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Extra resources for Statistical mechanics, deformation, ultrasonic spectroscopy
Sample text
13. 11 Equations of Internal Motion for the Molecules; Hydrodynamic and Brownian Forces (DPL, Sect. 5) To get the (statistically averaged) equations of motion for the beads, we multiply Eq. 5) by pf and integrate over all the momenta of molecule a. This gives, when use is made of Eqs. 1): a < '(5í | [[iî]] L L J J ¥« = - 12 9) for the distribution m momentum space have been used. For example, some kinetic theorists have used a Maxwellian distribution about the velocity v at the center of mass of the molecule. Another possible assumption is that of a skewed distribution, in which additional empirical parameters are introduced so that the smoothed Brownian motion force may be stronger in the chain backbone direction than in the transverse directions [9, 20a, 20b, 20c, 21], [DPL, Sect. 7]. This idea has been proposed for describing the restricted motion of polymer chains in concentrated polymer solutions and in undiluted polymers. 6), an equation for the configurational distribution function 4^, is obtained. The resulting equation is often referred to as the diffusion equation for 4*^. Often, however, we prefer to work with the center-of-mass vector rx and the connector vectors Qf By using Eqs. 13), we can obtain from Eq. 1) the following two equations (for models m which all bead masses, m, and friction coefficients, £, are identical). 3) In which the A]k = "LvBJVBk, are the elements of the Rouse matrix (see Eq. 12). The DJki = I,vBJVBkvBvi are the elements of a matrix which, as far as we know, has never been discussed.