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This article studies equilibrium singular configurations of gels and addresses open questions concerning gel energetics. We model a gel as an incompressible, immiscible and saturated mixture of a solid polymer and a solvent that sustain chemical interactions at the molecu- lar level. We assume that the energy of the gel consists of the elastic energy of its polymer network plus the Flory-Huggins energy of mixing. The latter involves the entropic energies of the individual components plus that of interaction between polymer and solvent, with the temperature dependent Flory parameter, χ , encoding properties of the solvent. In particular, a good solvent promoting the mixing regime, is found below the threshold value χ = 0.5, whereas the phase separating regime develops above that critical value. We show that cavi- ties and singularities develop in the latter regime. We find two main classes of singularities: (i) drying out of the solvent, with water possibly exiting the gel domain through the bound- ary, leaving behind a core of exposed polymer at the centre of the gel; (ii) cavitation, in response to traction on the boundary or some form of negative pressure, with a cavity that can be either void or flooded by the solvent. The straightforward and unified mathematical approach to treat all such singularities is based on the construction of appropriate test func- tions, inspired by the particular states of uniform swelling or compression. The last topic of the article addresses a statistical mechanics rooted controversy in the research commu- nity, providing an experimental and analytic study in support of the phantom elastic energy versus the affine one. 

In preparation for a more thorough study based on our own experimental work of the debonding of a thin film gel by stress concentration on the interface with a rigid substrate, in this article we revisit, from the viewpoint of the synergy between mathematics, exper- iments, and finite element simulations, the problem of the swelling of a thin rectangular polyacrylamide gel covalently bonded on the bottom surface to a glass slide. With meth- ods of the calculus of variations and perturbation theory we show that the solution to the corresponding zero-displacement boundary value problem converges, in the thin film limit, to a uniquely defined uniform uniaxial extension on the direction normal to the substrate. Both the experiments and the finite element simulations that we perform confirm that the amount of lateral swelling is very small, with a very good quantitative agreement between the two approaches. The proposed model of minimizing an energy functional comprising both a term for the elastic distortion and the Flory-Huggins expression for the entropy of mixing is thus experimentally and numerically validated, with parameters coming from ex- perimental measurements, including the initial polymer volume fraction of the hydrogel synthesized in the laboratory (which is taken as the reference configuration instead of the dry polymer).