Inertia-corrected budó treatment of dielectric relaxation in polar molecules: Application to the fir spectrum of acetonttrile and hexanone-2

Inertia-corrected budó treatment of dielectric relaxation in polar molecules: Application to the fir spectrum of acetonttrile and hexanone-2


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Volume 129, number 4


5 September 1986


W.T. COFFEY, P.M. CORCORAN and J.K. VIJ School of Engineering, Department of Microelectronics and Electrical Engineerink Trinity College Dublin, Dublin 2, Ireland Received 11 April 1986; in final form 31 July 1986

The theory of dielectric relaxation of an assembly of molecules with nearest-neighbour nally developed by Budo, is extended to include inertial effects. This yielda an orientational rise to Debye-type absorption in the microwave region and resonance absorption in the FIR ly similar to the itinerant~scillator model when the outer cage in that model carrier a dipole favourably with experimental observations on acetonitrile and hexanone-2.

dipole-dipole coupling, origicorrelation function that gives region. The results are formalmoment. The results compare

1. Introduction

Some time ago a detailed account was given by Bud6 [ 1] of dielectric relaxation in an assembly of molecules containing rotating polar groups. Recently, there has been renewed interest in the use of this model, when inertia is included in the equations of motion, to explain the experimentally observed absorption spectrum of polar liquids in the microwave (MW) and far infrared (FIR) regions [2-41. A full treatment of the dipole-dipole interaction between two polar molecules requires a numerical solution of the linear differential-difference equations arising from the corresponding two-particle Fokker-Planck-Kramers (FPK) equation [5,6]. However, when the two dipoles are equal in all respects the correlation functions arising from the FPK equation may be rewritten in terms of simpler correlation functions [7,8]. This allows the motion of two coupled dipoles to be described as the motion of a free rotator (eq. (13.11) of ref. [9]) and the motion of a particle in a cosine potential [IO,1 11. This uncoupling of the motion into two distinct and independent modes simplifies the problem considerably. We will refer to this case, i.e. when the two dipoles are equal in all respects, as the equal-dipole (ED) model. For a recent review see ref. [93. In earlier work [ 121 an itinerant-oscillator (IO) model was developed where it is assumed that any particular molecule of the liquid is surrounded by a cage of nearest-neighbour molecules. These are assumed to rotate as a rigid body about the central dipole. This earlier model suggests that a significant peak will not occur in the FIR region of the spectrum unlessthe moment of inertiaof the cage of moleculesis somewhatlargerthan that of the centraldipole. It is the purpose of this Letter to indicate briefly how, for the case of unequal dipoles (UD), the equations of motion of an IO-type model may be decoupled. Dipole-dipole coupling isfilly included and both dipolesare subject to random therma torques. This allows the various correlation functions of the model to be rewritten in terms of simpler correlation functions of the new uncoupled variables. It is also shown, how the UD model can produce a significant broadening of the polarisability in the FIR region. For the ED case we fmd that this effect is insignificant, contrary to previous indications [2]. The only restriction that has been imposed, in order to derive these results, is that the friction coefficients which act on the dipole and the cage are equal [ 131. Thus the IO model presented here is more general than any previously considered. 0 009-2614/86/$ 03.50 0 Elsevier Science Publishers B.V. (North- Holland Physics Publishing Division)


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2. Factorlsation of the UD model into normal modes Following [12] we shall confine our analysis to the 2-D case. This avoids the difficulties associated with a 3-D treatment of rotational Brownian motion [ 141. The central dipole is regarded as a disk, with moment of inertia I, and dipole moment cl, free to rotate about an axis passing through its centre and perpendicular to the plane of its surface. The cage of neighbours is conceived of as an annulus, with moment of inertia I2 and dipole moment B2, concentric with, and coplanar to, the inner disk (dipole). Note that in ref. [ 121 the contribution to the polarisability from the dipole moment of the annulus was neglected. In the present model both disk and amrulus experience random torques, I’,(t) and r,(r), respectively, and are subject to rotational friction due to the surroundings. The only constraint imposed on our model is that both disk and annulus are subject to the same friction coefficient (per umt rotational inertia), 0. The equations of motion, following the removal of a steady electric field, are [13] I&

+ bQ,

+ ‘v’($1 - $2) = F,(t),



+ PIzi,

- “(@1 - 92) = Q(O,


where #1 and #2 are the angles that cl and B2 make with an arbitrary unit vector e. Y is the potential energy of interaction and for dipole-dipole coupling between permanent dipoles V(& - 0,) = -25

cos($ - 4,)s

V’($ - 92) = 2vu s’“(41 - 4,).


