Cole-Cole plots: Methods for unbiased parameter estimation

49

Cole-Qlle Plots: Methods for Unbiased Parameter Estimation S. Havriliak Jr. a and S. J. HavriIiak ’ = Rohm and Haas Co., Bristol Pa. IYOU

b i-hvriliak

Software Dcvclopmcnt

Co., Htx~i.ingdon Vnllcy Pa. PWM

1.1 INTRODUCTION

One of the laxring contributions made by ProA. F R. l-2. Cole [ !] is ZImethod of data representation which has become wideIy know12 as ‘Cole-C’qle plots’. In tiis method of dzta representation the imaginary p;ur (E*‘(U)) of the complex dielectric constant (E*(W)) is plotted against its real component (E’(W)). The plots a.~ generzliy made at constant temperature wilh frequency as a running v:tia5k:. There is no reason why tie F.lots canrot be made at constant frequency and temperature as a rJnr,ing v,?iablc. These Cole-Cole plots have become so colnrzon that rarely if ever do szien:ists me ation the fact ch2.r the imaginary axis has its sign reversed. In many of the ori giilai papers the !ocus of experimentai pcints were in the fourth quadrant bu: almost universally. tk locus is now in the first qucldrant. One of the advantages of these: Cole-Cole plots is tha; the shapes are quite simp!e and amenable to analytic forms. As a result or-the simple shape. the locus of experimental points are readily exrrapo!ated to obtain the instantaneous (E,) and equilibrium (E,,) values of +fxedielectric constant. Cole rp..:ognized that one of the simple shapes (circular arc) which is observed for a great many polar compounds takes the form: 11_ E*(o) -E, - = ( 1 + (icIXJU;: -l E,, -E, In this expression COis the radian frequensy oi measurement, t#, is the relaxatiun time and cx is a parameter of the relaxation function which in geometric terms represents the displacement of ;he arc’s cemer below the real axis. The parameter a has been the subject of muzh discussion and can be interpreted in terms of a

016i-7322/93/506.00

0

1993 - Elwicr

Scicncc Publishers

B.V.

All rights reserved

distribution of refaxation times or a parameter specifying the time dependent correlation function. Some time later Cole and co-workers [Z] found other sets of exFrimenta1 data that could be represented by the form:

In this expression, p is a parameter representing the skewness of the Cole-Cole plot at high frequencies. The parameters p can also be intevreted in terms of a distibution of relaxation times or a time dependent correlation function. Some time later the senior author of this paper and former student of Prof. R. CoIe recognized 1-13*thatthe dielectic relaxation processes of many polymers can be represented by Equation 3. The uniqueness of the Cole equations 1 an3 2 as well as the definition; of the parameters are retained in the fcrmalism of Equation 3. Although Equation 3 will be referred to in this work as the H-N function, it should always be recognized that when CIC = 1 the Colt: Cavidson function is implied and when p = 1 the Cole-Cole function is implied. E*(c9) -E_ E,, - Em

= { 1 + (iwrt,)“)

-!I

3.

Often, the parameters in Equation? i, 2. and 3 are detcrmirted by graphical techniques similar to those developed by Cole. The objective of this paper is to discuss methods for estimating the five parameters of Equation 3 from experimental data using rigorous statistical techniques. i. e. rr&icseE methods. There se several mascns .Nhy unbiased methods are impcrtant. First, each of the parameters in Equation 3 may depend on temperature and fcr this reason estimation of the temperature coefficients is also importr..nt.In a11there may be 19 or possib!y 15 parameters that are required tc:represent the relaxdtior. data. Estimating this large number of parameters from experimental data can easily degenerate into a polynomial curve fitting exercises wiLL little physical or mclecular significance left to the parameters or tieir temperature coefficients. On the other hand this large number of parameters indicates the theoretical richness that a c~ef~l analysis of the Cole-Cole plots is capable of providiilg a scientist. The second reason for using unhksed methods is that they are well defined and the results reproducible because personal judgement has been eliminated. This comes about because the model building procedures are objective and reproducible and not dependent on individual judgement. This means that given ary set of experimental data, two isolated analysts will derive the same set of parameters. The third reason is that unbiased methods when properly used, lead not only to estimates of the model standard error of estimate but to parameter confidence intervals The former quantity is similar in concept to experimental standard deviation. The latter quantity, i. e. parameter confidence intervals are absollstely crucial to a proper analysis and interpretatios,of the data. For example Equation

