Mechanics of sub-isostatic fiber networks at a finite temperature

Fibrous networks are a ubiquitous component of physiological systems, e.g., the interconnected collagen protein in the extracellular matrix (ECM). These networks are responsible for the mechanical stability of cells and tissues. It has been shown that a simple coarse-grained model of spring networks with bending interactions can capture the rheology of real biopolymers. Maxwell showed that the athermal spring networks with an average connectivity below a threshold (isostatic point) are unstable under small deformations. The experimental studies confirm that the average connectivity of real biopolymers is far below the isostatic connectivity. Under a finite applied shear strain, these sub-isostatic networks undergo a transition from a floppy to a rigid state at a critical strain that depends on the connectivity and geometry of the network structure. In the linear regime, on the other hand, sub-isostatic networks at finite temperature exhibit a non-zero shear modulus that, in contrast to entropic elasticity, has an anomalous dependence on temperature. Using a Monte Carlo method, we study this temperature-dependence elasticity and the corresponding critical exponents in sub-isostatic networks near their critical strain in both 2D and 3D models. Interestingly, our results will shed light on the analogy between this floppy-to-rigid phase transition and the zero-temperature criticality in quantum systems.