Inertial effects in the theory of dielectric and Kerr effect relaxation of an assembly of non-interacting polar molecules in strong alternating fields. II. The effect of higher-order terms in the distribution function

Chemical Physics 125 (1988) 99-118 North-Holland, Amsterdam

INERTIAL EFFECTS IN THE THEORY OF DIELECTRIC AND KERR EFFECT RELAXATION OF AN ASSEMBLY OF NON-INTERACTING POLAR MOLECULES IN STRONG ALTERNATING FIELDS. II. THE EFFECT OF HIGHER-ORDER TERMS IN THE DISTRIBUTION FUNCTION

W.T. COFFEY, S.G. McGOLDRICK and K.P. QUINN School of Engineering, Department ofMicroelectronics and Electrical Engineering, Trinity College, Dublin University, Dublin 2, Ireland

Received 18 February 1988

Formulae for the dielectric and Kerr effect responses for any size of the n x n matrix in the perturbation solution of the Kramers equation described in paper I of this series are given. The solution is carried out explicitly for n=2 and n= 3 for the linear dielectric response and for n = 2, 3 and 4 for the Kerr effect response of an assembly of dipoles having permanent moments only. This is a non-linear response. It is found that for small inertia1 effects the linear dielectric response and also the Kerr effect response for induced dipoles only (which is a linear response as far as solving the Kramers equation is concerned) is adequately described by the truncation n = 2. This corresponds to the use of a modified Smoluchowski equation. For the Kerr effect response for permanent dipoles it is shown from the n= 3 and n = 4 results that the n = 2 result - corresponding to the use of a modified Smoluchowski equation - does not provide an adequate description of the behaviour of the system. An explanation for this is sought in the work of Wilemski, Titulaer, Stratonovitch, Skinner and Wolynes.

1. Introduction

In the first paper of this series [ 1 ] we have outlined a perturbation method for the solution of the Kramers equation for an assembly of polar molecules free to rotate in two dimensions and under the influence of alternating fields. The perturbation procedure is implemented by writing down the Kramers equation for the Debye disk model [2] of a polar molecule under the influence of an ac field. This is solved by separation of the variables. The velocity-dependent part of the solution is expanded in Weber functions while the configuration part is expanded in circular functions, this yields a set of linear differential-difference equations for the separation coefficients, a;(t) which give the time behaviour of the distribution function. These are the Brinkman equations. They are a set of equations in the double differences, n, the order of the Weber functions and p the order of the circular functions, n ranges from 0 to 03 through integer values and p ranges from --co to co through integer values. a; ( t ) for n = 0 and p = f 1, f 2 give the ensemble averages appropriate to dielectric and Kerr effect relaxation respectively. These equations are solved [ 1] by supposing fl +z krwhere E is the amplitude of the applied field and p is the dipole moment of a member of the assembly, thus one may expand the distribution function, or equivalently the ai (t), in powers of the field strength yielding new sets of differential-difference equations correct to first and second order in the field strength respectively. These equations, as a consequence of applying the perturbation procedure, only involve terms like a:(t), a;(t) and al I (t) as forcing terms on their right-hand sides, the al 1(t) can be found from u:(t) by symmetry. Thus as a result of the perturbation the double matrix of the set of Brinkman equations is effectively reduced to a single one that involves the n dependence only, thus the n and p dependences may be systematically uncoupled. The resulting equations for u’f( t) etc. may then be solved by successively limiting the size of the n matrix in them, whence the desired ensemble averages can be found using the Fourier transform method described in ref. [ 11. 0301-0104/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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W.T. Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. I1

The differential equations ( (40), (50) of ref. [ 1 ] ) are valid for any size of the n-matrix. We have however restricted ourselves to a 2 x 2 n-matrix when solving them, and we have found that this yields the same results as those that may be obtained using a modification of the Smoluchowski equation originally discussed by Brinkman [ 31 and Sack [ 41 and later applied to rotational motion by Coffey [ 5,6]. This modified Smoluchowski equation has the advantage that it is a diffusion equation in configuration space only. It thus appears attractive to use for calculating the configuration space distribution function since for all but the most simple problems the solution procedure for the Kramers equation is beset with formidable mathematical difficulties [ 71. Moreover these difficulties are greatly magnified when the non-linear response, such as required for the Kerr effect for permanent dipoles, is taken into account. In order to circumvent these difficulties we have used the modified Smoluchowski equation to calculate the Kerr effect response for the sphere model of a polar molecule [ 11. This has been carried out for various types of driving field and the results are detailed in ref. [ 11. The equation has also been used by Alexiewicz [ 8 ] in a discussion of the response of the sphere model to a rectangular pulse. Despite the seeming attractiveness of the modified Smoluchowski equation as a means of calculating the configuration space distribution function directly, there has however been considerable criticism of it in the literature by Hemmer [ 91, Wilemski [ lo], Titulaer [ 111, Skinner and Wolynes [ 121 and Coffey [ 13,141. These criticisms centre on a truncation of the exact configuration space differential equation originally proposed by Brinkman. The problem has also been discussed by Stratonovitch [ 15 ] and in a neat form using projection operators by Gardiner [ 161. Since our perturbation scheme for the Kramers equation, when the size of the nmatrix is limited to 2x2 yields the same results as the modified Smoluchowski equation, since that equation has been severely criticised and since eqs. (40) and ( 50) of ref. [ 1] are valid for any size of the n matrix, it is important to consider the effect of increasing the size of the n-matrix to 3 x 3,4 x 4, etc., and then to see if the results so obtained agree with the 2 x 2 case. This is the main purpose of the present paper. We find that the modified Smoluchowski equation provides an adequate representation of the inertia-corrected linear dielectric response in that for the parameter values under consideration the 2 x 2 result agrees well with 3 x 3 etc. This will also hold good for the Kerr effect for molecules having induced dipole moments only since this to first order is a linear response. These conclusions are in accordance with the work of previous investigators Gross [ 17 1, Sack [ 18 1, Scaife [ 19 1, McConnell [ 71, whereby the 2x2 result essentially corresponds to the Rocard equation which is an accurate representation of the dielectric response for small inertial effects. This is not however true of the Kerr effect for permanent dipoles (a nonlinear response) where we find that the 2 x 2 approximation does not agree at all with those of the 3 x 3 and 4 x 4 approximations, which lie almost on top of each other. Indeed the Debye approximation, where inertia is entirely neglected, seems to furnish a more accurate representation of the permanent dipole Kerr effect, or first-order non-linear response, than does the modified Smoluchowski equation. This appears to corroborate the work of Hemmer [ 91, where for delta function initial conditions (albeit not those of the present problem), the Smoluchowski equation provides a more accurate approximation to the configuration space distribution function than does the mod$edSmoluchowski equation. It is interesting to note in connection with the present problem that the modified Smoluchowski equation predicts a change in sign of the dc component of the birefringence as the frequency of the imposed field is increased. This effect is not present when the size of the n-matrix is increased indicating that such a sign change is an artifact of the inadequate representation of the non-linear response, by a 2 x 2 truncation. In conclusion of this section we note that in the present treatment and in that of ref. [ 1 ] we ignore the effect of the hyperpolarizabilities b, y. A complete treatment of the problem would require the use of our methods combined with the approach described in ref. [20]. We reiterate that our purpose here is merely to ascertain how accurately the modified Smoluchowski equation may describe the non-linear response arising from permanent dipoles. The contribution of the induced moments in the square law approximation is treated in ref. 121 I.

