Journal of Molecular Liquids, 48 (1991) 9 9 - 1 0 9 99 E l s e v i e r Sc i ence P u b l i s h e r s B.V. , A m s t e r d a m

INERTIAL THEORIES OF DIELECTRIC RELAXATION IN LIQUIDS

JAHES HcCONNELL

School o f Theoretical Physics, Dublin Institute for Advanced Studies,

Dublin 4, Ireland

(Received I August 1 9 8 9 )

SUMWtARY Theories of dielectric relaxation in liquids that take due account of

effects of the inertia of the polar molecules are discussed, and their impilcations for the absorption spectrum are studied. A non-Harkovian theory is found to lead to satisfactory agreement with certain relaxation

experiments.

INTRODUCTION D u r i n g t h e t w e n t i e t h c e n t u r y t h e o r e t i c a l s t u d i e s o f r e l a x a t i o n p h e n o m e n a i n

liquids have continued since the Investlgatlons of Einstein on Brownlan motion

(ref. I). The classical studies of Debye [re/s. 2,3) were based on th~ neglect

of molecular mass in certain calculations. Nevertheless they continue to

provide valuable results for many physical processes- They also serve as

a useful point of comparison with inertial theories by considering the limits

of inertial results when the molecular mass is allowed to tend to zero.

The first serious attempt to provide a mathematically satisfactory inertial

theory of dielectric relaxation appears to have been due to Sack (ref. 4), but

he did not fully expound his theory at thlm stage. A valuable bibliography.for

dielectric relaxation In polar fluids has been provided by Galduk and Kalmykov

( r e f . 5 ) .

I n t h e f o l l o w i n g s e c t i o n t h e i n v e s t i g a t i o n o f d i e l e c t r i c r e l a x a t i o n b y

mathematical methods associated with the Langevin equation will be Introduced.

It is ~otod that such investigations may be performed either by solving

a differential equation, to bo called a Fokker-Planck equation, or by

employing a stochastic rotation operator. Explicit calculations will be

presented in the two subsequent sections. Then a recent theory of relaxation

based on the assumption that molecular collisions are not instantaneous will

be d e s c r i b e d ( r e f . 6).

D e d l c a t e d to Professor Save B r a t o s

0 1 6 7 - 7 3 2 2 / 9 1 / $ 0 3 . 5 0 ~ 1991 - - E l s e v i e r S c i e n c e P u b l i s h e r s B.V.

I00

LANGEVIN-TYPE THEORIES OF ROTATIONAL R£]_AXATION

Langevln-type theories of relaxation are based on the equations of

rotational Brownlan motion of a rigid molecule that is being tossed around by

a white noise driving couple, the rotational motlom being slowed down by

a frictional couple arising from the environment. A molecular coordinate frame

is specified by the three principal axes of inertia of the molecule through

its centre of mass. It Is assumed that the friction tensor and the inertial

tensor are simultaneously dlagonallzable, and this allows us to write down

the Euler-Langevln equations of motion (ref. 7, p. llO)

I,~I-(I2-13)~2w3=-~l~l+A1 (t)

Ia~2-[13-11}~3~l=-~2~2+A2(t)

I 3 ~ 3 - { I - I , a)~,~2=-~3w3+A3 (t)"

(11

In these equations 11. I2, I 3 are the principal moments of inertia. ~i" ~2" ~3

the corresponding components of angular velocity of the molecule, ~l" ~2" ~3

the coefficients of microscopic friction and A (t), A [t), A (t} I 2 3

the components of the random drlvlng couple. These are assumed to obey

the relation

<A (t)A (t')> - 2kT~i51 5(t-t') (2) J J "

where the angular brackets denote ensemble average for a steady state of

the physical system.

THE FOKKER-PLANCK EOUATION

There is a standard method of d e r i v i n g a Fokker-Planck equation

corresponding to a set of stochastic equations (ref. 7. section 6.1). Suppose

that at time t~O the molecule P~s angular velocity ~ in d3~ and orientation g N

In dg. where d E may be expressed in terms of Euler angles ~.~0W specifying

the orientation of the molecule with respect to a laboratory coordinate system

by

dg = sin~ da dE d~.

We denote by w(u, g. t; ~o' go } the conditional probability density that

the molecule has the above angular velocity and orientation, if at time zero

it had angular velocity ~0 and orientation go" Then w(~, g, t; ~0' g0 ) obeys

the Fokker-Planck equation (ref. 8, eq. {2.19})

101

3 3

aw + I J w ~ ~ ~k ~ -- -- ~ W -k ~ O, (3) a t ] 1 0 1 I ] / I ~ 8 ~ a

, . , Z__.. ] [ [ ] ] ] ] -1 i - t

where J1" J2" 3 3 are the infinitesimal generators of rotation of the molecule

and J, k. I is a cyclic permutation of I, 2, 3.

