Adding the Effect of Topological Defects to the Flory–Rehner and Bray–Merrill Swelling Theories

The Flory–Rehner and Bray–Merrill swelling theories are venerable theories for calculating the swelling of polymer networks and are widely applied across polymer materials. Here, these theories are revised to include cyclic topological defects present in polymer networks by using a modified phantom network model. These closed-form equations assume defect contributions to the swelling elasticity to be linear and additive and allow different assumptions regarding prestrain of larger loops to be incorporated. To compare to the theories, swelling experiments are performed on end-linked poly(ethylene glycol) gels in which the topological defects (primary and secondary loops) have been previously measured. Gels with higher loop densities exhibit higher swelling ratios. An equation is derived to compare swelling models independent of knowledge of the Flory–Huggins χ parameter, showing that the revised swelling models for loop defects are more accurate than both the phantom network model that neglects loops and the Bray–Merrill equation.