Reaction field in anisotropic dielectrics

Reaction field in anisotropic dielectrics K. H. Lau and K. Young Department of Physics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong (Received 3 June 1975) Drawn polymers are often characterized by dielectric anisotropy. This is determined by the anisotropy in the molecular polarizability, averaged over the distribution of polymer chains. Since each dipole in the dielectric is situated in a macroscopically anisotropic medium, corrections arise from the anisotropic cavity field and reaction field. The necessary reaction field correction is explicitly evaluated for an axially symmetric dielectric with negligible electronic polarizability and is found not to exceed 2-3%. This shows that the ad hoc anisotropic generalization of Onsager's equation used in the literature need only be corrected for the cavity field effect in the manner recently discussed.

INTRODUCTION A useful tool for determining the orientation function in a drawn polymer is the measurement of dielectric anisotropy. From the measured permittivi_'ty tensore0., one first obtains the molecular polarizability a¢, where denotes an average over the polymer chain distribution. Comparison with the polarizability a¢ of a single molecule yields the orientation function. Our purpose here i__sto examine in some detail the relation between e o.and a~/; this relation for the isotro1 pic case is the Onsager equation. Previous analyses of anisotropic data 2-4 have not used a proper anisotropic generalization of the Onsager equation. This is particularly worrying because (a) it is the anisotropy, rather than the average value, which is of main interest in polymer physics, and (b) the likely size of the error involved has not hitherto been estimated. We give a systematic formulation of dielectric theory in the anisotropic case and show that two effects need be taken into account: (a) the cavity field effect recently discussedS; and (b) the reaction field. The latter is the main concern of this paper, and the necessary correction is evaluated for an axially symmetric dielectric with negligible electronic polarizability. Our main result is that the reaction field correction turns out to be very small. It is fortunate, but not a p r i o r i obvious, that this should be so. The rest of the paper is organized as follows. In the next section, we consider the response of an isolated molecule to an external field and define the polarizability ai/. Next the response of a molecule in a medium is considered and a general expression is found for the permittivity el~. In both cases, the example of an axially symmetric material with negligible electronic polarizability serves to illustrate the main ideas. Finally, the reaction field is studied for this special case and the correction is estimated.

Here a is a unit vector specifying the direction of the permanent dipole and x is the position of the electron. The generalization to several electrons is straightforward. The thermal average of the dipole moment is: ~ui) = f lai exp (-t3U)d3xdf2 f exp (-/3U)d3xd~

(2)

Here g2 denotes the orientation of the unit vector a,43 = (kT)- 1 and U is the energy of the dipole. For an isolated molecule, U is taken to be of the form: U = A(a ) + ½ x i k o . x j - p ' E = U 0 - p "E

(3)

Here A(a) is the orientation energy of the permanent dipole, the symmetric matrix k# ~presents an anisotropic 'spring' binding the e~ctron and E is an external electric field. To first order in E, equation (2) may be written as:

=

(4)

where the polarization tensor cto. is: a# =

f ldila] exp (-13U0)d3xdg2 f exp (-flUo)d3xd~

(5)

A special case

We shall focus attention on a special case of equation (5) in order to bring out the essential physics. First, we assume the electronic contribution can be ignored, so that equation (5) reduces to: ct#. = ~pO 2 faia/ e x p (--13A)i:tg2

f exp (--/3A)d~2 MODEL FOR AN ISOLATED MOLECULE

(6)

Define General formulation

The molecular dipole will be assumed to be made of two parts: a rigid permanent dipole of magnitude Po and a bound electron of charge -e: la = pO a - e x

(1)

Q(~) = exp (-/3A)

(7)

which is just the Boltzmarm distribution function for the permanent dipole. Further assume that Q(~2) is axially symmetric about the chain axis (of the monomer molecule). If

POLYMER, 1976, Vol 17, January

7

Reaction field in anisotropic dielectrics: K. 14. Lau and K. Young the angle between the chain and the dipole a i s denoted by 7, Q(~2) may be written as:

