VOLUME 71, NUMBER 14 PHYSICAL REVIEW LETTERS 4 OCTOBER 1993 Amorphous Solid State of Vulcanized Macromolecules: A Variational Approach Paul M. Goldbart and Annette Zippelius* Department of Physics and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 (Received 26 May 1993) We present a microscopic theory of the transition to and properties of the amorphous solid state of a system of vulcanized (i.e., randomly crosslinked) macromolecules. Our approach invokes a variational hypothesis for the random static density fluctuations characterizing this solid state. The variational parameter is the inverse monomer localization length, which is zero in the liquid state and increases continuously as the mean number of crosslinks exceeds a critical value. The emergent solid is a homogeneous isotropic elastic medium, whose elastic moduli we compute near the transition. PACS numbers: 64.70.Dv, 61.41.4-e, 82.35.+t Introduction.—Vulcanization, i.e., the introduction of a sufficient number of permanent crosslinks at random into a melt or solution of macromolecules, causes the equilibrium state of the system to undergo a thermody- namic phase transition from a liquid state to an amor- phous solid state—the so-called rubbery state. In the liq- uid state, the macromolecules will, given sufficient time, wander throughout the container, and the system will respond viscously to an external static shear stress. By contrast, in the solid state at least a finite fraction of the macromolecules will spontaneously become localized in space, their monomers retaining forever a statistical association with a certain mean location. Because of the random nature of the crosslinks, this solid will be an amorphous one. It will respond rigidly to an external static shear stress by undergoing a static shear deforma- tion. Typical instantaneous microscopic configurations would not distinguish this amorphous solid state from the liquid state, although the correlations found in long temporal sequences of configurations would. The purpose of this Letter is to develop a statistical mechanical theory of the equilibrium state of a system of randomly crosslinked macromolecules. We focus, in particular, on the so-called vulcanization transition, and on the elastic properties of the resulting rubbery state. The theory addresses both solutions of macromolecules, for which collapse of the network is prevented by the presence of a good solvent, and melts. Our investigation is based on the model of randomly crosslinked macromolecules pioneered by Edwards and co-workers [1,2]. We adopt the spirit of Ref. [3], and construct a Ginzburg-Landau free energy in terms of the local static density fluctuations, which result from the random localization of the macromolecules and emerge at the transition to mark the onset of solidification. At the heart of the present approach is a variational hypoth- esis for the local static density fluctuations: we assume that in the solid state the monomers are localized at ran- dom spatial positions about which they exhibit Gaussian fluctuations characterized by a single length scale £ that is finite, whereas in the liquid state £ is infinite. The free energy is computed as a function of £, and the phys- ical value of £ is selected so as to make the free energy stationary. The principal results of this Letter are as follows. First, we find a continuous transition from the liquid state, in which £ -1 = 0, to the solid state, in which £~ 1 grows continuously from zero, as the mean num- ber of crosslinks [M] is increased beyond a critical value M c . For 0 < ([M] - M c )/M c < 1 we find ^ ~ 1 10 [[M]/M c — l) . Second, we construct the free-energy cost associated with static elastic distortions of the equi- librium state, and show that the liquid is a compressible fluid, with zero shear modulus, whereas the solid is a homogeneous isotropic elastic medium, compressible and resistant to shear. We find that the shear modulus S be- haves as S ~ £~~ 4 ~ ([M]/M c ~ l) near the transition, whereas the bulk modulus acquires a singular contribu- tion equal to 2/d times the shear modulus (in d dimen- sions) . Model and order parameter.—We consider a system of N identical macromolecules of arclength L and persis- tence length I, moving in a d-dimensional (hyper)-cubic volume V. The macromolecules are labeled by the index i = 1,..., TV, and the location in space of the monomer an arclength distance s from a certain end of chain i is given by the (d-dimensional) vector c^(s) (with 0 < s < L). In the absence of interactions the rms end-to-end distance of a single macromolecule is y/lL. We consider the ther- modynamic limit: N, V —> oo with N/V and VIZ fixed and finite. To ease the notation, we introduce dimension- less vectors, volumes, and arc lengths: c —> c/^L/d) 1 / 2 , V —• V/(lL/d) d / 2 , and s —• s/L. Also, we adopt units of energy such that ksT = 1. We model the system using the Edwards Hamiltonian [4-6] which, in the units we have adopted, is 2 hJ<> I ds I + r T Y, f ds f ds f 6(c i ( 8 )-c i ,{s')), (1) 2 iti , =1 Jo Jo where A 2 characterizes the effect of the (repulsive) ex- cluded-volume interaction between monomers and 6(c) is the d-dimensional Dirac delta function. We suppose that permanent crosslinks are introduced between a ran- 2256 0031-9007/93/71(14)/2256(4)$06.00 © 1993 The American Physical Society