Semidilute polymer solutions scaling theory

The first theories that implemented a proper balance of intramolecular interactions and conformational elasticity of the branches were developed by Daoud and Cotton [21] and by Zhulina and Birshtein [22-24]. These theories use scaling concepts (the blob model), originally developed by de Gennes and Alexander to describe the structure of semidilute polymer solutions [64] and planar polymer brushes [65, 66]. Here, the monomer-monomer interactions were incorporated on the level of binary or ternary contacts (corresponding to good and theta-solvent conditions, respectively), and both dilute and semidilute solutions of star polymers were considered. Depending on the solvent quality and the intrinsic stiffness of the arms, the branches of a star could be locally swollen, or exhibit Gaussian statistics [22-24]. 

Finally, analytic predictions for the osmotic pressure of polymers in good and theta solvents can be derived based on the Gaussian thread model, PRISM theory, and the compressibility route. The qualitative form of the prediction for large N is " pP °c (po- ), which scales as p for theta solvents and p " for good solvents. Remarkably, these power laws are in complete agreement with the predictions of scaling and field-theoretic approaches and also agree with experimental measurements in semidilute polymer solutions. ""... 

This review is intended to complement those of Cohen Stuart et al. (1986) and de Gennes (1987). The former details experimental techniques available for probing polymer-particle interactions and the lattice, i.e., mean field, theories that predict, via numerical solutions, segment-density profiles and interaction potentials. The latter constructs a simple and elegant picture of the same phenomena through scaling theories developed for semidilute solutions. 

Beyond the overlap concentration threshold, c>c = pN/lP, star polymers form a semidilute solution. Because of the fact that the arms in a star are stretched, the scaling theory [24] predicts that the properties of semidilute solutions of star polymers are distinctively different from those of linear polymers. When the polymer concentration c > c, a semidilute solution is envisioned as a system of closely packed and virtually non-interpenetrating (segregated) polymer stars. A further increase in polymer concentration leads to a progressive contraction of the coronae of the individual stars. This contraction results in an increase in the conformational entropy of the partially stretched star arms. 

Renormalization group theory (see, e.g., [35]) lies at the heart of this theory, justifying the use of scaling laws in the asymptotic limit, i.e., for infinitely long polymer chains and for dilute solutions. For semidilute solutions, however, this criterion is not so crucial because the polymer chains are overlapping and many properties, e.g., osmotic pressure, are independent of the chain length. 

The dynamics of polymer chains in semidilute solutions has been one of the central topics in polymer physics for more than ten years. Inspired by the scaling concepts advanced so eloquently by de Gennes and subsequent refinement of theory based on more rigorous calculations, many experiments have been reported. Unfortunately, several inconsistent conclusions have been drawn from many such experiments. It seems that the dynamic behavior of polymer semidilute solutions requires further investigation, at least from the experimental viewpoint. 

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, ... 

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