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Non-markovian effects in dielectric relaxation
Volume 55, number 2
NON-MARKOVIAN EFFECTS
CHEMICALPHYSICS LEITERS
15
April 1978
JN DIELECTRLC RELAXATION
F. HERMANS and E. KESTEMONT 223, Univemité Libre de Bruxelles,
Faculté des Sciences. CP 1050 Brussels, Belgùm
Receïved 19 October 1977 Revisedmanuscriptreceived3 January 1978
A generalized Nee-Zwanzïg equation is proposed to describe both low and high frequency dielectric relaxation. It is shown that the inertial effect is closely related to the non-markovïan character of the dïferential equation governing the time evolution of the dipole-dipole autocorrelation function. Comparison with ex~erirnental results is @en.
1. Introduction Sophisticated theorïes of dielectrïc relaxation have been formulated to descrîbe experimental data at low or medium frequenties (in the microwave range) [ l41 which give a correct treatment of the reactïon field acting on the dipoles. However they rely on the existence of an exponentiahy decaying dipole-dipole autocorrelation function[5] _ They cannot account for the important far ïnfrared absorption observed 6-101. Lobo et al. [ 111, generalizing the NeeZwanzig equation by including the inertia; terms, gave a formda whïch has the correct hmitïng behavïour at both high and low frequenties. Other approaches have been made to account for the inertial effect [12-16]_ The memory function formalism and the Mori expansïon have extensively been used [17-lg]_ in Thaisframework, Evans and Evans [ 191 pointed out the diffïculty to approximate in a satisfactory way both low and high frequency data. Harp and Beme [23] studied in great detail different farms of memory functions in connection with results obtained by computer experiment% These authors poïnted out that none of the approximate memories adequately represents the long time behaviour of their ‘experimental” memories. Unfortrmately they made no attempt to compare their results with truc experimental data. The aim of thîs note is to see if, starting from a general integro-differential equation for the dipole-dïpole
autocorrelation functîon and inserting the required asymptotic behaviour, one can obtain a solution whïch, with a limited number of parameters, describes correctly the experimental data avaïlable.
2. Formahsm We are ïnterested in the variation ~6th frequency of the absorption coeffkient (Y(O) of a sample of dielectric permittivity E(O). We start wïth the Nee-Zwanzig equation [4] (h ereafter referred to as NZ), for polarïzable molecules with permanent electric moment u_ q-J [E(O)
ddCq)
-
L
1 r24a)
-combo
+ ca 1
f Gd
= .@(-dy/dt)
_
(1)
Here E,,, is the vahre of the real part E’(O) of the permittivity [E(O) = E(u) + iY’(o)] at infmite angular frequency w; eO is the static permittivity; O(-dr/dt) stands for the Laplace-Fourïer (LF) transform of minus the time derïvative of the microscopic dipoledipole autocorrelation function, defmed by 7(f) = (P(o)-
a(r))/W(o))
>
where ( ) represents an ensemble equilibrium average in the absente of extemal electric field. Once 7(t) is known, one obtains a(o) from [20]
305
Volume55, number2
CHEMICAL
(Y(w) = (2t’2/c)W(e’)1’2
{[l
+ (e”/&]
1’2 -
131’2 , (2)
where c is the velocity of lïght in vacuum. It ís relevant here to quote the macroscopic equation gîven by Glarum 1213 [e(o)
- G.,1/Ce0- eJ = 2 (+Wdt)
,
(3)
where @(t) ís the autocorrelation function of the total electric moment of the sample. This.equation is correct in the limit of varüshïngly smalt dipole moment. Fq. (1) goes back to the Onsager equation at zero frequency. The probiem is to íìnd a satisfactory form of the relevant autocorrelation function. Nee-Zwanzïg made the assumption that -G?(4) can be dïvïded ïnto two parts, one of them being related to the dielectric friction CD the other hnked to the motion of the molecules in the absente of electric dìpole ïnteractions, so _@(+)
= {[fi(+O)]
-’
- iwSD(o)/2kT)-l
15 April 1978
PHYSICS LETPERS
(4)
(k is the Boltzmann constant, T the absolute temperature). The calculation of Nee-Zwanzig leads to
(5)
solution is then the exponential decaying function 70 (t) = eer/’ ,
(9)
which in turn gives the NZ equation. It has been shown by Finsy and van Loon [S] that for l,l,ltrichloroethane (TCE) the NZ equation gives the best fit to experimental data on the low frequency wïng of the spectra. Thus a reasonabie requirement seems to be that ra(t) goes to the solution (9) when the tïme t is long compared to the characteristic time rc of the kemel K. (we fellow here closely an argument of Résibois [24] .) This gives USa fust equation for K. Indeed, in the long time limit (r > T > T=) at ro(O = -Yo
s
K(t?dt’
s 0
K(f’)dr’=
l/r
_
= - j ~(t - t’)ro(f’)dt’
_
(6)
0
Introducing the LF transform of K(t) defmed by
(11)
One of the simplest farms of K is the decaying exponent tial. Taking this wîth (1 l), K is completely determined:
K(r) = ( l/nc)
emfirc .
