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Dielectric friction and ionic mobility in polar liquids and liquid crystals
DIELECTRIC
11 March 1983
CHtiMICALPHYSrcS tETTER.8
Volume 95, number 4J
ERICTION AND IONIC hlOBILITY
IN POLAR LIQUIDS AND LIQUID CRYSTALS
J.B. HUBBARD, R-F. KAYSER Titermop~zysics Division, :Vafioml Bureau of Standanfs, Washingfon, D-C_ 20234, USA
and P.J. STILES Cltemistry Departmenx. hlacquarie University, Norrh Ryde, NS. W, Ausrralk;r 21 I3
Received8 December 1982
We introduce3 continuum model of dielectric friction on an ion in a polar liquid_This model couples hydrodynamic motions of the polarized solvent to dielectric relaxation by both rotational diffusion and transIationa1diffusion of solvent molecules. We show that in solvents with sufficiently Ions dielectric relasation times, translationaldiffusion is the dominant relasation mechanism. We compare our predictions with esperimental data on ion mobilitics in nematic liquid CQSta1.S
The role of dielectric friction in determining the motion of an ion in a polar liquid has received considerable attention over the last few years. The basic idea is that an ion migrating through a polar solvent creates a transient non-equilibrium polarization in the vicinity of its trajectory. and the energy dissipated by the relaxation of this polarization is associated with a frictional force on the ion in addition to the viscous force. The relaxation can occur either locally through rotational diffusion (dipole reorientation), or nonlocally through translational diffusion, which should be included when the spatial gradients in the non~qu~ibr~urn polarization are sufficiently large. Continuum approaches to the problem of calculating ionic mobilities in “simple” polar liquids have. until now, dealt exclusively with the orientational relaxation mechanism and its coupling with viscous hydrodynamics. However, in nematic liquid crystals for example, highly anisotropic correlations may lead to unusually large (and anisotropic) dielectric relaxation times, and in this case Meyer and de Gennes [ 1] have called attention to the possible existence of an important relaxation mechanism involving translational diffusion. The role of translational motions in simple polar liquids has also been emphasized in a 0 009-2614~~3~000~0000~~
03.00 0 1983 North-boded
molecular theory of dielectric friction recently developed by Wolynes and Colonomos [2,3] _ We have recently formulated an electrohydrodynamic model of dielectric friction which includes relasation of the orientational polarization by both rotational and translational diffusion_ Details of this analysis will be presented in a forthcoming paper: we present here some of the main iesults. We have shown that the decay of the orientational (Debye) polarization, Pr,, in an inhomogeneously polarized stationary fluid may be described by the following generalization of the Debye formula:
aPD/ar = 7D1(XDE - pD)
-m’(xDE -
pD),
where E is the local electric field, & = (e. - e,)/4a is the susceptibility for orientational polarization (e8 and E, are the low- and high-frequency
dielectric
constants), rD is a Debye relaxation time, and D is a phenomenological coefficient associated with translational diffusion_ Microscopic considerations suggest that D might be approximated by the self-diffusion coefficient of a solvent dipole, bur strictly speaking collective motions should also be included_ If we now allow for hydrodynamic motion of the fluid, symmetry arguments based on the invariance of the dissipa399
11 March 1983
CHE%fICALPHYSICS LE?PTERS
Volume 95, number 4.5
tion with respect to rigid body motions dictate that the simplest generalization of eq. (1) is 14-61
by simple scaling arguments.] If the ion radius is then large enough to ensure that R” % Rk, it can be shown [S] that the ionic drag coefficient becomes
aIJ~,iaf+cu-v)P,+~~~x(vxu)
[ = 47rnR + ‘2 (z’e2/DR)(q, = P*/rL, - DV’P*.
\vhcrc the “polarization
(2)
deficiency”
P* is defined
by
I’* = Q,E -- PD.
(3)
ad u is the velocity
Iield of the fluid. In the absence cri incrrial effects and dielectric saturation [7]. I lubbmd md Onsager (HO) have shown [4.5] that rbc linearized equations of motion for an incompressihlc solvent may be written I)‘?% = Vjr .- +Eu X curl P* - +Eo div P*.
(4)
u.I~txc7) is the fluid shear viscosity. p is the “electrohydrodynamic” pressure and E. is the electric field
\vl~en u = 0 (static fluid). Instead of tackling the formidable task of finding a general expression for the drag on an ion of radius I; iron the coupled equations (.2) and (4). in this paper \ve restrict our attention to limiting cases. In the regime where DT~/R’ is very small. translational diffusion is negligible compared to rotational diffusion and WC recover the I10 theory. Iiere. the ionic drag dqm~ds on the relative magnitudes ofR and the HO 1W~I11 I:,,,
=
I-_%‘(E(,
-
’
E,)T,)/16zqE(j)
l/4
.