W.T. Coffey et al. /Inertial effectsin dielectricand Kerr effectrelaxation.II

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2. Arrangement of the paper

The paper is arranged as follows: First we write down the differential equations (40) and ( 50) of ref. [ 11, which, when solved, give the time behaviour of the linear dielectric and Kerr effect responses. We then show how these equations may be solved for any size of the n-matrix, by means of Fourier transform methods. Next we give explicit results for the case where inertial effects are negligible (the Debye approximation), then the results for the n-matrix limited to 2 x 2 (these have already been given in ref. [ 1 ] ) , 3 X 3, and 4 X 4. The linear dielectric response in all cases contains a term which is in phase with the applied field E cos wt and a term which is 90” out of phase with it. The coefficient of the in-phase term, which is the real part of the complex polarizability is plotted versus log,,w for (i) the Debye approximation, (ii) limiting the size of the n-matrix to 2 X 2 and (iii ) to 3 x 3. It is found that for all values of the parameter y= kT /Is’ less than approximatelyo. 1 (corresponding to small inertial effects) that the 2 x 2 and 3 x 3 approximations yield the same results. The 2 x 2 result is that which is obtained using the modified Smoluchowski equation. Thus in the linear response approximation for moderate values of y the modified Smoluchowski equation seems to provide an adequate description of the behaviour of (cos 19). Again we note that the 2x2 approximation is equivalent to describing the complex polarizability by the Rocard [ 221 equation. The in-phase part of the complex polarizability is shown in figs. l-3 below for various values of y in the 2 x 2 and 3 x 3 approximations in order to illustrate the points made above. Note that y plays the role of the Q factor so that small values of y correspond to a low-Q system, that is the heavily damped approximation. We now proceed to the Kerr effect response of permanent dipoles. The differential equation ( 50) of ref. [ 11, describing the Kerr effect response is already much more complicated than that describing the linear dielectric response, since that equation is driven by every term which occurs in the linear dielectric response. In general the terms which occur in the response for an applied field E cos wt are a dc term whose magnitude is frequency dependent, which represents rectification of the applied field by the system, and terms in the second and higher even harmonics of the applied field. The calculation, just as in the linear dielectric case, is carried out for the Debye approximation and the limitation of the n-matrix to 2 x 2,3 x 3 and 4 x 4, respectively. In order to compare the various approximations the magnitude of the dc term is plotted against frequency in each case. For values of y less than about 0.1, the Debye approximation and 3 x 3 and 4 x 4 limits never become negative, while the 2 x 2 approximation on the other hand crosses the frequency axis at a critical frequency w= JkTII, then decreases and increases to zero again. Thus as is borne out by figs. 4-l 1 the Debye result seems to provide a better approximation to the behaviour of the system as far as the Kerr effect is concerned than the 2 x 2 truncation. Since the 3 x 3 and 4 x 4 results lie almost on top of each other one may conclude that they provide a reasonable approximation to the actual behaviour. We note that the Rocard expression, for the birefiingence, eq. (7%) of ref. [ 11, also never crosses the frequency axis and has behaviour indistinguishable from the Debye approximation. An explanation for the failure of the modified Smoluchowski equation to provide an adequate description of the non-linear response is sought in the objections to that equation made by Hemmer [ 91, Wilemski [ lo], Titulaer [ 111, etc. Their objections to the use of that equation are described later and are also summarised in ref. [ 27 1. Finally, it is shown in the appendix how the Brinkman [ 23 ] equations may be derived by using the characteristic function of the velocity distribution function. This eliminates the need to know the properties of the Weber functions and it is shown how the Weber function and the characteristic function methods both lead to the same set of differential-difference equations.

