By Stuart A. Rice
This sequence offers the chemical physics box with a discussion board for severe, authoritative reviews of advances in each quarter of the self-discipline. quantity 131 contains chapters on: Polyelectrolyte Dynamics; Hydrodynamics and Slip on the Liquid-Solid Interface; constitution of Ionic beverages and Ionic Liquid Compounds: Are Ionic drinks real beverages within the traditional Sense?; Chemical Reactions at Very excessive strain; Classical Description of Nonadiabatic Quantum Dynamics; and Non-Born Oppenheimer Variational Calculations of Atoms and Molecules with Explicitly Correlated Gaussian foundation services.
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Extra resources for Advances in Chemical Physics, Volume 131
18 m. muthukumar where the proportionality factor Â is to be determined below from a consideration of equilibrium. The correlation function h½Rðs; tÞ À Rðs0 ; t0 Þ2 i follows from its Fourier transform as ð ð dq 1 dq0 1 do h½Rðs; tÞ À Rðs ; t Þ i ¼ À1 2p À1 2p À1 2p ð1 do0 ÀiðqÀq0 ÞsÀiðoÀo0 Þt 0 0 0 ½e À 2eÀiqs À iot þ iq s þ io t À1 2p 0 0 ð1 2 0 0 0 ~ ðq; oÞ Á R ~ Ã ðq0 ; o0 Þi þ eÀiðqÀq Þs ÀiðoÀo Þt hR ð85Þ Substituting Eq. (83) in Eq. (85) and by performing q0 and o0 integrals, we obtain ð1 dq h½Rðs; tÞ À Rðs ; t Þ i ¼ 2 À1 2p 0 0 2 ð1 do À1 2p 0 Â½1 À cos½qðs À s0 ÞeioðtÀt Þ ^ ^ ðqÞq ðqÞq ðio þ 3kB ‘T1DðqÞ ÞðÀio þ 3kB T‘1DðqÞ Þ ð1 dq Â‘1 ðqÞ ¼ ^ ðqÞq2 2p 3kB T D À1 !
B. Translational Friction Coefﬁcient The translational friction coefﬁcient ft of the polymer chain at inﬁnitelyPdilute solutions is calculated by equating the net force acting on the chain, À i hri i, 0 0 and Àft R_ , where R_ is the net drift velocity of the center of mass of the chain X i hri i ¼ ft R_ 0 ð98Þ 20 m. muthukumar If there is no velocity ﬁeld in the absence of polymer (F ¼ 0), ri follows from Eq. (43) as X hri i ¼ hR_ i i À hGðRi À R j Þ Á rj i ð99Þ j6¼i 0 Assuming that R_ i is the same as the center-of-mass velocity R_ for all i and invoking the preaveraging approximation, we get 0 hri i ¼ R_ À X hGðRi À R j Þihrj i ð100Þ j6¼i With the assumption of uniform chain expansion due to potential interactions, we have ( ) 1 1 hGðRi À Rj Þi ¼ ð101Þ 6pZ0 jRi À Rj j 1=2 1 1 ð102Þ ¼ Z0 6p3 ‘‘1 ji À jj Deﬁning hrj i fi R_ 0 ð103Þ Eq.
90Þ 19 polyelectrolyte dynamics The mean-square displacement of a labeled monomer is 0 2 h½Rðs; tÞ À Rðs; t Þ i ¼ 2 jtÀt0 j dq ‘1 ðqÞ À tq 1Àe p q2 ð1 0 ð91Þ When hydrodynamic interaction dominates, tq $ qÀ3n , so that 0 2 h½Rðs; tÞ À Rðs; t Þ i $ ð1 dq qÀð2nþ1Þ ð1 À eÀq 3n jtÀt0 j Þ ð92Þ 0 $ jt À t0 j 2=3 ð93Þ The incoherent dynamical structure factor is given by Sinc ðk; tÞ ¼ expðÀk2 h½Rðs; tÞ À Rðs; t0 Þ2 i=6Þ 0 2=3 $ expðÀk jt À t j 2 ð94Þ Þ where k is the scattering wave vector. Sinc ðk; tÞ is therefore of the form exp½Àðk3 jt À t0 jÞ2=3 ð95Þ so that the decay rate is proportional to k3 .
Advances in Chemical Physics, Volume 131 by Stuart A. Rice