Entangled systems

This section presents results of the space-time analysis of the above-mentioned motional processes as obtained by the neutron spin echo technique. First, the entropically determined relaxation processes, as described by the Rouse model, will be discussed. We will then examine how topological restrictions are noticed if the chain length is increased. Subsequently, we address the dynamics of highly entangled systems and, finally, we consider the origin of the entanglements. 

Entanglement in open systems is a fascinating concept that is not yet fully understood. It is essential for QIP, yet it is also the origin of decoherence. The interplay between system-system entanglement, system-probe entanglement and system-bath entanglement on non-Markovian timescales is an intriguing field that deserves more attention. 

Highly entangled systems, especially those of narrow molecular weight distribution, are characterized by a set of relaxations at long times (terminal relaxations) which are more or less isolated from the more rapid processes. The modulus associated with the terminal processes is called the plateau modulus G°,. Because t]0 and depend on weighted averages over H(x), their values are controlled almost completely by the terminal processes. These experimental... 

Unlike Williams, Chikahisa attempts to deal directly with entangling systems. The result is a viscosity expression which is not far from that observed experimentally, but the form is unfortunately more dependent on a series of intermediate assumptions about the nature of the friction forces than on the basic transport theory itself. Although not implausible, the assumptions are nevertheless arbitrary and lacking in theoretical justification. 

All the above theories attribute the enhanced friction in entangled systems to extra motions. In a crude way they speak to the main issue, which is the loss of relaxation pathways due to connectedness in the chain environment. However, all are rather arbitrary they select certain types of motion for examination and exclude others, while in fact the motions must be cooperative and interdependent. In addition, the theories have features which appear to be incorrect, or at least inconsistent, even within the limited realm of motions examined. 

The failure of the Rouse theory was attributed to the pathological nature of medium motions in entangled systems, and not any special defect in the Rouse representation of the polymer chain itself. For Rouse chains in a deforming continuous medium, the frictional force depends on the systematic velocity of the bead relative to the medium. The frictional force on a bead is therefore a smootly... 

Such behavior is qualitatively understandable in terms of partial disentanglement in steady shear flow. In highly entangled systems (cM>gM )Je0 is of the form (Section 5) ... 

Thus, one can choose from the two possibilities to simplify the system (3.1). We are convinced, that the approximation of independent chains appears to be a very good initial approximation. The situation appears to be similar to a situation in dilute solutions discussed in the previous chapter. However, in contrast to the case of dilute solutions, the correlation times of the surrounding medium cannot be neglected for entangled systems. The initial phase of the theory might be found to be rather formal but the justification for every theory regarding physics eventually rests on the agreement between deductions made from it and experiments, and on the simplicity and consistency of the formalism. Comparison with experiment will be discussed in Chapters 5, 6, 9 and 10. 

Particular cases of dynamic equation (3.11) were investigated by Ronca (1983) and by Hess (1986, 1988) who apparently did not know about previously published results. They made unsuccessful attempts to describe dynamics of macromolecule in an entangled system without the second dissipative term which is connected with the internal resistance forces. One can see in subsequent chapters that the properties of polymer melts cannot be understood correctly without this term. The importance of the internal resistance term was recognised by Pokrovskii and Volkov (1978b) after the first attempt to tackle the problem (Pokrovskii and Volkov 1978a). 

In the case of the bulk polymer, the requirement of self-consistency of the theory states that the relaxation time r can be interpreted as a characteristic of the whole system. Properties of the system will be calculated in Sections 6.3.2 and 6.4.3, which allows one to estimate relaxation time r and quantity y. It will be demonstrated that, for weakly entangled systems (2Me < M < 10Me), the quantity x has the self-consistent value... 

For strongly entangled systems (M > 10Me), the requirement of self-consistency is fulfilled identically, while the quantity x is connected with the intermediate length (see Section 5.1.2, formula (5.8)), or (as we shall see in Section 6.4.4, formula (6.55)) with the length of the macromolecule between adjacent entanglement Me, that is... 

In the last case, the parameter has the meaning of the ratio of a length of macromolecule between adjacent entanglements to the length of the macromolecule (see Section 5.1.2). The parameters t, x, , and Me appear to be equivalent for the strongly entangled systems. One of these parameters is used to describe polymer dynamics in either interpretation. 

The parameter x is always small for entangled systems. Due to the above written results, the self-consistent values of the quantity can be approximated, for M/Me > 2, as... 

This quantity has value of zero for non-entangled systems and increase with increase in the length of macromolecules. As for external force, there is a slight difference in resistance, when the particle moves along the chain or in a perpendicular direction, but, in this subsection, this effect is neglected for simplicity. 

For the systems of long macromolecules (strongly entangled systems), the requirements of universality and self-consistency allow us to write practically identical asymptotic relations (5.17) and (6.53) between the parameter y, introduced in Section 3.3.1, and the ratio E/B, which allows us to write for this case... 

For the weakly entangled systems, one can expect, that the ratio E/B, that is the parameter of internal viscosity is small. It can be demonstrated in Section 4.2.3, that transition point from weakly to strongly entangled systems occurs at E B. To describe these facts, one can use any convenient approximate function for the measure of internal resistance, for example, the simple formula... 

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. 

Now the dependencies of the phenomenological parameters B and E on the concentration of polymer c can also be given. From the above relations, it follows, for example, that for the strongly entangled systems... 

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