Capillaries in alginate gel as an example of dissipative structure formation

Capillaries of uniform diameter (8-300 mu m) are formed in a gel growing during diffusion of Me(2+) metal ions into a sol of Na alginate. The alginate chains are cross-linked by the metal ions. The transition from sol to gel is limited by diffusion and occurs in a propagating front. Our investigations provide evidence that the dissipative process of structure formation in this front is caused by friction between the contracting alginate chains and the surrounding solution, leading to a pattern of hydrodynamic flow similar to Rayleigh-Benard convection. This flow pattern is mapped on the growing gel and leads to a structure of parallel capillaries filled with aqueous solution. We present a reduced mathematical model describing the onset of a spontaneous structure formation. The model includes the hydrodynamics of the system formulated by a Navier-Stokes equation. In this equation the frictional force between the contracting alginate chains and the solution plays the role of an external force. A second basic equation describes the binding of alginate molecules to the gel front. The model shows that pattern formation occurs above a critical value of the contraction velocity. This critical value and the spacing of the resulting pattern depend on a number of parameters such as viscosity, chain density, thickness of the contraction zone, friction coefficient between contracting chains and surrounding solution, diffusion constant of alginate and concentration of the sol.