1

Q(~2) = ~ ~ QIPI(COST) 1

(8)

Here 0' is the polar angle between the chain and the draw direction. The distribution (15) is normalized to unity if f0 = 1. With these definitions in hand, ~9(~2) can be evaluated: Q(~2) - f Q(~)f(~2')df2'

Putting these into equation (6) gives:

~uo2 c~# - 47rQ0

l

4Tr

Ql faia/Pl(COST)d~2

In a frame where the z-axis coincides with the chain, a# takes the diagonal form: a¢. = a(6# + vr///)

~

QtfIPl(cosO)

(16)

(9)

(10)

where we have used equations (14) and (15) and the orthogonality of the spherical harmonics. Comparing with equation (8), we see that (21 has been replaced by Qlfl/(2l + 1). Thus if we parametrize the average polarizability as:

where 7/O.is the matrix (matrices will be denoted by ^) fl =

('

)

1

c~# = c~(8# + v~7¢) (11)

-2 Thus a = tr ~/[3 is the average value (over the principal directions) of the polarizability and v is a measure of the anisotropy. Taking the trace of equation (9) and noting that a 2 = 1, we get: ot =/~/a02/3

(12)

which is the classical result of Langevin and Debye 6 , here found to be unmodified so long as we talk about the average. For the anisotropy we obtain: v-

1

Q2

(13)

(17)

then if= a, which is physically obvious. The average anisotropy vcan evidently be obtained from equation (13) with the replacement QI ~ Qlfl[(21 + 1):

v-

1 Q2f2/5_ 5 Qofo

1 Q2f2

(18)

25Qofo

Again l > 2 is not involved. The proportionality to f2 is physically reasonable: it expresses the fact that no matter how anisotropic each molecule is, there will be no average anisotropy v if the molecules are randomly oriented. THE MOLECULE IN A DIELECTRIC MEDIUM

5 Qo It is to be noticed that ai/is insensitive to Ql with l > 2; therefore measurement of the polarizability will yield no information on the higher Ql. This comes as no surprise since a#. is a rank two tensor and therefore involves only l = 0 a n d 2.

Averaging over the chain distribution Let us now transform to a frame suitable for macroscopic discussion: we take the z-axis to be the draw direction (rather than the chain direction). In this frame, let the direction of the dipole be ~ and that of the chain be ~ ' , with the angle between them being 7. Then equation (8) becomes, by the addition theorem for spherical harmonics*.

a(~2) = ~ at Ylm*(~') Ylm(~2) Im 2 l + 1

(14)

We are interested not so much in Q(~2) as in its average value ~)(g2) over the chain distribution. Let the chains be distributed according to f(~2'); this is, of course, the orientation function one wants to study. Let f(~2') be axially symmetric about the draw direction:

General formulation We next consider a substance with N such molecules or monomers per unit volume. Our treatment will draw on the method of Fr6hlich 7. In contrast to the case of the isolated molecule, it will be necessary to consider interaction between molecules. We consider a sphere of radius R containing one molecule; this molecule will be treated microscopically while the rest of the medium will be treated macroscopically as a dielectric with permittivity e0.. An alternate but equivalent method is to consider a larger sphere containing many molecules but to neglect the short range interaction between these molecules. Thus we are not concerned here with Kirkwood's generalization s of the Onsager equation to include short range effects. Now the energy of the rn~olecule in question will still have the intrinsic pieces A(a) + Vzxik#x/ of molecular origin. In addition, there will be the energy used to set up the dipole: --} #

f ~'d~

(19)

0 ....}

where F is the electric field ~ the origin, other than that due to the dipole itself, i.e. F i s defined by:

1 Z flPl(cos 0')

~ n ' ) = U~ t

....~ ....~

-1E (41r) 1/2

1 (2l + 1) 1/2 flYlo(Ft')

* We shall not explicitly indicate the I2 dependence of Q.