(12) ís then
J?(-+O (t)) = [ 1 - iwr(1 - iwrc)] -1 .
do) -
c?3
1 = (1 - iurT1)(l - iw2)
K(t)
eiw t dt ,
(7)
0
eq. (6) gives straightforwardly the well known result JX-+~(~))
= K(w)/ [-ia
+ K(o)]
-
(8)
The markov%.nlimit of (6) is obtaïned if K is chosen to be a 6 function of time with amplitude l/~. The 306
(14)
’
-1 withq +T2 =rand$ + r2-1 = Tc . Eq_ (14) implies that o(t) is a differente of decaying exponentials e(r) = (Tl - 72)-l(r71 e -ft71 - ~~ e 4%
K(o) = /
(13)
When (13) is used forI?(-&)wïth (3) one obtaîns the Rocard 1141 and Powles [15] equation
eo -Go
a~o(tyaf
(10)
so that
The LF transform of ‘ra(f) As pointed out by Wyllie [lol, P(-fo) needs not be independent of frequency. It is preciscly a plausible form of r. (t) we are seeking for. Let us assume the foltowing ïntegro-dîfferential equation for ai to held. (We can take it as providing a deftition of the memory kemel K(t) [22,23] _)
3
0
)-
W)
In the familiar Cole-Cole [25] representation of dielectrïc data, thïs leads to a curve lying outsïde the classical Debye semi-circle in marked contrast with experimental results; (15) nevertheless presents the important characterïstic to have the correct short time expansion (P(r) a 1 -t2/2T172 .
(16)
Volume 55, number2
CHEMICAL PHYSICS LETTERS
IS April 1978
When tbis is compared to the exact result given by Gordon 1261 for e(t) one obtains
S(-+,)
r1 r2 = 1JZk-T ,
whïch aside a factor rr U2 is identical with (13); at high frequenties on the other hand
(17)
where 1 is the relevant moment of inertia. The applicability of (14) to far infrareä absorption bas been thoroughly discussed by Davies et al. [7] and by Bimbaum and Cohen [ 161. l’he mti drawback of (12) is of course that it is not an even function of thne as it should. To overcome this we try a gaussian memory exp(-t2/& (sec ref. [23])_ Following the same lines as above, and if rc is much lower than r, one readily obtains K(t)
= (21771i277c) eBPI+
.
K(w)
is now given by [27]
K(“)
=$
e-w2+4 [
+2i mw ..
= -(ilnTqz(;
UTc/
e -w%$/4
et2dt
s
n
3
,
”
WTJ .
= [1+ 7r1&7/2
($ orc)]
(20)
-l
-
(21)
Once more, one sees that taking account of the nonmarkovian aspect of (6) leads to a frequency dependent “relaxation thne”. This effect appears necessary to account for the return to trausparency of pok liquids at high frequency, a requirement not met with Debye type relaxation. Eq_ (2 1) with (l), (4) and (5) is our generalization of the NZ equation
(22)
lr112&JT(q) - %j WJ) - EO1-l = [ l+z(&Tc)
-
~o[2E(W)+e~]
1
l
Thïs choice of K(t) implïes
rrc = I/n l12kT.
(23)
Two limíting cases are of interest: at frequenties such thatWrcG1 z(+ccur,j = iir 112 - ; Urc , hence
-’
,
Z($&JT,) * -2/WTc
(25)
(26)
and wïth the help of (23)
fi(-io)
a (-021/2kT)-1.
(27)
This last expression combined with (4) and (5) gives the NZ result including the inertial term at very high frequenties.
3. Comparïson wïth experimental results and discussion
Hence .e(-5,)
- iwr&r “‘)]
(18)
(19) wbich is closely related to the plasma dispersion function [28] Z K(u)
= [l - iw(l
(24)
In order to compare (22) with the experimental results, we go over to the absorption coefficient ar(w). The function 2 is computed Ging power series and asymptotic expansions as given by Fried and Conte [28] and Frïed et ai. 1291. On the basis of the best fit on low frequency data Finsy and van Loon [5] give r = 3.84 ps usîng e. = 7.082 and E, = 2.137 at 25°C [30]. We use the data of L.eroy and Constant [3 l] for a(w) at the same temperature. Let USconsider T to be fuced by the low frequency behaviour of tbe system (in this case 3.84 psj. If one assumes the inertia moment 1 to be known independently then T= is fixed by relation (23). From microwave data [32] I is calculated to be 3.54 X íO-45 K m2, then Ti = 0.126 ps. From these values, (Y(W) is computed. The curve obtained shows a maximum of about 22.5 Neper cm-’ located at 13 cm-l , to be compared to i&e experimental maximum of 24 Neper cm-l at 30 cm-l . We have plotted in fig. 1 ar(w) from 10 to 100 cm-l together with the experimental values of Leroy and Constant_ Althougb the agreement is not very good, the overall behavìour is correctly gïven by our formula. At thïs point it is worth to mention the diffïculty to obtaïn reliable ïnfonnation about experimental values of the absorption coeffìcient. Davies et al. gïve for (Ya maximum of 30 Neper cm-l at 29 cm-: for TCE at 20°C whereas more recently, Pourprix et al. [8] essentially confïrm the values of Leroy and Constant with a maximum of 25 Neper cm-l at 30 cm-l als0 at 20°C. The calculated curve relies entirely on the value of 307
acV)
iv
2Q_
15 April 1%‘8
CHEMICALPHYSICS LETTERS
Volume 55. number 2
1OQ. cm-’
50
Fig. 1. Absorption coeftïcïent a as a functïon of the wavennmber Ü.--generaliaed NZ equationr r ref. [S] , I ref. :32] ; ~eneralized NZ equation: 2 parameter adjustment on far infrared data; X experimental points of Leroy and Constant. r obtained by a fit of the NZ formula to the microwave data only, and on the value of I determined independently. In vïew of the uncertaïnty in I(Davies et ai. give as much as 20%) we found it interest@ to try a best fit on the far ïnfrared data taking 7 and ~c as parameters. The fit is made usïng a least mear, square criterion, mGmizing
wïth respect to the k parameters ei. Nis the number of angu!ar fre uencïes o- at which II is known. The de9 viation s = [S I(N - k)] Ij2 îs also computed. We took ten values of a from 10 to 100 Table 1 gives the values of T, rc aud s obtained respectively with T. rc fmed; T fuced to its low frequency value, ~c adjusted
cm-t.
