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Low-connectivity polymer networks were recently found to behave auxeticallywhen subjected to small tensions, that is, their Poisson's ratio $\nu$ becomesnegative. In addition, for specific state points, numerical simulationsrevealed that diamond-like networks reach the limit of mechanical stability,exhibiting values of $\nu = -1$, a condition that we define hyper-auxeticity.This behavior is interesting per se for its consequences in material science,but also appealing for fundamental physics because the mechanical instabilityis accompanied by evidences of criticality. In this work, we deepen ourunderstanding of this phenomenon by performing a large set of equilibrium andstress-strain simulations in combination with a phenomenological elasticitytheory. The two approaches are found to be in good agreement, confirming theabove results. We also extend our investigations to disordered polymer networksand find that the hyper-auxetic behavior also holds in this case, stillmanifesting a similar critical-like behavior as in the diamond one. Finally, wehighlight the role of the number density, that is found to be a relevantcontrol parameter determining the elastic properties of the disordered system.The validity of the results in disordered conditions paves the way to anexperimental investigation of this phenomenon in real systems, such ashydrogels.

Auxetic polymer networks: The role of crosslinking, density and disorder