We note that some ambiguities exist in the literature regarding the definition of the random torques, r1 (t) and I’,(t). If we define a parameter ar = (kB T/1)1/2, then the effect of resealing the noise torques is merely to rescale CLIn this Letter we will follow the defmition given by Marchesoni et al. [2], that is

q(o) q(f)) =4tiu kgqui6(t), i,j

= I, 2.


The normal modes of the system described by eqs. (1) and (2) can be found by introducing the new variables [13,151 X


[email protected]





where x describes the motion of the centre of rotational inertia, while 11gives the relative motion of the two dipoles. Eqs. (1) and (2) may now be rewritten in terms of the x and 11variables as (I1 +12)ji+ (I1 + z2)Pk = r,(t) + r,(f),

(6) (7)

It is also convenient to write the original @I and G2 variables in terms of the new x and TIones, i.e.


$2=x-a2’7, a2=11+12,




where we also note that a1 t a2 = 2. We will next indicate how explicit correlation functions may be obtained in 376

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Volume 129. number 4

terms of the new X and 7)variables. The discussion, however, will be confined to a calculation of the mean dipole moment for the UD model. A more detailed treatment is given in ref. [ 131.

3. Calculation of the mean dipole moment in the harmonic approximation For the case of unequal dipoles the mean dipole moment is given, quite generally, by M(t) = ((nr(O)*e) (m(O*e))O = ( [Irl cos $I (0) + cc2cos #,(O)l b1 cos dJ1 (0+ P2cos G2(Ol ).

=&cos61(o) cos$l(t))o + 2/y.f2hx4$(0)

cos#2(t))o + p;(cos~2(o) cos#2(t))o.


In the case where the friction coefficient (per unit rotational inertia) acting on both dipoles is equal this becomes, in terms of our X and Q variables [ 131 M(f) = kos X(O)COSx(t)>, {p; [(COUrlI)(0)cosal~(t))O + (sinUlr)(0) sinalt(t))u ] + 2/.5fi2 [(c0sa1r)(0)c0sa2t7(t))() - Ginalq(0)sina2q(t))r-J

+ p; [(cosu2g(0) cosu2rl(t))o + knu2’I(0) sinu2?l(f))o I],


where both x and tl are independent random variables. In deriving eq. (11) we have also used the fact that (cos x(O) X cos X(r)>u= (sin x(O) sin x(t)>,,. The (cos x(O) cos x(t)& term is, by analogy with the ED case (cf. [5] , pp. 90 et seq.) 2k,T

Cx(t) = (cos x(0) cos x(t)>0 = i exp -1

@t- 1 [email protected])



(1, +3M2

These results are perfectly general and hold for any type of interaction potential. We now consider the harmonic approximation where -2 I$ cos 2~ m -2 V,( 1 - 4n2/2). Introducing

we find that [ 131 <(m(O)*e)

42 (m(r)*e)>o= kexp -uz F @t - 1 t eeflt) If exp -a; $ [ 1 - x(t)]) ( )[ (

+ 2Erlcc2exp -3(u: + fri) $ exp ( ) (





If the two dipoles are equal in all respects then ccl = cc2= p and u1 = u2 = 1 so that eq. (13) reduces to (u2

( [email protected] -

<(m(O)*e) (m(f).e)>o = 4p2 exp -

1 t e--I))cosh(-&x(t))exp(--$,


which is the ED result in the harmonic approximation. The full non-linear solution of eq. (11) will be the subject of a future publication [ 16] .


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4. Calculation of the polahability/pemittivity The complex polarisability, o(w), which is proportional to the permittivity *, is given by [ 171

eiwr dt

= a(O) - io J

<(m(t$e) (m(t)*e)+, eiwf dt.



