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.
Read or Download Advances In Polymer Science Vol 125: STATISTICAL MECHANICS, DEFORMATION, ULTRASONIC SPECTROSCOPY (Advances in Polymer Science) PDF
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Extra info for Advances In Polymer Science Vol 125: STATISTICAL MECHANICS, DEFORMATION, ULTRASONIC SPECTROSCOPY (Advances in Polymer Science)
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.