Length and Time Scales of Entanglement and Confinement Effects Constraining Polymer Chain Dynamics
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1227-JJ02-01
Length and Time Scales of Entanglement and Confinement Effects Constraining Polymer Chain Dynamics Rainer Kimmich1 and Nail Fatkullin2 1 University of Ulm, 89069 Ulm, Germany ([email protected]). 2 Kazan State University, Kazan 420008, Tatarstan, Russian Federation ([email protected]). ABSTRACT With characteristic time constants for polymer dynamics, namely τ s (the segment fluctuation time), τ e (the entanglement time), and τ R (the longest Rouse relaxation time), the time scales of particular interest (i) t < τ s , (ii) τ s < t < τ e , and (iii) τ e < t < τ R will be discussed and compared with experimental data. These ranges correspond to the chain-mode length scales (i) A < b , (ii) b < A < d 2 / b , and (iii) d 2 / b < A < L , where b is the statistical segment length, d is the dimension of constraints by entanglements and/or confinement, and L is the chain contour length. Based on Langevin-type equations-of-motion coarse-grained predictions for the meansquared segment displacement and the spin-lattice relaxation dispersion will be outlined for the scenarios “freely-draining”, “entangled”, and “confined”. In the discussion we will juxtapose “local” versus “global” dynamics on the one hand, and “bulk” versus “confined” systems on the other. INTRODUCTION The atomistic description of chain dynamics in polymer fluids fails because of the excessive number of internal degrees of motional freedom coupled in a non-linear way. In particular, distinct valence bond restrictions exist so that fluctuations of rotational isomerism are only possible with essentially fixed valence bond angles and lengths. Therefore, one usually refers to coarse-grained treatments of phantom model chains consisting of freely-jointed statistical segments (or subchains). The mean squared end-to-end distance of the random coil of such a chain is Ree2 = bReemax , where Reemax is the end-to-end distance of the fully stretched chain, and b is the root-mean squared end-to-end distance of a statistical segment. For flexible polymers, b typically corresponds to the contour length of 3 to 6 monomers. The length scale of this sort of coarse-grained description is therefore A >> b . Under such conditions, linearized equations of motion can be established with average solutions in the form of superimposed relaxation modes. The time constants (or relaxation times) of these modes depend on the length of the chain stretch affected by the “wave length” of the mode. The shortest relaxation time corresponds to the length of a statistical segment, b , and will be called τ s . The coarse-grained length scale is consequently intimately related to the coarsegrained time scale t >> τ s . In the following we will discuss coarse-grained solutions for polymer melts in three different scenarios: i) macromolecules freely draining in a viscous medium, ii) “entangled” polymers the dynamics of which is not only hindered but also retarded by neighboring polymers, iii) macromolecules confined in pores of a solid matrix. (Note that the dimension effecti
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