Because the absorption coefficient, gQ(w), is given by gQ(w) = we”(o), the quantity we are most interested in calculating is o”(w), i.e.

d’(w) = Re w i

<(m(O)*e)(m(t)*e)>o eiwr dt

((m(0)*e)(m(t)*e)lo cos wt dt,



where <(m(0)*e)(m(t)*e))o is a real function of time and Re denotes the “real part of’ a complex quantity. An analytic expression was given recently by Coffey [4] for the ED model. This involved expanding the higher transcendental functions in eq. (14) in a Taylor series and truncating these after the quadratic terms. From the complexity of the resulting expression it is evident that little, if any, progress is to be made by straightforward analytic calculation of the polarisability. Accordingly, we have evaluated the integral expressions in eq. (16) using fast Fourier transform (FFT) methods [ 181. We shall give a detailed account of the procedures involved elsewhere [ 161. For the purposes of this communication it will suffice to give a short account of how the full non-linear pola&ability may be calculated in the ED case. In the ED case, as was indicated in the introduction, the mean dipole moment may be written in terms of the X and r) variables as (I1 = 12) MQ)



(cos x(0) cosx(t)), Gmsv(0)cosr)(t))(J

(cos2 x(O)>,

(cos2 q(O)>,


where M(t) has been normalised by its value at t = 0. The (cos X(O)cos x(t))o/(cos2 x(O)>, average is simply the motion of a free rotator and may be written as exp [-([email protected])@t - 1 + e-m)] . On the other hand, the (cos ~(0) X cos q(t))o/o/ (cos2 ~(0))~ is then evaluated at a set of discrete times corresponding to those of the solution set of (cos a(O) cos q(t)jo/ (cos2 ~(0))~ and the two are multiplied together. The resulting discrete time sequence for M(t)/M(O) is then inverted back into the frequency domain using, once again, FFT techniques. This procedure yields o”(w). A discrete time sequence for M(t)/M(O) may also be evaluated from eq. (14) and transformed into the frequency domain by the same means. The two are compared in fig. 1. From fig. 1 it seems reasonable to conclude that: (i) the FIR region of the polarisability spectrum does not exhibit a distinct peak, but a small shoulder; and (ii) the harmonic approximation to M(t), given by eq. (14), appears to be a reasonable approximation to the full non-linear solution. Conclusion (ii), cited above, leads us to expect that a numerical Fourier transformation of eq. (13) should in* Seesection 6.3.1and, in particular eqr. ( and ( of ief. [S]. 378

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Fig. 1. The full nontiear polarisabiIity (solid curve) compared with the harmonic approximation (dashed curve) given by eq. (14)intheEDcate:[email protected]=l;y=2.

dicate, at least in a qualitative fashion, how the polarisability spectrum will be affected if the moments of inertia of the two dipoles are no longer equal. Accordingly we show, in fig. 2, this approximate polarisability spectrum calculated for two different values of the ratio 12/11. This shows significant broadening of the spectrum in the FIR region, developing into a distinct peak when12/11 = 8.

5. Comparison of the model with experiment

To match the model with experiment we have normal&d the dipole moment by its value at t = 0 and applied our FFT technique to obtain the normal&d polarisability. If the approximations given in [5] are valid then o”(o) = E”(W). We have used recent data taken in both the MW and the FIR region of the spectrum [20]. Fig. 3a shows a comparison with experimental measurements on pure acetonitrile. In this case we do not expect the approximations to be fully valid. Despite this, the model still produces a close titting to the data. Fig. 3b shows a comparison for a dilute solution of hexanone-2 in cyclohexane. The data have been divided by the mole fraction so as to extrapolate the measurements to full concentration [20]. In this case the approximations in [5] are valid. To obtain a fitting of the model with experimental data it is useful to note the following points:

Fig. 2. (a) Polarisability in the harmonic approximation for the UD case with a = @= 1; 7 = 2 and 12 /II = 4. The dashed lines show the behaviour of the equivalent Debye processes; (b) as for (a) with &/II = 8.


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5 September 1986


Fig. 3. (a) The model (solid curve) fitted to experimental data (stars) for acetonitie in both the Mwand the FIR regions:Q= 4.8; p = 11; 7 = 25;z*/z1 = 12; es - eon ~‘32; (b) The model (solid curve) fitted to data (solid dots) for hexanone-2 in cyclohexane; a= 3S;p= 12;7= 20;12/11 =6;eS -e_= 7.65.

