Onset of criticality in hyper-auxetic polymer networks

Abstract

Against common sense, auxetic materials expand when stretched or contract when compressed by uniaxial strain, being characterized by a negative Poisson's ratio ν. The amount of perpendicular deformation in response to the applied force can be at most equal to the imposed one, so that ν = −1 is the lowest bound for the mechanical stability of solids, a condition here defined as "hyper-auxeticity".In this work, we numerically show that ultra-low-crosslinked polymer networks under tension display hyper-auxetic behavior at a finite crosslinker concentration. At this point, the nearby mechanical instability triggers the onset of a critical-like transition between two states of different densities. This phenomenon displays similar features as well as important differences with respect to gas-liquid phase separation. Since our model is able to faithfully describe real-world hydrogels, the present results can be readily tested in laboratory experiments, paving the way to explore this unconventional phase behavior.The mechanical response of a material subjected to uniaxial strain in the direction orthogonal to the deformation is quantified via the Poisson's ratio ν, defined as the negative ratio between transverse and longitudinal deformation.For the most common three-dimensional materials ν is positive, so that these expand (contract) in response to a compressive (extensional) strain. This situation is schematically illustrated in Fig. 1(a). On the contrary, auxetic materials are characterized by negative values of ν, meaning that they become thicker perpendicularly to the deformation axis, as shown in Fig. 1(b). Auxetic behaviour has been so far reported in a large variety of systems, including foams, polymers, fibers, tendons and crystals [1][2][3][4][5][6][7]. Recently, a strong research interest has been devoted towards auxetic metamaterials in which the elastic properties can be tailored by geometrical design [8][9][10] or by pruning methods [11].Besides geometrical reasons, a negative ν can also be obtained by exploiting critical behavior and phase transitions, as in the case of ferroelastic materials in the vicinity of the Curie point [12,13].Within linear elasticity theory [14], the appearance of a negative ν can be related to a decrease of the bulk modulus K with respect to the shear modulus G, namely to an isotropic softening of the material. A vanishing K echoes the divergence of the isothermal compressibility occurring at a gas-liquid critical point. However, the presence of a finite shear modulus, as found in polymer networks such as hydrogels, may induce a negative ν. Pioneering evidence of a negative Poisson's ratio has been reported for these systems close to the so-called volume phase transition [15][16][17]: in this case, a variation in temperature changes the affinity of the polymer to the solvent, favouring monomer-monomer aggregation, in full analogy with the gas-liquid critical point, but with the additional constraint of infinite connectivity.Another thermodynamic parameter that influences the netw...