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Polarization diffusion and dielectric friction in polar liquids. II
1. Introduction Several years ago Hubbard and Onsager (HO) [l] extended Debye’s formula for diekctric rehucation in polar liquids [2] to ahow for coupling with hydrodynamic flow_ De Gennes [3] had suggested earlier that Debye’s ‘relarration mechanism,~ based on IoeaI rotational diffusion of polar moIecuIes~ could be supplemented by non-local transIationaI diffusion of polar molecules down the gradient of the non-equiIibrium polarization field. We recentIy developed this concept and proposed a simple generalization of Debye’s equatiotr to allow for both relaxation channeis {4]_ This formahsm, which is appIicabIe whenever the gradients in the.otientational pblarization are sufficientIy large (eg in the vicinity of a solvated ion) has already been used by van de; Zwan and Hyn&s [5] to. anaiysez a variety of problems in the are&of solution kinetics_ A substantial reduction in dielectric friction has been found in several contexts [S] and the polarization charge in the vicinity. of an ion whose charge is suddeniy quenched can decay much faster than expected from the Debye charmel alone
[41-
Here, we extend our preIimit&~
analysis [4] of
0301_0104/85/S0330 Q PIskier Seiet&PubIishers (North-HoIIand~ Physics Publishing Division)
-he impact of the translation reIax&ion channel on the limiting mobihties of ions in polar solvents_ The ion’s motion creates a transient non-equilibrium orientational polarization in the xkinity of its path. Born’s notion [6] that the energy dissipated by the reiaxation of this polarization is associated with a frictionaI force on the ion which enhances the viscous friction, has recently experienced considerable development r/l at both the continuum [1,4,S-121 and moIeeuIar 113-151 Ievek Whereas mokcuIar theorics of pohuization dynamics in the vicinity of ions have emphasized the translational motions of solvent moIecules_ previ-. ous continuum models [1,8-121 -assumed that non-equihbrium orientational~ ~poIari+ation relaxed Sorely *tit& the -Debye char&I and included solvent translation only insofar as dieIe+ic i-eIaxation is coupkd to hydrodynamic flow. The present two-channel continuum model [4] clearly helps to reconciIe the -macroscopic and microscopic descriptionsIn section 2 we review the equations of motion and the corresponding dissipation ~function. We demonstrate tbac with appropriate electrohydk. dynamic. boundary conditions, the equations of motion can be deduced from the dissipation funcB-V_
8
P-I.
S*&_
_I_Ef_
tion through the principle of least dissipation_ The actual caIcuIation of the friction. which is reIativeIy straightforward when only a singie rclaxation channel is avaiIabIe [1,4.8-121 becomes more formidable when both channeIs operate simuhaneously. For this regime, section 3 demonstratei how the fundamentaI equations and boundary conditions can be manipulated to define the probIem mathemakaIIy and to provide a convenient computational algorithm. FinaIIy, in section 4, we compare the ionic frictional coefficient when both channels for diekctric rektxation are operating with the earlier rest&s of Hubbard and Onsager [l] based on the Debye channel alone
Z Ekctrohydrodynamics polarized diekcttic fItrId
of an inhomogeneousljc
Conservation of Iinear momentum continuum of density p takes the form pdo/dz = v -S,
in a fhrid
where d/dt denotes the convective derivative and S is the sum of the hydrodynamic and electric (Maxwell) srress ?ensors fl]_ When the steady state version of *Ihisequation is Iinearized in the hydrodynamic velocity ZJof a viscous. incompressible fItid we obtain the HO equation [1] qvsi=vp-$dE,x(v-XP’)-$E,(v-P*)* (2&I (2-2b:l
in which g is the shear viscosity, p the dynamic pressure, E, the electrostatic field when u = 0 and P*, the pohuization Udeficiency’* * [l]- is a measure of the non-equihbrium polarization in the soIvent_ It is d=&imedin terms of the electric field E and orientationai polarization Pr, as p=x,,E-I’,,,
(23
where, xn = (co - c,)_/4a is the susceptibility fur orientational poIarization_ The generakation~ of the Debye ~equation appropriate to an &homogcneousIy pokiiized dielec, =. Van derZaarr P,=-P_
and Hyna
[a d&ire
a +k%ition
tric fluid is [4]. apn/i3r -I- 0’ VP, = P’/rD
‘excess”_
-I- $Pn x( V x 0)
- Dv’P*.
(2.4)
Here, or, is the Debye reIaxation~.timeof the solvent and D is a phenomenologicai coefficient associated with translational diffusion Microscopic considerations [4] suggest that D might be approximated by the self-difiusion coefficient of the solvent molecules_ We also introduced a poIaiization flux whose divergence in an quiescent tensor J_. fluid yieIds the polarization current associated with transIationaI diffusion of dipolar molecules Eq_ (24) requires this translational component of the totaI flux to be given by (2-5) Jzw= Dvp’The rate at which energy is dissipated in unit volume of the fluid is then given by the dissipation function. 9=
(2-l I
v -o=o,
-.
Hu&&nd)PosiuiamiondifjiUiOll in poinrG+f.S
$!.,* f $J = 27I(
voy: ( vo)’
c~~‘[~P’~~/~~+D(vP~):(vP*)].
(2-h)
in which the superscript -s” indicates the symmetric part of the velocity gradient Vv_ It is reassuring that the dissipation function (2_6), in conjunction with the ekctrohydrodynamic boundary condition 141, ii-(f[z;Po--~~tI(P,-c)]
+DvP*}
=O.
(2.7)
obtained from our generaked Debye equation (2-4) can be used to derive the equation of motion (22a) from the principle of minimum cncrgy dissipation- The gIobaI dissipation rate, ///-4-a3r. is minimized with -respect to the hydrodynamic velocity field u subject to the_ incompressibility constraint V -u=O and the condition that the variation 60 in u vanisheson the boundaries of the fluid_ We show in the appendix that
#JS$f_
d3r
md
= -
$.iJ6r;[
_ rud
+&X(V
Eo(v-Pf)
XP*)-V(i-&]
--1.
d3r_
~.
