Correlated Dilution of Filamentous Networks Leads to Reentrant Rigidity Percolation

Networks of stiff, cross-linked polymers are widespread in biological tissues. A remarkable phenomenon exhibited by such tissues is an abrupt increase in elastic moduli as a result of a comparatively small increase in polymer density. Biopolymer network models commonly consist of an isotropic and homogeneous arrangement of bonds with harmonic stretching and bending stiffness, with a certain fraction of bonds removed at random. Whereas dilution of bonds has customarily been spatially uniform, many real tissues, such as articular cartilage, exhibit alternating dense and sparse regions. To capture spatially varying density, we modify the dilution of bonds to introduce structural correlation, so that a bond is more likely to be retained if neighboring bonds have already been retained, following recent work on colloidal gels. With greater structural correlation, we initially find a network reaches a non-zero shear modulus with fewer bonds, but that for high correlation, rigidity demands even more material be retained than in the case of spatially uniform dilution. We again find a peak in the measure of non-affinity of network displacement to coincide with rigidity percolation, but observe that this peak becomes higher and broader with increasing correlation.