to the far infrared data and r and ‘i-, allowed to vary. Also quoted in table ? is the value of I/2kT deduced with the help of (23). We gïve the same information when using our form ofZ!(-+o), eq_ (21), together with the “macroscopic equation” (3). I[t appears that the generalized equation gives good results in the three cases, the calculated vahes of 1/2kT dífferïng no more than 5% from the experimentally deterrninad vaíue. However, the fit on the low frequency wing of the spectrum is not so good when r is entirely determined by the high frequency values of Q (two parameters adjustment). This is not surprïsing because, ïfexperïmental results are @en in terms of absorption, Iow frequency data represent a very small part of the spectrum_ The different trials we did gave essentialïy the same fit. When 7 and rc as given by table 1, are used to compute E’ and e” in the microwave region, Finsy-van Loon’s value for r gives decidedly the best agreement, experimental points being alrnost undïstìnguíshable of the calculated curve (figs. 2 and 3), whereas our best fit based on far infrared data only, giyes a not too bad adjustment for E ‘(fg. 2) and for e” (fig. 3) but with a slight sbïft to higher frequenties of the dispersion and maximum absorption. On the contrary tbe macroscopic equation although gïving in the far infrared regïon essentially the same deviation is defmitively unable to represent correctly the permittivity at centimeter and millimeter wavelengths, The computed value of Ij2kT on the basis of tbis equahon seems far too high. As a second test for our formula, we take chloroform (CHCQ for which permittivity vahres are available
Table 1 Parameter estünates and devïations s obtaIned by fitting the generalizcd NZ equation or the “macroscopic” equation to the experïmental data ti the far ïnfrared; r and T= are given in ps, s is in Neper crn-l and f/2kT in sz
TCE
CHCl3
308
generalized NZ equation r, sc fixed r fured 2 parameters “macroscopic” equaticn 2 parameters gereralized NZ equation r, rc fixed 7 fwed 2 parameters “macroscopic” equation 2 parameters
T
Tc
S
10251/2kT
3.84 3.84 3.49
0.126 0.120 0.130
2.6 2.7 2.4
4.3 4.1 4.0
4.55
0.129
2.5
5.9
4-40 4.40 2.25
0.079 0.014 0.101
7.6 7.3 2.8
3.1 OS 2.0
2-74
0.101
2.8
2.8
Volume55, number2
CHEMICALPHYSICS LETTERS
Fig. 2. Reaipart of the pemxittïvïtyas a function of the logarithm of the frequency. Same syznbols as in fg. 1 and --“macroscopic equatïon”; X experimental Finsy and van Loon.
10.
ll.
points
of
12.
Fig. 3. Jmaginarypart of the permittivity as a function of the logarithmof the frequency.Samesymbolsas in fig. 2.
cver aU the frequency range [9] _ A fïrst step was to determine in the same way as Finsy and van Loon the low frequency value of the relaxation time 7. To do this we minhnize the function
15 April 1978
experimental data for CHC13 _With these two characteristic times rand rc the far infrared absorption coeffìcient 01is 50% too low! Thls is an unexpected result obtained by fitting low frequency data only. When the two parameters are allowed to vary, and adjustment is made on ar(w) from 10 to 100 cm-1, one obtzins a quite good fit with 7 = 2.25 ps and 5rc= 0.101 ps. The calculated value of 1/2kT is 35% too low. However the discrepancy is then moved to low frequency. We have to conclude that, although the non-markovian character of eq. (6) is essential to recover the correct high frequency behaviour of the absorption, nevertheless, the generelized NZ equation appears unable to account for the various experimental effects observed, reasonable agreement being however obtained for TCE. In our opinion, this could be due to the fundamental hypothesis on which the Nee-Zwanzig theory is based, hypothesis rxpressed by relation (4). It is indeed not clear that short range effects related to yo and dielectrlc friction can be treated at the leve1 of e(--+) as additive properties. It could be that such an hypothesis is more valid for TCE than for trichloromethane, the dielectric friction CD for the former being more lmportant by at le& a factor of two at medium and high frequenties. Refercnces [l] B.K.P. Sxife, Proc. Phys. Sec. (Londen) 84 (1964) 616. [2] E. Fatuzzo and P.R. Mason,Proc. Phys. SOC.(Londen) 90 (196”) 741.