3. The linear dielectric response Referring to eq. ( 10 1) of ref. [ 1 ] which is

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W.T. Coffeyet al. /Inertial effectsin dielectricand Kerr effect relaxation.II

i V. cos wt -m’

we wish to calculate from this set the quantity a?“’ (t) since 1 a?“‘(t)+ 2

(cos8(t))=

andad

-

a0

[a?“‘(t)]*, O(O)

isalwaysrealsothat

bj’~(t)=Boa~‘)(t)+Co~,

a8c0j

I

dm

(2)

=2A9

(cosO(t))

issimplya?“‘(t)/ao

(1)

O(O) . Eq. ( 1) may be written in the matrix form

i V. cos ot

(3)

where B. is an n x IZsquare matrix and Co is an n-row column matrix. Note that B. and C, are infinite matrices since the index of the Weber functions goes to infinity. If we now Fourier transform eq. (3) we find that .~T(i(2n)[s(w-o)+s(w+Q)l},

iQA$“(L2)=BOA~“(B)+Co

(4)

where AI’)(Q)=

7 ai’) (t) exp( -ii&) --oD

dt .

(5)

Rearranging eq. (4) we find that A$“(SZ)=(iQI-Bo)-‘Co

[6(w-8)+6(0+51)].

(6)

The inverse Fourier transform of this is 1 iV,n 2rrWT__

O” f

(&!I-B,)-‘Co[6(m-L2)+6(w+Q)]exp(iGt)dQ.

Now m I

(7)

CO

f(X) &x-u)

dX=

--to

s

f(x) &a-x)

d.x=f(a)

,

(8)

-al

so that

al’)(t)=

’ ivo 4~~

[(iwl-B,)-‘COexp(iwt)+(-iol-B,)-’Coexp(-iwt)].

(9)

By inspection of (9)) the right-hand term in that equation is the complex conjugate of the left-hand term and so it simplifies to al”(t)=

2 ‘ykTRe[

(iol-B,)-’

C,exp(iwt)]

.

(10)

W.T. Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

103

Note that cx= iJkTII but for computational convenience is treated as a real quantity within the B,, matrix. Eq. ( 10) allows the linear dielectric response to be calculated to any desired degree of accuracy by successively increasing the size of the matrix B,, thus automatically increasing the size of (iwl- Bo)- ‘. One then picks off the a?(‘)(t) element in order to get the dielectric response. If we start by llmiting the size of the B,, matrix to 2 x 2, one then finds as described in ref. [ 1 ] that

Bo=(_: 1;)

(11)

so that (iol-Bo)-‘=

1 iw(io+p)-a2

(iw+8 -_(y

--J.

(12)

Hence the dielectric response is from (2) p2E (kT/Z-w’) cos ot+opsin 21 (kT/Z-m2)2+ (w/3)’

M(t)=p(COSe(t))=

cot

up”‘(t) =0(o). a0

(13)

The real and imaginary parts of the complex polarizability, a(o)=a’(w)-i&(w),

(14)

corresponding to this, are 1- W%//3 (l-0%//3)*+ (wr)2 ’

ff’ (w) -= a’ (0)

WT

d(w) -=

(1 -w2r/j3)‘+

Q’ (0)

(wr)2 ’

(15)

(16)

where r= c/kT

(17)

is the Debye relaxation time for rotation in two dimensions and /3=C/Z,

a’(0)=p2/2kT.

(18)

If I= 0 we regain the Debye result 1 a’ (0) ~a’ (0) = l+(wT)2

(19)

a!” (0) -a’(O)

(20)

WT = l+(wr)2’

These are the results for the Debye approximation and the 2 x 2 limit. For purposes of computation it is useful to introduce the parameter y=kT/Zfi2

(21)

and the normalised frequency X=W?

(22)

W T. Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

104

x 6

8

lo

12

14

LOG,Od

Fig. 1. Real part of normalised complex polarizability versus log,,o for ~~0.05. 1 is the Debye result, eq. (25); 2 is 2x2 result, eq. (23) and 3 is 3x3 result, eq. (28). The Debye relaxation time is taken as 3 x 1Oe9 s. Note the close agreement between 2 x 2 and 3 x 3 approximations.

Fig. 2. As fig. 1, for y=O.l. Note the close agreement between 2 X 2 and 3 X 3 approximations.

so that eqs. (15), (16), (19) and (20) become 1 -x2y a!’ (xl m = (1 -x2y)Z+x2

’

a!“(x) cu’o

= (1 -x&+x2

’

a’ (x)

1

o!“(x) -=a’(O)

1:x2,

(23) (24) (25)

-=1+x23 o’ (0)

(26)

so that y provides a measure of the inertial effects. Since the response is linear, eq. ( 1) need not be solved in order to determine (Y’(co) and CX”(0) for higher values of the n-matrix. All one has to do is to calculate successive convergents of the continued fraction expression for (Y[ 18,191 namely (27)

LOGloo

Fig. 3. As fig. 1, for y=O.16.

