Polymer Gels: Frozen Inhomogeneities and Density Fluctuations Sergei Panyukov † and Yitzhak Rabin* Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Received January 29, 1996; Revised Manuscript Received August 5, 1996 X ABSTRACT: We present 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. The complete solution of the statistical mechanics of this model is given, and the monomer density correlation functions are calculated for neutral gels in good and Θ solvents. The theoretical scattering curves are compared to the results of small angle neutron scattering and light scattering experiments, and new experimental tests of our theory are proposed. 1. Introduction Recently, we presented a comprehensive statistical mechanical analysis of the Edwards model of gels, formed by instantaneous cross-linking of semidilute polymer solutions. 1 The model takes into account both excluded volume interactions between the monomers and the random character of the process of cross-linking but neglects permanent entanglements. 2 We showed that the thermodynamic conditions (qual- ity of solvent, degree of cross-linking, and monomer concentration) under which the network was prepared determine the statistical properties of its disordered structure. The inhomogeneous distribution of cross- links has a characteristic length scale which depends on the conditions of preparation and can vary from microscopic to macroscopic dimensions, depending on whether the gel was prepared away from or close to the “cross-link saturation threshold”. 1 We found that for each choice of thermodynamic conditions, a given net- work has a unique state of microscopic equilibrium in which the average position of each cross-link and of each monomer is uniquely determined by the thermodynamic parameters. When these parameters (temperature, quality of solvent, degree of swelling, forces applied to the boundary of the gel) change, the new balance of elastic and excluded volume forces produces a new state of equilibrium. We calculated the monomer density correlation func- tions (correlators) which can be directly measured in scattering experiments. The total structure factor can be represented as the sum of two terms: the correlator of static inhomogeneities which characterizes the sta- tistical properties of the inhomogeneous equilibrium density profile of the gel, and the correlator of thermal fluctuations about this equilibrium. The presence of static inhomogeneities gives rise to the observed sta- tionary speckle patterns in light scattering from gels. 3 When the gel is stretched, the anisotropy of the inho- mogeneous equilibrium density profile leads to en- hanced scattering in the stretching direction and to the appearance of butterfly patterns in isointensity plots. 4 The conclusion that the butterfly effect arises due to the effect of stretching on static inhomogeneities and cannot be attributed to the distortion of thermal fluctuations agrees with that of previous investigators. 5,6 Although the above theory provides a complete solu- tion of the statistical mechanics of polymer gels, it has several drawbacks, the most important of which is its mathematical complexity. The theory uses replica field theory methods which are unfamiliar to the majority of people in the polymer community. It is desirable to develop a more intuitive approach which would capture all the main physical ingredients of the complete theory and yet would not require the use of advanced methods of mathematical physics. Furthermore, although we have presented the complete formal solution for the density correlators, explicit analytic results were ob- tained only in the long and the short wavelength limits. Thus, we were unable to provide a quantitative descrip- tion of the interesting phenomena associated with the transition from liquid-like to solid-like behavior, which takes place on length scales of the order of the monomer fluctuation radius R (the typical length scale over which a monomer fluctuates about its mean position in the network). A more complete, even if approximate, de- scription is clearly necessary in order to understand the physics of this intermediate regime and to compare our predictions to the results of neutron and light scattering experiments across the entire range of wavelengths, from several angstroms to microns. Finally, since our earlier work 1 was based on a particular model of polymer gels, it was difficult to distinguish between universal results which apply to all types of polymer networks and those which are specific to instanta- neously cross-linked gels. The goal of the present work is to construct a phenomenological theory of density fluctuations and static inhomogeneities in randomly cross-linked polymer gels. In doing so, we are guided by the insights provided by our exact replica field theoretical results, which were not available to previous investigators who attempted to cope with this problem. 5-7 In section 2 we introduce the mean field free energy which governs small deviations from an arbitrarily swollen and stretched reference state of a polymer gel. The excluded volume (osmotic) part of the free energy is given by the standard quadratic expression in the monomer density F. The entropic part is written as the sum of solid-like and liquid-like contributions. 8 The solid-like contribution to the entropy is a functional of the displacement field u and depends on the anisotropic modulus of the deformed network and on the random force in the network (both the modulus and the corr- elator of the random force are calculated in the Ap- pendix). The liquid-like entropy depends on the con- tribution to the density, F liq , which comes from the † Permanent address: Theoretical Department, Lebedev Physics Institute, Russian Academy of Sciences, Moscow 117924, Russia. X Abstract published in Advance ACS Abstracts, October 1, 1996. 7960 Macromolecules 1996, 29, 7960-7975 S0024-9297(96)00164-7 CCC: $12.00 © 1996 American Chemical Society