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A method for analyzing molecular motion of a branched long chain polymer wherein the position coordinate, the orientation vector, the number of monomers in the constituent elements of a tube and counterpart tube constituent elements forming entanglement are time-developed over a unit time defined by expression 7 according to the Langevin equation defined by expression 1 on the basis of the reptation theory with regard to an inter-tube constituent element constituting each polymer produced by coarse-graining the polymers of a mixture of the branched long chain polymer material to be analyzed according to the entanglement molecular weight (step 201), the number of monomers in the constituent element of the tube is time-development according to expression 5 (step 202), the number n of monomers in the constituent element of the tube at each end of a polymer is monitored, the constituent element of the counterpart tube forming entanglement is altered according to expression 8, and geometrical position exchange is caused to take place between the entanglement and a branch point by the number of constituent element of the tube at a branch part belonging to the branch point

A method for analyzing molecular motion of a copolymer comprising a step for time-developing the constitutional element of an associated tube forming positional coordinates, an orientation vector, number of monomers in the constitutional elements of a tube and an entanglement over a unit time defined by expression 7 according to Langevin equation defined by expression 1 with regard to an inter-tube constitutional element constituting each polymer produced by coarse-graining the polymer of a mixture of a copolymer material to be analyzed based on the inter-entanglement molecular weight based on reptation theory (step 201), and a step for time-developing the number of monomers in the constitutional element of tube according to expression 5 (step 202) wherein the number n of monomers in the constitutional element of tube at each end of a polymer is monitored, the constitutional element of an associated tube forming an entanglement is altered according to expression 8, and geometrical positional exchange is caused to take place between the entanglement and a branch point by the number of constitutional elements of a tube at a branch part belonging to the branch point.

In a method of analyzing the molecular motion of a mixture comprising different polymer materials, the polymers of the mixture comprising different polymer materials to be analyzed are roughly visualized depending on the molecular weights among tangling points according to the reptation theory. Based on the intertube factor constituting each polymer thus formed, the position coordinate, the alignment vector, the number of monomers in the tube factor and the tube factor of the counterpart forming the tangling are time developed per the unit time as defined by the formula 7 in accordance with Langevin formula defined by the formula 1 (step 201). Then the number of monomers in the tube factor is time developed in accordance with the formula 5 (step 202) and the number (n) of monomers in the tube factor at each terminus of the polymer is monitored. In accordance with the formula 8, the tube factor of the counterpart forming the tangling is altered.

A method for analyzing the molecular motion of a polymer compound which comprises forming, according to the reptation theory, tube−constituting elements composing each polymer molecule through a macroscopic operation based on an entanglement molecular weight with respect to the polymer molecule of the polymer compound, subjecting the position coordinates and orientation vectors of the tube−constituting elements, the numbers of monomers in the tube−constituting elements, and the tube−constituting elements playing opposite at the formation of entanglement, to time development according to Langevin equation over the unit time defined by the formula (4) (step 201), subjecting the numbers of monomers in the tube−constituting elements to time development according to equation (2) (step 202), monitoring the number (n) of monomers in the tube−constituting element positioning in each terminal of the polymer molecule, and changing the tube−constituting elements playing opposite at the formation of entanglement according to the formula (5).

Two-dimensional simulations of the Brownian dynamics of polymers in a grid of topological obstacles were carried out in this study. A first model made use of the exact expression for the free energy of a chain with fixed ends interacting with an obstacle and can be used to derive the elastic force acting on the subchains of a Rouse-like model. With this model we calculated the diffusion coefficient of chains of different lengths constrained by the fixed obstacles. A second model, less rigorous but more efficient computationally, was used to simulate the chain dynamics in fast flows. In such a case, the obstacles were convected according to the shear gradient. During the simulation, the configurational changes of the chain constrained by the obstacles could be monitored and the stresses which developed were calculated. These simulations reveal important effects of the conformational fluctuations at high shear rates

In this paper we derive a few analytical results for the case of polymeric solutions flowing past a spherical obstacle. The calculations are meant to ascertain the possible importance of flow-induced migration (i.e., of changes in concentration), making use of the, recent two-fluid theory of Doi and Milner. In order to obtain solutions in closed form, the problem is suitably linearised; in particular, we have considered both the case of Newtonian behaviour of the polymeric liquid flowing past the sphere, and that of a linearly elastic polymeric gel perturbed by a localised force. The results, though confirming migration effects, indicate that, in these linear limits at least, the concentration change decays quadratically with the reciprocal distance from the centre of the perturbation, i.e., no long ''wake'' is predicted. Another outcome of these calculations refers specifically to the viscous case. The classical Stokes result is recovered only as a singular solution of the two-fluid problem

We present a new multi-chain Brownian dynamics simulation of a polymeric network containing both crosslinks and slip-links (entanglements). We coarse-grain at the level of chain segments connecting consecutive nodes (cross- or slip-links). Affine displacement of nodes is not imposed; rather, their displacement as well as sliding of monomers through slip-links is governed by force balances. The stress-strain response in uniaxial extension is compared with the slip-link theories of Ball et al. (Polymer, 22 (1981) 1010) and Edwards and Vilgis (Polymer, 27 (1986) 483), and with the molecular-dynamics simulations of Grest et al. (J. Non-Cryst. Solids, 274 (2000) 139). Qualitative agreement both with the Mooney-Rivlin expression and with the stress upturn at large strains confirms the role of entanglements in explaining departure from the classical theory of phantom chain networks. However, quantitative agreement with data is satisfactory at low strains only, and the observed discrepancy at larger strains suggests possible refinements of the model. Additivity of free energy contributions of crosslinks and entanglements used in several molecular theories of rubber elasticity is confirmed by the simulation results.