8

POLYMER, 1976, Vol 17, January

(15)

~-r ~ ~ ~ r ) = - ~ - - F'r + O(r 2)

(20)

as r -+ O, where ~bis the electrostatic potential. The linearity of the dielectric can be used to write ff as the sum of two

Reaction field in anisotropic dielectrics: K. H. Lau and K. Young -->

terns respectively proportional to the external field E and to/a:

so that comparison with equation (28) gives: (30)

elk - 8ik = 4nNZijXjk

(21)

Fi = X#Ej + Yi/la]

or in matrix notation: The first term, known as the cavity field, is the field at the origin when the external field is E and the cavity contains no dipoles:

~ - i = 4nNZX

(31)

where: 4)(r)~-E.r

r~oo

~ r ) ~ - (XijEj)r i + O(r 2)

r~ 0

(22)

The second term, known as the reaction field, is the field at t~e origin when there is no external field, but with a dipole /a in the cavity: ¢(r) -+ 0

r ~ oo - ( Yo.la/)ri + O(r 2)

(o(r) -+

r~ 0

(23)

The symmetric matrices X~7 and YO. are completely def'med .by equations (22) and (23) and will be functions of the macroscopic permittivity G0.. For reference, we quote the well-known isotropic result: 3G

=- XO" 2 e + l

2

r;;

---R

(24)

6 O.

e--1

3 2e+l

8n e--1 60.=--N-3 2e+l

8i]

flail1j exp (-13U1)d3xdf2 Zi] = 13 f exp (-/3U1)d3xda

Comparison with equation (5) shows that Z 0. differs from ~0" by the replacement U0 ~ U1. Physically this just means that the presence of the dielectric modifies the configuration energy, and hence the response to an external field. It is precisely the difference between Z and & that concerns us here. It will in fact be instructive for the reader to evaluate equation (32) for the isotropic case (where A = constant; X, Y are given by equations (24) and (25) and kij = k6#) and thus show that equation (31) reduces to the usual form of the Onsager equation. Indeed such a derivation of the Onsager equation avoids the.introduction of various ambiguous concepts such as the 'effective dipole'.

Spec&l case Again, to simplify matters, we ignore the electronic contribution, then equation (32) becomes:

(25) Z#'=~lao 2

where in equation (25) we have made the usual assumption ( 4 1 r / 3 ) R 3 N = 1. We are of course concerned here with the anisotropic case. The form of Xi! for this more complicated situation has recently been found and its effect discussed s. The rest of the paper will concentrate on the effect of Y#. If we now substitute equation (21) into equation (19), we get: #

-

F ' d l a = - # i X # E i - ½l~iY#la i

(26)

o

(27)

This can now be inserted into equation (2), which is val~ in general, to yield the following result to first order in E: fgila/exp (-13U l)d3xd~2 Q~i) =/3 f exp (-/3U1)d3xd~2 XjkEk

(33)

1) 11 +ur/]

(34)

with 7} defined as in equation (11). Here:

U = A(a ) + ½ x i k # x ] - ½uiYifl.t] - laiXi]Lj = U 1 - laiXijlzj

f aiaj exp (-13A) exp (½13/~02aiYo'aj ) dr2 f exp(_t3A)exp(½131aoZaiYo.aj)d ~

Note that in the isotropic case aiYi/a/~x a 2 = 1 is independent of ~2 and hence can be dropped. Thus the reaction field has no effect in the isotropic limit when the electronic contribution is ignored; it is a specifically anisotropic effect. This is in fact.0hysically obvious: in the isotropic case, the reaction field ER is necessarily in the same direction as the rigid d__~po~, t~erefore the torque tending to orient the dipole: r = ~ x ER~vanishes. When the medium is anisotropic and Y¢ 4= Y60., E R is no longer parallel to/J, so there is a non-zero torque. The matrix Yij is of course a function of ei/. We consider a case where eij is symmetric about the z-axis: - i =(e-

so that the total configuration energy is now:

(32)

e = try?/3 is the average value (over the principal directions) of the permittivity, while: u=

(28)

(35)

Gxx + ~Vy -- 2ezz

6 ( e - 1)

(36)

is a measure of dielectric anisotropy*. We have calculated Recall that the permittivity tensor e 0- is defined by: 4nNQai) = (elk - 5ik)E k

(29)

* The parameter u here corresponds to u m ref 5.

