The Mesoscopic Theory of Polymer Dynamics
The theory presented in this book explains in a consistent manner all dynamic effects (diffusion, neutron scattering, viscoelasticity, optical birefringence) observed in very concentrated solutions and melts of linear polymers from a macromolecular point
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Introduction: Macromolecular Systems in Equilibrium
Abstract The general theory of equilibrium and non-equilibrium properties of polymer solutions and melts appears to be derived from the universal models of long macromolecules which can be applied to any flexible macromolecule notwithstanding the nature of its internal chemical structure. Although many universal models are useful in the explanation of the behaviour of the polymeric system, the theory that will be described in this book is based on the coarse-grained model of a flexible macromolecule, the so-called, bead-andspring or subchain model. In the foundation of this model, one finds a simple idea to observe the dynamics of a set of representative points (beads, sites) along the macromolecule instead of observing the dynamics of all the atoms. It has been shown that each point can be considered as a Brownian particle, so the theory of Brownian motion can be applied to the motion of a macromolecule as a set of linear-connected beads. The large-scale or low-frequency properties of macromolecules and macromolecular systems can be universally described by this model, while the results do not depend on the arbitrary number of sites. In this chapter, the bead-and-spring model will be introduced and some properties of simple polymer systems in equilibrium are discussed.
1.1 Microscopic Models of a Macromolecule One says that the microstate of a macromolecule is determined, if a sequence of atoms, the distances between atoms, valence angles, the potentials of interactions and so on are determined. The statistical theory of long chains developed in considerable detail in monographs (Birshtein and Ptitsyn 1966; Flory 1969) defines the equilibrium quantities that characterise a macromolecule in a whole as functions of the macromolecular microparameters. To say nothing about atoms, valence angles and so on, one can notice that the length of a macromolecule is much larger than its breadth, so one can consider the macromolecule as a flexible, uniform, elastic thread with coefficient of elasticity a, which reflects the individual properties of the macromolecule V.N. Pokrovskii, The Mesoscopic Theory of Polymer Dynamics, Springer Series in Chemical Physics 95, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-2231-8 1,
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1 Introduction: Macromolecular Systems in Equilibrium
(Flory 1969; Landau and Lifshitz 1969). Thermal fluctuations of the macromolecule determine the dependence of the mean square end-to-end distance R2 on the length of macromolecule M and temperature T which is, we assume, measured in energy units. If M T a R2 =
2M a . T
(1.1)
The last relation shows that a long macromolecule rolls up into a coil at high temperatures. The smaller the elasticity coefficient a is, the more it coils up. Another name for the model of flexible thread is the model of persistence length or the Kratky-Porod model. The quantity a/T is called the persistence length (Birshtein and Ptitsyn 1966). One can use another way to describe the long macromolecule. One
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