Friction coefficient Zimm model

The theoretical prediction of these properties for branched molecules has to take into account the peculiar aspects of these chains. It is possible to obtain these properties as the low gradient Hmits of non-equilibrium averages, calculated from dynamic models. The basic approach to the dynamics of flexible chains is given by the Rouse or the Rouse-Zimm theories [12,13,15,21]. How-ever,both the friction coefficient and the intrinsic viscosity can also be evaluated from equilibrium averages that involve the forces acting on each one of the units. This description is known as the Kirkwood-Riseman (KR) theory [15,71 ]. Thus, the translational friction coefficient, fl, relates the force applied to the center of masses of the molecule and its velocity... 

One of the most successful models for gel electrophoresis is the reptation theory of Lumpkin and Zimm for the migration of double-stranded DNA (Lumpkin, 1982). An in-depth discussion can be found in Zimm and Levene (1992) for a synopsis see Bloomfield et al. (2000). The velocity v of a charged particle in a solution with an electric field E depends on the electrical force Fei = ZqE, in which Z is the number of charges and q is the charge of a proton, and the frictional force l fr = —fv, in which/is the frictional coefficient. At steady state, these forces balance and the velocity is v = ZqE/f. The electrophoretic mobility fi is the velocity relative to the field strength, fi = vE = Zq/f. 

The balance of these forces for dilute solutions of polymer molecules in a small-molecule solvent was first described by PE Rouse [5 and BH Zimm [6. Here is a simple version of their models. Assume that the polymer molecule has been deformed by the applied shear to have an end-to-end distance x = X(). The elastic retractive force per molecule in a d-solvent is given by Equation (32.17), /clastic = -3kTx/Rj, in terms of the mean-square end-to-end length Rj = Nb = 3RqI2. The viscous force exerted per polymer chain on the surrounding medium is /viscous = where 5 is the friction coefficient (see Equation (18.41)) and v = -dxjdt is the velocity of retraction in the direction that opposes the original applied force. You can express Equation (33.29) as a differential equation ... 

These bead-spring models of Rouse and Kirkwood-Riseman-Zimm suffer from the artificiality of the beads and springs. The bead friction coefficient is an ad hoc phenomenological coefficient. This should arise naturally from the frictional forces coupling the polymer and solvent directly with the continuous version of the chain without beads and springs. 

The Zimm model rests upon the Langevin equation for over-damped motion of the monomers, i.e., it applies for times larger than the Brownian time scale Tb 2> OTm/where is Stokes friction coefficient [12]. On such time scales, velocity correlation functions have decayed to zero and the monomer momenta are in equilibrium with the solvent Moreover, hydrodynamic interactions between the various parts of the polymer are assumed to propagate instantaneously. This is not the case in our simulations. First of all, the monomer inertia term is taken into account, which implies non-zero velocity autocorrelation functions. Secondly, the hydrodynamic interactions build up gradually. The center-of-mass velocity autocorrelation function displayed in Fig. 9 reflects these aspects. The correlation function exhibits a long-time tail, which decays as (vcm(t)vcm(O)) on larger time scales. The... 

The first indication that the preaveraging technique has a substantial effect on the calculation of the friction coefficients of star pol3miers is found in the calculation of the g-dependence of Dq, i.e., the contributions of internal modes [90,91]. Zimm showed that preaveraging of the hydrodynamic interactions in the Kirkwood-Riseman model was largely to blame for the discrepancy between ggp and experimental values of gjj [57]. Further refinements of the Monte Carlo simulations with thermodynamic interactions to represent 0-solvent conditions or good solvent conditions combined with non-preaveraged hydrodynamic interactions lead to close agreement between calculated and... 

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