POLYMER, 1976, Vol 17, January

9

Reaction

field in anisotropic dielectrics: K.

H. Lau and K. Young

Yo"in terms of e and u, using a method similar to that in ref 5, with the result: Y# = R32 e - 1 2e+l

( 6# + u 3 1 - 102e+l

7/# + O(u 2) )

molecular parameters embodied in ~)(Z). It remains to investigate how a non-zero A affects equation (45).

(37)

It is anticipated that the O(u 2) contribution will be negligible; this is the case with X#, which we have previously calculated to O(u 2) s Insert equation (37) in equation (33) and note that the 6# piece in Y0' affects only the overall normalization and may be dropped. Thus:

The reactionfield correction We expand the exponential in equation (45) and also insert equation (16) to get:

y Qlfl (-2Au) n ~n "7" ~ + l ~ .n f P2(Z)n+lPl(Z)dZ (46)

u=-

~n ~

l

Z¢ =/3po 2

f aiaiQ.(~2) exp (.4uairli/al)df2 f~(g2) exp (Auairl#ai)d~2

(38) Since A < 0.05 and lul < 1, we terminate the series at n = 1 :

where we have made the replacement, exp ( - 3 A) = Q ~ Q, as discussed earlier and where A stands for: A-

3 3/a.02

e-1

10 R 3 (2e + 1) 2

(39)

Since the calculation will be carried only to first order in the anisotropy u, we only need A to zero order in u, i.e. for the isotropic case. In that case, equation (30) is:

e_l=4nN31a02 3

3e 2e+l

(40)

u-

NO - 2AuN1

(47)

D O - 2AuD 1 where:

£

Oth

No = Z Ie2(Z)et(Z)dZ = Qj: I 2l+1 25

N1 = ~l ~Qlfl fP2(Z)2PI(Z)dZ

so that comparing equations (39) and (40) gives:

2

4

4

- 5 Qofo + ~5 Q2f2 + 31b _-z-:_Q4f4

1 (e-l) 2 A -

Qlfl (-2Au) n 21~-1 ~ .n fP2(z)net(Z)dZ

(41)

I0 e(2e + 1)

DO = ~l ~Qtfl fPl(Z)dZ = 2Qofo

Note that A does not exceed 0.05. Let us parameterize Z 0. as:

= ~(i + w~)

(42)

D1 = ~I ~Qlfl fP2(Z)Pt(Z)dZ = ~-~ 2 Q2f2

(48)

then solving for w from equation (38) gives: w=-

fP2(z)a(z)

exp (-2auP2(Z))dZ

f Q(Z) exp (-2A uP2(Z))dZ

Notice that since (43)

fP2(Z)nPI(Z)dZ

where we have used:

airlift/= - 2 P

2 (cos 0) = -2P2(Z)

(44)

For reference note that the usual treatment in the literature 23 ignores the anisotropy in the reaction field and corresponds to putting A = 0. Provided that neither the reaction field correction (owingto A ¢: 0) nor the cavity field correction (owing to X ~:X1) is too large, the two corrections may be independently applied. The cavity field correction has already been discussed s, so in this paper we take .~" to be isotropic as in equation (24) and concentrate on the reaction field. Thus comparing anisotropies in equation (31), we find u = w. Hence equation (43) now reads: u=-

fP2(Z)Q.(Z) exp (-2A uP2(Z))dZ f-Q.(Z)exp (-2A uP2(Z))dZ

10

POLYMER, 1976, Vol 17, January

(49)

