Dynamics of a Macromolecule in an Entangled System

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In this chapter, a system of entangled macromolecules in fluid state, that is a concentrated solution or a melt of polymer, will be considered. Every macromolecule in the investigated system can move among the others macromolecules, exchanging neighbours and remaining the integrity of each individual macromolecule unaffected. It allows introducing the mesoscopic approximation, which deals with the motion of a single macromolecule in an effective medium, created by the neighbouring macromolecules. One can note that the tradition of the mesoscopic approach begins with the first work on concentrated polymer solutions (Ferry et al. in J. Appl. Phys. 26:259–362, 1955), in which some specifying hypotheses about the environment of the probe macromolecule were formulated. Some earliest approaches to the problem are developed by Edwards with collaborators (Scolnick and Kolinski in Adv. Chem. Phys. 78:223–278, 1990). One of the hypotheses ascribes the properties of a relaxing medium to the environment of a probe macromolecule (Edwards and Grant in J. Phys. A: Math. Nucl. Gen. 6:1169–1185, 1973). This idea was developed later into the theory, based on the non-Markovian stochastic equation. An alternative hypothesis was an assumption about the tube and reptation motion of macromolecules (Doi and Edwards in The Theory of Polymer Dynamics (Oxford University Press, Oxford), 1986). Now, one can see that both the first and the second hypothesis reflect the reality, and the theory, which will be exposed here, can be considered as a reconciliation of the alternative approaches. In this chapter, a unified model of macromolecular dynamics can be formulated as the Rouse model of a chain of coupled Brownian particles in the presence of a random dynamic force. It came to a consistent theory of the phenomena and constitutes a phenomenological frame within which both the results of empirical investigations and the results of microscopic, many-chains approaches can be considered. The mesoscopic approach reveals the internal connection between phenomena and provides more details than a strictly phenomenological approach.