Order parameters, mesoscopic polymer

Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallization proceeds by the formation of spherical clusters, the process can be characterized by a coordinate y, which may represent for instance the number of monomers in a cluster, its radius or even a global-order parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. 

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... 

The formation of liquid-crystalline phases by covalent rigid, wormlike, and segmented chains has been extensively described [50]. Anisotropy and orientation are characterized at the molecular level by the order parameter and at the mesoscopic level by director orientation. In the case of supramolecular polymers orientation and growth may occur according to the following mechanisms ... 

Orientation induced by elongational flow field has been shown to promote the formation of nematic order at concentration below the critical value in the absence of flow [94]. For covalent systems, the effect of local ordering (described by the order parameters) was shown to be not as dramatic as that occurring over a mesoscopic scale due to director orientation. It is known that application of flow, electrical, and magnetic fields may lead to perfectly ordered single LCs. Theoretical analysis for supramolecular polymers has so far been restricted to the relatively simpler case of helices in magnetic and electrical fields [46]. For polarizable helices in a quadruple field, and for helices with a permanent dipole in a dipolar field, growth enhancement was predicted as more pronounced... 

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. 

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