-(2X)
This expression supplements -the Weli-kn0~ ’ .‘drodynamic result [16]
hyi
..-.
If X is the Lagrange multipiier for the incompressibility condition v -o= 0. least dissipation is en-. sured when
M( SS-
Xv
.At &fficientIy--hqe.dista&& (+ &);$=o_m- t&e’: -ion, the electric force den&y Fa .can’ i+ng-g&ted, a&d eq_ (22a) reduces t&-the NavierStok&:-. equation$ich i%satisfied by _. -1’ : I f(f)=-1
The frictionai coefficient J;. fohows standard hydrodynamic resuIt ,
from
the.
(3.3) _’ and can also be -caIcuIated vi+ ‘the dissipation : function from
fluid
IV’
xl”*)--vp]
.‘._.>-(3.+
<-= 4r;zc,RlI,
-SD) d?r = 0,
i.e. when
+&x(v
+K,(R/+Ks(R/f)‘__
(2-11)
d3r=0,
where p = X -I- SE,, - P’_ The equation of motion (2.2a) is therefore completely consistent with the dissipation function (2.6) if the boundary condition (27) is satisfied_ We aIso note that a formaI proof, similar to that in the appendix, can be given for the correspondence
=
$-- +i
d3r_
The vector fieI& c(r) and P*(r) ikhikh determine the coefficients K, and K3 in the asymptotic .form (3.2) of the velocity field can be regarded as soIutions of the coupled differentiaI equations (22) and (2_4)_ We invoke the standard hydrodynamic boundary conditions
F(C) u(r)-
is finite for aII --u
as
r>
(35a)
R;
rdco,
(3Jb)
abeady impIicit in (32). and between the global electric dissipation and the rate at which work is performed by the dissipative electric body force (2-13)
F~=fE,(V-P*)~~E,x(vxP*). from the equation of motion (22a)_
F-o=0
frictionalcoefficientofa transIatingion
We now consider the steady motion of a spherical ion of radius R. and charge ze through an inftite dielectric fIuid_ If, reIative to the ion. the velocity of- the fluid at infinity is the constant vector z/k - Uz^ oriented along the polar (5) axis of a spherical poIar coordinate system, the divergence-free velocity field for incompressible flow around the ion ca.rr be ‘e.xpr&sed in terms_ of the scalar function, J(r), defined by -.
o=fvx[rXClj(r)j_
_.
-.
..
(3.1)
r=R,
(35c)
which ensures that the soixnt cannot penetrate the ion_ The remaining boundary conditions are electrohydrodynamic in character_ We assume that the polarization deficiency due to the fIow vanishes at infinity: P*( r)+O
3,The
when
as r--,cx)_
(3-6)
The surface force, SS,; defined as the ?Iiscontinuity in the shearing stress ‘S, at the~ionic surface, has been shown [4,10] to take the form * SS,=S,(hyd)++(P,*E,--EE,P,*),
‘m
(3-B)
in which the fieIdsP* and E are those in the fluid just outside the ion. To first order in the relative
* In pint I (see ret
[4j) the factor of 4s;jn cq. (55) sboukt be replaced by $; and in cq_ (53) the arpersaipc (f &r ~-) asociataiuithca+~~shddbcmersaiinsi~ :-
velocity this Surface force reduces to
condition(3_6).taketheforn~~
.-
ss,.=s,(iqd)
- fEm_,;,f.
(3_7b>
k-IF(r)
=Ai(r)
S,(hyd)
= $E,_,P;_
‘-
(3_7c)
Provided that the normal component of the polarization flux through the ionic surface aIso vanishes we have the additional dectrohydrodynamic boundary condition (2_7), where, for a spherical ion the unit normal fi = F_ We turn now to the solution of the differential equations (22) and. (24) subject to the boundaty conditions (2.7). (35). (3.6) and (3-7~). If the y*poIarization deficiency** in the dielectric fluid is written as P’(r.
8) = P,‘(r)
cos 8 i-t- P;(r)
sin 88,
(3-Q
:.“.
:
di
_tn(rj/cb(r)c(r)
’ I.
dr.
--b(r)Lsn(r)~&ri
(3112a)
:-. and G(r)
= Br-’
e-“’
- r -* sinh kr p(kt)ml I--
l r-’
e-h
rRl(kt)-i
evrrdit)
dr
sinh kt d(z)
dr, (3_12b)
where a(t)
= [(kt)-*
sinb kr
+ 3(kr)-3] wsh kt,
-3(kt)-2 b(r) = e-“[(X-r)-’
(3.12~)
-I- 3(kt)-‘+
3(kf)-3].
the curl of (22a) becomes
(3-126)
r=f”‘~(l-)i_8r’f”(r)~88r=f”(r)-s8rf’(r) = -/3(4r-V(r)
- ;r[F’(r)--(r)]),
(3-9)
= D&J;(r)
+ PBf(r)]/rexoU*
G(r)=Dc,[P:(r)-2Pg(r)l/zex,,U
(3.12e)
d(r)=;rf”(r)+-f’(r).
(3.12f)
(3lOa)
u,(r)
(3.10b)
uq(r) =1”‘(r), u,(r)
and $ = z2e’xJ&D_
(3_1oc)
If k = I/(DrD)‘n, the scalar components of the steady-state version of the vector equation (24) for the pokuization deficieucy can be-regged as
=%fl(r)++r(r)-3r-‘/(r)
(3_lla)
and
=f(r).
r+(r)
=f’(r).
Us(r) =f”(r),
us(r) = F(r).
