the cation, so that the effective diffusivity is about an order of magnitude
higher than that expected from molecular diffusion alone. Our theoretical
predictions are in excellent agreement with our laboratory measurements of the
growth of a manganese hydroxide membrane in a microfluidic channel, and this
enhanced transport is thought to have been needed to account for the
bioenergetics of the first single-celled organisms.
