Statistical Mechanics of Polymers

From about 1910 onwards, physical chemists began studying the characteristics of polymer solutions, measuring such properties as osmotic pressure, and found them 

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Edwards, S. F. Statistical mechanics of polymers with excluded volume. Proc. Phys. Soc. (London) 85, 613 (1965). 

Kuchanov, S., New results in weak segregation theory of block copolymers, in "Statistical Mechanics of Polymers New Developments" (K. Binder, A. R. Khokhlov, V. A. Ivanov, S. I. Kuchanov Eds.), Macromolecular Symposia Vol. 252, pp. 76-89 (2007)... 

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

Vologodskii, A.V., Lukashin, A.V., Frank-Kamenetskii, M.D., Anshelevich, V.V., 1974. The knot problem in statistical mechanics of polymer chains. Sov. Phys. JETP 39 1059-1063. 

The techniques of functional integration are naturally introduced into the study of the statistical mechanics of polymer systems because they are ideally suited to the description of simple generic models of flexible and stiff ideal polymer chains. Thus we discuss ideal polymer chains to introduce the simple generic models which then naturally lead to the use of functional integrals. 

The formulation of the foundations of the statistical mechanics of polymers in bulk is just in its infancy. A number of questions of principle still require careful attention. This review can then only be of a state-of-the-art nature, and only a few very simple applications can be considered. In the next section, a simple model is considered which serves as a zeroth-order model for a system of polymers in bulk. This model, which employs the Wiener integral formulation for flexible chains, has great pedagogical value since it introduces some of the formidable problems to be encountered in any statistical mechanical description of polymers in bulk. This model then naturally leads to a discussion of the nature of statistical mechanics for systems with internal constraints (Section IX). 

This review of the main theoretical concepts in the statistical mechanics of polymers is far from being complete. We had to omit a number of important topics like polymer-magnetic analogy, topological properties of polymers, helix-coil transitions and other secondary chain stmctures, the effects of chirality on polymer stmctures, polymer adsorption, and polymers in two dimensions. 

B. Smit, S. Karaborni, and J. Siepmann, Macromol.Symp. 81, 343 (1994) (paper presented at the First International Conference on the Statistical Mechanics of Polymer Systems, Theory and Simulations, Mainz, Germany Oct 4 6, 1993). 

Polymerization, gelation and vulcanization have been discovered as new subjects in statistical mechanics. They have been embedded in the theory of critical phenomena and statistical mechanics of polymers in general (see, for example, refs. 3,4 and 5), and many new discoveries have been made. The new developments in theoretical physics of these subjects initiated by the classical... 

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