51

3 has 5 parameters while Equations I. and 2 have four. “The fit is adequate”, thcagh often cited is not a sufficient description of the resulis. The question now arises, “Is the fifth parameter necessary or important and how do we decide ?*’ Statistically, this is a typical question and is answered by askzag, ” Is 1 in the confidence inta-vals of a or p. This question can be answerzd rigorously by estimating paramcte: confidence intervals. In the present case ‘1 is chtiser, becatise if either oc cr p = 1 *hen we have either Equation 2 or 1, respectively. If neither a or p have 1 in their confidence intervals, then Equation 3 must be used. In this paper we address the questions posed in *the previous paragraph by discussing the foIlawing topics in sequence. First we review a hypothetical (biecea) experiment followed by an analysis of da-a in Section : .2 to point out some serio;ls flaws in these procedures. Then we discuss ii1Section I.3 z unb&reti method for a single parameterestimation because the procedure is simpie to visualize since only two dimensional space is required_ The method is expanded to the case for n pa-ameters (n-dimensionni space Zs now required) in section l-4. The concept of poofed data sets is developed in this section. Paroling of data peltits us to combine experimental data obtained at temperatures too high or LOOlow to be analyzed successfully at constant temperature. NIX:, the results for an unbiased constaX temperature analysis are reviewed in section 1.5. In Section 1.6, the residuals arc carefully scrutinized to determine if there are any systemztic deviations that would lead one to suspect breakdown of Equation 3. We then analyze a pooled data sets to determine how they effect ;-egression results in Section 3.7. Temperatures are included in the pooled data sets that could not be antlyzd at constant temperature. As a result the temperature range is increased substantially. It is well known that the real and imaginary parts of the complex dielectric constant are closeiy related quantities through thz Kronig - Kramers relationship (Section 1.8). The regression criteria are set up to analyze the dielectric con&tnt as a complex quantity, or as the absolute value, the real value, the dielectric loss or the loss tangent. Except for the loss tangent, the ether methods lead to the same set of parameters for the mode! equations_ Even analysis of the loss taugent yields dynamic paramezers &at are consistent with the other results At present a sorry state of affairs exists in the dielectric literature because cunre Citing techniques are receiving widespread ux as the performance of personal computers is improved. Results from these analysis are reported without citing the condition for convergence, the model standard error of estimate or the pzameters confidence intervals. This practice is equivalent to reporting dielectric exsrimental results without citing the specific dielectric equipment that was used. or the design of the dielectric cell, or to what extent temperature wiw controlled, or the methods used for specimen preparation. This paper reviews the best or unbtied methods that should be used in the analysis of dielsc+sic data.

52

1.2 AN UNBIASED

VIEW

OF THE DIELZXTRIC

EXPERIMF,NT

An unbiased observer would see the experiment proceed in four steps. First, the independent experimental variables, frequency (O ) and temperature (T) are set; these varinblcs can be refen-ed to as perturbations. Second, after the experimental conditions have reached equiiibrium, the two dials, one for cnpncitancz and one for conductance (sometimes this dial is replarcd by a dissipation factor dial) are adjusted until the bridge is balanced. The dependent experimental variables can be referred to as responses. The values for the system perturbatitins and responses are then recorded in a tabular form, see Table I, using one line for each observation. The experiment can proceed in any g!nnber of ways, but two are most common: temperature is ‘::ept constant and the frequency is varied (i. e. isotherma! measurements) or; frequency is kept constant and temperature is varied (i. e. isochronal measurements [43 >. The zmbiased observer wollld insist that this procedure builds in certain biases and that a better experimental design is one in which the perturbations are varied using a random scheme. Random is not to be confused with haphazard. In any case, step three involves repeating the previous two steps over a frequency - temperature range th:tt brackets the entire relaxation process. While it is clear that the r’ielectric experiments described above consists of two perturbations, the number of responses is not so clear. How these responses should be counted is one of the subjects addressed in this paper. The point is an important one because the application of regression techniques [S] to the determination of dielectric relaxation parameters requires an assumption about the number and nature of responses. Finally, the fourth step in this procedure is the analysis of the data to determine the Cole-Cole plot parameters. Almost always, but not necessarily, these parameters are determined from Cole-Cole plots drawn at constant lemperature. The parameters are then plotted against temperature to determine thermal coefficients such as the temperature dependence of E,, or ti.&e slope of a log(relaxation) time - reciprocal temperature Absolute (i_ e. activation calergy). At this point in the analysis, some scatter in the parameter plots become appalcnt and the scientist often ‘wonders if it is possibie tojug& thepurarnefcrs a bit to improve the fit’. A cautious scientist might even re-calculate the complex dielectric cclgstant from the parameters and Equation 3. The computed results arc often put in a ~bular form such as columns E’,(O) and E”=(W) in Table I. ? he scientist could, and almost always does make small ?d+tments to the parameters until 11~is satisfied with the fit and the scatter in the temperature plots. This procedure, no matter how tedious or well intended depends on the scientist who is carrying out the procedure and for this reason it will be referred to as a biased analysis, Table I lists some of the experimental differences for the hypothetical experiment

