A Macromolecule in an Entangled System

2 A Macromolecule in an Entangled System Diffusive Mobility of a Macromolecule 

The mobility of a macromolecule, constrained by other macromolecules, can be also calculated as (5.1). In the linear approximation, the zeroth normal co-ordinates of the macromolecule (equation (4.1), at z/jj = 0) define diffusive mobility of macromolecule. The one-sided Fourier transform velocity correlation function is determined by expression (4.15), so that we can write down the Fourier transform 

Multiplying this expression by tult and integrating with respect to uj from —oo to oo, we find 

We use the formula to write down the general expression for the mean square displacement of a particle in an arbitrary viscoelastic liquid 

Turning to the particular memory functions (3.15), one finds for this simple 

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To describe the behaviour of a macromolecule in an entangled system, we have introduced the ratio of the relaxation times x and two parameters B and E connected with the external and the internal resistance, respectively. These parameters play a fundamental role in the description of the dynamical behaviour of polymer systems, so that it is worthwhile to discuss them once more and to consider their dependencies on the concentration of polymer in the system. 

To describe the behaviour of a macromolecule in an entangled system, the mesoscopic theory has introduced the correlation time r and two parameters B and E connected with the external and the internal resistance, respectively. 

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

Dynamics of a single macromolecule in an entangled system is defined by the system of non-linear equations (3.52)-(3.54), containing some phenomenological parameters, which will be identified later. 

The right-hand side of equation (125) has to be reduced to a divergent form. To transform the second term into the required form, we use the dynamic equation (122) which, in this general form, is valid for a macromolecule both in a viscous liquid and in an entangled system. After summing over all the particles of the macromolecule and averaging, one can write for each macromolecular coil... 

The length of a macromolecule between adjacent entanglements Me is used as an individual characteristic of a polymer system. Table 1 contains values of Me for certain polymer systems. The more complete list of estimates of the quantity Me can be found in work by Aharoni (1983, 1986). One can compare expressions (6.52) and (6.54) for the value of the modulus on the plateau to see that the length of a macromolecule between adjacent entanglements Me is closely connected with one of the parameters of the theory... 

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula... 

The mesoscopic approach gives an amazingly consistent picture of the different relaxation phenomena in very concentrated solutions and melts of linear polymers. It is not surprising the developed theory is a sort of phenomenological (mesoscopic) description, which allows one to get a consistent interpretation of experimental data connected with dynamic behaviour of linear macromolecules in both weakly and strongly entangled polymer systems in terms of a few phenomenological (or better, mesoscopic) parameters it does not require any specific hypotheses. 

With regard to the mechanical reactirai of a polymer network to a stress applied, it is important that loose ends of macromolecules in a network structure are as shmrt as possible and/or their concentration is low. As these ends mostly extend out of the lamellas of crystallites then, while crossUnking is taking place in an amorphous phase and with the simultaneous presence of crystallites, a network with small loose ends should be formed. The crosslink junctions stabilize the natural molecular network (entanglements and crystallites), and every chain in the system is potentially elastically operative and can contribute to the stress in a tensile experiment [33]. The stabilization effect of chemical crosslinks on entanglements and crystallites may be the direct cause of observed differences in the determination of the amount of chemical crosslinks from mechanical property measurements and sol-gel analysis of the cross-linked polymer. 

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