Stability and anomalous entropic elasticity of subisostatic random-bond networks

Generic Elasticity of Thermal, Underconstrained Systems

Athermal (i.e., zero-temperature) underconstrained systems are typically floppy, but they can be rigidified by the application of external strain, which is theoretically well understood. Here and in the companion paper [C. T. Lee and M. Merkel, Phys. Rev. E 110, 064147 (2024)PRESCM2470-004510.1103/PhysRevE.110.064147], we extend this theory to finite temperatures for a very broad class of underconstrained systems. In the vicinity of the athermal transition point, we derive from first principles expressions for elastic properties such as isotropic tension t and shear modulus G on temperature T, isotropic strain ϵ, and shear strain γ, which we confirm numerically. These expressions contain only three parameters: entropic rigidity κ_{S}, energetic rigidity κ_{E}, and a parameter b_{ϵ} describing the interaction between isotropic and shear strain, which can be determined from the microstructure of the system. Our results imply that in underconstrained systems, entropic and energetic rigidity interact like two springs in series. This also allows for a simple explanation of the previously numerically observed scaling relation t∼G∼T^{1/2} at ϵ=γ=0. Our work unifies the physics of systems as diverse as polymer fibers and networks, membranes, and vertex models for biological tissues.

Mechanical criticality of fiber networks at a finite temperature

At zero temperature, spring networks with connectivity below Maxwell's isostatic threshold undergo a mechanical phase transition from a floppy state at small strains to a rigid state for applied shear strain above a critical strain threshold. Disordered networks in the floppy mechanical regime can be stabilized by entropic effects at finite temperature. We develop a scaling theory for this mechanical phase transition at finite temperature, yielding relationships between various scaling exponents. Using Monte Carlo simulations, we verify these scaling relations and identify anomalous entropic elasticity with sublinear T dependence in the linear elastic regime. While our results are consistent with prior studies of phase behavior near the isostatic point, the present work also makes predictions relevant to the broad class of disordered thermal semiflexible polymer networks for which the connectivity generally lies far below the isostatic threshold.

Thermalized connectivity networks of jammed packings

Jammed packings of repulsive elastic spheres have emerged as a rich model system within which the elastic properties of disordered glassy materials may be elucidated. Most of the work on these packings has focused on the case of vanishing temperature. Here, we explore the elastic properties of the associated connectivity network for finite temperatures, ignoring the breaking of bonds and the formation of new ones. Using extensive Monte Carlo simulations, we find that, as the temperature is increased, the resulting spring network shrinks and exhibits a rapidly softening bulk modulus via a cusp. Moreover, the shear modulus stiffens in a fixed volume ensemble but not in a fixed pressure ensemble. These counter-intuitive behaviors may be understood from the characteristic spectrum of soft modes near isostaticity, which resembles the spectrum of a rod near its buckling instability. Our results suggest a generic mechanism for negative thermal expansion coefficients in marginal solids. We discuss some consequences of bond breaking and an apparent analogy between thermalization and shear.

Critical behaviour in the nonlinear elastic response of hydrogels

In this paper we study the elastic response of synthetic hydrogels to an applied shear stress. The hydrogels studied here have previously been shown to mimic the behaviour of biopolymer networks when they are sufficiently far above the gel point. We show that near the gel point they exhibit an elastic response that is consistent with the predicted critical behaviour of networks near or below the isostatic point of marginal stability. This point separates rigid and floppy states, distinguished by the presence or absence of finite linear elastic moduli. Recent theoretical work has also focused on the response of such networks to finite or large deformations, both near and below the isostatic point. Despite this interest, experimental evidence for the existence of criticality in such networks has been lacking. Using computer simulations, we identify critical signatures in the mechanical response of sub-isostatic networks as a function of applied shear stress. We also present experimental evidence consistent with these predictions. Furthermore, our results show the existence of two distinct critical regimes, one of which arises from the nonlinear stretch response of semi-flexible polymers.

Viscoelastic properties of marginal networks in a solvent

Polymer networks at the margins of mechanical stability are known to be highly sensitive to applied forces and fields and to exhibit an anomalously large resistance to deformation. In this paper, we study the effects of hydrodynamic interactions on the behavior of marginal networks using a hybrid molecular dynamics and multiparticle collision dynamics simulation technique. We examine how the filament and solvent properties affect the response of marginal networks to shear. We find that the stiffening of the network shows a stronger dependence on the shear frequency when hydrodynamic interactions are present than when they are not. The network shear modulus scales as G'∼ω(α(c)), with a critical stiffening exponent α(c) that can be controlled by varying the relative concentrations of the network and the solvent. Our results show that this arises due to the solvent aiding the relaxation of the network and suppressing the network nonaffinity, with the system deforming more affinely when hydrodynamic interactions are maximized.