It is clear that, unless the molecule has some special symmetry,

the solution of {3) will be extremely difficult. Sack [ref. 9) soIved

the equation for molecules that are either spherical or linear. We shall

consider only the case of a spherical molecule. The Fokker-Planck equation |s

then ( r e f . 10, eq. (9}}

Ow + V e Ow v~ - v~ Ov 0 v e 8--t D-O + cotO v z - ~-- s i n O 0---0

(4)

I n t h i s e q u a t i o n I I s t h e m o m e n t o f i n e r t i a o f t h e s p h e r i c a l m o l e c u l e . ~ I s

t h e c o e f f i c i e n t o f m i c r o s c o p i c r o t a t i o n a l f r i c t i o n , 0 a n d ¢ a r e E u l e r a n g l e s

w h i c h w i t h a n a n g l e X s p e c i f y t h e o r i e n t a t i o n o f t h e m o l e c u l e w i t h r e s p e c t t o

a l a b o r a t o r y s y s t e m o f c o o r d i n a t e s , a n d

v e 6, v¢ s i n e . £ + c°s°

T h e m o l e c u l e h a s a c o n s t a n t e l e c t r i c m o m e n t p a l o n g a d i a m e t e r w h o s e E u l e r

a n g l e s a r e O, ¢ a n d t h e r e I s a n e l e c t r i c f i e l d o f i n t e n s i t y F i n t h e d i r e c t i o n

o f 0 = O. S a c k s o l v e d ( 4 ) u n d e r t h e a s s u m p t i o n t h a t ~ F ~ kT a n d t h a t i n

the solution terms involving powers of higher order than the first in uF/(kT)

are neglected. Thus he worked in a linear approximation. If ~(~) is

the complex polarizability for orlentatlonal polarization, Sack's result is

expressed by a continued fraction:

3 - ~" ~ - - ~ -~ 1 - ~ + 2~f I + +

l 1 I z+i " 1 + Ira" + ~'~'(2 + Ira"

+ 4~" I + . . . • I 1 }-1

where

( 5 )

kT/

ga

The c o n t i n u e d f r a c t i o n a r o s e from a Bet o f recurrence r e l a t i o n s ( r e f .

{ 6 )

9.

eq . ( 3 . 1 7 ) ) .

102

An alternative discussion of (4) leads to another set of recurrence

relations and these are solved by a matrix method (ref. I0). The final result

i s :

~ ( ~ ) 2 k T k T I = L + C2 •

O0

( 7 )

where

Ln, = (i(o'+2n+l)3nl (8]

23 ( 2n + 2 2n + 3 "~ = ~ - - 1 , 1 ÷

~4nl - i ~ " + 2n ( ~ + I ~ " + 2n + 2J 5h i

(n + l}[2n ÷ 5)6 +i, l

I'.~' + 2 n + 2 (9 )

and ~" is given by ( 6 ) - On substitution of (B) and (9) into (7) It Is readily

deduced that

q ( ~ ) _ 2~

~(0) i~'(l~" + I) iw" (i~" + l) 2 I~" + 2

+ 2~ 3 ( ~.( 4 + 12 + ( 1 0 )

i~'(l~" + l] a i~')2(I~" + 1) I~" (I~" + 1)(I~" + 2)

9

( i ~ " + 1 ) ( I ~ ' + 2 ) 2

,0 } ÷ +

( l u " + 2 ) z ( i ( a " + 3 )

where ~ is defined in (6). There is no difficulty in extendlnE the right h a n d

side of (I0) to higher orders in the dimensionless constant ~. This is

probably the most convenient way to express ~(~) in a power series.

So far we have been dealing entirely wlth orlentatlonal polarlzatlon. In

the general case where translational and induced polarizations a r e also

present and we are concerned with polar molecules in a weak solutions of

a nonpolar liquid we make In (5), (7) and (I0) the replacement

, . 1111

where c(w) is the complex permlttlvlty, ~ is c(O) and c is the value of c(w)

for frequencies so high that the orientational polarization is negllglble

(ref.7, chap. 2). The absorption coefficient a(~) is then expressed In terms

103

o f c(w} b y the relation

21/20 a(~} ~ - - (Ic(~) I _ £, (s})ire , (12)

C

where c'(~) is defined b y

c ( ~ } - c " ((.,} - t o " C , . , }

a n d g ' ( w ) , c " { e } a r e b o t h r e a l .

