Coarse-grained models for diffusion in oil-filled hydrogel microbeads

Diffusion of digestive enzymes in oil-filled hydrogel microbeads is a highly complex process which is difficult to model, particularly for systems with high volume fractions of incorporated nano-droplets. In this paper coarse-grained models for this process are compared. The results show that the interplay between adsorption at the oil-water interface and diffusion through the matrix of the bead can lead to a front-like motion of the enzyme. This motion can be described by combining the mass balance for the enzyme with a Maxwell-Cattaneo type equation for the mass flux vector. Solutions of the resulting partial differential equation show that when τ<<td (where τ and td are characteristic times for adsorption and diffusion) the time evolution of the enzyme concentration is identical to the profile calculated using Fick's law. For τ>>td and time t≤τ the enzyme migrates through the hydrogel as a sharp front. The position of the front changes linearly with time, and this corresponds well with findings of a recent experimental study (van Leusden et al. (2018), Food Hydrocolloids, 85, 242–247). The effects of poly-dispersity of the interior oil droplet phase were described using a multi-mode generalization of the Maxwell-Cattaneo model, and the results show that a widening of the droplet size distribution leads to smoothing of the front. The results show that this level of coarse-grained modelling can capture the dynamics of these complex systems quite accurately.