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We present a pedagogical review of the swelling thermodynamics and phase transitions of polymer gels. In particular, we discuss how features of the volume phase transition of the gel's osmotic equilibrium is analogous to other transitions described by mean-field models of binary mixtures, and the failure of this analogy at the critical point due to shear rigidity. We then consider the phase transition at fixed volume, a relatively unexplored paradigm for polymer gels that results in a phase-separated equilibrium consisting of coexisting solvent-rich and solvent-poor regions of gel. Again, the gel's shear rigidity is found to have a profound effect on the phase transition, here resulting in macroscopic shape change at constant volume of the sample, exemplified by the tunable buckling of toroidal samples of polymer gel. By drawing analogies with extreme mechanics, where large shape changes are achieved via mechanical instabilities, we formulate the notion of extreme thermodynamics, where large shape changes are achieved via thermodynamic instabilities, i.e. phase transitions., 40 pages, 22 figures, Submitted to Nanotechnology

In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state. In this random solid state, particles are permanently but randomly localized in space, and a rigidity to shear deformations emerges. Owing to the permanence of the random constraints, this phase transition is an equilibrium transition, which confers on it a simplicity (at least relative to the conventional glass transition) in the sense that it is amenable to established techniques of equilibrium statistical mechanics. In this Paper I shall review recent developments in the theory of random solidification for systems obeying permanent random constraints, with the aim of bringing to the fore the similarities and differences between such systems and those exhibiting the conventional glass transition. I shall also report new results, obtained in collaboration with Weiqun Peng, on equilibrium correlations and susceptibilities that signal the approach of the random solidification transition, discussing the physical interpretation and values of these quantities both at the Gaussian level of approximation and, via a renormalization-group approach, beyond., Comment: Paper presented at the "Unifying Concepts in Glass Physics" workshop, International Centre for Theoretical Physics, Trieste, Italy (September 15-18, 1999)

We generalize the statistical mechanical theory of vulcanization to the case of D-dimensional polymerized manifolds. Starting from a continuum model of self-avoiding manifolds, we study the effects of introducing random crosslinks between monomers on (in general) different manifolds. As for the case of linear polymers, one observes a continuous phase transition from a fluid to an amorphous solid state, characterized by a finite fraction of localized monomers. We compute this fraction, as well as the typical localization length near the transition.

When a sufficient density of permanent random crosslinks is incorporated into a system of macromolecules, the system undergoes a continuous equilibrium phase transition from a liquid to an amorphous solid state. In this solid state, a certain fraction of monomers are entirely delocalised. The remaining fraction (i.e. the gel-fraction) are localised about random mean positions, and have random r.m.s. displacements (i.e. localisation lengths). A microscopic mean-field theory of this so-called vulcanisation transition is presented, in which the gel-fraction and statistical distribution of localisation lengths are determined self-consistently. A scaling form for the distribution of localisation lengths, valid for all near-critical crosslink densities, is obtained, and it is found that both the fraction of localised monomers and the typical inverse localisation length vanish continuously at the transition.

We present a statistical-mechanical analysis of a system of randomly cross-linked macromolecules, focusing on the emergence of random static-density fluctuations as the solidification transition is approached. In mean-field theory the problem can be formulated in terms of a single macromolecule in an effective random potential. The distribution of the random potential, which must be calculated self-consistently, determines all moments of the static-density fluctuations. We find that the solidification transition occurs when an infinite network of cross-linked macromolecules is formed, and that a sequence of nonlinear density response functions diverges simultaneously at this transition.

Normal-metal films overlaid with arrays of superconducting islands undergo Berezinskii-Kosterlitz-Thouless (BKT) superconducting transitions at a temperature $T_{BKT}$. We present measurements of T$_{BKT}$ for arrays of mesoscopic Nb islands patterned on Au films for a range of island spacings $d$. We show that $T_{BKT} \sim 1/d^2$, and explain this dependence in terms of the quasiclassical prediction that the Thouless energy, rather than the superconducting gap, governs the inter-island coupling. We also find two deviations from the quasiclassical theory: (i) $T_{BKT}$ is sensitive to island height, because the islands are mesoscopic; and (ii) for widely spaced islands the transition appears to lead, not to a superconducting state, but to a finite-resistance "metallic" one.

Classical vector spins with quenched random Dzyaloshinskii-Moriya interactions and uniaxial anisotropy are discussed using replica mean-field theory. Spin-glass order is predicted below a critical temperature and the Parisi replica symmetry-breaking scheme is found to yield a stable solution with the properties of conventional spin glasses.