Effects of the growth kinetics on solute diffusion in porous films

For the development of porous materials with improved transport properties, a key missing ingredient is to determine the relations between growth kinetics, structure, and transport parameters. Here, we address these relations by studying solute diffusion through three-dimensional porous films produced by simple deposition models with controlled thickness and porosity. We simulate growing films with competitive aggregation rules that incorporate lateral aggregation (relative rate proportional to $p$), which leads to pore formation, and surface relaxation ($1-p$) that favors compaction. By connecting a solute source at the basis and a drain at the top outer surface of the films, we extract the effective diffusion coefficients from steady-state simulations. We find that for a given film thickness, the larger the $p$, the larger the effective porosity and diffusivity, but the smaller the tortuosity. Keeping constant growth conditions (i.e., same $p$), the increase of thickness always leads to an increase of effective porosity, but a nontrivial behavior in the diffusivity occurs: for $p\leq 0.7$, the diffusion coefficient is larger in thicker films; this is accompanied by a decrease of the tortuosity with the thickness, thus indicating that growth continuously improves pore structure for diffusion. Microscopically, such a result is associated with narrower distributions of local solute currents at higher points of films. Particularly for $p\geq 0.9$, in which film porosity is $\sim 0.65-0.7$, the tortuosity is between $1.3$ and $2$, increases with the thickness, and has maximal changes near $25\%$. Pairs of values of porosity and tortuosity that we obtain here, for the thickest films, agree with those experimentally measured in some porous electrodes. Noteworthy, the increase of film thickness is generally favorable for diffusion in their pores, and exceptions have small losses in tortuosity.