Thermodynamic properties, renormalization

The correlation length is also found to scale in a power-law fashion, and it becomes very large at the transition temperature. One of the most significant results of renormalization group theory is to show that the behavior of the correlation length in the critical region is the basis of the power-law singularities observed in the other thermodynamic properties. 

Wertheim has also introduced a second renormalization or 2-R approximation, but this development does not directly enter the computation of e, although it is relevant to Wertheim has shown that for key thermodynamic properties, the 2-R results differ only slightly from those given by the 1-R theory.)... 

This book offers a detailed account of the renormalization group theory. The book contains an excellent review of many experimental data on the thermodynamic properties of polymer solutions. 

In recent years, studies of solutions of polymer blends and of copolymers have aroused a substantial theoretical and experimental interest. This is motivated by both numerous applications and more fundamental issues concerning the usefulness of the scaling and universality concepts to describe the thermodynamic properties and the phase transitions in these systems. In this lecture, chain interactions in dilute and semidilute solutions are reviewed and it is discussed how and when the interactions between chemically different monomers lead to a macroscopic phase separation in the case of ternary polymer A-polymer B- solvent systems and to a mesophase formation in diblock-copolymer solutions. The important conclusion is that due to both the overall monomer concentration fluctuations (excluded volume effects) and the composition fluctuations, the classical Flory theory often fails. This requires the use of the renormalization method and of scaling concepts to give a correct description of the phase diagrams and the critical phenomena observed in these complex systems. We give only here a brief outline, a complete review has been published elsewhere, ... 

The quasiharmonic approximation studied in Sects.5.2,3 gives reasonable results for the thermodynamic properties of crystals in which the anhar-monicity is weak and the force constants are renormalized by thermal expansion only. In crystals with very strong enharmonic interactions, however, this approximation breaks down. Examples are the vibrations in rare-gas solids, in particular solid helium, soft modes in ferro-electric phase transitions and melting processes. For these cases a method has been developed, the self-consistent harmonic approximation (SCHA), which allows a qualitative description of the effects of strong anharmonicity. 

Llovell, F. P mies, J.C. Vega, L.F. (2004). Thermodynamic properties of Lennard-Jones chain molecules Renormalization-group corrections to a modified statistical associating fluid theory. /. Chem, Phys, 121,10715-10724. 

In this framework, optimizing the division scheme of the potential seems to allow us to approach such uniqueness [65]. By searching through the function space, one hopes to find a unique representation B = B(j ) after renormalization. Such endeavors are beyond the scope of research in the field of SCIETs. The logic is as follows If the closure is exact, it must necessary satisfy all known thermodynamic and structural conditions. The reverse question, the sufficiency condition, requires more care. If such a functionality provides, a posteriori, accurate results for all properties, though not proof of unique functionality, it is favorably disposed to it. 

Freire and coworkers [285,286] studied the case of miktoarm star copolymers of the type AxBf x, where f is the total functionality of the star copolymer. The conformational characteristics of these kinds of molecules were investigated as a function of molecular weight and number of the different branches, as well as the thermodynamic cross interactions between the arms and the solvent medium. Calculations based on the renormalization group and Monte Carlo methods allowed the estimation of the dimensions of each arm and of the whole molecule and the mean square distance between the two centers of mass of the different homopolymers. From these estimations different expansion factors relative to the homopolymer precursors could be calculated (Fig. 2). Different degrees of agreement were obtained by the two methods depending on the property under consideration. 

---