E. A. S. CAVELL AND P. C. KNIGHT
1656
Acknowledgment. We thank Professor Stanley Bruckenstein and Mr. Peter Daum for helpful discussions. This work was supported by the National
Science Foundation (Grant No. GP-3830) and the United States Army Research Office, Durham, North Carolina (Contract No. DA-31-124-ARO-D-308),
On the Analysis of Dielectric Dispersion Data into Two Relaxation Regions
by E. A. S. Cave11 and P. C. Knight Department of Chemistry, The University, Southampton, England Accepted and Transmitted by The Faraday Society
(November l o 91966)
A series of values of normalized permittivity
(7’) and loss (7”) have been calculated by means of the ColeCole equations with four different values of the distribution parameter (a). On the assumption that these values can also be represented as a sum of two superposed Debye dispersions, bhe appropriate relaxation times have been determined from them by the usual numerical method for a series of values of the Debye dispersion The sum of the squared deviations (8) between values of 7’ and 7” calculated amplitudes, C1 and (1 - (71). by means of the Cole-Cole equations and those obtained for the “best” corresponding pair of Debye dispersions for each value of C1 considered has been evaluated. I t appears that as a increases, the minimum value of X also increases, although the sensitivity of S to changes in the value of C1 decreases significantly as (Y approaches zero. A graphical method based on these calculations for obtaining preliminary estimates of the relaxation times of the Debye type dispersions has been outlined.
Introduction The dielectric dispersion data of a surprisingly large number of pure liquids and of dilute solutions of simple polar molecules in nonpolar solvents can be represented within the limits of experimental error by means of the Debye relation.’ I n the majority of cases, however, the expressions of Cole and Cole,2and Davidson and Cole,* which incorporate an empirical distribution parameter, have commonly been employed and various suggestions have been advanced to account for the need to employ these modified relations. F r o h l i ~ hfor , ~ example, considered that deviations from Debye behavior might arise from a distribution of orientational activation energies within a system, and Glarums has shown that a unidimensional defect diffusion model could quantitatively account for an apparent distribution of relaxation times. Smyth and his collaborators,6-* on the other hand, seem to prefer to analyze their own experimental data, where necessary, into two or more relaxation processes, associated with dipoles of varying sizes and characterized by single relaxation times. I n certain cases, e.g., benzyl chloride, rotation of a polar group, about the bond connecting the group to the remainder of the molecule, can make a characteristic contribution to the over-all dielectric dispersion in addition to that made by rotation of the molecule as a whole. In these circumstances the observed permittivity (E’) and loss (E”) The Journal of Physical Chemistry
can often be represented as the sum of the appropriately weighted terms for each relaxation process) i e .
In eq 1 and 2, eo and e, are the low and high frequency limiting values of E’, w is the angular frequency (2nf), C1and Cz are the dispersion amplitudes such that C1 C z = 1, and T I and r 2 are the relaxation times characterizing the Debye type dispersions concerned. I n the most favorable case for which C1 = Cz, separate absorption maxima are not observed9unless n / r 2 > (3 2 4 ‘ 2 ) . For many practical cases, therefore, in
+
+
(1) P. Debye, “Polar Molecules,” Chemical Catalog Co., New York, N. Y., 1929, p 94. (2) K. 8. Cole and R. H. Cole, J . Chem. Phyls., 9, 341 (1941). (3) D. W. Davidson and R. H. Cole, ibid., 19, 1484 (1951). (4) H. Frohlich, “Theory of Dielectrics,” Oxford University Press, London, 2nd ed, 1958, p 92. (5) 9.H. Glarum, J . Chem. Phys., 33, 639 (1960). (6) K. Higasi, K. Bergmann, and C. P. Smyth, J. Phys. Chem., 64, 880 (1960).
(7) K. Bergmann, D. M. Roberti, and C. P. Smyth, ibid., 64, 665 (1960).
(8) F. K. Fong and C. P. Smyth, ibid., 67, 226 (1963). (9) D. W. Davidson, Can. J . Chem., 39, 571 (1961).