W.T. CO&I et al. /Inertial efJcts in dielectricand Kerr egecf relaxation.II

105

The first convergent of this is the Debye result. The second is eqs. (23) and (24) and the third yields a’(x) cu’o

2(2-3x2y)+x2y[(2+3y)-x2y21 = (~-~x~Y)~+x’[ (2+3y)-x2y2]*

a”(x)

x{2[(2+3y)-x2y2]-y(2-3x2y)} (2-3~~y)~+x~[(2+3y)-x~y~]~

m=

(28)

’

(29)

*

This procedure is entirely equivalent to limiting the size of the matrix in ref. [ 1 ] and then extracting a?“)(t) [ 23 J, Limiting the size of the B0 matrix to 2 x 2, 3 x 3, etc., corresponds to calculating the second-, third- and higher-order convergents of (27 ). We do not give results for the fourth convergent as it can be seen from the figures below that the second and third convergents approach each other very closely for typical values of y which for small inertial effects would be in the range O-O.1 as exhaustively discussed by Gross [ 1’71and Sack [ 18 I. Thus the second convergent provides an adequate approximation to the actual behaviour of the system for the relevant y values. By way of ~lustration of our results we show the behaviour of ty’ (0) versus log,,@ for various values of 1: in figs. l-3. It is evident that an acceptable approximation to the behaviour is yielded by the second convergent except in the case where y= 0.16, where there is significant deviation of the second from the third convergent. This value of y is, however, rather larger than the critical one 0.05 suggested by Sack [ 181 and normally y would not be as large as 0.16. Hence we are justified in saying that for the linear ~espu~e, the second convergent or equivalently the 2 x 2 approximation to the BO,or transition matrix of the system, provides an adequate description of the behaviour of a (0). It also indicates that the modified Smoluchowski equation may also be used to describe the linear response for moderate y. Having discussed the linear response we shall now examine the dynamic Kerr effect for permanent dipoles.

4. The Kerr effect for permanent dipoles or first-ordernon-linear response The Kerr effect response is got by solving (appendix B of ref. [ 1] ) the matrix differential equation 0

~~~=~

‘g

1:;

which may be written ~12’(t)=B,a~2f(t)+Cl(t)

i V0cos ot --=

~~~~

lI a?(‘)(t) af(‘)(t) ... a;-‘(‘)(t) ...

i V, cos cot

2JIk5T’

(30)

(31)

where 0

G (l)=

apt’)(t) at(‘)(t) ... a;-‘(‘)(t) ... 1.

(32)

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W.T. Coffey et al. /Inertial effectsin dielectricand Kerr effectrelaxation.II

In order to simplify eq. ( 3 1) , we introduce the skew diagonal matrix [ 24 ]

E=

0 0

1 0

0 1

0 0

... 0 ..#

(33)

and also we note that

B, =[-k

‘K

In,

(34)

:-‘).

In order to utilise eq. ( 33 ) , in the solution of ( 3 1) , we observe that EAj” (a) =cl (8) ,

(35)

where A[ ’ ) (a) is the Fourier transform of a f ’ ) (t ) and C, (a) is the Fourier transform of C1 ( t). Thus, utilising the shifting theorem of Fourier transformation, namely 9{exp( -iot)f(t)}=F(O+w)

(36)

,

we find that (3 1) becomes (al-8,)

A$“(&!)=

[EAI”(iLw)+EA{“(8+o)],

(37)

or utilising eq. ( 6 ) , (iQl-B,)A$2’(J2)=-

+ [i(Q+w)

I-BO]-’

$${E{[i(B-w)

I-B,]-‘C,[6(52-20)+S(n)]

C0[8(Q)+8(J2+2c0)]}},

(38)

whence Ai2’ (a) = - $-$(iQl-B,)-‘{E{[i(G-o) + [i(Q+w)

I-Bo]-’

I-B,]-‘Co[6(8-20)+6(8)]

Co[6(9)+d(L2+2co)]}}.

(39)

We can now invert this equation into the time domain just as we did for the linear response. We first note that

in performing the inverse Fourier transformation we have to evaluate integrals of the form 1 O” F,(Q) F2(s2&o) 8(Q) exp(iSlt) dS2 % s -ca

(40)

and 1 m I;,(Q) F,(G!fo) 5 s -ca

6(8+20)

exp(iQt) ds2.

Both these integrals may be evaluated using

(41)

W.T Coffey et al. /Inertial effects in dielectric and Kerr eflect relaxation. II

107

co

s

f(x-a)

6(x) dx=f( -a)

(42)

--oo

and co

f(x)

j

6(x-u)

dx=

--a,

7

f(x)

&a-x)

cLx=f(a)

.

(43)

--a,

It is convenient when evaluating the inverse transform to consider the terms in 6(Q) and 6(52+ 20) separately. The term in 6(a) when inverted to the time domain is - &-B,)-YEN

-iol-BO)-‘C,-,+(iwl-BO)-‘Co]},

(44)

which simplifies to -~*(-B,)-lERe[(iwl-B,)-lC,]. This is a frequency-dependent

(45) dc term. The terms in 6( Q+_20) yield

- &~{(i2Wl-B,)-*[E(iwl-B,)-‘C,,exp(i2wt)] + (-i201-B,)-‘[E(-iwl-B,)-‘C,,exp(-i2c&)]},

(46)

which further simplifies to - $$e{(i2wl-B,)-1[E(iwl-B0)-1C,

exp(i2c&)]}

(47)

since the two terms are complex conjugates of one another. By inspection of (47) we see that it is the second harmonic of the impressed field, which has a frequency-dependent amplitude, determined by the size of the nmatrix. (All the formulae above for the steady state response, including eq. ( lo), may be readily adapted using the Laplace transform to calculate the transient response to a suddenly applied field.) The dc term which represents rectification of the applied field is simple to plot and we will calculate it in the 2 x 2, 3 x 3 and 4 x 4 limits, of the system matrices. The second-harmonic term will be given in the 2 x 2 and the 3 x 3 limits in order to avoid excessive calculations. Consider the dc term for the 2 X 2 limit, we have

and

c0= thus

( O> a0O(O)

’

(49)

W.T. Cofey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

108

and (-B,)-L=

&( ;i 2;).

