Mathematical models for phase transitions in biogels

It has been shown that reversible and irreversible phase transitions of biogels can be represented by epidemic models. The irreversible chemical sol–gel transitions are modeled by the Susceptible-Exposed-Infected-Removed (SEIR) or Susceptible-Infected-Removed (SIR) epidemic systems whereas reversible physical gels are modeled by a modification of the Susceptible-Infected-Susceptible (SIS) system. Measured sol–gel and gel–sol transition data have been fitted to the solutions of the epidemic models, either by solving the differential equations directly (SIR and SEIR models) or by nonlinear regression (SIS model). The gel point is represented as the “critical point of sigmoid,” defined as the limit point of the locations of the extreme values of its derivatives. Then, the parameters of the sigmoidal curve representing the gelation process are used to predict the gel point and its relative position with respect to the transition point, that is, the maximum of the first derivative with respect to time. For chemical gels, the gel point is always located before the maximum of the first derivative and moves backward in time as the strength of the activation increases. For physical gels, the critical point for the sol–gel transition occurs before the maximum of the first derivative with respect to time, that is, it is located at the right of this maximum with respect to temperature. For gel–sol transitions, the critical point is close to the transition point; the critical point occurs after the maximum of the first derivative for low concentrations whereas the critical point occurs after the maximum of the first derivative for higher concentrations.