ANALYSIS OF DIELECTRIC DISPERSION DATA
1657
which 1 < 7 1 / 7 2 < 5 and C1 # C2, a single broad dispersion results. The analysis of experimental data to yield the two discrete relaxation times may be carried out graphi~ally,~** but generally the most convenient arid reliable procedure is to regard C1, r l , and r 2as adjustable parameters and then to calculate e’ and e‘’ by means of eq 1 and 2 for an appropriate range of numerical values by means of an electronic computer. The set of values of C1, 71, and r 2 chosen is that set for which the sum of the squared deviations ( S ) is a minimum. S is commonly defined by means of an expression such as eq 3, sometimes refined by the insertion of weighting factors to allow for experimental uncertainties. # €1
= z(:(E’calcd
- jet’
==
em
E’obsd)2
+
[eo
+
Z(Q’’ca1cd
- emI/[l
- e”obsd)2
+ (jWTo)l-al
maximum value. Equations 5 and 6 have been employed in the present paper to calculate a series of values of normalized permittivity (q’) and loss (7”) for wro = 1/3, 1, and 3 and for four different values of a. These results constitute our “observed” data for the purposes of eq 3. The values of 0 7 0 employed were chosen so that the resulting values of loss would lie symmetrically on the corresponding Cole-Cole plot and hence would simulate typical experimental data. q’ =
(€’
(eo
-
e,)
%{l -
(3)
-
Em)
}
sinh [(l - a ) In wr0] cosh [(l - a) In ~ 7 ~ sin 1 (a7r/2) .
+
(5)
(4)
Some authors seem to imply that the Cole-Cole relation (eq 4) on the one hand and two superposed Debye expressions (eq 1 and 2) on the other may be used as alterna,tive representations of their experimental data,7~8~10 although the relations concerned are evidently not equivalent representations. I n a recent review,” Smyth has stated explicitly that if two relaxation times are close together, it may be difficult to distinguish their effect from that of a distribution of relaxation times around a single, most probable value, Le., from dielectric behavior described by eq 4. Experimental values of permittivity and loss are usually considered to be subject to uncertainties of least f1% and &3%, respectively. It is conceivable, therefore, that within the limits of experimental error of this order, either formulation may be used in a purely empirical manner to represent the frequency dependence of e‘ and e“. The object of the present paper was primarily to determine the magnitude of the numerical discrepancies which exist between values of permittivity and loss calculated by means of the ColeCole equation with selected values of TO and a and those calculated from eq 1 and 2 employing the “best” values of the parameters, C1, 71, and Q, obtained by the computational procedure mentioned above. I n this way, it should be possible with experimental data of sufficiently high precision to decide which of the two alternatives is the better representation. I n those cases where the precision is such that this is not possible, the nature of the system investigated and the possible physical significance to be attached to the derived parameters concerned will probably determine the method of representation chosen.
Calculations I n order to calculate permittivity and loss for several different frequencies, the Cole-Cole relation shown in eq 4 may be transformed into eq 5 and 6 by separation of the appropriate real and imaginary terms. The parameter 70 is defined by the condition that WOTO = 1, where wo is the angular frequency a t which e’‘ has its
cos (aa/2) cosh [(l - a) In wrO] sin (a7r/2)
+
An I.C.T. 1909 computer was then programmed to obtain optimum values of r1/n and 7 2 / 7 0 for a range of fixed values of C1. This was achieved by varying rl/ro and T ~ / Tin~steps of 0.025 between suitable limits determined by the magnitude of C1 selected. The optimum values of r1/70and T ~ / T O are those values which satisfy the condition of a minimum in S , expressed now in terms of normalized permittivities and losses. Values of 7’oalcd and qttCalcd were obtained by means of appropriately modified versions of eq 1 and 2 for each of the three values of m0 quoted above. This procedure was repeated for a series of values of C1 ranging from 0.15 to 0.85 in steps of 0.05. Figures 1 and 2 illustrate the manner in which the optimum values of r 1 / 7 ~and ~ ~ respectively, / 7 ~ vary with the dispersion amplitude (Cl). Figure 3 shows the relationship between the magnitude of C1 and the sum of the squared deviations ( S ) obtained when optimum values of ~ 1 / o and 7 2 / ~ o are employed in the calculation of normalized permittivity and loss by eq 1 and 2.
