Dynamics of a single entangled chain

The subject of this paper is limited to the dynamics of a single entangled polymer chain, just as in the original de Gennes paper on reptation. In a melt or concentrated solution, the dynamics of any chain would be affected by the motion of surrounding polymers (by constraint release), and this effect has to be self-consistently taken into account. In order to do that, one has to start from a reliable model of the single-chain dynamics, such the reptation model, Doi s fluctuation theory,or the repton modeldescribed in the present paper. 

In Chapter 3, we used the Rouse model for a polymer chain to study the diffusion motion and the time-correlation function of the end-to-end vector. The Rouse model was first developed to describe polymer viscoelastic behavior in a dilute solution. In spite of its original intention, the theory successfully interprets the viscoelastic behavior of the entanglement-free poljuner melt or blend-solution system. The Rouse theory, developed on the Gaussian chain model, effectively simplifies the complexity associated with the large number of intra-molecular degrees of freedom and describes the slow dynamic viscoelastic behavior — slower than the motion of a single Rouse segment. 

The QELS method can be utilized to evaluate Z)(oo) of a single polymer in concentrated solutions. To this end, chemically labeled chains are dispersed at a very low concentration in a given concentrated solution of unlabeled chains of the same or different species and 5(fc, r) data for the labeled species are analyzed by the method established for dilute solutions. QELS measurements on concentrated solutions in which polymer chains entangle with one another usually reveal a decay of 5(fc, r) much faster than that observed in dilute solutions. This decay is associated with the relaxation of local concentration fluctuations, and defines a dynamic quantity called the cooperative diffusion coefficient and discussed in Chapter 7. 

An alternative idea proposed by Edwards and Freed is to consider the motion of a single chain in an effective medium which includes the effect of the other chains. The property of the effective medium is determined self-consistently from the single chain dynamics. Though this method fails to describe the entanglement effect appropriately, it indicates an important aspect of the hydrodynamic interaction in the concentrated system, which is the screening of the hydrodynamic interaction. 

Our current understanding of the dynamics of the highly entangled state is based on the concept of reptation. This picture is rigorously correct for the system that has been presented, i.e., a single chain in a... 

Show that a stress relaxation modulus of an entangled but non-concate-nated melt of rings on the basis of the single chain dynamic modes described in Problem 9.31 is... 

Though the tube model is successful, our present understanding of the dynamics in entangled systems is still incomplete. Agreement between theory and experiments is not yet complete as we shall discuss later. More seriously, the tube model does not describe all aspects of the dynamics it describes properties which depend on a single chain... 

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