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By Stuart A. Rice
This sequence offers the chemical physics box with a discussion board for severe, authoritative reviews of advances in each zone of the self-discipline. quantity 131 comprises chapters on: Polyelectrolyte Dynamics; Hydrodynamics and Slip on the Liquid-Solid Interface; constitution of Ionic drinks and Ionic Liquid Compounds: Are Ionic drinks actual drinks 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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Explicit formula for Á is derived in Ref. 48. Its Fourier transform is of the form Vk Ák ¼ k 1 þ kcV 2 2l ð222Þ k where lk is a functional of Ák and l1 , and Vk is the Fourier transform of V in Eq. (24). In the high salt limit, Ák becomes wc ð223Þ Ák ¼ w þ 2 k2 x2 =ð1 þ k2 x2 Þ k À Á where x2 ¼ l2 =c w þ wk2c and wc ¼ 4pz2p e2 =EkB T. In the low salt limit, Ák 37 polyelectrolyte dynamics TABLE I The Dependencies of Radius of Gyration Rg , Static Correlation Length x, Hydrodynamic Screening Length xH , Viscosity Z, Self-Translational Diffusion Coefficient D, Cooperative Diffusion Coefficient Dc , Coupled Diffusion Coefficient Df , and Electrophoretic Mobility m on c and N for Various Regimes of Polyelectrolyte and Salt Concentrations Salt Quantity Level 6R2g L ¼ l1 x xH ZÀZ0 Z0 D Dc m Dilute (Zimm) h qffiffiffiffiÀ Ái2=5 1=5 3 B w þ 4pl High 3l43 2p N l k2 2=3 p4pl ffiffi B Low Nl 2 6p5=2 l Concentrated (Entangled) À Á1=4 À1=4 À3=2 B w þ 4pl c l l k2 1=6 pffiffi pffiffi2=3 4plB 3 6 2 cÀ1=2 lÀ1=2 l l p 4 Â À ÁÃÀ1=2 5=4 3pffiffi B À1=4 À3=4 À1=2 B c l l 6c w þ 4pl w þ 4pl k2 Þ k2 4 p pffiffi1=3 qffiffiffiffiÀ ÁÀ1=6 À24pclB ÁÀ1=4 6 2 3 4plB ðclÞÀ1=2 16 l l2 p 3p3=4 ffiffi p High Rg $ N À3=5 Low Rg $ N High 1 Low 1 High À Á3=5 6=5 4=5 B c w þ 4pl l N k2 1 2 2 3 24 c l l1 L $ c5=4 N c4:25 N 3:4 Low clB l2 N 2 1 2 2 3 24 c l l1 L $ pffiffiffi cN c1:7 N 3:4 High 8 ffiffi kB T p 3 p 6pZ0 Rg ÁÀ1=4 À3=4 À1=2 B w þ 4pl c l k2 2=3 À ÁÀ1=6 4plB p ffiffi p8ffiffi p ðclÞÀ1=2 l 1 p2ffiffi 3=4 p3 3p $ N À3=5 À 6 2 $ c0 N À1 cÀ1=2 N À2 High D kB T xH 6pZ0 x xþxH $ c3=4 N 0 Low D kB T xH 6pZ0 x xþxH $ c1=2 N 0 High N0 QD kB T Z0 k2=3 N0 Z0 z2p Nc High N À3=5 1 þ z2 cþ2c s c Low kB TxH 3pZ0 R2g N0 1:17 ðkRg Þ2=3 cÀ1=2 cÀ7=5 N À2 8 ffiffi kB T p 3 p 6pZ0 Rg $ N À1 cÀ1=2 1 ffiffi $ Np c kB TxH 3pZ0 R2g Low Low Df Semidilute (Rouse) $ N 0 cÀ1=2 $ N 0 c0 pffiffi c 4pQcm Ek2 N $ cþ2cs QD kB T 4pQcm Ek2 N $ N 0 c0 becomes Ák ¼ wc k2 x4 =ð1 þ k4 x4 Þ ð224Þ where xÀ4 ¼ wc c=l2 .
Naturally, in the o ! 0 limit, results of Table I are recovered. Only the high frequency limit is mentioned now. 1. Dilute In infinite dilute solutions, the intrinsic viscosity [Z] is given by [3, 4, 61] ½Z ¼ N RT X tj MZ0 j¼1 1 þ iotj ð306Þ where R, T, M, and tj are, respectively, the gas constant, absolute temperature, molecular weight (proportional to the number of Kuhn segments N per chain), and relaxation time of the jth Rouse–Zimm mode. The intrinsic storage [G0 ] and loss [G00 ] moduli are ½G0 ¼ N o2 t2j RT X M j¼1 1 þ o2 t2 j 00 ½G ¼ N RT X otj M j¼1 1 þ o2 t2 ð307Þ j In the high frequency limit, the intrinsic viscosity follows from Eq.
00 ¼ 0, so that For k ! 0, it follows from Eq. (159) that A m¼ Q ft ð167Þ Since ft $ Rg $ N for k ! 0, and Q ¼ Nzp e, m $ N0 For kRg ) 1, we get from Eqs. (156) and (166), " # Q 1 b þ m¼ N Z0 ‘ðk‘Þ1=nÀ1 ð168Þ ð169Þ where b is a known numerical coefficient [50]. Since Q ¼ Nzp e, m is independent of N in the high salt limit also. Furthermore, since n ’ 3=5 for high salt solutions, we expect m$ 1 Z0 k2=3 when hydrodynamic interaction dominates. ð170Þ polyelectrolyte dynamics 29 Summarizing, the electrophoretic mobility of a flexible polyelectrolyte chain in infinitely dilute solutions is given by Eq.