modified rocard relation for complex permittivity

Physica 105.4 (1981) 593-600 0 North-Holland Publishing Co.

MODIFIED ROCARD RELATION COMPLEX PERMITTIVITY

James MCCONNELL

FOR

School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 4, Ireland

Received 24 June 1980 Received in final form 6 October 1980

It is shown that to a high degree of accuracy the complex relativity permittivity for a gas dielectric or for a liquid dielectric in very dilute solution in a nonpolar solvent is given theoretically for symmetric polar molecules by the modification of a relation, which essentially goes back to Rocard. When the molecules are linear or spherical, the modified relation is particularly simple. There is no simple relation for complex permittivity when the polar molecules are asymmetric. Thus in comparing the theory of rotational Brownian motion with the results of dielectric relaxation experiments it would seem advisable at the present stage to focus attention on experiments performed with symmetric polar molecules, preferably spherical or linear.

1. Introduction

The Debye theory of dielectric relaxation leads to a curve for absorption of electromagnetic rays as a function of their frequency which has a plateau at high frequencies.‘) On the other hand the experimental absorption curves attain a maximum and then decrease steadily towards zero2*3r4). Since in the Debye theory the effects of the inertia of the polar molecules are neglected, an obvious approach towards resolving the above discrepancy is to examine the implications of inclusion of inertial effects in the calculation of the absorption coefficient a(w). This coefficient is expressible in terms of the complex relative permittivity E(O) by the relation’)

a(w) = ?(I E(O) 1 - E’(w))“*,

where the relative permittivity E’(O) and the loss factor E”(W) are given by

E(W) = E’(W) - k”(w), and

( E(W) 1 = {(E’(t.# + (E”(o))*}“?

The complex relative permittivity for a gas dielectric or for a very dilute solution of a liquid dielectric in a nonpolar solvent is obtainable from the Kubo relation’.‘)

m E(O) - Em

= 1 - io Es-Em I

((n(0) * n(t)))e-“” dt. 0

(1)

593

594 JAMES MCCONNELL

In this equation l s is the static relative permittivity, E, is the relative permittivity for frequencies so high that dipolar polarization is no longer effective, n(t) is a unit vector in the direction of the dipole moment of a molecule of the dielectric and the angular brackets denote ensemble average for the relaxation of the molecule in the thermal motion of its environment.

Expressions for ((n(0) - n(t))), w ere inertial effects are included, have been h calculated in recent years for a spherical molecule6,‘~*) and for a linear molecule’). However, when these expressions are substituted into (I), the integration cannot be performed easily. A fresh approach may be based on the stochastic rotation operator’) and the use of the Krylov-Bogoliubov method of solving non-linear differential equations’091’). If R(t) is the rotation operator that brings axes fixed in the molecule from their orientation at time zero to their orientation at time t, it is easy to see that

the OO-element of R(t) in the three-dimensional representation having as basis vectors the spherical harmonics Yr,-r, Y1,o, Y,,,. Hence (1) is expressible as

cc E(W) - Em

= 1 - iw Es - Em I

(R(t))& eei”* dt. 0

(2)

In the following four sections we shall consider separately the cases where the polar molecule is modelled as a spherical rotator, linear rotator, asymmetric rotator and symmetric rotator.

2. Spherical rotator model

We take each polar molecule to be a sphere with moment of inertia I about a diameter and we suppose that in the thermal motion a frictional couple IB times the angular velocity acts on the molecule. It has been found that’*)

(R(t)) = [I + ?.I*(1 - eeB’) + -y*{J*[f - (Bt + l)eeB’ - a e-2B1] + (577; _ e-al + ; e-zBt I) + y3{J2[y - (iB*t* + 2Bt + l)eeB’ - (!Bt + l)e-*” - 4 e-3B’]

+ (J’)‘[$ - (Bt + $e-” + (Bt + i)e-2B1 f a ee3”]

+ (J2)3[d - 4 ewB’ + 1 e-2B1 - d em3”]} + . . .]

xexp[- yB{l+!y +$y2+(!-QJ2)-y3+. . .}J*tl, (3)

where I is the identity operator, J* is the square of the rotation operator (.JX, .I,, Jz) and y = kT/(IB*), a small dimensionless quantity whose order of

MODIFIED ROCARD RELATION FOR COMPLEX PERMITTIVITY 595

magnitude is 0.01. The value of (R(t))& is obtained from (3) on replacing I by unity and J* by 2 in the right hand side. When this value is substituted into (2), we deduce that

E(O) - Em 2 ~- 6% - Em l+G’+iw’ I

312 -- I+$+iw’-(l+G2+io’)‘+2+G’+io 1

8 2 ~- 1+G’+iw’-(1+G’+iW’)2-(1+G’+iw’)’

5 512 519 +2+G’+iw’+(2+G’+ior)‘-3+G’+iw’ (4)

where

G’=2y(l+$y+$y2+~y3+...) and ~‘=olB. (5)

