Scaling Analysis of Polymer Dynamics
In dilute solutions, diffusion of a single chain can be described by a non-draining mode of the coil. In the concentrated bulk phase, diffusion of a short chain can be described by a free-draining mode of the bead-spring Rouse chain, while diffusion of a
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Scaling Analysis of Polymer Dynamics
5.1
Simple Fluids
In this part, we start with the basic laws of molecular motions in simple fluids, to learn the scaling analysis of polymer dynamics, followed with polymer deformation and polymer flow. The trajectories of Brownian motions of hard spherical molecules can be analogous to random walks. As we have leant in Chap. 2, the mean square end-to-end distance of a random walk is proportional to the number of steps, i.e. ~ n. The threedimensional mean-square displacement of particles in Brownian motions is also proportional to the motion time t, as ¼ 6 Dt
(5.1)
where D is the diffusion coefficient. The discovery of such a law in the Brownian motion of the molecular particles is actually one of Einstein’s milestone contributions in 1905 (Einstein 1905). Accordingly, the characteristic time is defined as the moving time of a particle through a distance of its own size, as t
R2 D
(5.2)
In simple fluids, the external driving forces on a small particle are equilibrated with the friction due to the collisions with its surrounding medium. Therefore, the total frictional force f is proportional to the activated constant velocity v of the moving particle with respect to its surrounding medium, as f ¼ zv
(5.3)
where z is the friction coefficient. This law of fluid dynamics is similar to the Newton’s second law for the external force proportional to the acceleration. W. Hu, Polymer Physics, DOI 10.1007/978-3-7091-0670-9_5, # Springer-Verlag Wien 2013
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5 Scaling Analysis of Polymer Dynamics
According to the fluctuation-dissipation theorem in Brownian motions (Nyquist 1928), both the driving forces and the frictional forces on a particle are initiated by the collisions of the surrounding particles with the thermal energy kT. Accordingly, we have the Einstein relationship (Einstein 1905) kT ¼ Dz
(5.4)
On the other hand, the frictional forces are induced by the viscosity of fluids. The Stokes law reveals the relationship between the friction coefficient z and the viscosity (Stokes 1851), as given by z ¼ 6pR
(5.5)
Therefore, one can obtain the Stokes-Einstein relationship as D¼
kT 6pR
(5.6)
One can measure the viscosity and the diffusion coefficient to determine the so-called hydrodynamic radius as Rh
kT 6pD
(5.7)
This quantity of sizes reflects the effective volume-exclusion range of the moving particle interacting with its surrounding particles. For a single polymer chain in a good solvent, the theoretical hydrodynamic radius Rhtheo can be defined as 1 Rtheo h
1 X1 < > n2 i6¼j rij
(5.8)
where < . . . > is an ensemble average, and n is the number of monomers in the polymer (Des Cloizeaux and Jannink 1990). Such a theoretical definition makes the hydrodynamic radius close to the radius of gyration of the polymer coil. However, as we have introduced for polymer solutions in the previous chapter, the hydrodynamic radius of an anisotropic coil could be larger than its static radius of gyration. Thus from the dynamics point of view, the actual critica
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