(i) The frequency of the FIR peak is given approximately by OFm a 2(a7/a1)1/2 in the underdamped c&e where 7 % 0. This is the natural frequency of oscillation of the system. There is also a small shoulder at twice this frequency. In the ED case the main peak disappears completely and the frequency of the FIR resonance becomes WFIR = 4(h~~)~‘~ (ii) The position


of the MW (Debye) peak is approximately oMw =_(a&/@)(1 - ork/S2) where aeq = (a2a2)‘i2. Details have been given in the discussion preceding eq. (2.23) of r&f.'B] . (ii) The height and shape of the FIR peak depend on the ratio of the moments of inertia of the cage and th& dipole,12/11, and also on the friction coefficient, p. (Effectively they depend on the Q-factor of the system, that is Q = [2Vo& t 12)//321112]lj2.) (iv) The ratio of the dipole moments, cl and c2, may also be varied. However, to reduce the number of physical parameters, we assume p. = c 2. Thus the dipole moment of the central dipole is assumed equal to the nett dipole moment of its surroundings. (v) If eq. (13) is divided by its value at t = 0 and transformed by FFT methods we obtain the normalised permittivity (polarisability). This must be multiplied by es - e, to give the experimental values of E”(O).

6. Conclusions The model which we have outlined in this Letter is a natural generalisation of the ED (Bud6) model [l-4,6-9] . We have shown how approximate permittivity spectra may be obtained and that these compare favourably with experimental data in both the MW and the FIR region. Thus the analytic results outlined in this Letter, when used with simple FFT programs, should provide a useful working model for the analysis and interpretation of experimental spectra. Finally, we mention that analytic results have been given recently [2,4] to account for non-linear effects due to the shape of the cosine potential in the ED model. These may be incorporated into the present model without difficulty enabling the temperature dependence of oFTR to be predicted. However, the effect of such modifications on the polarlsability is less noticeable than the effect of allowing the inertias of the two dipoles to become unequal.


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hnowledgemellt The analysis in terms of normal modes was suggested ,by Professor F.J.M. Farley. We also thank B.K.P. Scaife, F. Marchesoni, C.J. Reid and F. Boland for usef$discussions. PC thanks TCD and the Irish Department of Education for fmancial’support.

References [l] A. Bud6, J. Chem. Phys. 17 (1949) 686. [21 F. Marchesoni, J.K. Vij and W.T. Coffey, Z. Physik B61 (1985) 357. [3] F. Marcheroni, Chem. Phyr. Letters 112 (1984) 315. [4] W.T. Coffey, Chem. Phya. Letters 123 (1985) 416. [5]1H.D. Vollmer, Z. Physik B33 (1979) 103. [6] W.T. Coffey, M.W.Evans and P. Grigolini, Molecular diffusion and spectra Iwiley-Interscience, New York, 1984). [7] H. Risken and H.D. Vollmer, Mol. Phys. 46 (1982) 1073. [8] W.T. Coffey, C. Rybarsch and W. Sch&r, Chem. Phyr. Letters 92 (1982) 245. [9 ] W.T. Coffey, in: Advances in chemical physics, Vol. 65, ed. M.W.iEvans (Wiley-Interscience, New York, 1985). [lo] C.J. Reid, Mol. Phys. 49 (1983) 331. [ll] W.T. Coffey, C.Rybarrch and W. S&r&x, Phys. Letter: A88 (1982) 331. [12] J.H. Calderwood and W.T. Coffey, Proc. Roy. Sot. A356 (1977) 269. [ 131 W.T. Coffey, P. Corcoran and M.W.1Evans, Proc. Roy. Sot A, to be published. [ 141 J.R. McConnell, Rotational Brownian motion and dielectric theory (Academic Press, New York, 1980). 1151 F.J.M. Farley, private communication. [ 161 W.T. Coffey and P. Corcoran, to be published. [17] B&P. Scaife, Complex pezmittivity (English Univ. Press, London, 1971). [ 181 E.O. Bringham, The fast Fourier transform (Prentice-Hall, $nglewood cliffs, 1974). [19] R.C. Singleton, Comm. ACM 10 (1967) 647; Documentation on the /IMSLsubroutine FFTRC. [20] J.K. wj, unpublished work (1985).