Dynamics at the sol-gel transition

Static pair correlation functions are not very sensitive to the establishment of long range connections. This implies that dynamic parameters, such as the diffusion coefficient co p, ate not very sensitive to the establishment of a weak gel phase. 

A more interesting mechanical parameter is the viscosity of the reaction bath, when the fraction p of connected bonds increases and becomes very close to the critical value Pg. Clearly the viscosity will diverge when = Pg - p — 0—i.e., when we arrive at very large clusters. 

In Chapter V we related one mechanical parameter of the gel phase (the elastic modulus E) to certain electrical problems on a percolation network. A related analogy can be established for the viscosity of the sol phase. Here, we assume that the bonds (with probability p) correspond to superconducting links. The voltages X at both ends of such a link must be equal Xi = Xj. On the other hand, when the pair if) is not linked (probability 1 - p), we assume that there is a capacitance Cg between (() and O ) This corresponds to a current 

As explained in Chapter V, the mechanical viewpoint amounts to taking Xi as the displacement of monomer (0 and as the force due to (0 and acting on 0)- Eq. (Vn.40) gives a force proportional to the relative velocity and correctly describes viscous effects between clusters of all sizes (for very small, point-like clusters, eq. (VII.40) does lead to the Navier-Stokes function in a fluid of monomers). 

Returning to the electrical problem, when we increase p toward the threshold value pg, we expect that the macroscopic dielectric constant, or the polarizability (a) of the system will tend to diverge with a certain exponent s 

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