Hans-Helmut Kohler

Capillaries in alginate gel as an example of dissipative structure formation

Capillaries of uniform diameter (8–300 μm) 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.

Theory of capillary formation in alginate gels

The formation of capillaries in alginate gel is a dissipative process coupled with hydrodynamic flow in the immediate neighborhood of the front of gel formation. The hydrodynamic flow is due to the contraction of the alginate chains resulting from the crosslinking reaction. As shown earlier, capillary formation only occurs above a critical value of the contraction velocity. In this paper a complete theoretical model of the ratio of the actual to the critical contraction velocity is presented. The theoretical predictions are compared with experimental data for the formation of copper alginate gel from solutions of sodium alginate and copper dichloride. The ratio of the actual to the critical contraction velocity is described as a function of the bulk concentrations, the diffusion coefficients, the properties of the alginate molecules and the rate constant of copper alginate complex formation. For capillary formation to occur, the rate constant must be neither too small nor too large. In agreement with experimental data the model predicts that capillary formation is restricted to a finite time interval and will not take place if the copper concentration is too low or the alginate concentration is too high.

A modified Poisson–Boltzmann equation: I. Basic relations

Abstract The fundamental electrostatic and thermodynamic equations governing the local balance approach for the description of a charged interface in solution are revised. Special attention is given to a detailed thermodynamic analysis of space charge regions being subject to an electric field. The equilibrium conditions of the components of the system are derived, without approximations, in terms of their (electro-)chemical potentials. Combined with Poissons equation they yield a fundamental set of self-consistent local balance differential equations as a general basis for further detailed modelling computations. Comparison with literature shows that work done in the field of the Poisson–Boltzmann approach is often based on incorrect or oversimplified equations.

Coupling of diffusion and reaction in the process of capillary formation in alginate gel

Abstract The formation of capillaries in a gel obtained by cross-linking alginate chains with divalent metal ions is caused by a microscopic hydrodynamic circular flow in the immediate neighborhood of the gel formation front. This flow is due to the contraction of the alginate chains accompanying the cross-linking reaction. In an earlier paper we have shown that capillary formation will occur above a critical value of the contraction velocity. The critical contraction velocity is determined by a number of parameters, such as the coefficient of friction between the contracting chains and the surrounding water, the viscosity of the fluid and the coefficient of convective transport. We show here that the latter depends on the gradient of the alginate concentration profile in front of the contraction zone. To establish a quantitative theory, the diffusional transport of the alginate chains and the divalent metal ions towards the contraction zone is investigated. The diffusive mass transport is described theoretically by a diffusion-reaction-model that is confirmed by the experimental data obtained for the alginate concentration profile in a long tube. As a final result the coefficient of convective transport is represented as a function of the bulk concentrations and the effective diffusion constants of the alginate and the metal ions.

A singlet reference interation site model theory for solid/liquid interfaces Part II: Electrical double layers.

The previously established singlet reference interaction site model (SRISM) theory for the calculation of the fluid structure in the vicinity of a plane impenetrable interface is renormalized for the application to electrical double layers. In combination with the HNC and KH closures, the equations are solved numerically for a 1 M electrolyte solution adjacent to a charged wall with varying surface charge densities. The wall-solvent and wall-ion density distributions as well as the profiles of the electrical field and the electrical potential are compared to computer simulation results. Reasonable agreement is obtained.

A modified Poisson–Boltzmann equation: II. Models and solutions

Abstract The properties of the charged interface between a dielectric particle and a surrounding aqueous electrolyte solution are calculated numerically over a wide range of surface charge densities for plane, cylindrical and spherical geometries. As a basis for the calculations, we present detailed models for the partial molar volumes, the dielectric permittivity and the activities of the components. These models are combined with a generalized set of local balance thermodynamic and electrostatic differential equations derived in the first part of this series. The influences of volume effects, dielectric saturation, polarization and self-atmosphere potentials on surface potential and electrostatic energy of a charged particle are investigated. Deviations from the ordinary Poisson–Boltzmann theory become very important at surface charge densities above 0.2 C m −2 . Quite generally, self-atmosphere potentials are of minor importance. The most important correction of the ordinary Poisson–Boltzmann equation is due to dielectric saturation in combination with the volume effect. It is found that the electrostatic potential, the electric field and the concentration of the counterion near a charged surface strongly depend on the excluded volume of the counterion. This leads to a distinct counterion sensitivity of the Gibbs energy of the system. Assuming a positively charged surface, competition between the counterion pairs Cl−/Br− and Cl−/SO42− is investigated. For sufficiently high surface charge densities it is found that, in the immediate vicinity of the surface, the smaller Br−-ion displaces the larger Cl−-ion and the Cl−-ion, in turn, displaces the larger SO42−-ion, although the latter is divalent.

Orientation of chain molecules in ionotropic gels: a Brownian dynamics model

As is known from birefringence measurements, polysaccharide molecules of ionotropic gels are preferentially orientated normal to the direction of gel growth. In this paper the orientation effect is investigated by means of an off-lattice Brownian dynamics model simulating the gel formation process. The model describes the integration of a single coarse grained phantom chain into the growing gel. The equations of motion of the chain are derived. The computer simulations show that, during the process of integration, the chain is contracting normal to the direction of gel growth. A scaling relation is obtained for the degree of contraction as a function of the length parameters of the chain, the velocity of the gel formation front and the rate constant of the crosslinking reaction. It is shown that the scaling relation, if applied to the example of ionotropic copper alginate gel, leads to reasonable predictions of the time course of the degree of contraction of the alginate chains.