Molecular theory of strain hardening of a polymer gel: Application to gelatin

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Abstract
The elasticity of gelatin gels at large deformation has been measured for various experimental conditions. The general pattern is that stress increases with strain in a nonlinear way up to the point where the gel fails. To interpret this nonlinear stress increase, we studied a number of molecular models by Monte Carlo simulation and by mean-field methods. The effect of finite polymer length is studied via the FENE model (finite extensible nonlinear polymer connections) and via the exact statistics of Kramers’ model (chains of freely rotating stiff rods) for a small number of elements per chain. To investigate the effect of fractal connections, the end-point distribution that comes forward from scaling theory has been generalized to arbitrary fractal dimension. Finally we studied a heterogeneous network model: connections formed by rods and coils. We also discuss the consequence of microphase separation. Combining experiment and theory we conclude the following: (i) The elastically active network connections in gelatin are most certainly not Gaussian. (ii) Strain hardening in gelatin can be attributed to either: (a) finite polymer length (the chain length between connection points should be some 2.5 times the persistence length), or (b) a fractal structure of the polymer strands (the fractal dimension should be roughly df=1.3–1.5), or (c) the presence of both stiff rods and flexible coils (the length of the rods should be 1.4–4.4 times the radius of gyration of the coils). (iii) Models b and c describe the experimental data significantly better than model a. From a single parameter (the fractal dimension) the fractal model correctly describes (1) the nonlinearity of the stress–strain curve, (2) the scaling of Young’s modulus with polymer concentration, (3) the scaling of neutron scattering intensity with wave number, and (4) it predicts the scaling exponent of the linear dynamic modulus with frequency in the glassy transition zone (no experimental data available). The experimental parameters found for the rod+coils model suggest a Rouse diffusion controlled growth mechanism for the rods. Although the theory presented here is applied to gelatin, its formulation is quite general, and its implications are also relevant for other strain hardening polymer gels.

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