Reptation theory: geometrical and topological aspects

Abstract

SUMMARYThis paper discusses topological and geometrical aspects of reptation theory which are common to all versions of reptation theory. These are: the postulated existence of the tube, the functional relationship between the tube diameter a and the polymer/monorner density p, the crossover from the Rouse to reptation regime. Statistical mechanics of the geometrically confined polymer chain is reanalyzed by careful separation of the diffusive motion of the chain into the longitudinal and transversal parts. Connection between old results and the new formalism is established. It is shown that the longitudinal motion resembles that known for directed polymers. This provides a source of the effective rigidification of the reptating chain's backbone thus facilitating the viscosity exponent to be larger than 3. The transversal motion is also reanalyzed. It is shown that the diffusion on the Bethe lattice used before to describe the transversal (planar) motion (conformational statistics) of the trapped chain is actually the diffusion on the universal covering of the corresponding Riemannian surface. This fact allows to reanalyze the tube stability using topological arguments. Detailed numerical comparison of the obtained new theoretical results with available experimental and Monte Car10 data is provided. Very good agreement between theory and experiment is found. It is also shown that the emerging physical picture of the tube destruction is isomorphic to that which was developed earlier with the help of the quantum Hall effect analogy (J. Phys. I 4, 843 (1994)). Remarkable connections between the reptation theory and the theory of quantum chaotic/mesoscopic systems are established thus making the reptation theory part of the more general theory of quantum chaotic systems.

Constitutive equations for weakly entangled linear polymers

“…Apart of empirical justification, the mesoscopic effective-field approach itself is needed in proper microscopic justification. The characteristics of reptation motion can be also deduced from geometrical and topological aspects of macromolecular dynamics [36,37], so that the parameters of the theory eventually could be linked with details of structure of entangled systems. One can believe that the developing methods [14,[34][35][36][37][38] can be helpful to bring a microscopic justification of a single-macromolecule equation in the entangled systems.…”

“…The description of phenomena can be improved, if one takes into account the terms of the first and higher orders at simplification of relaxation Eqs. (36) and (37). In this way, as it was shown in [27], one can get the system of constitutive relations of the first order according to parameter , which contains two coupled relaxation equations.…”

“…In virtue of applicability of relaxation equations (36) and (37) (˛2 1/ at the limit value of = 0.3), only one or two modes, which determine the main contribution to viscoelasticity ought to be used for description of flows. One can expect that a reasonable approximations to many-mode constitutive relations can be get at reducing the number of modes to minimum.…”

“…Every mode of the relaxation equations (36) and (37) contains two relaxation processes. So, one can keep one relaxation equation from each relaxation branch only and have the two relaxation equations to be considered:…”

Spray applications: Part III. Assessment of viscosities at high shear rates and dynamic uniaxial extensional viscosities on fan nozzle air sprayability

“…14). The explanation for this may lie in reptation theory, where the more flexible high M v POE molecules are thought to be trapped in an array of obstacles (the low M v HEC molecules that are present in significantly higher concentration than the high M v HEC in the POE blends, and the POE chains are forced to move in a worm-like (reptative) fashion 19 (Fig. 15).…”