[3] D.D. Klug, D.E. Kranbuehland W.E. Vaugban, J. Chem.
with respect to parameters ei, where ecalc ( {ei), wj) is the calculated value of the dielectric permittivity at the known angular frequency wi- We used the sixteen iìrst values quoted by Goulon et al. from 0 to 29.7 cm-l _ EOis taken to be 4.719 and e_ = 2.23. The fit is made using the original NZ equation. One obtains T = 4.40 ps with a dimensionless deviationfequal to 1 .l X 10-3. Finsy-van Loon found a deviation of 1.5 X 10e3 for TCE. If we take for e_, the value e’ extrapolated in the far infrared, i.e. 2.15, the fit is slightly better: f = 0.94 X 10-3 and 7 = 4.23 ps. IJsQ T = 4.40 ps íínd 1= 2.54 X 10A5 K m2 1321, (23) gives rc = 0.79 ps. Same results as for TCE are given in table 1. It is lmmediately clear that our formula is not at all suited to describe
Phys. 50 (1969) 3904. [4] T.W. Ne.- and R. Zwanzig, J. Chem. Phys. 52 (1970) 6353. [S] R. Fins) and R. van Loon, J. Chem. Phys. 63 (1975) 4831. [ó] Y. Leroy, E. Constant, C. Abbar and P. Desplanques, Advan. Mol. Relaxation Processes 1 (1967) 273. [7] M. Davïes, G.W.F. Pardoe, J.E. Cbamberlaïn and H.A. Gebbie, Trans. Faraday Sec. 64 (1968) 847. [Sl B. Pourprix, C. Abbar and D. Decoster, Compt. Rend. Acad. Sci. (Paris) 272 B (1971) 1418. [9] J. Gouloin, J.-L. Riva3,J.W. Flemïng, J. Chamberlain and G.W. Chautry, Chem. Phys. Letters 18 (1973) 211. [ 101 G. WyJJïe, in: Dielectrïc and related molecular processes, Vol. 1 CTbeChemical Society, London, 1972). [ 1 l] R_ Lobo, JE. Robinson and S. Rodriguez. J. Chem. Phys. 59 (1973) 5992. [ 121 F. Bliot and E. Constant, Cbem. Phys. Letters 18 (1973) 253.
[13] E. Kestemont, J. Phys. C9 (1976) 2651.
309
Volume 55. numbei 2
CHIIMICAL PI-IYsIcs LETIERS
(14J Y. Rocard. 1. Phys- Radium 4 (1933) 247[ 15J LG. Powles, Trans. Faraday Sec. 44 (1948) 802. 1161 ~~grnbauxu and ER. Cohen, 3. Chem. Phys. 53 (1970) Il7J H. h&i, Progr. Theoret- Phys- 33 (1965) 42R [ 181 6;%ot aud E. Cous+at. (hem. Phys. Letters 29 (1974) 119 J GJ. Evaos and MW. Evaos, Chem. Phyr Letters 45 (1977) 454. f20J G-W- Cban-, Sub~e~r spectrascopy (Academic Press, New York, I971). [21] S.H. Glanun, 3. Chem. Phy~. 33 (1960) 1371. [22J P. Résiiois and M. de Leener, ClassicaI timory of fluids (Wiiey-Interscience, New York, 1977). [23J GD. Harp and BJ. Beme, Phys. Rev. 29 (1966) 255; A2 (1970) 975; BJ- Beme and GD. Harp, Advan. Chem. Phys. 17 (1970) 63.
15 April 1978
P. Résïïis, in: Physks of many particle systems, ed. E. MeeronGordon and Breac&,Londoa, 1966). [2!G]KS. Cole and RW. Col@,4. Chem, P&ys.9 (1941) 34L [24J
1261 R.G. Gordon, 3. Chem. Phys_ 41(1964) 1819. 1271 M. Abrarnowita and I.A. Steguu. Handbook of mathematical fúnctions (Doveq, New York, 1965). 1281 B.D. Fried and S.D. Conte, Tbe plasma diirsion functiou (Academic Press, New York, 1961). 1291 Ba. Fried, C.L. IIedrick and J. McCnue, Phys. FIuids 11 (1968) 249. f30 J R. Finsy and R- vao Loon, 1. Pítys. Chem. 80 (1976) 2783. [31 J Y. Leroy aud E. Constant, Compt. Rend. Acad. Sci. (Paris) 264B (1967) 533. (321 N.S. Lojko and Y. Beers, 3. Res. Na& Rut_ Std. US 73 A (1969) 233.