(5)
whcrc :c* is the ion charge. If R’ B R;‘io the friction sorfticient takes the form (we use t!te slip boundary ilmdition throughout) [4.5] < = -!nrlR + &z’,’ IJlIf Wil~ll dent crff\ 3 = li.h’l
I?
<
/&
31111 aioprs
q,/R3.
[(eu - e,)&
(6)
the friction becomes indepenIhe Stokes-like form [4.5]
7$2,,0.
(7)
In 111~’ opposite limit where DT,/R” is very IarSe. 11~ predominant mechanism for polarization relaxaZion is translational diffusion. Further analysis [S] shows that if. in addition_ the inequality (DT~)’ -%RftO holds. then the ionic drag involves a diffusion iLYl~lh I\,,
jl‘hc
-loo
=
1 -I [rc-(co
GISS
-
~,)/7]4D]
yz
_
in which (DT, j’ Q R$,
w
cannot
be analyzed
- E,)/E; _
(9
On the other hand, for small ions we have R2 4 R& and with the condition (DT~)~ >>Rio, < takes the fomi [S] i_=const.X
nRD.
(10)
One should note that the condition (DT~)~ 2= Rio is equivalent to both Rh
Volume 95, number 4,s
CHEMICAL PHYSICS LETTERS
than in the HO theory, so that the inclusion of translational diffusion should improve the agreement with experiment_ A more striking example of the difference between dielectric friction dominated by either translational or rotational diffusion is provided by the behavior of the limiting conductances through the nematic-isotropic phase transition. Dielectric relaxation times in various nematogens are often observed to be 100 times larger in the nematic phase than in the isotropic *, and if the friction is dominated phase [2,10-121 by rotational diffusion, this implies a substantial discontinuity in the conductance as we pass through the clearing point. However, it has been observed that the conductivity as well as the translational self-diffusion
coefficient exhibits little change on passing through the clearing point [ 131, and this is in accord with a relaxation process dominated by translational motions. Data on D [ 14]$, TV and ho for salt-liquid crystal systems are rather sparse: and this is especially the ’ Dielectric relaxation times for 4,4’-pentylcyanobiphenyl (PCB) have been reported in ref. [ 101, viscosities for PCB are given in ref. [ 111, dielectric studies of 4,4tl-heptylcyanobiphenyl (HCB) are given in ref. [ 121. * Self-diffusion mefficicnts for 4’.4-methosy-cIj benzylidene (D-XIBCA). 4,4’-pentylcyanobiphenyl(5CB) and (D-5CB) have been reported in ref. [ 141.
11 March 1983
case near the transition point. Further measurements would be of considerable help in understanding the role and nature of dielectric friction and ionic motion in liquid crystals.
References [ 11 P-G. de Gennes, The ph$+s
of liquid crystals (Ciarcndon Press, Oxford, 1974). [2] P-G. Wolynes, J. Chem. Phys. 66 (1978) 473. 131 P- Colonomosand PG. Wolynes, J. Chem. Phys. 71 (1979) 2644. [41 j_ H&bard and L. Onsager, J. Chem. Phys. 67 (1977) 4850. 151 J-B. Hubbard. J. Chem. Phys. 68 (1978) 1649. 1’51J-B. Hubbard and R-F. Kayser, J. Chem. Phys. 74 (1981) 3535. I71 J.B. Hubbard and R-F. Kayser, Chem. Phys. 66 (1982) 377. 181 PJ_ Stiles, J-B. Hubbard and R-F_ Kayser, J. Chem. Phys. to be published_ I91 R. Hirino. J. Chem. Phys_ 74 (1981) 3016. 1101 P-G. Cummins, D.A. Dunmur and DA. Laidler, hlol. Cryst. Liquid Crvst. 30 (1975) 109.
1111 F. Kiry and P- hiartinot;. J. P&s. (PSs) 38 (1977) 153.
I121 hl. Davies, R. hioutnn. A.H. Price. Bf. Beevers and G. Williams. J. Chem. Sot Farads\, II 72 (1976) 1447. 1131 R. Hkino. private communication. 1141 A_J_ Leadbetter, F.P. Temme, A. Heidemann and W.S. Howells, Chem. I’hys. Letters 34 (1975) 363.
