Renormalization-group approach to the vulcanization transition

The vulcanization transition-the cross-link-density-controlled equilibrium phase transition from the liquid to the amorphous solid state-is explored a nalytically from a renormalization-group perspective. The analysis centers on a minimal model which has previously been shown to yield a rich and info rmative picture of vulcanized matter at the mean-held level, including a co nnection with mean-field percolation theory (i.e., random graph theory). Th is minimal model accounts for both the thermal motion of the constituents a nd the quenched random constraints imposed on their motion by the cross-lin ks, as well as particle-particle repulsion which suppresses density fluctua tions and plays a pivotal role in determining the symmetry structure land h ence properties) of the model. A correlation function involving fluctuation s of the amorphous solid order parameter, the behavior of which signals the vulcanization transition, is examined, its physical meaning is elucidated, and the associated susceptibility is constructed and analyzed. A Ginzbug c riterion for the width tin cross-link density) of the critical region is de rived and is found to be consistent with a prediction due to de Gennes. Int er alia, this criterion indicates that the upper critical dimension for the vulcanization transition is 6. Certain universal critical exponents charac terizing the vulcanization transition are computed, to lowest nontrivial or der, within the framework of an expansion around the upper critical dimensi on. This expansion shows that the connection between vulcanization and perc olation extends beyond mean-field theory, surviving the incorporation of fl uctuations in the sense that pairs of physically analogous quantities (one percolation related and one vulcanization related) are found to be governed by identical critical exponents, at least to first order in the departure from the upper critical dimension land presumably beyond). The relationship between the present approach to vulcanized matter and other approaches, su ch as those based on gelation-percolation ideas, is explored in the light o f this connection. To conclude, some expectations for how the vulcanization transition is realized in two dimensions, developed with H. E. Castillo, a re discussed.