Link between Alginate Reaction Front Propagation and General Reaction Diffusion Theory

Abstract

We provide a common theoretical framework reuniting specific models for the Ca(2+)-alginate system and general reaction diffusion theory along with experimental validation on a microfluidic chip. As a starting point, we use a set of nonlinear, partial differential equations that are traditionally solved numerically: the Mikkelsen-Elgsaeter model. Applying the traveling-wave hypothesis as a major simplification, we obtain an analytical solution. The solution indicates that the fundamental properties of the alginate reaction front are governed by a single dimensionless parameter λ. For small λ values, a large depletion zone accompanies the reaction front. For large λ values, the alginate reacts before having the time to diffuse significantly. We show that the λ parameter is of general importance beyond the alginate model system, as it can be used to classify known solutions for second-order reaction diffusion schemes, along with the novel solution presented here. For experimental validation, we develop a microchip model system, in which the alginate gel formation can be carried out in a highly controlled, essentially 1D environment. The use of a filter barrier enables us to rapidly renew the CaCl(2) solution, while maintaining flow speeds lower than 1 μm/s for the alginate compartment. This allows one to impose an exactly known bulk CaCl(2) concentration and diffusion resistance. This experimental model system, taken together with the theoretical development, enables the determination of the entire set of physicochemical parameters governing the alginate reaction front in a single experiment.

Microdrop Printing of Hydrogel Bioinks into 3D Tissue‐Like Geometries

“…In order to understand the mechanics of gelation and to optimize this alginate system for successful 3D printing, we started with a reaction‐diffusion model19 and applied it to alginate gelation using experimentally validated physicochemical parameters describing the gelling diffusion front as Ca 2+ ions diffuse into ungelled alginate 20, 21. For 50 μm alginate droplets printed onto gelatin substrates loaded with 10 m M Ca 2+ at 22 °C, the model predicts that the droplet/substrate interface is 50% gelled after 1.5 s and that the droplet is 90% gelled after 5 s (Figure 1C), suggesting that alginate gelation occurs on a much longer timescale than spreading.…”

Biopolymer gels with “physical” cross-links: gelation kinetics, aging, heterogeneous dynamics, and macroscopic mechanical properties

“…Such a dip in the scattered intensity may reasonably be attributed to the presence of a thin layer before the advancing gel, with a thickness of the order of a few hundred microns, where the concentration of alginate is sensibly lower than in the bulk. The physical origin of such a "depletion layer", which has apparently been observed in other measurements of alginate gelation, 37,38 is not easy to spot. Tentatively, we suggest that it could stem from a mechanism that is not included in the M-E model: syneresis, namely a progressive shrinking over long times of the structure by solvent ejection, which is a common and extensively studied phenomenon in polymer gels.…”

Depth-controlled laser-induced jet injection for direct three-dimensional liquid delivery

“…2b-d) rather than the gelation of the alginate-based ink in the Ca 2+ -rich gelatin substrate. Analyzing the respective penetration time of the jet and gelation time of the alginate-based ink reveals indeed that the injection occurs over a time shorter by several orders of magnitude, ~ 3 μs, than the diffusion [35] of calcium ions over 6 μm, ~ 30 ms. Similarly, the geometry of micro-injections shown in the imaging cross-sections of Fig.…”