(51)

Thus -$&(-B,)-’

ERe[(iol-BO)-‘C,,]=u~(2)(t) v&zp

=

8Z2[ (kT/Z-w’)‘+

1

(w/Q21

(52)

2kT/Z ’

where the bar denotes a time average, i.e. we are only considering the dc term. Since (COS2e(t))=a~(2)(t)/a~‘o’

(53)

because [a;‘2’(t)]*=u;‘2’(t),

(54)

we find that (‘OS 2e(t))=

J’;(kT/Z-w2) 16kTZ[ (kT/Z-w’)‘+

(55)

(wp)2]

and the complete expression for (cos 28(t) ) is as in ref. [ 11, (cos28(t))

G = 8Z2[ (kT/Z-w2)2+ +

(w/?)~]

[(?-w2)(&+ 1*

2[4(kT/Z-w2) cos2wt+2w/lsin2wt] 16(kT/Z-w2)2+4(wj3)2

sin2wt-2w/3cos2wt] ws2[4(kT/Z-w2) 16(kT/Z-w2)2+4(w/3)2

> (56)

On inspection of (56) we see that the constant term, just like the real part of the complex polarizability, vanishes at w2=kT/Z,

(57)

decreases and then increases to zero again, which indicates that the constant component of the birefringence due to polar molecules having permanent dipole moments can go negative. This is at variance with the work of Rocard [ 221 ( (75~) of ref. [ 1 ] ) which suggests that the birefiingence due to permanent dipoles goes like 1 [l+(w/~)2](l+w2T2)

1 = 1+w2r2+w‘V/p2

(58)

This never goes negative. We also note the Debye approximation where (59)

so that the dc birefringence decays like the real part of (Y(W),just as in the 2 X 2 inertial case. In order to see if the 2 x 2 approxi.mation provides an adequate description of the behaviour of the system, we now go to the 3 x 3 limit of the B. and B, matrices.

W.T. Coffey et al. /Inertial efects in dielectric and Kerr effect relaxation. II

109

In the 3 x 3 limit E+;

Gj

It)

(60)

and -2cY -p

0 -4CY

-2cY

-2p

(61)

so that with s=iw W-&)-I=

(s+B)(s+2/3)-2a2 -a(s+2/3) (Y2

det(s;_B,)

-a(s+2P) s(s+28) --(YS

2a2

(62)

where det(sl-Bo)=s[(s+B)(s+2j?)-2a2]-a2(s+2/3)

(63)

and (-B,

)-I=

8a2-2fi2 4a/3 -4a*

bp

4@ 0 0

-8~’ 0 4a2

(64)

Also 0 co = 1 up 00

)

(65)

(66) whence O(O) = - 8Lu2i(i2+b2)

ERe[(iwl-Bo)-‘Col and thus al’)(t)=

Vgzg’“’ ~ 16ZkTj?(a2+b2)

2fi2a-2aw2+5/3bw Qo2 ’ - 2j3bw

(68)

whence a;“‘(t)=-

where

v; 161kTB(a2+b2)

(2aw2-5j3bw-2/32a),

(69)

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W. T. Coffey et al. /Inertial eflects in dielectric and Kerr effect relaxation. II

a= -/3(3w2+2cy2)

(70)

and b=o(

-w2+2/32-3ci2)

.

(71)

In the same way the time-dependent part of a !(2) (t) is in terms of its cos 2mt and sin 2wt components aqC2)(cos 2rN) =

Vi$agC0)[4Acy2(3w2 -2/32) -28Ba2j30]

a:(2)(sin 20t) =

8kTZ(A2+B2)

,

V$a$Y”)[4B~2(3c02 -2/32)+28Acy2/30] 8kTZ(A2+B2)

,

(72)

where A=4[-2~~+0~(14P~+30~~)+~~(27~y~B~-28~-18~y~)+4~~8~],

(73)

B=4j3~[15w4+w2(37a2-9/92)+6~2(30!2-/!?2)]

(74)

and Vo=@.

(75)

The most singular feature of the 2 x 2 approximation is that the dc birefiingence changes sign at o= JkTII. We now examine whether the dc component of the 3 X 3 approximation shows a comparable change in sign. In order to investigate this we write the numerator of this expression (eq. (69) ) in terms of the parameter kT y= p

(76)

and set the numerator = 0, so that w4+4w2B2(~!y+1)+484y=O.

(77)

Eq. (77) has no real roots, so that the constant part of the birefringence does not go negative in contrast to the 2x2 result. In order to ascertain whether or not this behaviour is an artefact of the truncation we must now proceed to the 4 x 4 approximation.

5. The 4 X 4 approximation for the Kerr effect In order to examine whether or not there is satisfactory agreement between the 3 X 3 and 4 x 4 approximations it will be sufficient to compare the dc terms only, as the algebra involved in calculating the second-harmonic term becomes very tedious when one goes to the 4 x 4 approximation. Referring to eq. (44) we find that for the 4 X 4 approximation

(78)

111

W. T. Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

0 -2ix 0

1=

0

i

6

-2cY -2cu

0 -4a -28

0 0 -6ct

-p 0

-2a

-38 1

(79)

and thus (A-B,)-‘=

1 S4+6/9S3+ ( llj12-6a2) S3+6S2/3+S(

llp2-5cr2)

S2+j?(6B2-

14a2)S+3a2(a2-2j12)

-cys2-5j!?iYs

+3/3(2/3’-3a2)

2a2s+6a2P

-6a2

+3a((~‘-2/3’)

-aS2-5aj?S-6a~2+3a2

-2cxs2-6~~s

s3+5/3s2

6sa ’

+3s(2j12--‘) X

ff2s+3a2/3

-as2-3jks

s3+4s2p

+s(3jY2-cY2)-3a2p -ff3

Sff2

-crs2-j?o!s+&

- 3as2-

3cq3s

+3a2 ~~+3j3s~+s(2/3~-3a~) -2pa2

(80) (-B,)-‘=

1

6a,(2cu2-p2)

12a2(2a2-p2)

6a2/9 -4a3

0

0

0

- 6a2j?

