Polymer chains, dynamics ideal

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

The study of dynamics of a real polymer chain of finite length and containing some conformational defects represents a very difficult task. Due to the lack of symmetry and selection mles, the number of vibrational modes is enormous. In this case, instead of calculating the frequency of each mode, it is more convenient to determine the density of vibrational modes, that is, the number of frequencies that occur in a given spectral interval. The density diagram matches, apart from an intensity factor, the experimental spectmm. Conformational defects can produce resonance frequencies when the proper frequency of the defect is resonating with those of the perfect lattice (the ideal chain), or quasi-localized frequencies when the vibrational mode of the defect cannot be transmitted by the lattice. The number and distribution of the defects may be such... 

Attempts have been made to identify primitive motions from measurements of mechanical and dielectric relaxation (89) and to model the short time end of the relaxation spectrum (90). Methods have been developed recently for calculating the complete dynamical behavior of chains with idealized local structure (91,92). An apparent internal chain viscosity has been observed at high frequencies in dilute polymer solutions which is proportional to solvent viscosity (93) and which presumably appears when the external driving frequency is comparable to the frequency of the primitive rotations (94,95). The beginnings of an analysis of dynamics in the rotational isomeric model have been made (96). However, no general solution applicable for all frequency ranges has been found for chains with realistic local structure. 

Multiarm star polymers have recently emerged as ideal model polymer-colloids, with properties interpolating between those of polymers and hard spheres [62-64]. They are representatives of a large class of soft colloids encompassing grafted particles and block copolymer micelles. Star polymers consist of f polymer chains attached to a solid core, which plays the role of a topological constraint (Fig. Ic). When fire functionality f is large, stars are virtually spherical objects, and for f = oo the hard sphere limit is recovered. A considerable literature describes the synthesis, structure, and dynamics of star polymers both in melt and in solution (for a review see [2]). 

The proposed basis for the nature of the ideal elasticity exhibited by the family of protein-based polymers using the generic sequence (GXGXP) , where X is a variable L-amino acid residue, became very controversial. The adherents to the classic (random chain network) theory of rubber elasticity took great exception to oiu proposal that the damping of internal chain dynamics on extension gave rise to entro-pic elasticity (for more extensive treatment of this controversy, see Urry and Parker. ). The physical basis for the different (even heretical, to some) mechanism of near-ideal elasticity provides insight into the remarkable biocompatibility of elastic protein-based polymers. 

So far we have paid attention mostly to dilute solutions, c < c, in which polymer chains are more or less separated from each other. Chapter 2 focused on thermodynamics, and Chapter 3 focused on dynamics. These solutions were mostly ideal. We also learned how the concentration c might change the thermodynamics and dynamics, as represented by the osmotic pressure and the diffusion coefficient, from those in the ideally dilute solutions. 

Abstract This introductory chapter provides a brief (textbook-like) survey of important facts concerning the conformational and dynamic behavior of polymer chains in dilute solutions. The effect of polymer-solvent interactions on the behavior of polymer solutions is reviewed. The physical meanings of the terms good, 9-, and poor thermodynamic quality of the solvent are discussed in detail. Basic assumptions of the Kuhn model, which describes the conformational behavior of ideal flexible chains, are outlined first. Then, the correction terms due to finite bond angles and excluded volume of structural units are introduced, and their role is discussed. Special attention is paid to the conformational behavior of polyelectrolytes. The pearl necklace model, which predicts the cascade of conformational transitions of quenched polymer chains (i.e., of those with fixed position of charges on the chain) in solvents with deteriorating solvent quality, is described and discussed in detail. The incomplete (up-to-date) knowledge of the behavior of annealed (i.e., weak) polyelectrolytes and some characteristics of semiflexible chains are addressed at the end of the chapter. 

In this section a series of experimental NMR studies based on the techniques described above will be compared with predictions of the model theories. Of course, any model is based on idealizations approaching reality only under certain conditions. The objective of the first few paragraphs will therefore be to demonstrate the rich variety of phenomena that can influence polymer dynamics. It will be elucidated under what circumstances the essence of the model theories and their predicted limits comes to light. We will first consider three basic features of polymer dynamics, namely the three dynamic components governing molecular motions of polymer chains, the dynamics of chain-end blocks in contrast to the central segment block, and how free volume and voids influence dynamics and the appearance of NMR measur-ands. 

In dilute solutions in a theta solvent, flexible polymer chains are in an isolated, ideal random coil state. For description of the dynamics in such solutions, Zimm model subdivides the chain into subchains (cf Figure 2), represents the friction and elasticity of the subchain by a bead (friction center of the subchain) and the entropic springs conneaing the beads, respectively, and analyzes the motion of this bead-spring chain in the presence of hydrodynamic and thermal... 

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