Gaussian connecting chains

Ideal or Perfect Networks The lUPAC Commission on macromolecular nomenclature defines a perfect network as a network composed of chains all of which are connected at both of their ends to different junction points [7]. If a perfect network is in the rubbery state, then, on macroscopic deformation of the network, all of its chains are elastically active and display rubber elasticity. An ideal or perfect network can also be defined as a collection of individual Gaussian elastic chains (linear... 

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. 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

The chains of typical networks are of sufficient length and flexibility to justify representation of the distribution of their end-to-end lengths by the most tractable of all distribution functions, the Gaussian. This facet of the problem being so summarily dealt with, the burden of rubber elasticity theory centers on the connections between the end-to-end lengths of the chains comprising the network and the macroscopic strain. 

The calculation of g for Gaussian uniform star chains was carried out by Zimm and Kilb (ZK) [83]. They used a modified version of the dynamic Rouse theory including preaveraged HI (in the non-draining limit) that considers the particular connectivity of units consistently with the star architecture. This ap-... 

Fig. 3.1 Bead-spring-bead model of a Gaussian chain as assumed in tbe Rouse model. Tbe beads are connected by entropic springs and are subject to a frictional force where v is the bead velocity and fo the bead friction coefficient...

It is seen that

characteristic behavior suggests that the molecular shape of PBLG in the mixed solvent studied does not differ very much from swollen spheres of randomly coiled polymers at stages where the helical fraction is less than about 0.6. In this connection, it is worth recalling from Chapter C, Section 2.b that the dimensional features of a polypeptide remain close to Gaussian at such stages of helix-coil transition, provided the chain is sufficiently long. 

The equations for copolymers are a little more complicated but can be reduced to similar expressions, as will be shown later in this chapter. Moreover, if Gaussian statistics is obeyed for the subchains connecting two chain elements j and k, we have... 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

Most synthetic polymers in which the monomer units are connected via single bonds have rather flexible chains. The bond torsion energy is relatively small and the units can rotate around their bonds [14,30,31]. Each molecule can adopt a large number of energetically equivalent conformations and the resulting molecular geometry is that of a statistical coil, approximately described by a Gaussian density distribution. This coil conformation is the characteristic secondary structure of macromolecules in solution and in the melt. It is entropically favoured because of its... 

From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. 

The first of these, proposed by Martyna, Tuckerman, and Klein (MTK), was based on the notion that the variable py, itself, has a canonical (Gaussian) distribution exp(- 3p /Q). However, there is nothing in the equations of motion to control its fluctuations. MTK proposed that the Nose-Hoover thermostat should, itself, be connected to a thermostat, and that this thermostat should also be connected to a thermostat. The result is that a chain of thermostats is introduced whereby each element of the chain controls the fluctuations of the element just preceding it. The equations of motion for such a thermostat chain are ... 

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