NON-MARKOVIAN EFFECTS
CHEMICALPHYSICS LEITERS
15
April 1978
JN DIELECTRLC RELAXATION
F. HERMANS and E. KESTEMONT 223, Univemité Libre de Bruxelles,
Faculté des Sciences. CP 1050 Brussels, Belgùm
Receïved 19 October 1977 Revisedmanuscriptreceived3 January 1978
A generalized Nee-Zwanzïg equation is proposed to describe both low and high frequency dielectric relaxation. It is shown that the inertial effect is closely related to the non-markovïan character of the dïferential equation governing the time evolution of the dipole-dipole autocorrelation function. Comparison with ex~erirnental results is @en.
1. Introduction Sophisticated theorïes of dielectrïc relaxation have been formulated to descrîbe experimental data at low or medium frequenties (in the microwave range) [ l41 which give a correct treatment of the reactïon field acting on the dipoles. However they rely on the existence of an exponentiahy decaying dipole-dipole autocorrelation function[5] _ They cannot account for the important far ïnfrared absorption observed 6-101. Lobo et al. [ 111, generalizing the NeeZwanzig equation by including the inertia; terms, gave a formda whïch has the correct hmitïng behavïour at both high and low frequenties. Other approaches have been made to account for the inertial effect [12-16]_ The memory function formalism and the Mori expansïon have extensively been used [17-lg]_ in Thaisframework, Evans and Evans [ 191 pointed out the diffïculty to approximate in a satisfactory way both low and high frequency data. Harp and Beme [23] studied in great detail different farms of memory functions in connection with results obtained by computer experiment% These authors poïnted out that none of the approximate memories adequately represents the long time behaviour of their ‘experimental” memories. Unfortrmately they made no attempt to compare their results with truc experimental data. The aim of thîs note is to see if, starting from a general integro-differential equation for the dipole-dïpole
autocorrelation functîon and inserting the required asymptotic behaviour, one can obtain a solution whïch, with a limited number of parameters, describes correctly the experimental data avaïlable.
2. Formahsm We are ïnterested in the variation ~6th frequency of the absorption coeffkient (Y(O) of a sample of dielectric permittivity E(O). We start wïth the Nee-Zwanzig equation [4] (h ereafter referred to as NZ), for polarïzable molecules with permanent electric moment u_ q-J [E(O)
ddCq)
-
L
1 r24a)
-combo
+ ca 1
f Gd
= .@(-dy/dt)
_
(1)
Here E,,, is the vahre of the real part E’(O) of the permittivity [E(O) = E(u) + iY’(o)] at infmite angular frequency w; eO is the static permittivity; O(-dr/dt) stands for the Laplace-Fourïer (LF) transform of minus the time derïvative of the microscopic dipoledipole autocorrelation function, defmed by 7(f) = (P(o)-
a(r))/W(o))
>
where ( ) represents an ensemble equilibrium average in the absente of extemal electric field. Once 7(t) is known, one obtains a(o) from [20]
305
Volume55, number2
CHEMICAL
(Y(w) = (2t’2/c)W(e’)1’2
{[l
+ (e”/&]
1’2 -
131’2 , (2)
where c is the velocity of lïght in vacuum. It ís relevant here to quote the macroscopic equation gîven by Glarum 1213 [e(o)
- G.,1/Ce0- eJ = 2 (+Wdt)
,
(3)
where @(t) ís the autocorrelation function of the total electric moment of the sample. This.equation is correct in the limit of varüshïngly smalt dipole moment. Fq. (1) goes back to the Onsager equation at zero frequency. The probiem is to íìnd a satisfactory form of the relevant autocorrelation function. Nee-Zwanzïg made the assumption that -G?(4) can be dïvïded ïnto two parts, one of them being related to the dielectric friction CD the other hnked to the motion of the molecules in the absente of electric dìpole ïnteractions, so _@(+)
= {[fi(+O)]
-’
- iwSD(o)/2kT)-l
15 April 1978
PHYSICS LETPERS
(4)
(k is the Boltzmann constant, T the absolute temperature). The calculation of Nee-Zwanzig leads to
(5)
solution is then the exponential decaying function 70 (t) = eer/’ ,
(9)
which in turn gives the NZ equation. It has been shown by Finsy and van Loon [S] that for l,l,ltrichloroethane (TCE) the NZ equation gives the best fit to experimental data on the low frequency wïng of the spectra. Thus a reasonabie requirement seems to be that ra(t) goes to the solution (9) when the tïme t is long compared to the characteristic time rc of the kemel K. (we fellow here closely an argument of Résibois [24] .) This gives USa fust equation for K. Indeed, in the long time limit (r > T > T=) at ro(O = -Yo
s
K(t?dt’
s 0
K(f’)dr’=
l/r
_
= - j ~(t - t’)ro(f’)dt’
_
(6)
0
Introducing the LF transform of K(t) defmed by
(11)
One of the simplest farms of K is the decaying exponent tial. Taking this wîth (1 l), K is completely determined:
K(r) = ( l/nc)
emfirc .