= G(r),
us(r)
z+(r)
= F(r), (3.13)
= G’(r).
in the dependent variables. the fourth-order equation (3.9) together with the two second-order equations (3.11) become equivalent to the foIIowing set of eight fit-order differential equations u:(r)
r’F’*(r)tZrF(r)-(k’r’+6)F(r)
r%“(r)
C(f)=~ff”(f)+~f’(f)-3r-~~(r).
With the further changes
where F(r)
1..~-
oc’
where Ei_, = x/co R” is the radial field in the stationary fIuid_ The appropriate generalization of _ the hydrodynamk hound.ary condition describing perfect siipping is therefore
-.
= 5 cii(r)uj(r) i-1
(i=
l-S)_.
(3-14)
The twenty-two non-zero coefficients are
.-
c,,=c,=c,=c,=c,=l; i- ZrG’(r)
= -
- k’r’G(r)
-r(r)_
=41
(3llb)
GeneraI solutions to (3.11) satisfying the boundary
=
-$rr6,
~c,~~8r-3+L
_+-‘+*/+-4,
=43
=
c45
e$@(k2r-3
2/3r-5,
=44X _
&-9,
&
-$Jr-‘, &
L
...-
-~
: f@-4,
~.
c;.=kqb(Ri)‘rka(Ri)
Although’thIs
in standard form. half of *he boundary conditions are at -the ionic surface and the other half apply at infinity. A simple algorithm for solving this twopoint boundary value probiem can bc obtained by exploiting the linearity of the system-and the continuity of the eight components of u(r) acrosS the surface-r = R,. beyond which the -flow is essentiallystoke&n_ An arbitrary solution of the firstorder system (3.14) is provided by .the linear combination H(r)=
5
i-1
(316)
&P(r);
of the fundamental set of column vectors &‘(r) sarisf~~~(3.14)andtheinitialconditions &>(R,)L&'*
(3.17)
in which the elements of the unit vectors e(i) are given by ejir = Sii and Sii is the Kronecker delta_ Numerical techniques such as the Runge-Kutta procedure can be used to solve the system (3-14) for the r&“(r), subject to the initial conditions (3.17), over the restrictedintervalR G r G R,. Only the u(i)(R) imd be retained_.Clearly, the eight coefficients Ci must be selectedso that all bo-undary conditions. on u_and P* -aresatisfied. Four of the eight conditions have aheady been used to obtain the asymptotic form (32) for u(r) and the expres-. sions (3-12) for P(r)_.. Continuity of all eight components of u-across the sphericalsurface r.= R, permits the Cj to be expressed in terms of K,;K,, AandEWefind . . -. -.. C, = 1 + KJUZ,’ c;=
.
seiof first-order equations is now
+ K3i3RF3,
-&R;2--3K3R’R;i,
C,+,I&;‘+
pK,RjR;‘,
C, = -&RR;4
- 60K3R3i+,
_.
]- (3S8a)
.:._
.I- 1(3:18b)
--~--
:., (518+
-.-~
‘(3.1&i)
x
\<-Tb(i)c,(i
1R~(kf)-%-‘rd&j 1. ,-_
C, = B(r-’
e-k’)R,.+(~-’
30 x
1R,
(kr)l’
dt, sinh &r);i,
e’*‘dm(t)
dt_
(3.18~) .:
-(3_1Sh)
Here. c,(t) and J=(t); the asymptotic forms. of c(r) and ci(r) at Iarge distanti from the ion, .: follow from (3.2) and (3.12) aS 1 -~ ~.:. f K,RI-2
C,(f)
= -3Q-r
d=,(t)
= 3K3R3t-‘_
-F&k3i~E4),
(3,lSi) (3.2s;)
coefficients K,, K3, A and B .a&~unreadily determined from the four remaining boundary conditions at the ionic surface. The reduced forms of the surface conditions (~SC), (3.7~) and the radial and tangential components of (27) are
The
(3.19a)
q(R)=% u~(R)+QRLI~(R)--~~R-~[II~(R)-u,(R)]
=o.
(3.19b) u~(R)-$R-~u~(R)=O,
(3_mj
u,(R’)
(3.196)
t+R-‘u,(R)=O_
On substituting the coefficients (3-18) -and the fundamental set of vectors, r@(R), at the ionic surface into (116) and.then& i& (3.19)we obtain four equations, linear in K,, K3, A and B, which can be solved straightforwardlyfor Ki_ ~The. frictional coefficient 3 follows directly from eq. (3.3).
FrietionaJcoefficients for univalentions of various~sizes in acetoni&e and~wakrat25"C we& computed using -the sohfent prop&ties- listed ‘in
table II The phenomcnoIogicaI scriiing
the rate at whicIi
coeffrkient.D,
the solvent
dc-
Table 3 Dependence of t& axration factor A (A = &,/j~Rq) for diekcuic friction on univaht ions. in acetonitrileand uater at ts=‘C. on the pohrization diffusion cocfftien: D
polarization
dcwn its own gradient, ha& been taken. to coefficient of the solvent in table 2, which compares frictionaI coeffi+nts k-cm the present thee-y with those from the HO modei The coefficient D is fiied for a particrdar sokent at a given temperature and pressur e. but we cannot be. sure that D is well approximated by the sokent’s self-diffusion coefficient_ For this reason D has been ailowed to vary and in table 3 we examine the sensitivity of the friction to this parametcz_ Table 2 shows that the addition of the transhttional channel for polarization decay causes the friction on a migrating ion to dechne markedIy_ in some cases to onIy 50% of its vaIue in the HO limit where the polarization diffusion coefficient, D, vanishes. Tabk 3 suggests that as D increases and dielectric relaxation becomes faster the net friction decreases monotonicahy from its value in the HO limit SimiIar conclusions have been obtained by diffuses
the seEdiffusion
be
Table 1 Phykal propaxks of amonic@Ie and of axrcr at 25’C azd standard pressure
!50IVcnt aeconicrik
-iKttcr
c0
Cm
1o=q (gem-’
5-1)
36.0 a* LO r’ 0341 *I 78.3”’ 52” 0.890=’
=’ From rcf_ [81.
b’ From ref. [In
lo=, (s)
IOSD (cm’ s- 1 )
39 -’ 82 -’
4.34 b’ 231e
=’ From ref. [18l_
Table 2 Corrrctioo factorsA (A = &+/&Rq) ior didaxric friction OP uni~-a!entions in acetonicriIeand water at 2S°C Water radius
b.
