Molecular theory of the yield behavior of a polymer gel: Application to gelatin

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Abstract
A microscopic model for the endpoint separation dependent scission rate of a polymer connection in a network is developed. The predicted dissociation rate is proportional to the exponential of the bond force, which is in line with experiments. This atomistic description is thereupon incorporated in a mesoscopic theory to describe strain hardening and failure of physical gels. The resulting theory has been analyzed by a new numerical algorithm, which is some 100 to 1000 times faster than the algorithm described in the literature. We applied this theory to gelatin. To arrive at the correct nonlinear rheologic behavior of the gel, the non-Gaussian nature of the polymer endpoint distribution has to be taken into account. There are four important physical quantities that describe the nonlinear rheology of gelatin. For relatively small shear strain (0<γ<1), stress increases nonlinearly with strain when a gel is deformed (strain hardening). The strain at which the gel ruptures (the yield strain γy) increases quite slowly with shear rate: γy∝γ̇0.05. When experiments are carried out at different shear rate, we find a linear correlation between yield stress and yield strain. Finally it is observed that the yield strain decreases with increasing strain hardening. These four observations are all covered by the theory up to quantitative accuracy. The interpretation that comes forward from this work is that the nonlinearity of the stress–strain curve for γ<1 is correlated to the strain at which the gel yields. The reason for this correlation is that both effects are dominated by a non-Hookean force–distance relationship of the polymer connections. On the one hand side, this function directly causes the upturn of the stress–strain curve. On the other hand, the rate by which polymer connections break is proportional to the exponential of this force. Therefore a nonlinear force–distance relationship leads both to a nonlinear stress–strain relation and to an early and sudden yield behavior.