and calculated values as well as their just discussed. In Table I the subscripts

53

e, c and r represent experimental. calcuktted and difference. Hereafter the differences !isted in Table I will be referred to as residuals and are defined as follows: the real residual is simply E’,(O) = e’,(co) -E’,(O) while the loss residual is given by E”,(O) = E”,(O) -E”,(O) . The second and third columns of Table I can be collected into a matrix = E, containing the experimental quantities that is two columns wide, one for real and the other for the imaginary components and n rows long, one for each frequency-temperature combination. Columns 4 and 5 can be collected into a similar matrix defined as the predicted matrix, P while columns 6 and 7 can be collected into a matrix defined as the residual matrix = R. These matrices, as are the columns in Table I, related to each other by:

R=E-P

4. TASLE

I

COIUPAKISON OP ISXI’I4tIMENTAI~ AND CALCULATED COhll’LWC DlELIX’TRlC CONSTANTS AND Tl11:IR RESIDUALS

- 156.9 -156.9 -156.9

201000 302ooo 50HocKl

4.203 3.893 3.520

0.981 1.017 o.!xi2

4.2oc1 3.873 3.488

0.98 1 ;;;g

0.003 :::i:

0.000 0.027 0.044

In other words Equation 4 is a matrix statement of Table I and therefore follows the rules of matrix algebra. If we define a residual matrix transpose as RT , the regression function , d , may be defined as folIows: d = determinant (R’R)

5.

Statistically [5] it can be shown that minimizing d by GJjusting the parameters of Equation 3 (see the Appendix for the separated Equations Al through A4) gives the ‘best fit’ parameters. Equations A 1 through A4 are generally referred to as fitting functions. The umbkzsed objective is to find the set of fitting function parameters that yield a minimum in Equation 5. This minimization is called an u&Sasedparameter estimation, technique or method because given the same data set any two scientists no matter how isolated from each other they are, will arrive at the same set of parameters because the minimum criteria has been set. In other words an rtnbiassd rnirknization procedure is reproducible and not dependent on analyst. Intuitively, the scientist was trying to obtain the same goal by trial and errur and using his judgement as a criteria for convergence. Another advantage of

unbiased techniques is that confidence intervals fcr the parameter also be made. It is convenient to define a scaled determinant as:

a=”

estimates

can

6. DOF

In this expression DOF is the degrees of freedom. DOF is defined as DOF= N-P-l, where= N is the number of observations (for the present case the number of rows in Table I, and P is the number of parameters in the regrssion expression The scaled determinant is useful because it is independent of the num’ber of observations and parameters.

.I

n

DIELECTRIC

INCREMENT(p)

Since the best choice of model pzza,meters are those that minimize the scaled determinant 6. we shall now discuss the minimization with respect to one of the fitting function parameter P. A plot of the scaled determinant, 6, exhibiting a minimum with a systematic change in the H-N parameter P is given in Figure 1. The minimum in 6 represeats the best choice in P _ We represent the vtiation of 6 with P by a Taylor series. ht P,, in Figure 1 be the point about which a TayLr series is expanded, P, be the value that minimizes the scaled determinant and P, + A,Fr be the value of Pthat minimizes the truncated Taylor series representation (to include up to second order terms) of the scaled determinant. We now ask how to find the point P, that minimizes the scaled determinant. An arbitrary function such as 6(P) that represents the

55

dependence of the scaled determinant on parameter P can ‘be approximated by a trtincated Taylor series (tts) which includes terms up through second order in the displacement A. 7.

To minimize the Taylor series representation of the scaled determinant we tie the derivative of Equation 7 with respect to A and set it equal to zero. This leads to: 8.

which Ieads to the definition of A,,,,#(the vaiue that minimizes the Taylor series representation of the scaled determinant). Ial’) ‘7 1p=p, A rpr= -&.,, I I cfP2I,.=,,

9.

Equation 9 permits the estimation of the distance from PC,to the minimum, P,, if the first and second derivatives of the fitting function are known. The first derivatives of the fitting function are listed in the Appendix at theend of this paper. The second derivatives are complicated and their evaluation is best left to approximate methods [S]. Another feature of unbiased parameter estimation is that the uncertainty of determining the minimum can be estimated [5]. This is in fact the confiden_ce intervaI for the parameter estimation, i. e. a, . This quantity is always determined during linear least square fitting. but essentially omitted from all non-linear fitting and reporting of dielectric relaxation data. Qmission of a convergence statement. i. e. 6 and a confidence interval parameter O, is equivalent to not specifying dielectric equipment and some statement of experimental error associated with the temperature and electrical measurements. 1.4 GENERALIZED MATION

CRITERIA

FOR

UNBIASED

PARAMETER

ZSTI-

In the previous section we discussed the minimization of 5 with respect to one H-N parameter at a single temperature. The general procedure for setting up

56

regression problems is to use the Z-perturbation N-response formalism defined by Equation 9. In this case, Table I incIudes temperature as one of the perturbations or experimenta variables.

where R(I) = E -P(Z)

11.

