Sergei Panyukov

Statistical physics of polymer gels

Abstract This work presents a comprehensive analysis of the statistical mechanics of randomly cross-linked polymer gels, starting from a microscopic model of a network made of instantaneously cross-linked Gaussian chains with excluded volume, and ending with the derivation of explicit expressions for the thermodynamic functions and for the density correlation functions which can be tested by experiments. Using replica field theory we calculate the mean field density in replica space and show that this solution contains statistical information about the behavior of individual chains in the network. The average monomer positions change affinely with macroscopic deformation and fluctuations about these positions are limited to length scales of the order of the mesh size. We prove that a given gel has a unique state of microscopic equilibrium which depends on the temperature, the solvent, the average monomer density 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 calculate the RPA density correlation functions that describe the statistical properties of small deviations from the average density, due to both static spatial heterogeneities (which characterize the inhomogeneous equilibrium state) and thermal fluctuations (about this equilibrium). We explain how the deformation-induced anisotropy of the inhomogeneous equilibrium density profile is revealed by small angle neutron scattering and light scattering experiments, through the observation of the butterfly effect. We show that all the statistical information about the structure of polymer networks is contained in two parameters whose values are determined by the conditions of synthesis: the density of cross-links and the heterogeneity parameter. We find that the structure of instantaneously cross-linked gels becomes increasingly inhomogeneous 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 and the short wavelength limits and an exact expression for the total static structure factor, valid for arbitrary wavelengths, is obtained for 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 thermodynamic experiments. Finally, we discuss the limitations and the possible extensions of our work.

Small angle neutron scattering studies on structural inhomogeneities in polymer gels: irradiation cross-linked gels vs chemically cross-linked gels

A comparison of network structure in a solvent was made for two types of poly(N-isopropylacrylamide) gels cross-linked by chemical reaction with N,N 0 -methylenebisacrylamide (BIS) (chemical gels) and by g-ray irradiation (g-ray gels). The cross-linking density dependence for these gels was examined by small angle neutron scattering (SANS). The SANS results indicated an increase of frozen inhomogeneities with an introduction of cross-links for both chemical and g-ray gels. However, it was found that the effect of cross-linking is much stronger in the chemical gels than in the g-ray gels. The differences in the structure were successfully interpreted by a statisticalmechanical theory of gels proposed by Panyukov– Rabin (Phys. Rep. 269 (1996) 1). The degree of polymerization between cross-links, N, was a decreasing function of cross-linking content for both types of gels, while that for the g-ray gels was a weak function of irradiation dose. Quantitative analyses on BIS concentration and g-ray dose dependence led to an experimental evidence of the existence of cross-linking saturation threshold. q 2002 Elsevier Science Ltd. All rights reserved.

Phase diagram of microphase-separated multiblock copolymers

The behavior of a random AB multiblock copolymer melt with Markovian sequence of monomers has been studied. We demonstrate that, in contrast to ordinary disordered solids, the quenched randomness does not destroy the long-range periodic order: In the Brazovskii approximation it is shown that thermodynamic fluctuations in the critical point give rise to an incommensurate lamellar structure whose period is greater than the average size of a block. This structure appears as a result of the first-order phase transition from the homogeneous phase, and these two phases coexist with one another in the finite temperature range. We calculate the phase diagram of such a multiblock copolymer near the critical point and show that, depending on the average block size, it can include the regions with different symmetry of the superlattice (lamellar, hexagonal and bcc) and the regions of coexistence of the corresponding microphase-separated phases. The parameters of these phases are calculated both in the mean field approximation and taking into account the fluctuation effects. We demonstrate that the period of the superstructures varies continuously with the temperature variation.

Microphase separation in multiblock copolymers

The phase diagram of a multiblock copolymer containing domains of coexistence of phases having different superlattice symmetries with a density of monomers of a given type that varies periodically in space. The parameters of such superlattices are calculated in the mean-field approximation, and it is shown that their wave vector varies continuously with the temperature.

The effect of thermodynamic fluctuations on the formation of superstructures in random heteropolymers

The behavior of a random AB block copolymer with the same fraction of monomers of each type near the critical point has been studied. In the Brazovskii approximation it is shown that thermodynamic fluctuations give rise to a lamellar structure whose period is greater than the size of a block. The structure appears as a result of a first-order phase transition. The parameters of this transition are calculated and the region of coexistence of the disordered and lamellar phases is found.

Microstructure and phase diagrams of polymer gels

Recently we developed [Panyukov and Rabin, Macromolecules 29 (1996) 7960] a phenomenological theory of randomly cross-linked polymer networks, based on the separation of solid-like and liquid-like degrees of freedom and taking into account the frozen inhomogeneity of network structure. We calculated the scattering spectra of weakly charged, randomly cross-linked polymer gels in good, poor and in Θ solvents [Panyukov and Rabin, Macromolecules 29 (1996) 8530; Rabin and Panyukov, Macromolecules 30 (1996) 301]. For some values of the thermodynamic parameters, the competition between poor solubility, electrostatics and network elasticity leads to the divergence of the structure factor at a wave vector q∗, signaling the onset of microphase separation in the gel. Depending on the choice of thermodynamic parameters, the characteristic wavelength 1/q∗ varies from microscopic to macroscopic length scales.

Recent developments in the theory of polymer gels

SummaryWe review the results of a recently developed theory of randomly cross-linked polymer gels. The theory is based on the exact statistical mechanical solution of the Edwards model which takes into account both the frozen disorder of network structure and excluded-volume effects. Predictions are made for the behavior of individual network chains, the density correlation functions which are directly measured by neutron and light-scattering experiments and for the thermodynamics of gels. We now have a complete statistical description of polymer gels in good solvents, ranging from monomer length scales to the continuum limit.

Buckling of spontaneously twisted ribbons

We study the effect of spontaneous twist on the buckling instability of thin rectilinear elastic filaments with non-circular cross-section (ribbons). We find that the presence of twist in the stress-free reference state increases, albeit non-monotonically, the critical load at which buckling takes place. Stability with respect to buckling increases with the asymmetry of the cross-section and saturates at twice the Euler value for an untwisted beam. The classical Euler result is recovered, independent of twist, for filaments with circularly symmetric cross-section.