R 6) 1.0 15
20 25 3.0 4-o 5-O
l-75 126 1.1: i-05 1.03 I.010 x004
z26 1.48 1.19 LO9 I_04 1.014 1_006
131 1.10 L_lx 1.02 I-01 1_003 l_aOl
1_72 122 1.08 1.03 l-02 I_905 Lo02
superscriptHO rcfax to the Hubbard-Onsager which does not imorporatc poIarization diffhsion.
m’ l-be
moc!d
_
10’ D
A &acemnicrile)
(cm2s-1)
R-IA
R=Za
R=IA
226 l-96 l-78 166 158 I51 134
l-19 1.13 l-11 -l_lO l-09 l-08 1.06
l-72 133 123 l-18
0.0
20 4.0
6-O 8.0
10-O 20.0
van der Zwanand
A <=-) R=2.& -1-08 1.00 1.03 1.03
l-15
I-02
I-13 1M
1.02 1.01
.
Hynes [5] who examined diekctric friction on a charge moving along the axis of an infinite cylindrical cavity surrounded by a dielectric Experimental data for Iimiting conductivities of ions in weakIy associated pohu solvents show that the HO frictional coefficients underestimate the drag due to the solvent, so the incorporation of polarization diffusion actually broadens the gap between predicted and experimental conductivities Nevertheks, when sIip boundary conditions are used. the present model based on spatially homogeuneous solvent properties qualitatively reproduces the observed [Sj minimum value when the frictional coefficient is pIot:ed as a function of ionic radius_ Refatively simpIe continuum models of didectric friction still emerge when the assumption of a spatially homogeneous solvent is relaxed [12]. A recent examination of the effect of solvent electrostriction on ionic mobilit& [19] shows that the enhanced solvent density in the vicinity of the ion increases the viscous drag of weakIy associated solvents quite appreciably and therefore counter: acts the effect of poIa.rization diffusion AIthough sirnpie -continuum treatments may successfuIIy model rather general perturbations of the static and dynamic solvent properties they. disregard more specific factors, such as. s4ucture making/ breakiug and solvent residence times_inthe coordination shell [20,21), which caninfIuenCethe IocaI viscosity [21.U]. Such models cannot therefore be expected to provide p&tkuIarly accurate descriptions of frictionaI coefficients for smaII ions.. Prob-
We wish to thank Dr_ B_M_N_ Clarke for helpful dkcussions on computationai aspects of this work
= --/1/[2(~v):(P*S0);(SoV):~~P*) -(S&):(P*&)]
Appenais From (2-S) we set that the rate at which ekctriCal m.ergy is dissipated in the fluid is
+ D( VP*) : ( VU’*)]
d3r. (A-l)
With the aid of the boundary conditions (27) Su = 0 the surface integral in the identity,
SJl(VP*) =-
I//
t ( vSP*)d3r P*
l
v=SP*
+j-jP’-(B-v)SP* -is seen to
JI_S$.=
and
d3r d’r,
(A-2)
vanishand(A.l)canberecastas
d3r
~x~‘JJJP* -( 76%~ - DV’SP*) d3r. (A-3)
From the steady-&e version of our eq_ (2.4) for the polarization deficiency, the integral on the right becomes So-v)E,+EOx(~xS~)]-
d3r_
d3r,
which reduces straightforwardly to (2.8) when the eIectrostatic conditions v xE,= V l E,=O_ are applied.
[I]
J_ Hubbard 4850;
and L
Onsager.
J_ Ghan
Phys
67 (1977)
J-B- Hubbud. J. Chun Ph_ys 68 (1978) X49_ [2] P. Dcbyc Polar mokczdcs (Dover. New York. 1929)_ [3] P-G- de Games. The ph>sics of Ec+d crjsmls (-on Press. oxford. 1974)_ [4] J-B. Hubbard, RX_ Kqser and P-3. Stiks, Chan Phys_ Letters 95 (1983) 399; P-7. Stiles and J-B. Hubbard. Cham Phys 84 0384) 431 (pan 1). (51 G. van dcr Zand J.-f’_ Hyns, Physica 121A (1983) 227; Ghan Phy% Letters lOl(l983) 367_ [q hi. Fkm, Z Physik X(1920) 22L f7l P-G_ Wolynes. Ann Rev_ Phys_ Ghan 31(1980) 345. J-B_ Hubbard and P-G_ WoIynq [8] D_F_ Emns. -I-_-I-J. Pbyx Cbani 83 (X979) X69_ [9] JB_ Hubbard and R_F_ Ka>wr, J_ Clan_ Pb>x 74 (1981) 3535_ [IO] B-U- FcIdcrhof, MoL Phys. 48 (1983) 1093: 49 (1983) 449_ [ll] E Nomk, J. Chcza Phys 79 0983) 976. [12] J.B. Hubbard 377:
and R.F_ Kay%
*an.
Phyx_ 66 0982)
PI StiIcs. J-B. Hubbard and RF. Kayxcr. J_ Ghan. Phys n (1982) 6189_ [13] P-G_ Wolynes I Chui Ph>x 68 (1978) 473_ 1141 P- ~onomos and P-G- Wolyne J. (3hem Phys 710979)
[1!5l it!&aIef
zad P-G_ Wolynes, 470; 78 (l983) 4145.