When w’ e 1, the sums in the square brackets of (4) are of the order of magnitude of their first terms. When o’= 1, the order of magnitude of all the sums is unity. When w’ z=- 1, the sums are all of order w’-‘. It therefore seems reasonable to suppose for all values of o’ that the terms in the braces with ascending powers of y are progressively smaller and that each such term is of order y relative to the preceding one. We now agree to omit the terms proportional to y2, y3,. . . , thus obtaining

E(W) - 60 = G’( 1 + G’ + iw ‘) - 2y iw’ Es - ECC (G’ + iw’)( I+ G’ + iw’)

G’(l + G’) + y2 io’ = (G’ + iw’)( 1 + G’ + io’)

from (5) on neglecting higher order terms in G’. Hence in this approximation

(6)

To put this in a form that would be more suitable for comparison with dielectric eXperhentS we define a friction time rF as B-‘, so that 0’ = WTr, and we write

IB 2kT = TD,

the relaxation time that occurs in the Debye theory13). We note that

I 7F = 277D = 2kTTD. (7)

596 JAMES MCCONNELL

Then we deduce on employing (5) that approximately

4m) - E- = I+ fr iw7F Es - Em (1 + iw [rn - &I)( 1 + iorF[ 1 - 2~1)’ (8)

It is found in experiments on dielectric relaxation2) that yorr is of order of magnitude unity only for frequencies well above the range of validity of the classical statistical mechanics which has been employed in the derivation of (8). Hence for consistency we do not consider such frequencies, and so we replace (8) by

E(W) - Em = 1 Es - Em

I (1 + iu[TD-4TF])(1 + iwm)’

(9)

In a different but equivalent study of inertial effects on orientational polarization’4) a relation was obtained that may be expressed in the present context as

E(O) - Em = G'( 1 + G’) E~-Em (G’ + iw’)( 1 + G’ + iw’)

iw’G’2{$(G’ + io’) + i} - (G’ + iw’)( 1 + G’ + io’)2(2 + G’ + io’) -. . ‘.

If the series on the right hand side is truncated after the first term and if we neglect quantities proportional to y@rF, we again obtain (9).

Rocard”) investigated the effect of the inertia of polar spherical molecules on orientational polarization. His investigations were inadequate, since they were based on a Brownian motion differential equation which does not exist. However, he deduced a result for polarizability which in our notation would be obtained by taking real parts in the equation

e(u) - em = 1 (10) es-em (1 + iwrn)( 1 + iorr)’

We shall refer to this as the Rocard relation. We may regard (9) as providing a correction to (10); eq. (9) is a modified Rocard relation.

3. Linear rotator model

When the polar molecule is a linear rotator, we denote by I its moment of inertia about an axis through its centre and perpendicular to the molecule. Moreover we suppose that the frictional couple about this axis is IB times the corresponding component of angular velocity. Then employing the results of


ref. 9 and the method of ref. 12 we find that (3) is replaced by

(R(t)) = [I + -y.I’(l - eeB’) + y2{J2[i - (2Bt + 2)e-” - f e-2B’]

+ (J2)2[1 - e-*’ + $ e-2B’]} + . . .]exp[ - YB( 1 + y + $y’ + . . .)J2t].

It will then follow that

E(W)-em G” . =--1W’

2 2 Es-E, G” + iw’ ([

y G”+iw’- l+G”+io’ 1 7

+Y2 G”+iw’- [ l+G’+io’-(l+G’+iW’)2+2+G’f+iw’ 1 1 +‘*’ ’ with

G”=2y(l+y+!y2+...).

Proceeding as above for the sphere we obtain

E(W) - Em = 1 Es - Em (l+i~[7~-~7~1)(1+iw7~)

in place of (9). Equation (11) is the modified rotator.

4. Asymmetric rotator model

(11)

Rocard relation for the linear

An equation for the complex polarizability due to the rotational Brownian motion of asymmetric polar molecules has been given by Ford, Lewis and McConnell”). Neglecting quantities that are of significance only for frequencies in the quantum region we may express their result in terms of complex relative permittivity by

40) - l m = 02(4 + D3 + B2) + D3@2 + D3 + B3)

Es - Em D2+D3+B2+iW 4+D3+B3+io

03(4 + 0 + B3) a(4 + 0 + Bl)

4+D,+B,+ict,+D,+D,+BI+io

+ D,(D, + D2 + Bd + 02(0 + 4 + B2)

D1+D2+Bl+io D,+Dz+Bz+iw (12)

Coordinate axes labelled 1, 2, 3 have been taken through the centre of mass and in the directions of the principal axes of inertia, the corresponding moments of inertia being II, 12, 13. The moment about the ith axis of the frictional couple is supposed to be IiBi times the ith component of the angular

598 JAMES MCCONNELL

velocity of the molecule. The direction cosines of the dipole axis are nt, n2, n3,

D, = D\” + D\*’ (13)

D(1) = kT I 131 (14)

112B2B3(B2 + B3) - BdB: + B2B3 + B:) 2 2

B,B2B3@2 + B3)

+IBz(B2+B3)-2B:+IB3(B2+B3)-2B: (12 - 41*

* B,B2B:(B2 + B3) 3 B,B:Bs(B2 + B3) - I,B:(Bz + B3) ’ I etc*

It is seen that 0:” is a small correction to 0:” of relative order kT/(IB*), where I and B denote generic values of the moments of inertia and frictional constants. Equation (12) has not the structure of a modified Rocard relation.