0

4ff3

(81)

(82)

(83)

whence 0 ERe[(iol-Bo)-‘Co]=-&-+

O(O)

a~z[w~+3(a~-2/3~)] -5jko2a+ob[3(2/?2-a2)-w2]

(84)

accw2 - 3@wb where this time a=w4-w2(

ll/?2-6a2)+3a2(a2-2fi2)

(85)

and

b=6w/13-

14w/k2-6w3f3,

(86)

W. T. Coffey et al. /Inertial effects in dielectric and Kerr efect relaxation. II

112

(-B,

Co]

)-I E Re[ (iwl-B,)-’

a0O(O) = 12a*(2cr*-B2)(a*+b2) i

3/3(/I*-6cu*) 6a(2a2-/I*) 6a2/? -4a3

~cx(~cx~--~*) 0 0 0

(87)

and thus ~a;‘*‘(t)=-

161kr(2;f(0;;(a2+b2)

+ b[03( -2/3)+w(

{a[w2(-2a2-ll~2)+6a4-15a2~2+6~4]

17p3-4a2/3)]}.

(88)

Substituting for a and b, O(O) v;

u4’2’(f)=-

161kr(2;;_~2) (u*+b*)

[06(/32-2~2)+c04(-6~4-7cy2/V2+13/34)

+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(89)

Eq. (89) is the de term in the 4 x 4 approximation. The second-harmonic term can be evaluated using our general equation (47) but we do not consider it necessary to do this in order to study the convergence of the various approximations. 6. Numerical comparison of the various approximations The results of numerical computations of the dielectric response have been discussed in section 3 and are in agreement with the conclusions reached by Sack [ 18 1, Scaife [ 19 1, Gross [ 17 ] and McConnell [ 7 1, thus we need not discuss them any further here. It is evident on inspection of figs. 4- 11 that the 2 X 2 limit provides a particularly poor approximation to the

z.

1

6

e

L.-

(

10 LOGlOO

12

14

Fig. 4. Debye equation (59) 2 x 2 equation (55), 3 X 3 equation (69) and 4x4 equation (88) limits versus log,,+ for y= 10e4. Here, and in the following figures, a!(“) (t) is normalised by its initial value at(O) and his taken as 2X lOI* rad/s. On this scale it is impossible to distinguish the high-frequency behaviour which is determined by the various approximations.

i

10

n

12

13

1L

LOGlO 0

Fig. 5. Magnified version of high-frequency part of fig. 4. 1 is the Debye result (59), 2 is 2X2 approximation (55), 3 and 4 are 3 x 3 and 4 x 4 approximations (69) and ( 88). For this value of y the convergence is achieved at the 3 X 3 approximation. Note the poor approximation provided by the 2 x 2 limit which is worse than that provided by the Debye result, gained from the Smolochowski equation. Note also, how the 2 x 2 approximation produces a negative component of dc birefringence at high frequencies.

113

W. T. Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

x

6

10 LOG,,, (3

12

14

Fig. 6. As fig. 4, y=O.Ol. Note again the wide deviation of the 2 x 2 limit from the 3 x 3 and 4 x 4 limits which lie on top of each other, again showing that satisfactory convergence is obtained at the 3 x 3 approximation for this y value.

x

,,‘,,,.,.

6

I,‘,,‘,‘,,

L-..

10

'8

0

12

14

LOG,od

Fig. 8. As fig. 4, y=O.OS.

6

8

10

Fig. 7. Magnified version of figure 6 showing the high-frequency behaviour in more detail. Note again how the 2 x 2 goes negative.

0

10

n

12 LOG10 0

13

14

Fig. 9. High-frequency behaviour of fig. 8 in more detail.

12

14

LOG,Od

Fig. 10. As fig. 4, y=O. 1. Note that 3 x 3 and 4 x 4 limits are now beginning to diverge from each other as is expected for this value of y above which many terms in the n-matrix must be taken in order to ensure convergence.

LOG,od

Fig. 11. Magnified version of the high-frequency behaviour of fig. 10.

W.T. Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

114

behaviour of the (cos 20(t) ) averages for all values of y. Thus it appears that the 2 x 2 approximation which corresponds to using the modified Smoluchowski equation for the variation of the configuration space distribution function cannot adequately describe the dynamic Kerr effect for permanent dipoles, and by inference all higher-order averages (cos nt9(t) ) .

7. Conclusions An explanation for the bad approximation to the Kerr effect average provided by the 2 x 2 limit may be sought in the objections to the use of the modified Smoluchowski equation which have been made by Hemmer [ 91, Wilemski [ 10 1, Titulaer [ 111, Skinner and Wolynes [ 12 1. Their objections stem from a criticism of a truncation of a series expansion for the distribution function in powers of the inverse friction coefficient /I-’ first given by Brinkman [ 3 1. Brinkman derived a set of equations from the Kramers equation for the Laplace transform T( 8, s) of the density in configuration space. Further, if all terms on the right-hand side of this set of equations except the first one are ignored then the Laplace transform of the Smoluchowski equation with corrections namely

$(e,S)-f(&O)= iye, t)= - av/ae

&$

zp KE!

ae

0 - -we, zp > fuw

1,

(90) (91)

in our notation, is obtained. Hemmer [ 91 solved this equation for a free particle, i.e. F= 0 with the initial condition f(e, 0) =s(e-$)