Discussion The symmetrical character of a Cole-Cole plot requires the amplitudes of the two Debye dispersions derived from it to be equal, so that for all values of a examined here, S has its minimum value when C1 = 0.5. The symmetry of the plots themselves about C1 = 0.5, as shown in Figure 3, seems to be also a consequence of the symmetrical disposition on the ColeCole plot of the values of normalized permittivity and loss employed in the analysis. No great significance should therefore be attached to this symmetry. With actual experimental data, the symmetry would probably vanish while the minima in S would not necessarily (IO) W. F. Hassell and 8. Walker, Trans. Faraday Soc., 62, 861, 2695 (1966).
(11) C . P. Smyth, Ann. Rev. Phys. Chem., 17, 442 (1966).
Volume 79,Number 6 M a y 1968
E. A. S. CAVELL AND P. C. KNIGHT
1658
5.0
2.5
-
2.0
-
:1.5
-
4.0 *..
2 X
\
2
3.0
H
a
\
3
1.0
-
0.5
-
2.0
a=0.02
1.0
0.2
0.6
0.4
I
0.8
0.2
Cl
I
Figure 1. Variation of the optimum value of dispersion amplitude C1for various values of 01.
0.0
T1/To
with
-0.2
0.6
0.4
0.8
c1.
Figure 2. Variation of optimum value of 72/70 with dispersion amplitude C1 for various values of a.
occur at C1 = 0.5. A decreasing sensitivity of S to changes in the value of C1 as CY becomes smaller is also to be expected since as a tends to zero, both r l / r oand 7 2 / 7 0 tend to unity as shown in Figures 1 and 2, and the Cole-Cole circular arc reduces to a single Debye semicircle. The most significant feature of Figure 3 for present purposes is the increasing magnitude of the minimum value of S accompanying the increase in CY. It appears that if a minimum in S is employed as the sole criterion, then for the Cole-Cole relation and two superposed Debye expressions to be regarded as alternative representations of the same experimental data, the value of a should be less than 0.15. However, some regard should also be paid to the frequency distribution of the individual experimental results involved. This point is best illustrated by reference to Figure 4,in which a Cole-Cole circular arc for CY = 0.15, obtained by plotting loss against permittivity, is compared with the corresponding curve resulting from the superposition of the appropriate “best” pair of Debye semicircles. The divergence between the two loci is most marked for The Journal of Physical Chemistry
I
I
0.4
0.6
I
0.8
c1.
Figure 3. Dependence of the sum of the squared deviations ( 8 )on the dispersion amplitude C1for various values of 01.
I
I
0.2
I
I
I
0.6
0.4
I
1
I
0.8
9’.
Figure 4. Comparison of values of normalized permittivity (7’) and loss (7’)calculated by: (full line), eq 1 and 2 with C1 = 0.5, 71/70 = 2.10 and 72/70 = 0.475; and (broken line), eq 5 and 6 with a = 0.15 and 470 = l / 3 , 1, and 3.
values of W T O lying between 1/3 and ‘ / 6 and between 3 and 6. It is important, therefore, to ensure that an adequate proportion of the experimental results analyzed refer to frequencies corresponding to ihese ranges of wro. I n order to illustrate the way in which the magnitude of the maximum divergence between the two representations depends on CY, normalized permittivities and losses calculated with eq 5 and 6 for W T ~= 0.25 are compared in Table I with those calculated by means of eq 1 and 2 with C1 = 0.5 and the appropriate “best” estimated from Figures 1 and values of r 1 / r 0 and 7%/r0 2. This comparison indicates that with experimental data having precision of the order quoted above, it will not be appropriate to regard the two formulations as alternative empirical representations of these data if CY substantially exceeds 0.10, When circumstances are such that alternative representations are possible, the less specific Cole-Cole
ANALYSISOF DIELECTRIC DISPERSION DATA Table I : Comparison of Normalized Permittivity ($) and Loss (7") Calculated with Eq 5 and 6 for W 7 0 = 0.25 and Various Values of CY with Those Calculated with Eq 1 and 2 with CI == 0.5 and Appropriate Values of 7 1 / 7 0 and 7 2 / 7 0 from Figures 1 and 2 CY
calcd by eq 1 q r calcd by eq 5 qr
Aril
calcd by eq 2 calcd by eq 6
Aqrt
0 02 0 934
0.932 0 002 0 241 0.237
0.05 0.923 0:920 0 003 0.247 0.237
0 004
0.005
0.10 0.907 0.891 0.016
0.258 0.242 0.016
Table 11: Comparison of Literature Values of
71/70
Temp, Compound
O C
p-Dimethoxybenzene' p-Phen ylphenolb 1,4-Dia~etobenzene~ 1-Naphtholb 2-Naphthalene thiol' 1,4-Diacetoben~ene~ 1-Methoxynaphthalenea Anisole" a
Reference 12.
