Elasticity near the vulcanization transition

Signatures of the vulcanization transition—amorphous solidification induced by the random cross-linking of macromolecules—include the random localization of a fraction of the particles and the emergence of a nonzero static shear modulus. A semimicroscopic statistical-mechanical theory is presented of the latter signature, which accounts for both thermal fluctuations and quenched disorder. It is found (i) that the shear modulus grows continuously from zero at the transition, and does so with the classical exponent, i.e., with the third power of the excess cross-link density and, quite surprisingly, (ii) that near the transition the external stresses do not spoil the spherical symmetry of the localization clouds of the particles.