5. Symmetric rotator model

We next consider the special case of the model of the previous section where the polar molecule has an axis of symmetry, which we choose to be the third coordinate axis. On account of the symmetry the dipole lies along the third axis and we put

nl=n2=0, n3=l, 12=Z1, B2=B1.

Hence (12) reduces to

E(u) - Em = 2Dd2D,+ BJ Es - Em (20, + io)(2D, + B1 + io)

= (1 + io/2Dr)(l: iol(Bi + 201))’ (16)

where Di is given by (13), D{” by (14), and (15) becomes

-2B:+BiB3+B: (11 - 1312

B:Bs(B, + B3) - I,B:(B, + Bd ’ 1 (17) In eq. (16) we have

approximately, where we have put I,B1/(2kT) equal to 71). Moreover


with y = kT/(IIBi). Then on writing B;’ = TF,

since we may neglect the term proportional to y because it is insignificant for values of o belonging to the range of validity of classical statistical mechanics. We conclude that

1 Es - Em (1 + iW7n[ 1 - D\“/D\“])( 1 + iwF)

(18)

with D\” and Dp’ given by (14) and (17). This is the modified Rocard relation for the symmetric rotator.

We now check whether (18), which was deduced from (12), is consistent with the results (9) and (1 l), which were derived independently for the sphere and linear rotator. We must therefore examine whether (18) reduces to (9) and (11) for the sphere and linear rotator, respectively.

The case of the sphere presents no difficulty. It is seen immediately from (14) and (17) that for II = 13 = I, B1 = B3 = B

TDDP) DI”=

mkT _ 1 w - 47F3

by (7). This is what we require for (9). The case of the linear rotator needs more careful consideration. We have

from (14) and (17)

012’ kT ~(E&EJ_2.$+3.%+1_@&

H/J=IIB: I+$

(19)

For the linear rotator 4 6 11 and, if in addition we require that B3 & B, in such a way that IjB3 9 IIB1, we deduce from (19) that

Then on substituting into (18) we shall obtain (11). It is therefore possible to define a limiting process that will lead from the modified Rocard relation for the symmetric rotator to that for the linear rotator.

6. Conclusion

The first result of the foregoing investigations is that, when a dielectric consists of polar molecules with an axis of symmetry, the complex relative

600 JAMES MCCONNELL

permittivity is given to a first approximation by the Rocard relation (10). This equation has the advantage that its right hand side involves only one frictional parameter rr, which is deducible from rn by (7). The value of rn may often by obtained from an experimental Cole-Cole plot. To a second approximation the permittivity is given by a modified Rocard relation, which in the cases of spherical and linear rotators assumes the simple forms (9) and (11) having only the one frictional parameter rF but in the general case is given by (18) and so involves two frictional constants Br and B3. This would make difficult the comparison of theoretical and experimental results for the relative permittivity E’(O) and the loss factor E”(O), and consequently for dielectric absorption. Since eq. (12) involves the frictional constants BI, Bz, B3 and since there are no obvious ways of determining the values of these, either experimentally or theoretically, it seems that we cannot hope at present to institute accurate comparisons between theoretical and experimental values of E’(W) and E”(O) for dielectrics composed of asymmetric polar molecules.

References

1) J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, London, New York, 1980).

2) J. Goulon, J.L. Rivail, J.W. Fleming, J. Chamberlain and G.W. Chantry, Chem. Phys. Lett. 18 (1973) 211.

3) A. Gerschel, I. Dimicoli, J. Jaffre and A. Riou, Molec. Phys. 32 (1976) 679. 4) G.W. Chantry, IEEE Trans. Microwave Theory and Tech. (USA), Vol. MTT-25, NO. 6, (1977)

496. 5) R. Kubo, I. Phys. Sot. Japan 12 (1957) 570. 6) J.T. Lewis, J. McConnell and B.K.P. Scaife, Proc. R. Ir. Acad. 76A (1976) 43. 7) G.W. Ford, J.T. Lewis and J. McConnell, Proc. R. Ir. Acad. 76A (1976) 117. 8) Y. Pomeau and J. Weber, J. Chem. Phys. 65 (1976) 3616. 9) J. McConnell, Proc. R. Ir. Acad. 78A (1978) 87.

10) N.M. Krylov and N.N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton Univ. Press, Princeton, 1947).

11) G.W. Ford, J.T. Lewis and J. McConnell, Phys. Rev. Al9 (1979) 907. 12) J. McConnell, Physica lO2A (1980) 539. 13) P. Debye, Polar Molecules (Dover, New York, 1929). 14) G.W. Ford, J.T. Lewis and J. McConnell, Physica 92A (1978) 630. 15) M.Y. Rocard, J. Phys. Radium 4 (1933) 247.

modified rocard relation for complex permittivity

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