(92)

and showed that at short times f (0, t), calculated from the Smoluchowski equation approximates more closely the exact f (8, t) calculated from the Kramers equation than does that calculated from the modified Smoluchowski equation. Sack [ 41, however, pointed out that for the delta function initial condition the modified Smoluchowski equation is not valid at short times, and higher-order terms in Brinkman’s expansion must be included. Skinner and Wolynes [ 12 ] have considerably dispelled the confusion surrounding the modified Smoluchowski equation. Using a projection operator technique they showed that Brinkman’s complete expression fern 0, s) in the s domain, eq. (2.4.1.3) of ref. [ 21 is correct and that what is incorrect in his procedure is his method of inverting the Laplace transforms in his equation (2.4.1.3) of ref: [2] so as to produce the modified Smoluchowski equation. They then performed a consistent calculation in the sense that terms of the same order in the reciprocal of the friction constant in the set of Brinkman equations are all included in the inversion to the

time domain and find that the corrections in the time domain agree with these obtained by Wilemski and Titulaer. Thus in the inversion of Brinkman’s equations into the time domain, all terms of the same order in the reciprocal of the friction coefficient must be given equal weight. In the inversion procedure proposing the modified Smoluchowski equation (that equation is the leading term in (2.4.1.3) ) this has not been done, hence the confusion arising out of the use of that equation. In particular the reader is referred to eqs. (26) and (28) of ref. [ 12 ] where Skinner and Wolynes show that for both Brownian motion and BGK models the leading term in the expansion of the configuration space diffusion equation in inverse powers of the friction coefficient is the Smoluchowski equation, while the next term involves powers of the derivative higher than the second in contrast to the modified Smoluchowski equation. (The problem is also discussed at length by Gardiner [ 16 1, section 6.4, and by Stratonovitch [ 15 1, sections 11.1 et seq., note in particular eqs. (4.245 ) of ref. [ 15 ] and (4.108 ) of ref. [ 161.) Thus we conclude that the modified Smoluchowski equation cannot be used to describe the dynamic Kerr effect for polar molecules and higher-order non-linear effects such as the non-linear dielectric effect and effects arising from the hyperpolarizabilities. For a single applied field E cos ot the general equations (45 ) and

W.T. Coffey et al. /Inertial effects in dielectric and Kerr efect relaxation. II

115

(47 ) must be used, as they are true for all values of n. The procedure leading to (45 ) and ( 47 ) may be generalised to solve the problem of an applied double field El cos w, t+ E2 cos co2t as done in ref. [ 251 for I= 0. It may also be extended to the rotating sphere model as described by Sack [ 18 ] although the calculations become very difficult as the starting point is now the Kramers equation for the rotational Brownian motion of a sphere as given by Sack [ 18 ] and McConnell [ 7 1, which involves solving complicated sets of three-term recurrence relations. In view of the poor approximation to the non-linear response provided by the modified Smoluchowski equation the results given in ref. [ 1 ] which are based on that equation or on the 2 x 2 truncation of the B matrix should only be regarded as valid for the linear dielectric response of polar molecules, for the Kerr effect response for non polar-polarizable molecules which is also a linear response as far as solving the Kramers equation is concerned, and for the Kerr effect response of permanent dipoles when I= 0.

Acknowledgement We thank Mr. P.J. Cregg for considerable help in the preparation of the manuscript. Louth County Council is thanked for a grant to SGM. Mr. D. Simpson is thanked for diagrams.

Appendix. Alternative sets of differential-difference equations - another method for deriving the Brlnkman equations We have already indicated how the Brinkman equations may be obtained, in the time domain, from the Kramers equation by expanding the velocity-space distribution as a set of oscillator or Weber functions. A slightly easier and entirely equivalent method for obtaining this set of equations is to construct (following an idea originally due to Sack in 1957 [ 181) the characteristic functions of the velocity-space distribution. This leads to the same set of equations as those obtained using the Weber functions. The advantage of this approach is that, for problems involving interacting particles, the analysis is easier when the expansions are made in terms of characteristic functions. When the Weber functions are used we need to know the properties of the N-dimensional Hermite polynomials as described by, for example, Appell and KampC de Ftriet [ 261, where N is the number of interacting particles in the system. We begin by introducing a function, 0, which is the Pourier transform of the distribution function, YVin velocity space, thus cc 0,(e, U, t) =

I

W(&&t)exp(-iiu@d&(exp(-iu@)

(Al)

--co so that YV(8,& t)= &

5 @(O, u, t) exp(iu4) du . --m

The derivative of Wwith respect to 4 and the product, &-transform =iu@( 8, u, t) and

to

W.T Coffey et al. /Inertial effects in dielectric and Kerr effect relaxation. II

116

respectively. The Kramers equation may be transformed, using eq. (A. 1) to yield a partial differential equation for @, namely

(‘4.2) Since the potential, V, is assumed to be periodic we expand Vand CDas Fourier series. Thus

p=m

aqe,24, t)= -L 2r,=&, exPtip@ c%P(u, 0

T

where exp( -ip&

@(O, U, t) d&(exp(

-ipO) exp( -iu&)

,

-7l

Eq. (A.2) then yields, for a cosine potential v= - v, cos kfe

the set of linear partial differential equations: (A.3) which can be transformed into a set of linear differential-difference that @J u, t) may be expanded in powers of iu. Thus @p(u,t)=

,f b;(t) fl=O

(iu)n=
equations by supposing, following Sack,

exp( -ipS))

(A.4 1

and equating (A.3) and (A.4) b;(t)=

f

(exp( -i@)

(-b)n>

.

