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 molecular 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, \(\chi \), encoding properties of the solvent. In particular, a good solvent promoting the mixing regime, is found below the threshold value \(\chi =0.5\), whereas the phase separating regime develops above that critical value. We show that cavities 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 boundary, 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 functions, 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 community, providing an experimental and analytic study in support of the phantom elastic energy versus the affine one.

本文研究凝胶的平衡奇异构型，并解决有关凝胶能量学的开放性问题。我们将凝胶建模为固体聚合物和溶剂的不可压缩、不混溶和饱和混合物，在分子水平上维持化学相互作用。我们假设凝胶的能量由其聚合物网络的弹性能量加上混合的 Flory-Huggins 能量组成。后者涉及各个组分的熵能加上聚合物和溶剂之间相互作用的熵能，以及与温度相关的 Flory 参数\(\chi \)，编码溶剂的特性。特别是，促进混合机制的良好溶剂被发现低于阈值\(\chi =0.5\)，而相分离制度发展高于该临界值。我们表明空腔和奇点在后一种情况下发展。我们发现了两大类奇点：(i) 溶剂变干，水可能通过边界离开凝胶域，在凝胶中心留下暴露的聚合物核心；(ii) 空化，响应于边界上的牵引力或某种形式的负压，空腔可以是空的或被溶剂淹没。处理所有此类奇点的直接而统一的数学方法是基于适当的测试函数的构造，其灵感来自于均匀膨胀或压缩的特定状态。文章的最后一个主题解决了研究界的一个根植于统计力学的争议，