401
11 March 1983
CHtiMICALPHYSrcS tETTER.8
Volume 95, number 4J
ERICTION AND IONIC hlOBILITY
IN POLAR LIQUIDS AND LIQUID CRYSTALS
J.B. HUBBARD, R-F. KAYSER Titermop~zysics Division, :Vafioml Bureau of Standanfs, Washingfon, D-C_ 20234, USA
and P.J. STILES Cltemistry Departmenx. hlacquarie University, Norrh Ryde, NS. W, Ausrralk;r 21 I3
Received8 December 1982
We introduce3 continuum model of dielectric friction on an ion in a polar liquid_This model couples hydrodynamic motions of the polarized solvent to dielectric relaxation by both rotational diffusion and transIationa1diffusion of solvent molecules. We show that in solvents with sufficiently Ions dielectric relasation times, translationaldiffusion is the dominant relasation mechanism. We compare our predictions with esperimental data on ion mobilitics in nematic liquid CQSta1.S
The role of dielectric friction in determining the motion of an ion in a polar liquid has received considerable attention over the last few years. The basic idea is that an ion migrating through a polar solvent creates a transient non-equilibrium polarization in the vicinity of its trajectory. and the energy dissipated by the relaxation of this polarization is associated with a frictional force on the ion in addition to the viscous force. The relaxation can occur either locally through rotational diffusion (dipole reorientation), or nonlocally through translational diffusion, which should be included when the spatial gradients in the non~qu~ibr~urn polarization are sufficiently large. Continuum approaches to the problem of calculating ionic mobilities in “simple” polar liquids have. until now, dealt exclusively with the orientational relaxation mechanism and its coupling with viscous hydrodynamics. However, in nematic liquid crystals for example, highly anisotropic correlations may lead to unusually large (and anisotropic) dielectric relaxation times, and in this case Meyer and de Gennes [ 1] have called attention to the possible existence of an important relaxation mechanism involving translational diffusion. The role of translational motions in simple polar liquids has also been emphasized in a 0 009-2614~~3~000~0000~~
03.00 0 1983 North-boded
molecular theory of dielectric friction recently developed by Wolynes and Colonomos [2,3] _ We have recently formulated an electrohydrodynamic model of dielectric friction which includes relasation of the orientational polarization by both rotational and translational diffusion_ Details of this analysis will be presented in a forthcoming paper: we present here some of the main iesults. We have shown that the decay of the orientational (Debye) polarization, Pr,, in an inhomogeneously polarized stationary fluid may be described by the following generalization of the Debye formula:
aPD/ar = 7D1(XDE - pD)
-m’(xDE -
pD),
where E is the local electric field, & = (e. - e,)/4a is the susceptibility for orientational polarization (e8 and E, are the low- and high-frequency
dielectric
constants), rD is a Debye relaxation time, and D is a phenomenological coefficient associated with translational diffusion_ Microscopic considerations suggest that D might be approximated by the self-diffusion coefficient of a solvent dipole, bur strictly speaking collective motions should also be included_ If we now allow for hydrodynamic motion of the fluid, symmetry arguments based on the invariance of the dissipa399
11 March 1983
CHE%fICALPHYSICS LE?PTERS
Volume 95, number 4.5
tion with respect to rigid body motions dictate that the simplest generalization of eq. (1) is 14-61
by simple scaling arguments.] If the ion radius is then large enough to ensure that R” % Rk, it can be shown [S] that the ionic drag coefficient becomes
aIJ~,iaf+cu-v)P,+~~~x(vxu)
[ = 47rnR + ‘2 (z’e2/DR)(q, = P*/rL, - DV’P*.
\vhcrc the “polarization
(2)
deficiency”
P* is defined
by
I’* = Q,E -- PD.
(3)
ad u is the velocity
Iield of the fluid. In the absence cri incrrial effects and dielectric saturation [7]. I lubbmd md Onsager (HO) have shown [4.5] that rbc linearized equations of motion for an incompressihlc solvent may be written I)‘?% = Vjr .- +Eu X curl P* - +Eo div P*.
(4)
u.I~txc7) is the fluid shear viscosity. p is the “electrohydrodynamic” pressure and E. is the electric field
\vl~en u = 0 (static fluid). Instead of tackling the formidable task of finding a general expression for the drag on an ion of radius I; iron the coupled equations (.2) and (4). in this paper \ve restrict our attention to limiting cases. In the regime where DT~/R’ is very small. translational diffusion is negligible compared to rotational diffusion and WC recover the I10 theory. Iiere. the ionic drag dqm~ds on the relative magnitudes ofR and the HO 1W~I11 I:,,,
=
I-_%‘(E(,
-
’
E,)T,)/16zqE(j)
l/4
.