[!6]
K Lamb, Hj$ro$ynamics London. 1932).
RL I 78
Hurie and LA. (1982) W3_.
J. Chai (*&ige
Woolf. J. Chca~ Sk
Phus_ 78 (l&3) &iv_
Press,
FaradayT&s
_
14
Pf:
sfires
is_Hubbnrd /
Pci&hfim
diif/ioianin p&r
.Liqui&
-; :
[41-
Here, we extend our preIimit&~
analysis [4] of
0301_0104/85/S0330 Q PIskier Seiet&PubIishers (North-HoIIand~ Physics Publishing Division)
-he impact of the translation reIax&ion channel on the limiting mobihties of ions in polar solvents_ The ion’s motion creates a transient non-equilibrium orientational polarization in the xkinity of its path. Born’s notion [6] that the energy dissipated by the reiaxation of this polarization is associated with a frictionaI force on the ion which enhances the viscous friction, has recently experienced considerable development r/l at both the continuum [1,4,S-121 and moIeeuIar 113-151 Ievek Whereas mokcuIar theorics of pohuization dynamics in the vicinity of ions have emphasized the translational motions of solvent moIecules_ previ-. ous continuum models [1,8-121 -assumed that non-equihbrium orientational~ ~poIari+ation relaxed Sorely *tit& the -Debye char&I and included solvent translation only insofar as dieIe+ic i-eIaxation is coupkd to hydrodynamic flow. The present two-channel continuum model [4] clearly helps to reconciIe the -macroscopic and microscopic descriptionsIn section 2 we review the equations of motion and the corresponding dissipation ~function. We demonstrate tbac with appropriate electrohydk. dynamic. boundary conditions, the equations of motion can be deduced from the dissipation funcB-V_
8
P-I.
S*&_
_I_Ef_
tion through the principle of least dissipation_ The actual caIcuIation of the friction. which is reIativeIy straightforward when only a singie rclaxation channel is avaiIabIe [1,4.8-121 becomes more formidable when both channeIs operate simuhaneously. For this regime, section 3 demonstratei how the fundamentaI equations and boundary conditions can be manipulated to define the probIem mathemakaIIy and to provide a convenient computational algorithm. FinaIIy, in section 4, we compare the ionic frictional coefficient when both channels for diekctric rektxation are operating with the earlier rest&s of Hubbard and Onsager [l] based on the Debye channel alone
Z Ekctrohydrodynamics polarized diekcttic fItrId
of an inhomogeneousljc
Conservation of Iinear momentum continuum of density p takes the form pdo/dz = v -S,
in a fhrid
where d/dt denotes the convective derivative and S is the sum of the hydrodynamic and electric (Maxwell) srress ?ensors fl]_ When the steady state version of *Ihisequation is Iinearized in the hydrodynamic velocity ZJof a viscous. incompressible fItid we obtain the HO equation [1] qvsi=vp-$dE,x(v-XP’)-$E,(v-P*)* (2&I (2-2b:l
in which g is the shear viscosity, p the dynamic pressure, E, the electrostatic field when u = 0 and P*, the pohuization Udeficiency’* * [l]- is a measure of the non-equihbrium polarization in the soIvent_ It is d=&imedin terms of the electric field E and orientationai polarization Pr, as p=x,,E-I’,,,
(23
where, xn = (co - c,)_/4a is the susceptibility fur orientational poIarization_ The generakation~ of the Debye ~equation appropriate to an &homogcneousIy pokiiized dielec, =. Van derZaarr P,=-P_
and Hyna
[a d&ire
a +k%ition
tric fluid is [4]. apn/i3r -I- 0’ VP, = P’/rD
‘excess”_
-I- $Pn x( V x 0)
- Dv’P*.
(2.4)
Here, or, is the Debye reIaxation~.timeof the solvent and D is a phenomenologicai coefficient associated with translational diffusion Microscopic considerations [4] suggest that D might be approximated by the self-difiusion coefficient of the solvent molecules_ We also introduced a poIaiization flux whose divergence in an quiescent tensor J_. fluid yieIds the polarization current associated with transIationaI diffusion of dipolar molecules Eq_ (24) requires this translational component of the totaI flux to be given by (2-5) Jzw= Dvp’The rate at which energy is dissipated in unit volume of the fluid is then given by the dissipation function. 9=
(2-l I
v -o=o,
-.
Hu&&nd)PosiuiamiondifjiUiOll in poinrG+f.S
$!.,* f $J = 27I(
voy: ( vo)’
c~~‘[~P’~~/~~+D(vP~):(vP*)].
(2-h)
in which the superscript -s” indicates the symmetric part of the velocity gradient Vv_ It is reassuring that the dissipation function (2_6), in conjunction with the ekctrohydrodynamic boundary condition 141, ii-(f[z;Po--~~tI(P,-c)]
+DvP*}
=O.
(2.7)
obtained from our generaked Debye equation (2-4) can be used to derive the equation of motion (22a) from the principle of minimum cncrgy dissipation- The gIobaI dissipation rate, ///-4-a3r. is minimized with -respect to the hydrodynamic velocity field u subject to the_ incompressibility constraint V -u=O and the condition that the variation 60 in u vanisheson the boundaries of the fluid_ We show in the appendix that
#JS$f_
d3r
md
= -
$.iJ6r;[
_ rud
+&X(V
Eo(v-Pf)
XP*)-V(i-&]
--1.
d3r_
~.
-(2X)
This expression supplements -the Weli-kn0~ ’ .‘drodynamic result [16]
hyi
..-.