Gathering together terms of the same in order iu in (A.3) yields the set of differential-difference

W(t) dt

+n/?b;(t)-ip(n+l)

b,“+‘(t)+

@$’

[b;&(t)-b;;h(t)]-

$3b;-‘(t)=O.

equations:

(A.51

These are not the same as the Brinkman equations, their relation to them may be studied by introducing the transformations aye, 24,t) = qe,

U, t) exp

and noting that

and substituting (A.5) and (A.6) into (A.2) we find that

(‘4.6)

117

WT. Coffsvet al. /Inertial effectsin dielectricand Kerr effectrelaxation.II

=

__Bu[gexp( - $)- FuYex*( - g+ +exp( - F)]

)

where, upon cancelling out the exp ( - kTu2/2Z) terms, we have (A.7)

The advantages of this form of the diffusion equation over (A.2) have been discussed by Sack. From our viewpoint it simplifies the subsequent calculations considerably. Introducing the Fourier expansion

we find that

awp -p at

WP T$--pwP

kT

(

+

$%4(y,_,-y,+A4)+~u~

=o

(A.81

*

>

If we assume, as before, that vP may be expanded in powers of iu viz.

Y,(u, t) =

nzo c;(t) (iu)” ,

we find, by grouping together terms of the same order in iu, that dc”( t)

*

+/h;:(t)-ip

(n+l)

c;+*(t)+

Tc;-l(t)]

2z

[c;zL4t)--c;&(t)]=O.

This set of equations again differs slightly from those of the ug. It is not, however, very difficult to show the two sets are identical. One should first recall that the usual expansion of the distribution function, in terms of Weber functions expands the velocity-dependent part into functions of the argument. Thus, if we had transformed the velocity-dependent portion of the distribution function into, say, v space rather than u space, where vX Jl/kT= u, we should have obtained the standard differential-difference equations. Substituting vJllkT= u into (A.8),

a~p(vy0

at

+~flTv[yp_&,t)-yp+&,t)]+Payp;;t)=O

(A.lO)

and expanding v/Pin powers of iv as opposed to iu, dc”( t)

+---

+n&(t)-ipJkTII

[c;-‘(t)+(n+l)

c;+‘(t)]+

%@?[c;zL(t)-c;&(t)]

=O.

(A.ll)

These equations are the same as those for the a;, the apparent difference being that here the Fourier series expansion is in exp (ip0) while there the expansion is in exp ( - ip6). Now consider

W.T. Coffey et al. /Inertial effectsin dielectricand Kerr effectrelaxation.II

118

Then, kTu2

Thus we, u, 0) =_w)

.

References [ 1 ] W.T. Coffey and S.G. McGoldrick, Chem. Phys. 120 ( 1988) 1. [2] M.W. Evans, G.J. Evans, W.T. Coffey and P. Grigolini, Molecular dynamics ( Wiley-Interscience, New York, 1982). [ 31 H.C. Brinkman, Physica 22 ( 1956) 29. [4] R.A. Sack, Physica (1956) 917. [ 51W.T. Coffey, J. Phys. D 10 (1977) L83. [6] W.T. Coffey, Chem. Phys. Letters 54 (1978) 519. [ 71 J.R. McConnell, Rotational Brownian motion and dielectric theory (Academic Press, New York, 1980). [ 81 W. Alexiewicz, Acta Phys. Polon. A 72 ( 1987) 177. [9] PC. Hemmer, Physica 27 (1961) 79. [lo] G.J. Wilemski, Stat. Phys. 14 (1976) 153. [ 111 U.M. Titulaer, Physica A 91 ( 1978) 321. [ 121 J.L. Skinner and P.G. Wolynes, Physica A 96 (1979) 561. [ 131 W.T. Coffey, in: Molecular dynamics, M.W. Evans, G.J. Evans, W.T. Coffey and P. Grigolini (Wiley-Interscience, New York, 1982)~~. 111 ff. [ 141 W.T. Coffey, Mol. Phys. 39 (1980) 227. [ 15 ] R.L. Stratonovitch, Topics in the theory of random noise, Vol. 1 (Gordon and Breach, New York, 1963 1. [ 161 C.W. Gardiner, Handbook of stochastic methods, 2nd Ed. (Springer, Berlin, 1985 ). [ 171 E.P. Gross, J. Chem. Phys. 23 (1955) 1415. [ 181 R.A. Sack, Proc. Phys. Sot. B 70 (1957) 402,414. [ 191 B.K.P. Scaife, Complex permittivity (The English Universities Press, London, 197 1). [20] L. Hellemans and L. de Maeyer, Chem. Phys. Letters 129 (1986) 262. [21] W.T. Coffey, S.G. McGoldrick and P.J. Gregg, Chem. Phys., 125 (1988) 119. [22] M.Y. Rocard, J. Phys. Radium 4 (1933) 247. [ 231 W.T. Coffey, M.W. Evans and P. Grigolini, Molecular diffusion and spectra (Wiley-Interscience, New York, 1984) [also published in Russian (Mir, Moscow, 1987) 1. [24] W.T. Coffey, C. Rybarsch and W. Schroer, Compel 2 ( 1983) 9. [25] B. Kasprowicz-Kielich and S. Kielich, Advan. Mol. Relaxation Processes 7 ( 1975) 275. [ 261 P. Appell and J. Kampe de Feriet, Fonctions hypergeometriques et hypersphtriques, polynomes d’Hermite (Gauthier-Villars, Paris, 1926). [27] H. Risken, The Fokker-Planck equation (Springer, Berlin, 1984).