(12) ís then
J?(-+O (t)) = [ 1 - iwr(1 - iwrc)] -1 .
do) -
c?3
1 = (1 - iurT1)(l - iw2)
K(t)
eiw t dt ,
(7)
0
eq. (6) gives straightforwardly the well known result JX-+~(~))
= K(w)/ [-ia
+ K(o)]
-
(8)
The markov%.nlimit of (6) is obtaïned if K is chosen to be a 6 function of time with amplitude l/~. The 306
(14)
’
-1 withq +T2 =rand$ + r2-1 = Tc . Eq_ (14) implies that o(t) is a differente of decaying exponentials e(r) = (Tl - 72)-l(r71 e -ft71 - ~~ e 4%
K(o) = /
(13)
When (13) is used forI?(-&)wïth (3) one obtaîns the Rocard 1141 and Powles [15] equation
eo -Go
a~o(tyaf
(10)
so that
The LF transform of ‘ra(f) As pointed out by Wyllie [lol, P(-fo) needs not be independent of frequency. It is preciscly a plausible form of r. (t) we are seeking for. Let us assume the foltowing ïntegro-dîfferential equation for ai to held. (We can take it as providing a deftition of the memory kemel K(t) [22,23] _)
3
0
)-
W)
In the familiar Cole-Cole [25] representation of dielectrïc data, thïs leads to a curve lying outsïde the classical Debye semi-circle in marked contrast with experimental results; (15) nevertheless presents the important characterïstic to have the correct short time expansion (P(r) a 1 -t2/2T172 .
(16)
Volume 55, number2
CHEMICAL PHYSICS LETTERS
IS April 1978
When tbis is compared to the exact result given by Gordon 1261 for e(t) one obtains
S(-+,)
r1 r2 = 1JZk-T ,
whïch aside a factor rr U2 is identical with (13); at high frequenties on the other hand
(17)
where 1 is the relevant moment of inertia. The applicability of (14) to far infrareä absorption bas been thoroughly discussed by Davies et al. [7] and by Bimbaum and Cohen [ 161. l’he mti drawback of (12) is of course that it is not an even function of thne as it should. To overcome this we try a gaussian memory exp(-t2/& (sec ref. [23])_ Following the same lines as above, and if rc is much lower than r, one readily obtains K(t)
= (21771i277c) eBPI+
.
K(w)
is now given by [27]
K(“)
=$
e-w2+4 [
+2i mw ..
= -(ilnTqz(;
UTc/
e -w%$/4
et2dt
s
n
3
,
”
WTJ .
= [1+ 7r1&7/2
($ orc)]
(20)
-l
-
(21)
Once more, one sees that taking account of the nonmarkovian aspect of (6) leads to a frequency dependent “relaxation thne”. This effect appears necessary to account for the return to trausparency of pok liquids at high frequency, a requirement not met with Debye type relaxation. Eq_ (2 1) with (l), (4) and (5) is our generalization of the NZ equation
(22)
lr112&JT(q) - %j WJ) - EO1-l = [ l+z(&Tc)
-
~o[2E(W)+e~]
1
l
Thïs choice of K(t) implïes
rrc = I/n l12kT.
(23)
Two limíting cases are of interest: at frequenties such thatWrcG1 z(+ccur,j = iir 112 - ; Urc , hence
-’
,
Z($&JT,) * -2/WTc
(25)
(26)
and wïth the help of (23)
fi(-io)
a (-021/2kT)-1.
(27)
This last expression combined with (4) and (5) gives the NZ result including the inertial term at very high frequenties.
3. Comparïson wïth experimental results and discussion
Hence .e(-5,)
- iwr&r “‘)]
(18)
(19) wbich is closely related to the plasma dispersion function [28] Z K(u)
= [l - iw(l
(24)
In order to compare (22) with the experimental results, we go over to the absorption coefficient ar(w). The function 2 is computed Ging power series and asymptotic expansions as given by Fried and Conte [28] and Frïed et ai. 1291. On the basis of the best fit on low frequency data Finsy and van Loon [5] give r = 3.84 ps usîng e. = 7.082 and E, = 2.137 at 25°C [30]. We use the data of L.eroy and Constant [3 l] for a(w) at the same temperature. Let USconsider T to be fuced by the low frequency behaviour of tbe system (in this case 3.84 psj. If one assumes the inertia moment 1 to be known independently then T= is fixed by relation (23). From microwave data [32] I is calculated to be 3.54 X íO-45 K m2, then Ti = 0.126 ps. From these values, (Y(W) is computed. The curve obtained shows a maximum of about 22.5 Neper cm-’ located at 13 cm-l , to be compared to i&e experimental maximum of 24 Neper cm-l at 30 cm-l . We have plotted in fig. 1 ar(w) from 10 to 100 cm-l together with the experimental values of Leroy and Constant_ Althougb the agreement is not very good, the overall behavìour is correctly gïven by our formula. At thïs point it is worth to mention the diffïculty to obtaïn reliable ïnfonnation about experimental values of the absorption coeffìcient. Davies et al. gïve for (Ya maximum of 30 Neper cm-l at 29 cm-: for TCE at 20°C whereas more recently, Pourprix et al. [8] essentially confïrm the values of Leroy and Constant with a maximum of 25 Neper cm-l at 30 cm-l als0 at 20°C. The calculated curve relies entirely on the value of 307
acV)
iv
2Q_
15 April 1%‘8
CHEMICALPHYSICS LETTERS
Volume 55. number 2
1OQ. cm-’
50
Fig. 1. Absorption coeftïcïent a as a functïon of the wavennmber Ü.--generaliaed NZ equationr r ref. [S] , I ref. :32] ; ~eneralized NZ equation: 2 parameter adjustment on far infrared data; X experimental points of Leroy and Constant. r obtained by a fit of the NZ formula to the microwave data only, and on the value of I determined independently. In vïew of the uncertaïnty in I(Davies et ai. give as much as 20%) we found it interest@ to try a best fit on the far ïnfrared data taking 7 and ~c as parameters. The fit is made usïng a least mear, square criterion, mGmizing
wïth respect to the k parameters ei. Nis the number of angu!ar fre uencïes o- at which II is known. The de9 viation s = [S I(N - k)] Ij2 îs also computed. We took ten values of a from 10 to 100 Table 1 gives the values of T, rc aud s obtained respectively with T. rc fmed; T fuced to its low frequency value, ~c adjusted
cm-t.