Reference 13.
20 40 60 60 60
20 20 20
0.15 0.888
0.866 0.019 0.265 0.241 0.024
addition, according to Grubb and Smyth,12 the ratio of the amplitudes of the two dispersions involved, i.e. , Cl/(l - C,) should be equal to (p12/p22). In order to calculate the relaxation times r1 and r 2 from the appropriate experimental data, it may be more convenient in practice subject to the reservations noted above to evaluate the more readily obtainable Cole-Cole parameters T O and a, before attempting computer analysis. If T O and a are known and an estimate of C1 can be made, then Figures 1 and 2 may be used to obtain preliminary estimates of r1 and r2. It is obviously desirable to obtain initial estimates of
and 72/70 with Those Estimated from Figures 1 and 2
-
Literature values-----
r__(I!
0.04 0.07 0.08
0.09 0.10 0.11
0.12 0.15
c1
0.19 0.80 0.82 0.74 0.82 0.85 0.61 0.38
71/70
1.81 1.23 1.11 1.07 1.24 1.10 1.44 1.97
---Estimated
values---
72/70
TI/TO
72/70
0.79 0.25 0.30 0.43 0.17 0.31 0.50 0.72
2.20 1.25 1.23 1.35 1.25 1.20 1.65 2.58
0.85 0.35 0.32 0.37 0.25 0.22 0.45 0.58
Reference 8.
representation will probably be chosen unless there is independent experimental evidence for the existence of two independent relaxation mechanisms. If intramolecular orientation occurs in the system under consideration in addition to the normal molecular orientation process, then the existence of two relaxation times is obviously possible. Smyth" has reviewed the various types of intramolecular motion, which can in principle contribute to the total dielectric dispersion of a compound. Of the various possibilities considered, the commonest example of a second relaxation time seems to be that associated with the intramolecular rotation of a polar group about one of the principal axes of the molecule, when the direction of this axis does not coincide with the direction of the dipole moment of the group. I n this case, the component (PI) of the molecular dipole lying along the axis about which the group rotates will be associated with the over-all molecular orientation, which in turn will be characterized by the larger relaxation time (rl). The component (pg) of the group moment perpendicular to the axis of rotation of the group will be associated with the intramolecular rotation of the group corresponding to the smaller relaxation time (r2). I n
r1 and r2 as near as possible to the final values before computer analysis of the experimental data is attempted, and Figures 1 and 2 are suitable for this purpose. The dielectric dispersion data of a variety of organic compounds have been analyzed to yield two relaxation times. A selection of these compounds is given in Table 11,13in which reported values of the relaxation times are compared with those obtained by interpolation from Figures 1 and 2. I n some cases, literature values of T~ and r 2 were calculated by different procedures from the one employed in the numerical computations undertaken for this paper. However, it is evident that in most cases the estimated values of 7 1 / 7 0 and r 2 / r o are within about 20% of those given in the literature, so that Figures 1 and 2 seem t o provide a convenient method for obtaining initial estimates required for computational purposes.
Acknowledgment. We thank the Science Research Council for a research studentship to P. C. Knight. (12) E. L. Grubb and C. P. Smyth, J . Am. Chem. SOL, 83, 4873 (1961). (13) F.K.Fong and C. P. Smyth, ibid., 85, 1565 (1963).
Volume 73, Number 6
M a y 1068