If X is the Lagrange multipiier for the incompressibility condition v -o= 0. least dissipation is en-. sured when
M( SS-
Xv
.At &fficientIy--hqe.dista&& (+ &);$=o_m- t&e’: -ion, the electric force den&y Fa .can’ i+ng-g&ted, a&d eq_ (22a) reduces t&-the NavierStok&:-. equation$ich i%satisfied by _. -1’ : I f(f)=-1
The frictionai coefficient J;. fohows standard hydrodynamic resuIt ,
from
the.
(3.3) _’ and can also be -caIcuIated vi+ ‘the dissipation : function from
fluid
IV’
xl”*)--vp]
.‘._.>-(3.+
<-= 4r;zc,RlI,
-SD) d?r = 0,
i.e. when
+&x(v
+K,(R/+Ks(R/f)‘__
(2-11)
d3r=0,
where p = X -I- SE,, - P’_ The equation of motion (2.2a) is therefore completely consistent with the dissipation function (2.6) if the boundary condition (27) is satisfied_ We aIso note that a formaI proof, similar to that in the appendix, can be given for the correspondence
=
$-- +i
d3r_
The vector fieI& c(r) and P*(r) ikhikh determine the coefficients K, and K3 in the asymptotic .form (3.2) of the velocity field can be regarded as soIutions of the coupled differentiaI equations (22) and (2_4)_ We invoke the standard hydrodynamic boundary conditions
F(C) u(r)-
is finite for aII --u
as
r>
(35a)
R;
rdco,
(3Jb)
abeady impIicit in (32). and between the global electric dissipation and the rate at which work is performed by the dissipative electric body force (2-13)
F~=fE,(V-P*)~~E,x(vxP*). from the equation of motion (22a)_
F-o=0
frictionalcoefficientofa transIatingion
We now consider the steady motion of a spherical ion of radius R. and charge ze through an inftite dielectric fIuid_ If, reIative to the ion. the velocity of- the fluid at infinity is the constant vector z/k - Uz^ oriented along the polar (5) axis of a spherical poIar coordinate system, the divergence-free velocity field for incompressible flow around the ion ca.rr be ‘e.xpr&sed in terms_ of the scalar function, J(r), defined by -.
o=fvx[rXClj(r)j_
_.
-.
..
(3.1)
r=R,
(35c)
which ensures that the soixnt cannot penetrate the ion_ The remaining boundary conditions are electrohydrodynamic in character_ We assume that the polarization deficiency due to the fIow vanishes at infinity: P*( r)+O
3,The
when
as r--,cx)_
(3-6)
The surface force, SS,; defined as the ?Iiscontinuity in the shearing stress ‘S, at the~ionic surface, has been shown [4,10] to take the form * SS,=S,(hyd)++(P,*E,--EE,P,*),
‘m
(3-B)
in which the fieIdsP* and E are those in the fluid just outside the ion. To first order in the relative
* In pint I (see ret
[4j) the factor of 4s;jn cq. (55) sboukt be replaced by $; and in cq_ (53) the arpersaipc (f &r ~-) asociataiuithca+~~shddbcmersaiinsi~ :-
velocity this Surface force reduces to
condition(3_6).taketheforn~~
.-
ss,.=s,(iqd)
- fEm_,;,f.
(3_7b>
k-IF(r)
=Ai(r)
S,(hyd)
= $E,_,P;_
‘-
(3_7c)
Provided that the normal component of the polarization flux through the ionic surface aIso vanishes we have the additional dectrohydrodynamic boundary condition (2_7), where, for a spherical ion the unit normal fi = F_ We turn now to the solution of the differential equations (22) and. (24) subject to the boundaty conditions (2.7). (35). (3.6) and (3-7~). If the y*poIarization deficiency** in the dielectric fluid is written as P’(r.
8) = P,‘(r)
cos 8 i-t- P;(r)
sin 88,
(3-Q
:.“.
:
di
_tn(rj/cb(r)c(r)
’ I.
dr.
--b(r)Lsn(r)~&ri
(3112a)
:-. and G(r)
= Br-’
e-“’
- r -* sinh kr p(kt)ml I--
l r-’
e-h
rRl(kt)-i
evrrdit)
dr
sinh kt d(z)
dr, (3_12b)
where a(t)
= [(kt)-*
sinb kr
+ 3(kr)-3] wsh kt,
-3(kt)-2 b(r) = e-“[(X-r)-’
(3.12~)
-I- 3(kt)-‘+
3(kf)-3].
the curl of (22a) becomes
(3-126)
r=f”‘~(l-)i_8r’f”(r)~88r=f”(r)-s8rf’(r) = -/3(4r-V(r)
- ;r[F’(r)--(r)]),
(3-9)
= D&J;(r)
+ PBf(r)]/rexoU*
G(r)=Dc,[P:(r)-2Pg(r)l/zex,,U
(3.12e)
d(r)=;rf”(r)+-f’(r).
(3.12f)
(3lOa)
u,(r)
(3.10b)
uq(r) =1”‘(r), u,(r)
and $ = z2e’xJ&D_
(3_1oc)
If k = I/(DrD)‘n, the scalar components of the steady-state version of the vector equation (24) for the pokuization deficieucy can be-regged as
=%fl(r)++r(r)-3r-‘/(r)
(3_lla)
and
=f(r).
r+(r)
=f’(r).
Us(r) =f”(r),
us(r) = F(r).
= G(r),
us(r)
z+(r)
= F(r), (3.13)
= G’(r).
in the dependent variables. the fourth-order equation (3.9) together with the two second-order equations (3.11) become equivalent to the foIIowing set of eight fit-order differential equations u:(r)
r’F’*(r)tZrF(r)-(k’r’+6)F(r)
r%“(r)
C(f)=~ff”(f)+~f’(f)-3r-~~(r).
With the further changes
where F(r)
1..~-
oc’
where Ei_, = x/co R” is the radial field in the stationary fIuid_ The appropriate generalization of _ the hydrodynamk hound.ary condition describing perfect siipping is therefore
-.