to the far infrared data and r and ‘i-, allowed to vary. Also quoted in table ? is the value of I/2kT deduced with the help of (23). We gïve the same information when using our form ofZ!(-+o), eq_ (21), together with the “macroscopic equation” (3). I[t appears that the generalized equation gives good results in the three cases, the calculated vahes of 1/2kT dífferïng no more than 5% from the experimentally deterrninad vaíue. However, the fit on the low frequency wing of the spectrum is not so good when r is entirely determined by the high frequency values of Q (two parameters adjustment). This is not surprïsing because, ïfexperïmental results are @en in terms of absorption, Iow frequency data represent a very small part of the spectrum_ The different trials we did gave essentialïy the same fit. When 7 and rc as given by table 1, are used to compute E’ and e” in the microwave region, Finsy-van Loon’s value for r gives decidedly the best agreement, experimental points being alrnost undïstìnguíshable of the calculated curve (figs. 2 and 3), whereas our best fit based on far infrared data only, giyes a not too bad adjustment for E ‘(fg. 2) and for e” (fig. 3) but with a slight sbïft to higher frequenties of the dispersion and maximum absorption. On the contrary tbe macroscopic equation although gïving in the far infrared regïon essentially the same deviation is defmitively unable to represent correctly the permittivity at centimeter and millimeter wavelengths, The computed value of Ij2kT on the basis of tbis equahon seems far too high. As a second test for our formula, we take chloroform (CHCQ for which permittivity vahres are available
Table 1 Parameter estünates and devïations s obtaIned by fitting the generalizcd NZ equation or the “macroscopic” equation to the experïmental data ti the far ïnfrared; r and T= are given in ps, s is in Neper crn-l and f/2kT in sz
TCE
CHCl3
308
generalized NZ equation r, sc fixed r fured 2 parameters “macroscopic” equaticn 2 parameters gereralized NZ equation r, rc fixed 7 fwed 2 parameters “macroscopic” equation 2 parameters
T
Tc
S
10251/2kT
3.84 3.84 3.49
0.126 0.120 0.130
2.6 2.7 2.4
4.3 4.1 4.0
4.55
0.129
2.5
5.9
4-40 4.40 2.25
0.079 0.014 0.101
7.6 7.3 2.8
3.1 OS 2.0
2-74
0.101
2.8
2.8
Volume55, number2
CHEMICALPHYSICS LETTERS
Fig. 2. Reaipart of the pemxittïvïtyas a function of the logarithm of the frequency. Same syznbols as in fg. 1 and --“macroscopic equatïon”; X experimental Finsy and van Loon.
10.
ll.
points
of
12.
Fig. 3. Jmaginarypart of the permittivity as a function of the logarithmof the frequency.Samesymbolsas in fig. 2.
cver aU the frequency range [9] _ A fïrst step was to determine in the same way as Finsy and van Loon the low frequency value of the relaxation time 7. To do this we minhnize the function
15 April 1978
experimental data for CHC13 _With these two characteristic times rand rc the far infrared absorption coeffìcient 01is 50% too low! Thls is an unexpected result obtained by fitting low frequency data only. When the two parameters are allowed to vary, and adjustment is made on ar(w) from 10 to 100 cm-1, one obtzins a quite good fit with 7 = 2.25 ps and 5rc= 0.101 ps. The calculated value of 1/2kT is 35% too low. However the discrepancy is then moved to low frequency. We have to conclude that, although the non-markovian character of eq. (6) is essential to recover the correct high frequency behaviour of the absorption, nevertheless, the generelized NZ equation appears unable to account for the various experimental effects observed, reasonable agreement being however obtained for TCE. In our opinion, this could be due to the fundamental hypothesis on which the Nee-Zwanzig theory is based, hypothesis rxpressed by relation (4). It is indeed not clear that short range effects related to yo and dielectrlc friction can be treated at the leve1 of e(--+) as additive properties. It could be that such an hypothesis is more valid for TCE than for trichloromethane, the dielectric friction CD for the former being more lmportant by at le& a factor of two at medium and high frequenties. Refercnces [l] B.K.P. Sxife, Proc. Phys. Sec. (Londen) 84 (1964) 616. [2] E. Fatuzzo and P.R. Mason,Proc. Phys. SOC.(Londen) 90 (196”) 741.
