Gaussian chains scaling properties

The above expressions provide a universal description of the dynamics of a Gaussian chain and are valid for real linear polymer chains on intermediate length scales. The specific (chemical) properties of a polymer enter only in terms of two parameters N =Rl and The friction parameter is gov-... 

A polymer is a collection of molecules, called monomers, which interact with each other to form a long flexible chain. For example, a typical polymer like polyethylene consists of a chain of roughly 10 CH2 molecules. The large number of monomers allows for a statistical description of macroscopic properties which are independent of details on the monomer scale. The simplest mathematical model of a polymer chain is referred to as the Gaussian chain [6]. In this model the polymer is described by a position d-dimensional... 

Such a chain is called the Gaussian chain. The Gaussian chain does not describe correctly the local structure of the polymer, but does correctly describe the property on large length-scale. The advantage of the Gaussian chain as a model is that it is mathematically much easier to handle than any other of the models considered in Section 2.1. 

Neutron-scattering studies on polystyrene solutions have shown that while short-length scale correlations are perturbed, in the same manner as single chains in cood solvent, the large-length scale properties are indeed Gaussian. Daoud predicted that the mean squared end-to-end distance of the chains is swollen relative to a pure. melt but can be described simply by a swollen characteristic ratio Coo(( )... 

Assume that we are dealing with polymers in good solvents and in the semidilute solution. If r is a scale to measure, then the chain entanglement shows the following properties. At r >, that is, outside the blob, the repulsive interactions between monomers are screened out by other chains in the solution so that the whole chain is composed of blobs connected in an ordinary random walk without excluded volume effect. Overall, the chain follows Gaussian statistics. At r <, that is, within the blob, the chain does not interact with other chains, but there is a strong excluded volume effect. 

An attractive virtue of PRISM theory is the ability to derive analytic solutions for many problems if the most idealized Gaussian thread chain model of polymer structure is adopted. The relation between the analytic results and numerical PRISM predictions for more chemically realistic models provides considerable insight into the question of what aspects of molecular structure are important for particular bulk properties and phenomena. Moreover, it is at the Gaussian thread level that connections between liquid-state theory and scaling and field-theoretic approaches are most naturally established. Thus, throughout the chapter analytic thread PRISM results are presented and discussed in conjunction with the corresponding numerical studies of more realistic polymer models. 

The possibility of a rescaling of the representative freely jointed chain, as formulated by these equations, expresses an important basic property of ideal polymer chains, namely their self-similarity . Self-similarity here means that independent of the chosen length scale, i.e. the resolution, an ideal chain always exhibits the same internal structure, one for which all internal distance vectors are distributed like Gaussian variables. A change of the length scale leaves this structure s characteristics invariant. 

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