STATISTICAL PHYSICS OF POLYMER GELS

This work presents a comprehensive analysis of the statistical mechani cs of randomly cross-linked polymer gels, starting from a microscopic model of a network made of instantaneously cross-linked Gaussian chain s with excluded volume, and ending with the derivation of explicit exp ressions for the thermodynamic functions and for the density correlati on functions which can be tested by experiments. Using replica field t heory we calculate the mean held density in replica space and show tha t this solution contains statistical information about the behavior of individual chains in the network. The average monomer positions chang e affinely with macroscopic deformation and fluctuations about these p ositions are limited to length scales of the order of the mesh size. W e prove that a given gel has a unique state of microscopic equilibrium which depends on the temperature, the solvent, the average monomer de nsity and the imposed deformation. This state is characterized by the set of the average positions of all the monomers or, equivalently, by a unique inhomogeneous monomer density profile. Gels are thus the only known example of equilibrium solids with no long-range order. We calc ulate the RPA density correlation functions that describe the statisti cal properties of small deviations from the average density, due to bo th static spatial heterogeneities (which characterize the inhomogeneou s equilibrium state) and thermal fluctuations (about this equilibrium) . We explain how the deformation-induced anisotropy of the inhomogeneo us equilibrium density profile is revealed by small angle neutron scat tering and light scattering experiments, through the observation of th e butterfly effect. We show that all the statistical information about the structure of polymer networks is contained in two parameters whos e values are determined by the conditions of synthesis: the density of cross-links and the heterogeneity parameter. We find that the structu re of instantaneously cross-linked gels becomes increasingly inhomogen eous with the approach to the cross-link saturation threshold at which the heterogeneity parameter diverges. Analytical expressions for the correlators of deformed gels are derived in both the long wavelength a nd the short wavelength limits and an exact expression for the total s tatic structure factor, valid for arbitrary wavelengths, is obtained f or gels in the state of preparation. We adapt the RPA results to gels permeated by free labelled chains and to gels in good solvents (in the latter case, excluded volume effects are taken into account exactly) and make predictions which can be directly tested by scattering and th ermodynamic experiments. Finally, we discuss the limitations and the p ossible extensions of our work.