[3] D.D. Klug, D.E. Kranbuehland W.E. Vaugban, J. Chem.
with respect to parameters ei, where ecalc ( {ei), wj) is the calculated value of the dielectric permittivity at the known angular frequency wi- We used the sixteen iìrst values quoted by Goulon et al. from 0 to 29.7 cm-l _ EOis taken to be 4.719 and e_ = 2.23. The fit is made using the original NZ equation. One obtains T = 4.40 ps with a dimensionless deviationfequal to 1 .l X 10-3. Finsy-van Loon found a deviation of 1.5 X 10e3 for TCE. If we take for e_, the value e’ extrapolated in the far infrared, i.e. 2.15, the fit is slightly better: f = 0.94 X 10-3 and 7 = 4.23 ps. IJsQ T = 4.40 ps íínd 1= 2.54 X 10A5 K m2 1321, (23) gives rc = 0.79 ps. Same results as for TCE are given in table 1. It is lmmediately clear that our formula is not at all suited to describe
Phys. 50 (1969) 3904. [4] T.W. Ne.- and R. Zwanzig, J. Chem. Phys. 52 (1970) 6353. [S] R. Fins) and R. van Loon, J. Chem. Phys. 63 (1975) 4831. [ó] Y. Leroy, E. Constant, C. Abbar and P. Desplanques, Advan. Mol. Relaxation Processes 1 (1967) 273. [7] M. Davïes, G.W.F. Pardoe, J.E. Cbamberlaïn and H.A. Gebbie, Trans. Faraday Sec. 64 (1968) 847. [Sl B. Pourprix, C. Abbar and D. Decoster, Compt. Rend. Acad. Sci. (Paris) 272 B (1971) 1418. [9] J. Gouloin, J.-L. Riva3,J.W. Flemïng, J. Chamberlain and G.W. Chautry, Chem. Phys. Letters 18 (1973) 211. [ 101 G. WyJJïe, in: Dielectrïc and related molecular processes, Vol. 1 CTbeChemical Society, London, 1972). [ 1 l] R_ Lobo, JE. Robinson and S. Rodriguez. J. Chem. Phys. 59 (1973) 5992. [ 121 F. Bliot and E. Constant, Cbem. Phys. Letters 18 (1973) 253.
[13] E. Kestemont, J. Phys. C9 (1976) 2651.
309
Volume 55. numbei 2
CHIIMICAL PI-IYsIcs LETIERS
(14J Y. Rocard. 1. Phys- Radium 4 (1933) 247[ 15J LG. Powles, Trans. Faraday Sec. 44 (1948) 802. 1161 ~~grnbauxu and ER. Cohen, 3. Chem. Phys. 53 (1970) Il7J H. h&i, Progr. Theoret- Phys- 33 (1965) 42R [ 181 6;%ot aud E. Cous+at. (hem. Phys. Letters 29 (1974) 119 J GJ. Evaos and MW. Evaos, Chem. Phyr Letters 45 (1977) 454. f20J G-W- Cban-, Sub~e~r spectrascopy (Academic Press, New York, I971). [21] S.H. Glanun, 3. Chem. Phy~. 33 (1960) 1371. [22J P. Résiiois and M. de Leener, ClassicaI timory of fluids (Wiiey-Interscience, New York, 1977). [23J GD. Harp and BJ. Beme, Phys. Rev. 29 (1966) 255; A2 (1970) 975; BJ- Beme and GD. Harp, Advan. Chem. Phys. 17 (1970) 63.
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P. Résïïis, in: Physks of many particle systems, ed. E. MeeronGordon and Breac&,Londoa, 1966). [2!G]KS. Cole and RW. Col@,4. Chem, P&ys.9 (1941) 34L [24J
1261 R.G. Gordon, 3. Chem. Phys_ 41(1964) 1819. 1271 M. Abrarnowita and I.A. Steguu. Handbook of mathematical fúnctions (Doveq, New York, 1965). 1281 B.D. Fried and S.D. Conte, Tbe plasma diirsion functiou (Academic Press, New York, 1961). 1291 Ba. Fried, C.L. IIedrick and J. McCnue, Phys. FIuids 11 (1968) 249. f30 J R. Finsy and R- vao Loon, 1. Pítys. Chem. 80 (1976) 2783. [31 J Y. Leroy aud E. Constant, Compt. Rend. Acad. Sci. (Paris) 264B (1967) 533. (321 N.S. Lojko and Y. Beers, 3. Res. Na& Rut_ Std. US 73 A (1969) 233.