for I > n, by terminating the sum over n we automatically terminate the sum over I as well. We are now in a position to look at the properties of equation (47). Recall that the molecular polarizability determines Q2f2 by means of equation (17), but does not determine Q4f4, which appears in N 1. This is an interesting breakdown of classical dielectric theory, where 'normally' molecular polarizability determines the permittivity. In practice, however, this will not be an important effect, since the coefficient of Q4f4 is small. So in the rest of the discussion, we shall set Q4f4 to zero. This will be discussed in the Appendix. Ignoring Q4f4, and relating Q2f2 to v b y equation (I 8), equation (47) can be written as:

(45) u =

Thus we have derived an implicit equation relating u to

=0

1 +2Auv

(50)

Reaction field in anisotropic dielectrics: K. H. Lau and K. Young Solving for ~: v=u

[ (4 1 +A

2u2+-u 7

-

(51)

where O(A 2) terms have not been kept. In practice, one measures the anisotropy u = [exx + eyy - 2ezz]/[6(e - 1)] and determines vfrom it, so equation (51) is the experimentally useful equation. (The determination of the orientation function from v is standard and will not be discussed here; see for example ref 5.) The square bracket in equation (51) is the correction factor to be applied to the naive result v = u often used in the literature; the correction is of course proportional to A. In view of the bound A < 0.05, the correction never amounts to more than 2 - 3 % and is totally negligible at the present level of experimental accuracy. Thus the only significant correction, of up to 10%, comes from the cavity field effect s .

CONCLUSION Our main result is that the reaction field correction is small; if a correction is nevertheless desired, equation (51) may be applied. It is well to recall the limitations of the present calculation. First, we restricted our attention to axially symmetric systems. This is the most common situation in polymer physics, and in any case it is inconceivable that the qualitative conclusion of this paper will be altered if one removes this restriction. Secondly, electronic polarizability has been ignored. Thus we expect our result to be applicable in detail only if [e01 >2> I&o I (where the subscript denotes the frequency) or if we consider the dispersion A~ = gO -- ~o~, which should be due to the permanent dipoles alone. Whether the reaction field has any appreciable effect on the birefringence (i.e. the anisotropy in g=) is a separate question to be looked at.

3 Kakutani, H, J. Polym. ScL (A-2] 1970, 8, 1177 4 Phillips, P. J., Kleinheins, G. and Stein, R. S. J. Polym. ScL (A-2) 1972, 10, 1593 5 Lau, K. H. and Young, K. Polymer 1975, 16,477 6 Langevin,P. J. Phys. 1905,4,678; Debye, P. Phys. Z. 1912, 13,97 7 Frohlich, H. 'Theory of Dielectrics', Oxford Univ. Press, London, 1949 8 Kirkwood, J.G.J. Chem. Phys. 1939,7,911 Note added h~ proof Similar problems have been considered in the context of liquid crystals by Bordewijk (Physica 1974, 75, 146). We thank Dr Bordewijk for bringing this work to our attention.

APPENDIX Here we wish to justify the neglect of Q4f4, which, we stress again, cannot be calculated from the polarizability. If we keep the Q4f4 term in equation (48), (51) acquires an extra term: v=u

[(42)1 1 +A

2u2+ - u-+~ 7 5

(51')

where ~ = 4/315 ( Q J 4 / Q d o ) . By the positive o f f , we can deduce

tf41 = 9~ I ff(Z)p4(Z)d Z <<.9~ y f(Z)dZ = 9~ fo since IP4(Z)I ~< 1. A similar bound applies to Q4, hence

I~1 ~< 3 ~

35

REFERENCES 1 Onsager, k. J. Am. Chem. Soc. 1936,58,1486 2 Bares,J. Kolloid-Z. 1969, 239,552

Therefore the inclusion of ~ does not alter the fact that the correction will be quite small, on account o f A < 0.05.

POLYMER, 1976, Vol 17, January

11