= 5 cii(r)uj(r) i-1
(i=
l-S)_.
(3-14)
The twenty-two non-zero coefficients are
.-
c,,=c,=c,=c,=c,=l; i- ZrG’(r)
= -
- k’r’G(r)
-r(r)_
=41
(3llb)
GeneraI solutions to (3.11) satisfying the boundary
=
-$rr6,
~c,~~8r-3+L
_+-‘+*/+-4,
=43
=
c45
e$@(k2r-3
2/3r-5,
=44X _
&-9,
&
-$Jr-‘, &
L
...-
-~
: f@-4,
~.
c;.=kqb(Ri)‘rka(Ri)
Although’thIs
in standard form. half of *he boundary conditions are at -the ionic surface and the other half apply at infinity. A simple algorithm for solving this twopoint boundary value probiem can bc obtained by exploiting the linearity of the system-and the continuity of the eight components of u(r) acrosS the surface-r = R,. beyond which the -flow is essentiallystoke&n_ An arbitrary solution of the firstorder system (3.14) is provided by .the linear combination H(r)=
5
i-1
(316)
&P(r);
of the fundamental set of column vectors &‘(r) sarisf~~~(3.14)andtheinitialconditions &>(R,)L&'*
(3.17)
in which the elements of the unit vectors e(i) are given by ejir = Sii and Sii is the Kronecker delta_ Numerical techniques such as the Runge-Kutta procedure can be used to solve the system (3-14) for the r&“(r), subject to the initial conditions (3.17), over the restrictedintervalR G r G R,. Only the u(i)(R) imd be retained_.Clearly, the eight coefficients Ci must be selectedso that all bo-undary conditions. on u_and P* -aresatisfied. Four of the eight conditions have aheady been used to obtain the asymptotic form (32) for u(r) and the expres-. sions (3-12) for P(r)_.. Continuity of all eight components of u-across the sphericalsurface r.= R, permits the Cj to be expressed in terms of K,;K,, AandEWefind . . -. -.. C, = 1 + KJUZ,’ c;=
.
seiof first-order equations is now
+ K3i3RF3,
-&R;2--3K3R’R;i,
C,+,I&;‘+
pK,RjR;‘,
C, = -&RR;4
- 60K3R3i+,
_.
]- (3S8a)
.:._
.I- 1(3:18b)
--~--
:., (518+
-.-~
‘(3.1&i)
x
\<-Tb(i)c,(i
1R~(kf)-%-‘rd&j 1. ,-_
C, = B(r-’
e-k’)R,.+(~-’
30 x
1R,
(kr)l’
dt, sinh &r);i,
e’*‘dm(t)
dt_
(3.18~) .:
-(3_1Sh)
Here. c,(t) and J=(t); the asymptotic forms. of c(r) and ci(r) at Iarge distanti from the ion, .: follow from (3.2) and (3.12) aS 1 -~ ~.:. f K,RI-2
C,(f)
= -3Q-r
d=,(t)
= 3K3R3t-‘_
-F&k3i~E4),
(3,lSi) (3.2s;)
coefficients K,, K3, A and B .a&~unreadily determined from the four remaining boundary conditions at the ionic surface. The reduced forms of the surface conditions (~SC), (3.7~) and the radial and tangential components of (27) are
The
(3.19a)
q(R)=% u~(R)+QRLI~(R)--~~R-~[II~(R)-u,(R)]
=o.
(3.19b) u~(R)-$R-~u~(R)=O,
(3_mj
u,(R’)
(3.196)
t+R-‘u,(R)=O_
On substituting the coefficients (3-18) -and the fundamental set of vectors, r@(R), at the ionic surface into (116) and.then& i& (3.19)we obtain four equations, linear in K,, K3, A and B, which can be solved straightforwardlyfor Ki_ ~The. frictional coefficient 3 follows directly from eq. (3.3).
FrietionaJcoefficients for univalentions of various~sizes in acetoni&e and~wakrat25"C we& computed using -the sohfent prop&ties- listed ‘in
table II The phenomcnoIogicaI scriiing
the rate at whicIi
coeffrkient.D,
the solvent
dc-
Table 3 Dependence of t& axration factor A (A = &,/j~Rq) for diekcuic friction on univaht ions. in acetonitrileand uater at ts=‘C. on the pohrization diffusion cocfftien: D
polarization
dcwn its own gradient, ha& been taken. to coefficient of the solvent in table 2, which compares frictionaI coeffi+nts k-cm the present thee-y with those from the HO modei The coefficient D is fiied for a particrdar sokent at a given temperature and pressur e. but we cannot be. sure that D is well approximated by the sokent’s self-diffusion coefficient_ For this reason D has been ailowed to vary and in table 3 we examine the sensitivity of the friction to this parametcz_ Table 2 shows that the addition of the transhttional channel for polarization decay causes the friction on a migrating ion to dechne markedIy_ in some cases to onIy 50% of its vaIue in the HO limit where the polarization diffusion coefficient, D, vanishes. Tabk 3 suggests that as D increases and dielectric relaxation becomes faster the net friction decreases monotonicahy from its value in the HO limit SimiIar conclusions have been obtained by diffuses
the seEdiffusion
be
Table 1 Phykal propaxks of amonic@Ie and of axrcr at 25’C azd standard pressure
!50IVcnt aeconicrik
-iKttcr
c0
Cm
1o=q (gem-’
5-1)
36.0 a* LO r’ 0341 *I 78.3”’ 52” 0.890=’
=’ From rcf_ [81.
b’ From ref. [In
lo=, (s)
IOSD (cm’ s- 1 )
39 -’ 82 -’
4.34 b’ 231e
=’ From ref. [18l_
Table 2 Corrrctioo factorsA (A = &+/&Rq) ior didaxric friction OP uni~-a!entions in acetonicriIeand water at 2S°C Water radius
b.
