Entanglements polymer diffusion

The diffusion coefficients of entangled polymers in solution will most certainly depend on the viscosity of the medium and vice versa. It is reasonable therefore to expect that the diffusion coefficient would correlate well with the weight average molecular weight of the polymer. M is therefore used with equation (lO) giving... 

Restricted diffusion, correlated motion of spins, or any deviation from a free behavior of the molecules will result in a propagator shape different from a Gaussian one. A wide range of studies have dealt with such problems during the last two decades and NMR has turned out to be the method of choice for quantifying restricted diffusion phenomena such as for liquids in porous materials or dynamics of entangled polymer molecules. 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

Diffusion of Densely Entangled Polymer Chains by Reptation... 

Edwards SF, Grant JWV (1973) The effect of entanglements on diffusion in a polymer melt. J Phys A Math Nucl Gen 6 1169-1185... 

In the present paper, after a rapid presentation of the reptation model in its simplest version, in order to pinpoint the underlying hypothesis, we discuss the interest of complementary self diffusion and viscoelastic measurements, and present the currently available methods for measuring diffusion in entangled polymer systems. Then, results obtained on polydimethylsiloxane (PDMS), a model liquid polymer well above its glass temperature at room temperature will be described, and the consequences on the limits of the entangled regime as seen from diffusion measurements, compared to what is observed in rheometry, will be discussed. 

For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeavuc [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time Xj, proportional to M2, which can be considered as the Rouse time of an entangled polymer movii in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced Xj by an empirical combination of... 

W Graessley. Viscoelasticity and diffusion in entangled polymer melt. In M Nagasawa, ed. Molecular Conformation and Dynamics of Macromolecules in Condensed Systems pp (163-184). Amsterdam Elsevier, 1988. 

In entangled polymer melts this diffusion occurs along the contour of the tube, with the mean-square monomer displacement in space determined using Eq. (9.71) ... 

Entangled polymer dynamics 9.5.1 Relaxation times and diffusion... 

The limit in front of the ratio means that the time t has to be much longer than the longest relaxation time of the chain. The resulting diffusion coefficients obtained by Monte Carlo simulation of the Evans-Edwards model of entangled polymers are presented in Fig. 9.33(a). The diffusion coefficient decreases with the number of monomers in the chain. Another quantity that can be extracted from the Monte Carlo simulations of the Evans-Edwards model is the relaxation time of the chain. It can be defined as the characteristic decay time of the time correlation function of the end-to-end vector R[t)R 0)) exp( t/Trep). Figure 9.33(b) presents the results of such simulations. 

The primary mode of motion of a linear chain along its confining tube is reptation, first proposed by de Gennes. Reptation is a snake-like diffusion of a chain, as a whole, along the contour of its tube, with a Rouse curvilinear diffusion coefficient. The relaxation time of the melt is the time it takes the chain to reptate out of its original tube, called the reptation time Trep. The reptation time and the viscosity of entangled polymers are... 

Consider a small non-adsorbing spherical particle of diameter d diffusing in an entangled polymer solution of linear chains with N Kuhn monomers of length b, with volume fraction (p in an athermal solvent with solvent viscosity r s- Calculate the diffusion coefficient of the particle if its diameter is ... 

Jordan, E.A. Donald, A.M. Fetters, L.J. Klein, J. Transition from linear to star-B branched diffusion in entangled polymer melts. Polym. Prep. 1989, 30 (1), 63-64. 

Using scaling concepts, de Gennes [52] found that the diffusion coefficient, D, of a chain in the entangled polymer matrix depends on the molecular weight M as... 

Part I summarizes the main ideas of de Gennes, Doi and Edwards about tube models and reptation in entangled polymer systems. Attention has been limited to properties for which predictions can be made without invoking the independent alignment approximation macromolecular diffusion, linear viscoelasticity in the plateau and terminal regions, stress relaxation following a step strain from rest of arbitrary magnitude, and equilibrium elasticity in networks. 

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