Rigidity Transitions in Flexible Polymer Networks

Granular media composed of hard paritcles undergo a rigidity transition with increasing volume fraction. We show that networks of flexible polymers in good solvent described by the Edward's Hamiltonian undergo first-order rigidity transitions as the excluded volume interaction is increased. At the transition point, the network acquires a finite shear modulus, and the density of the network becomes anisotropic. This transition is dominated by the eigenvalue spectrum of the Laplacian matrix of the network.