Entangled Macromolecules

we can try to relate the above results to the experimental data on the viscoelasticity of concentrated solutions of polymers. For the systems of long macromolecules, the estimated values of parameter are small. Having used expressions (6.40) for this case, one can evaluate the terminal relaxation time of the system 

In the alternative case of large values of one can use the upper line of equation (6.40) to calculate the terminal relaxation time of the system, which coincides with the given relaxation time in order of magnitude 

The suspension of dilute macromolecular coils in a viscoelastic liquid is suitable for the interpretation of results on the viscoelasticity of concentrated systems with macromolecules, which are not long (M Me). This case was carefully investigated by Leonov (1994). He has confirmed the possibility of a self-consistent description for a system of very short macromolecules. 

Investigation of viscoelastic behaviour of linear polymer solutions and melts shows that there are universal laws for dependencies of the terminal characteristics on the length of macromolecules, which allows to interpret these phenomena on the base of behaviour of a single macromolecule in the system of entangled macromolecules (Ferry 1980, Doi and Edwards 1986). The validity of the mesoscopic approach itself rests essentially on the fundamental experimental fact that quantities that characterise the behaviour of a polymer system have a well-defined unambiguous dependence on the length of the macromolecule. 

The dependence of the characteristics on molecular weight was used for the classification of the systems (Ferry 1980 Graessley 1974 Watanabe 1999). 

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Chompff,A.J., Duiser,J.A. Viscoelasticity in networks consisting of crosslinked or entangled macromolecules. I. Normal modes and mechanical spectra. J. Chem. Phys. 45,1505-1514(1966). 

Entangled macromolecules, connected with weak van der Waals forces, make up a very concentrated polymer solution or a polymer melt. Mesoscopic approach considers the coarse-grained dynamics of a single macromolecule. The surrounding macromolecules are considered a reacting medium. 

The most advanced theories of relaxation phenomena in a system of entangled macromolecules is based on the dynamics of a single macromolecule. Dynamics of the tagged macromolecule is simplified by the assumption that the neighbouring macromolecules can be described as a uniform structureless medium and all important interactions can be reduced to intramolecular interactions. The dynamic equation for a macromolecule can be written as a modification of equation (2.1) for dynamics of macromolecule in viscous liquid... 

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approximation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables - the extra forces. This set of phase variables describes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible. 

There is a great difference between the relaxation behaviour of the system of entangled macromolecules and the relaxation behaviour of a macromolecule in a dilute system. Two relaxation branches have been shown to exist in a system of entangled macromolecules. 

In relaxation processes of the macromolecular coil to equilibrium, the competing mechanisms of mobility of particles are present simultaneously. However, in the region of weakly entangled macromolecules, relaxation occurs due to isotropic mobility of particles of the chain - the diffusive mechanism -... 

To calculate the characteristics of viscoelasticity in the framework of mesoscopic approach, one can start with the system of entangled macromolecules, considered as a dilute suspension of chains with internal viscoelasticity moving in viscoelastic medium, while the elastic and internal viscosity forces, according to equations (3.4)-(3.6) and (3.8), have the form... 

The system of entangled macromolecules becomes anisotropic when velocity gradients are applied, and one can assume that each Brownian particle of the chain moves in the anisotropic medium. The expressions for the discussed quantities (7.5) for case, when one can neglect the hydrodynamic interaction... 

Chompff AJ, Prins W (1968) Viscoelasticity of networks consisting of crosslinked or entangled macromolecules. II Verification of the theory for entanglement networks. J Chem Phys 48(l) 235-243... 

As one can see, the parallel-sided device taken as an example can disappear from vision to the naked eye either because it is broken into tiny pieces or fragments or because it goes into solution. None of these phenomena implies that macromolecules were degraded. The case of erosion where the disappearance of the polymeric matter occurs from the surface is a special mechanism that is often related to enzymatic degradation because enzymes are known for not being able to penetrate entangled macromolecules forming solid plastics. If the disappearance of a solid... 

Polymer viscoelasticity and rheology problems are discussed by Alexei E. Likhtman in Chapter 1.06. The emphasis is put on the molecular theory of rheological properties of the systems of entangled macromolecules and on the comparison of this theory with the results of real and computer experiments. 

The system of entangled macromolecules can exist in different physical states, depending on temperature. Further on, we will suppose that the temperature of the system exceeds the characteristic crystallisation and glass points, so that the system can be considered to be fluid. This is the case of concentrated polymer solutions or polymer melts. Any macromolecule in the system can only move as freely as its macromolecular neighbours allow it to. 

There are different approaches to the calculation of stresses in the system of entangled macromolecules [9, 34, 93, 94]. We consider the system to be a suspension of Brownian particles and start here to determine a expression for the stress tensor through correlation functions of the normal co-ordinates of the macromolecule, while we shall follow a method developed in the theory of liquids [95, 96] which was used by Pokrovskii and Volkov [55] in this case. 

The results from recent decades allow us to describe a picture of thermal motion of long macromolecules in a system of entangled macromolecules. The basic picture is, of coarse, a picture of thermal rotational movement of the interacting rigid segments connected in chains - Kuhn - Kramers chains. One can refer to this model as to a microscopic model. In the simplest case (linear macromolecules, see Sect. 3.3), the tensor of the mean orientation (e,ej) of all (independently of the position in the chain) segments can be introduced, so that the stress tensor and the relative optical permittivity tensor can be expressed through mean orientation as... 

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