Phase transitions, mesoscopic polymer

Such long times usually appear when the observed objects are large compared to single atoms, but are still mesoscopic, e.g. polymer molecules or aggregates of smaller molecules, or when dynamic processes at all length scales slow down in the vicinity of phase transitions or due to a glass transition. 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

The bilayer morphology of thin asymmetric films of may be unstable. A regularly corrugated surface structure of the films was ascribed to spinodal transition into a laterally phase separated structure, where the surface morphology depended on the polymer incompatibility and the interfacial interactions [347, 348]. Recently, the phase separation and dewetting of thin films of a weakly incompatible blend of deuterated PS and poly(p-methylstyrene) have been monitored by SFM [349, 350]. Starting from a bilayer structure, after 454 h at T= 154 °C the film came to the final dewetting state where mesoscopic drops of... 

Upon the transition from primary polymer architectures to secondary structural units at the mesoscopic scale interactions of solvent-solvent, solvent-polymer, polymer-polymer types are renormalized into effective interactions between sidechains and aqueous domains, as indicated in Fig. 3. These interactions control proton distribution and mobilities as well as the coupling between proton and water transport. At the mesoscopic level of the theory, the hydrophobic polymer phase formed by the backbones can already be considered as an inert, structureless matrix. 

The case of block copolymers is peculiar and deserves a specific development. In such a structure, the A and B blocks are connected to one another by a covalent bond, and their respective molar mass and composition can be varied independently. Being incompatible, A and B blocks tend to minimize their surface of contact but, contrary to the mere blends of two polymers they cannot phase separate to a macroscopic scale due to the bond which links them. Classical composition-temperature phase diagrams cannot be constructed for block copolymers as for the corresponding blends. Indeed the A and B blocks are forced to self-organize in domains of more reduced nano- or mesoscopic size. The transition from a homogeneous blend to a system composed of ordered phases as well as the size and the morphology of these ordered phases depend on two elements the product Xab " (X = total degree of polymerization) and the dissymmetry in size of the two blocks. 

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