THE STOCHASTIC ROTATION OPERATOR

The value of the complex polarizability ~(w) for orlentatlonal polarization

in the case of a polar molecule of any shape obeys the Kubo relation

=(~) = 1 - le <(n(O}-n(t)}>e -L~t dt,

O

{13)

where n(t) is a unlt vector in the dlrectlon of the dipolar axis (ref. 7

section 2.2). If ~" (t) is the anEle between n{O) and n(t). we see that N

<(n(O)-n(t})> = cos~" (t} = Pl(cos~" (t)) = Doe(a" (t),~" (t),~" (t}) , (14)

where D j denotes a W l g n e r function and ~" (t}, ~" {t). ~" (t) arc the E u l e r ab

angles that specify the orientation of the molecule at time t r~:latlve to Its

orientation at time zero (ref. 7, sectlon 7.3). We denote by O~(t} the operator

that describes the rotation of the molecule from its orientation at time zero

to its orientation at time t. Then

(a" (t).B'(t).~" {t}} R~ (t} (15} D •m ~ "n •

the m'm - element of the matrix representative of ~(t) in the 2J + I -

-dimensional r e p r e s e n t a t i o n with basis elements Y]=, where

s = -J, -J + l, ... J-l, J. We deduce from (13)-(15) that

= 1 - I <~{t)>uo e

o

d k . { 1 6 )

T h e v a l u e o f < ~ ( t ) > f o r a n a s y m m e t r i c m o l e c u l e w a s c a l c u l a t e d b y L e l c k n a m , o o

Gulssanl and Bratos for free rotation and for the rotational diffusion limit

(ref. 1 1 ) .

We see from (16) that dlelectrlc relaxation is closely related to ~(t),

104

which on account of the thermal motion of the environment is a stochastic

operator. An lnertlal theory of the rotational Brownian motion of an

asymmetric top was based on (I} and (2), and was applied to dielectric

r e l a x a t i o n (ref. 12)- I t was f o u n d t h a t

<et(t)~l(s}> = 0 (i ; I) (17)

] <~,(t)~ (s)> " exp ~ It - sJ + l I

Ill k

e x p l - + K - It-sl-o, , g I -sl

(IS)

where I, J, k is a cyclic permutation of I, 2. 3.

To introduce the rotation operator we recall that (ref. 7, section 7.3)

dR(t) dt = - i(~-~(t))R(t), (19)

where J l " J2" J3 are the lnflnlteslmal Eenerators of rotation associated wlth

the coordinate axes of (I}. Employing (17)-(19) and the method of averagln E

enables us after a lenEthy calculation to obtain the result (ref. 7, p. 194)

<R(t)> '= I + 1 - exp + ... x 2 i Z_c, [

1 , - ,1 [20}

x ex - D "1) + + P(kT) + ... t 2 2

s j t i l i ~ j C k I - - I l ~ l "

In this equation 0 is the identity operator, N

_ l _ [ j 2 j 2 j 2 j 2 / ~P = Z l 2 3 3 2j

D{I) _ kT t ~ t

and D (2) is a small correction to D (]) (ref. 7, (eq. (11, ! i

possible to proceed further and derive an expression for

However this wlll not be required for our present purposes.

(2:)

(22)

5.23)). It Is

R{t) {ref. ]3).

105

Let us employ ( 1 6 ) and (20) to calculate ~(w) for orlentatlonal

polarization (ref. 7 , section 12.5). We take the three-dlmensional

representation related to the molecular cartesian coordinate axes and given by

°°il 0 "I o - I 0 0 J = J l " 2

- I

2 It will follow that Jl'

0 J 3 ,= 0 .

0 0

j 2 j 2 a r e r e p r e s e n t e d b y d i a g o n a l m a t r i c e s a n d 2' 3

therefore, by (21), that P vanishes. A lengthy expression for ~{¢) is given by

eq. (12, 5.21) of ref. 7. When the polar molecule is a symmetric rotator , we

take the axis of rotational symmetry to be the third coordlnato axis. Then, if

we neglect D {2) and employ (22), we find to close approximation that 1

~ ( ~ ) 1 = ( 1 + l o - r ) ( 1 + i~ , rr ) "

D J

whore

( 2 3 }

~ l I ( 2 4 ) m - -

l

Equation (23) la essentially the same as the result derived in 1933 by Rocard

(ref. 14) in his attempt to take account of inertial effects on orlentatlonai

polarization. We shall therefore refer to (23) as the Rocard eguatlon.