(5)
whcrc :c* is the ion charge. If R’ B R;‘io the friction sorfticient takes the form (we use t!te slip boundary ilmdition throughout) [4.5] < = -!nrlR + &z’,’ IJlIf Wil~ll dent crff\ 3 = li.h’l
I?
<
/&
31111 aioprs
q,/R3.
[(eu - e,)&
(6)
the friction becomes indepenIhe Stokes-like form [4.5]
7$2,,0.
(7)
In 111~’ opposite limit where DT,/R” is very IarSe. 11~ predominant mechanism for polarization relaxaZion is translational diffusion. Further analysis [S] shows that if. in addition_ the inequality (DT~)’ -%RftO holds. then the ionic drag involves a diffusion iLYl~lh I\,,
jl‘hc
-loo
=
1 -I [rc-(co
GISS
-
~,)/7]4D]
yz
_
in which (DT, j’ Q R$,
w
cannot
be analyzed
- E,)/E; _
(9
On the other hand, for small ions we have R2 4 R& and with the condition (DT~)~ >>Rio, < takes the fomi [S] i_=const.X
nRD.
(10)
One should note that the condition (DT~)~ 2= Rio is equivalent to both Rh
Volume 95, number 4,s
CHEMICAL PHYSICS LETTERS
than in the HO theory, so that the inclusion of translational diffusion should improve the agreement with experiment_ A more striking example of the difference between dielectric friction dominated by either translational or rotational diffusion is provided by the behavior of the limiting conductances through the nematic-isotropic phase transition. Dielectric relaxation times in various nematogens are often observed to be 100 times larger in the nematic phase than in the isotropic *, and if the friction is dominated phase [2,10-121 by rotational diffusion, this implies a substantial discontinuity in the conductance as we pass through the clearing point. However, it has been observed that the conductivity as well as the translational self-diffusion
coefficient exhibits little change on passing through the clearing point [ 131, and this is in accord with a relaxation process dominated by translational motions. Data on D [ 14]$, TV and ho for salt-liquid crystal systems are rather sparse: and this is especially the ’ Dielectric relaxation times for 4,4’-pentylcyanobiphenyl (PCB) have been reported in ref. [ 101, viscosities for PCB are given in ref. [ 111, dielectric studies of 4,4tl-heptylcyanobiphenyl (HCB) are given in ref. [ 121. * Self-diffusion mefficicnts for 4’.4-methosy-cIj benzylidene (D-XIBCA). 4,4’-pentylcyanobiphenyl(5CB) and (D-5CB) have been reported in ref. [ 141.
11 March 1983
case near the transition point. Further measurements would be of considerable help in understanding the role and nature of dielectric friction and ionic motion in liquid crystals.
References [ 11 P-G. de Gennes, The ph$+s
of liquid crystals (Ciarcndon Press, Oxford, 1974). [2] P-G. Wolynes, J. Chem. Phys. 66 (1978) 473. 131 P- Colonomosand PG. Wolynes, J. Chem. Phys. 71 (1979) 2644. [41 j_ H&bard and L. Onsager, J. Chem. Phys. 67 (1977) 4850. 151 J-B. Hubbard. J. Chem. Phys. 68 (1978) 1649. 1’51J-B. Hubbard and R-F. Kayser, J. Chem. Phys. 74 (1981) 3535. I71 J.B. Hubbard and R-F. Kayser, Chem. Phys. 66 (1982) 377. 181 PJ_ Stiles, J-B. Hubbard and R-F_ Kayser, J. Chem. Phys. to be published_ I91 R. Hirino. J. Chem. Phys_ 74 (1981) 3016. 1101 P-G. Cummins, D.A. Dunmur and DA. Laidler, hlol. Cryst. Liquid Crvst. 30 (1975) 109.
1111 F. Kiry and P- hiartinot;. J. P&s. (PSs) 38 (1977) 153.
I121 hl. Davies, R. hioutnn. A.H. Price. Bf. Beevers and G. Williams. J. Chem. Sot Farads\, II 72 (1976) 1447. 1131 R. Hkino. private communication. 1141 A_J_ Leadbetter, F.P. Temme, A. Heidemann and W.S. Howells, Chem. I’hys. Letters 34 (1975) 363.
401