# Advances In Polymer Science Vol 125: STATISTICAL MECHANICS, by R.B. Bird, S.V. Bronnikov, C.F. Curtiss, S.Y. Frenkel, N.

By R.B. Bird, S.V. Bronnikov, C.F. Curtiss, S.Y. Frenkel, N. Hiramatsu, K. Matsushige, H. Okabe, V.I. Vettegren

This article examines advances in polymer technology, overlaying the parts of statistical mechanics, deformation and ultrasonic spectroscopy.

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Extra info for Advances In Polymer Science Vol 125: STATISTICAL MECHANICS, DEFORMATION, ULTRASONIC SPECTROSCOPY (Advances in Polymer Science)

Example text

When B is inserted into the general equation of change Eq. ~,(2x m~(~ ~ . 5) In order to put this in the form of Eq. 1) we begin by replacing f~i,just after the summation sign in the first line, by (i~~- v), where v(r, t) is the mass-average velocity of the fluid. F. B. B i r d Here, by virtue of Eq. ,k'~. ,j _ r)) 2 . , / r , - ~ 0 2 ~-,, t~Ju\ cry ~k~ (E(f:i~(e)off, A(rtti. 9) We now manipulate each of the "source terms" by using Eqs. 19-21) in order to obtain Eq. 1), with the heat-flux vector being the sum of four contributions: q = q(k) + q(O)+ q(a) + q(~).

In Sect. 4 an empirical expression is presented for these quantities, and some examples of solving the resulting "diffusion equations" are shown in Sect. 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 p~i and integrate over all the momenta of molecule ~. This gives, when use is made of Eqs. 1): mat~ d EEi'~]Y Tat = - mvX. 1) Next we replace the double bracket in the first term on the right side by [[(i-~,- u ~ ) ( ~ - u~)J]at, and add appropriate compensating terms; here and elsewhere we use the notation u~i(r~, t) = [[f~J]at for the average velocity of bead v.

10) Note carefully that the arguments of [ [p~] ]~ are the same as those of W~, namely r - R~, Q~, and t. Now for each of the two terms in the second form of Eq. 10) we make the Taylor expansion described in Eq. ~m~ t) d Q • R~[[r~]] ~F~(r,Q~,t)dQ ~ + ½VV: E~m~ jR~R~[[rd-1%(r, Q~, t ) d Q ~ + ... 3 Since it is only the divergence of p~v~ that appears in Eq. 8), one could add to the ( . . ) quantity in the last line of Eq. 8) any solenoidal vector function. However, it follows from the physical interpretation after Eq.