To study the implications of (23) for dielectric absorption we first make

the replacement (11) so obtaining

c ( u ) - c 1 m ( 2 5 }

e - c = (I + i u T ) ( 1 + l u T ) " • ~ D J

Then calculating the absorption coefficient a(~) from (12) we shall have

an absor ion curve whose qualitative behaviour is shown in FIg. I of ref. 6.

We see that the Debye plateau is no longer present, so that we have a return

to transparency. However the absorption peak of the experimental curve, that

is, the Poley absorption, is not obtained.

A THREE-PARAMETER THEGRY

Let us consider the present situation. We see that ( 2 5 ) contains the two

and ~ . Then (24) shows that p a r a m e t e r s T D J

I (26) TDTJ 2kT

F o r simplicity we shall now consider only spherical molecules- Equatlon (26)

lm the Hubbard relation In the f o r m r e l e v a n t to dielectric r e l a x a t i o n

e q . ( 5 . 1 8 ) o f r e f . 1 5 } .

106

T h i s relation holds not only for the Langevin-type mechanism but more

generally for a l l J-dlffusion models. Now it may be shown (re/. 16,

e q . (2.35}, (2.36)) that

2 k T M =-- (27) 2 I

= + - - {28) M4 3I 2 "

where M denotes the rth moment of the infrared spectrum and <N2> ls the moan F

square torque that acts on the molecule durlng a collision. In view of (26)

and (27) we may say that the Rocard model depends on the two Independent

parameters T and M . D 2

One may seek an explanat lon of the Poley absorption by cons truct ins

a theory with three independent parameters. Thls was done by Guillot and

Bratos ( r e / . 17) by taking the s e t

~a" M2' M4 {29}

They expressed their results in terms of TI, T 2. T a defined by

= M -I/z T -4 - M -- M 2 (30) T 1 = TD" T2 2 ' 3 4 2

T h e y t h e n c o m p a r e d t h e r e s u l t s o f t h e i r c a l c u l a t i o n o f t h e a b s o r p t i o n s p e c t r u m

b a s e d o n t h e M o r l t h e o r y w i t h e x p e r i m e n t s p e r f o r m e d b y D e s p l a n q u e s , C o n s t a n t

a n d F a u q u e m t m r g u e ( r e f . 18} o n 2 0 p e r c e n t s o l u t i o n o f b r o m o f o r m CHBr i l l 3

c a r b o n t e t r a c h l o r i d e CC1 a t 2 5 ° C . W i t h 4

T 1 = 6 - 1 9 " 1 0 - 1 2 S , T 2 -- 0 . 8 3 " 1 0 - 1 2 s , T 3 ~ 0 . 3 5 - 1 0 - 1 a s {31}

q u a l i t a t i v e a g r e e m e n t w a s o b t a i n e d a n d t h i s w a s a l l t h a t c o u l d b e e x p e c t e d .

s i n c e n o a c c o u n t h a d b e e n t a k e n i n t h e t h e o r e t i c a l c a l c u l a t i o n s o f t h e

influence of induced electric moments.

Another approach to the Poley absorption problem may b e made on recalling

that the Debye and Rocard models have one common feature, namely, they

presuppose that the molecule collisions are instantaneous. Denoting by J{t} N

the angular momentum of a molecule we wrlte

K j C t ) m < ( J C 0 ) - ~ ( t } ) >

I n o r d e r t o h a v e n o n - i n s t a n t a n e o u s m o l e c u l a r c o l l i s i o n s w e s u p p o s e t h a t t h e r e

e x i s t s a f i n i t e c o l l i s i o n t i m e T a n d w e a c c e p t a v a l u e o f K ( t ) t h a t i s c j

n o n e x p o n e n t l a l f o r t i m e s l e s s t h a n T . C o n s e q u e n t l y t h e s p e c t r a l d e n s i t y o f c

K j ( t ) w i l l b e n o n - L o r e n t z l a n i n t h e w i n e s . T o o b t a i n s u c h a K j { t } we a s s u m e

t h a t i t o b e y s a n e q u a t i o n

107

t

[~j(t) = -;IR(t-t')Kj,(t" )dr'. O

I t m a y b e d e d u c e d f r o m ( 3 2 ) t h a t ( r e f . 1 6 . p . 2 6 )

oo

I ~ ( -* t)dt = xj

o

a n d that

(3z)

(33)

<N2> ~ ( o ) = ~ .

We define T in terms of the memory function ~(t) by c

-1~R(t)dt T ~ R(O) c

o

whlch combined wlth (33) yields

-* = @~(0)T "Cj a"

From (27), (28) and (34)

M - k2 4 2

~(01 = M 2

Moreover (26) and (27) yield

-1 17 " MT D 2 J

C o m b i n i n g (30) with (36)-(38) we see that

f, ] 4 T 2 - T 3 I R ( 0 ) = _~z .