R 6) 1.0 15
20 25 3.0 4-o 5-O
l-75 126 1.1: i-05 1.03 I.010 x004
z26 1.48 1.19 LO9 I_04 1.014 1_006
131 1.10 L_lx 1.02 I-01 1_003 l_aOl
1_72 122 1.08 1.03 l-02 I_905 Lo02
superscriptHO rcfax to the Hubbard-Onsager which does not imorporatc poIarization diffhsion.
m’ l-be
moc!d
_
10’ D
A &acemnicrile)
(cm2s-1)
R-IA
R=Za
R=IA
226 l-96 l-78 166 158 I51 134
l-19 1.13 l-11 -l_lO l-09 l-08 1.06
l-72 133 123 l-18
0.0
20 4.0
6-O 8.0
10-O 20.0
van der Zwanand
A <=-) R=2.& -1-08 1.00 1.03 1.03
l-15
I-02
I-13 1M
1.02 1.01
.
Hynes [5] who examined diekctric friction on a charge moving along the axis of an infinite cylindrical cavity surrounded by a dielectric Experimental data for Iimiting conductivities of ions in weakIy associated pohu solvents show that the HO frictional coefficients underestimate the drag due to the solvent, so the incorporation of polarization diffusion actually broadens the gap between predicted and experimental conductivities Nevertheks, when sIip boundary conditions are used. the present model based on spatially homogeuneous solvent properties qualitatively reproduces the observed [Sj minimum value when the frictional coefficient is pIot:ed as a function of ionic radius_ Refatively simpIe continuum models of didectric friction still emerge when the assumption of a spatially homogeneous solvent is relaxed [12]. A recent examination of the effect of solvent electrostriction on ionic mobilit& [19] shows that the enhanced solvent density in the vicinity of the ion increases the viscous drag of weakIy associated solvents quite appreciably and therefore counter: acts the effect of poIa.rization diffusion AIthough sirnpie -continuum treatments may successfuIIy model rather general perturbations of the static and dynamic solvent properties they. disregard more specific factors, such as. s4ucture making/ breakiug and solvent residence times_inthe coordination shell [20,21), which caninfIuenCethe IocaI viscosity [21.U]. Such models cannot therefore be expected to provide p&tkuIarly accurate descriptions of frictionaI coefficients for smaII ions.. Prob-
We wish to thank Dr_ B_M_N_ Clarke for helpful dkcussions on computationai aspects of this work
= --/1/[2(~v):(P*S0);(SoV):~~P*) -(S&):(P*&)]
Appenais From (2-S) we set that the rate at which ekctriCal m.ergy is dissipated in the fluid is
+ D( VP*) : ( VU’*)]
d3r. (A-l)
With the aid of the boundary conditions (27) Su = 0 the surface integral in the identity,
SJl(VP*) =-
I//
t ( vSP*)d3r P*
l
v=SP*
+j-jP’-(B-v)SP* -is seen to
JI_S$.=
and
d3r d’r,
(A-2)
vanishand(A.l)canberecastas
d3r
~x~‘JJJP* -( 76%~ - DV’SP*) d3r. (A-3)
From the steady-&e version of our eq_ (2.4) for the polarization deficiency, the integral on the right becomes So-v)E,+EOx(~xS~)]-
d3r_
d3r,
which reduces straightforwardly to (2.8) when the eIectrostatic conditions v xE,= V l E,=O_ are applied.
[I]
J_ Hubbard 4850;
and L
Onsager.
J_ Ghan
Phys
67 (1977)
J-B- Hubbud. J. Chun Ph_ys 68 (1978) X49_ [2] P. Dcbyc Polar mokczdcs (Dover. New York. 1929)_ [3] P-G- de Games. The ph>sics of Ec+d crjsmls (-on Press. oxford. 1974)_ [4] J-B. Hubbard, RX_ Kqser and P-3. Stiks, Chan Phys_ Letters 95 (1983) 399; P-7. Stiles and J-B. Hubbard. Cham Phys 84 0384) 431 (pan 1). (51 G. van dcr Zand J.-f’_ Hyns, Physica 121A (1983) 227; Ghan Phy% Letters lOl(l983) 367_ [q hi. Fkm, Z Physik X(1920) 22L f7l P-G_ Wolynes. Ann Rev_ Phys_ Ghan 31(1980) 345. J-B_ Hubbard and P-G_ WoIynq [8] D_F_ Emns. -I-_-I-J. Pbyx Cbani 83 (X979) X69_ [9] JB_ Hubbard and R_F_ Ka>wr, J_ Clan_ Pb>x 74 (1981) 3535_ [IO] B-U- FcIdcrhof, MoL Phys. 48 (1983) 1093: 49 (1983) 449_ [ll] E Nomk, J. Chcza Phys 79 0983) 976. [12] J.B. Hubbard 377:
and R.F_ Kay%
*an.
Phyx_ 66 0982)
PI StiIcs. J-B. Hubbard and RF. Kayxcr. J_ Ghan. Phys n (1982) 6189_ [13] P-G_ Wolynes I Chui Ph>x 68 (1978) 473_ 1141 P- ~onomos and P-G- Wolyne J. (3hem Phys 710979)
[1!5l it!&aIef
zad P-G_ Wolynes, 470; 78 (l983) 4145.
[!6]
K Lamb, Hj$ro$ynamics London. 1932).
RL I 78
Hurie and LA. (1982) W3_.
J. Chai (*&ige
Woolf. J. Chca~ Sk
Phus_ 78 (l&3) &iv_
Press,
FaradayT&s
_
14
Pf:
sfires
is_Hubbnrd /
Pci&hfim
diif/ioianin p&r
.Liqui&
-; :