Tj ~ ' Tc = a [~ " T:

For the values of T,. T 2, T 3 in (31} we have from (39)

T = 6. 19-10-12s, T = 0. 1-10-12s, • = 0.2-I0-12s. D J C

Let us now assume a particular form of R(t), namely,

~(t) = ~(0)e-t/xc.

which clearly satisfies (35). On substitution of (41) into (32)

that

Kjlt) ~ 31kT k2 _ I - k2

It Is

[34)

( 3 5 )

{36)

(37)

(3S)

(39)

( 4 o )

( 4 1 )

f o u n d

( 4 2 )

108

where

1 -- ( 1 -- 4 K ) 1 / 2 1 + ( 1 - 4K) *re k = k =

1 2 T " 2 2 " r c o

a n d

K , - I R ( O ) ' r 2 - - - c

( 4 3 )

T

_ c ( 4 4 )

T J

by (36).

On substituting from (40) Into (44) we see that

K = 2

and this shows that k and k defined in (43) are complex. It is found that I 2

the far infrared spectrum resultlng from (42) is in close agreeme, t wlth

the theoretical absorptlon curve of Gu111ot and Bratos. Moreover a study of

the infrared spectrum shows that llbratlonal motion occurs at the frequency at

whlch Poley absorption appears In the Infrared spectrum (ref. 6). Thus it

seems that the postulate of a finite collision tlme explains at least

qualltntlvely for the experiment of Desplanques et al. the return to

transparency and the Poley absorption together wlth the llbratlonal effect,

for which a cage model had been previously proposed (ref. 19).

C O N C L U S I O N

It has been shown that the dlfflculty of the Debye plateau may be avoided

by including molecular Inertlal effects In a Markovlan theory assoclated wlth

the Rocard equation. In order to obtain both Poley absorption and return to

transparency It was found adequate for a set of experiments on bromoform

dlssolved In carbon tetrachlorlde t o discard the Harkovlan assumptlon and so

allow a finite tlme for the colllslon of a polar molecule wlth the molecules

of the solvent.

REFERENCES

I . A. E l n s t e l n • Annln. Phys. (4) 17 (1905) 549-560. 2 . P . D e b y e , P h y s . Z . 1 3 ( 1 9 1 2 ) 9 7 - 1 0 0 . 3 P. Debye, Ber. Dr. Phys. Ges. 15 (1913) 7 7 7 - 7 9 3 ; t r~mslated in :

The Co l l ec ted Papers o f Peter J.W. Debye, I n te rsc lence Pub l lshers , New Y o r k , b ) n d o n • 1 9 5 4 .

4 . R . A . S a c k , K o l l o l d z e l t s c h r l f t 1 3 4 ( 1 9 5 3 ) 1 6 . 5 . V . I . G a i 5 u k a n d Y . P . K a l m y k o v . J . M e l e e . L 1 q . 3 4 ( 1 9 8 7 ) 1 - 2 2 2 . 6. A.I. Burshteln and J. McConnell, Physlca AIS7 (1989) 933-954. 7. J. McConnell, Rotational Brownlan Motion and Dielectric Theory, Academlc

Press• London, 1980. 8. P.S. Hubbard, Phys. Rev. A6 (1972) 2421-2433. 9. R.A. Sack, Prec. Phys. Soc. BT0 (1957) 414-426.

I0. J. HcConnell, Prec. R. Ir. Acad. 77A (1977) 13-30. Ii. J.C. Lelcknam, Y. Gulssanl and S. Bratos, J. Chem. Phys. 68 (1978)

3 3 8 0 - 3 3 9 0 .

109

12. G.W. Ford, J.T. Lewis and J. McConnell, Phys. Rev. AI9 (1979) 907-919. 1 3 . J . McConnell, PhysIca 111A (1982) 85-113. 14- Y- Rocard, J. Phys. Radium 4 (1933) 247-250. 15. P.S. Hubbard, Phys. Rev. 131 (1963) 1155-1163. 16. A.I. Burshtein and S.I. Temkln, Spectroscopy of Mo]ecular Rotation

In G a s e s a n d Llqulds (Nauka, Novoslblrsk, 1982}, (in russian). 17. B. Guillo£ and S. Brmtos, Phys. Rev. A16 (1977) 424-430. 18. P. Desplanques, E. Constant and R. Fauquembergue, in: Molecular Motions

in Llqulds, ed. 3. Lascombe, Van Reldel, Dordrecht, 1974, 133-149. 19- N.E. Hill, Pmoc. Phys. Soc. 82 (1963) 723-727.