Once more R(1) is the residual matrix and R+(I) is the transpose of ‘P(l). The task at hand is to minimize the regression function S(Z), defined by Equation 10 with respect to al1 alIowed variations in the fitting function parameter list, I. This is done by expanding s(1) in terms of its Taylor series (truncated to include second order terms). This is given by: &(I +A) = @I) i- G - A+;A+.SA

12.

where A is the displacement matrix. G is the first derivative matrix of 6(r) with respect to the parameters the fitting function parameter list and S is the second derivatives of 6(r) with respect to ail parameters defined in 1. Minimizing Equation 12 with respect to all allowed displacements S and solving for A gives A= -Y’G

13.

In order to solve this Equation it is necessxy to know the first and second derivatives of Equations Al thrcugh A4 wi?h respect to the parameters. The first derivatives are listed in the Appendix of this paper. Watts [S] developed an approximate form for S“ (Hessian matrix> so that the second derivatives of 6(Z) need not be tabulated. Havriliak and W7atts developed their regression routines in a commercial software package provided by SAS” [6] referred to as PROC MATRIX. Later in this paper we shall discuss some other routines available in SASti that require :, 31~ first derivatives. Havriliak anti 3 atts 173. [S] determined the H-N parameters for a number of materials in the following way. The first and second columns of E contain the dielectric constants and losses, respectively measured at all frequencies and temperatures. Combining all of the constant. data temperature data sets will be referred to as pooled datu sets. There are a number of advantages in pooling data sets and these will be discussed this paper. Briefly data that cannot be regressed by the constant temperature method because it is to either frequency extreme, i.

57

e. high frequency or low frequency end can be included in the pooled data set. Introduction oi’ such data in the pooled set leads to a halanced data set since an

equal amount of low and high temperature data can be included. P(I) in Equation 14 contains the predicted values corresponding in GCc&~J. The matrix P(lj is generated using the I-I-N function first column of P(l) has the following elements.

I

pitw).1(I) = Real E_(T) +

Et,CT) (1

The Nx2 matrix to the elements as follows. The 14.

Gm

+ (iwr D(T))m)m

In this expression Real means take the leaI part of the expression. column has the folIowing elements. P 1(W.I).,(I) = Imagina

4

(1 I- (itm,,(T)y-pn

In this expression Imaginary means take the imaginary fitting function paranicter list, I = (ipZs,

c,..cs,q.

contains the coefficients that describe eters E,,, E_,,T(,,cx, and p. SpecificalIy

15.

&0(T) -L(T)

E,(T) +

The second

1

part of the expression.

The

16.

3s

the temperature

depende:

ze of the param-

:

q,(T) = I, + C,T + D,T=

17.

E,(T)=Z~+C,T+D,T=

18.

logu,) (T) = f3 + CJ 1000/K)

+ D,T’

19.

a(T) = I4 + C,T + D4T’

2G.

P(T) = Z5+ C,T + D,T’

21.

AI1 the parameters are quadratic functions of temperature except for log, . T3e assumed form for log, has been discussed [9] from the point of view o:Seuche’s [ 1Of jumping model. This form for the rate process has the potential of determining the segmental jumping vofume. Other forms can be used such as the \‘ogel [I ! ] Tammann [I21 function. The Vogel-Tammenn function interprets rate piot curvature 2s a temperature at which the rate is infinite. For the present this choice is unimportant. A better set of model equations from the point of numerical

58

considerations is to introduce the notion of a centering temperature To - This temperature is chosen to be in the center of the experimental temperature range and has the effect of reducing the size of intercepts of the n-mdcl equations. 22. =a* =II+C,-(T--T,)+-D,.(T--T,)2 &-00=u-C*-(T-TT,)+D2.

(T-T”)*

23.

25. p =

I.5 CONSTANT

pcntaue

mixtures

I, -I-

c, - (T

TEMPERATURE

26.

ANALYSIS

reported by Denney [13] and an urzbiased study constant temperature, which has a historica precedence, proceeds by determining the parameters first at a number of temperatures and then plotting fhe Farameters as a function of temperature to determine my dependence. In Figure 2 we have given a plot of the experimental and predicted E’(O) with log(frequency) for i: number of temperaturesfor the 33.3 mole 9%isoamyl bromide solution. These temperatures represent the highest, lowest and

reported in [ 141 _

originally

- T,,)

Analysis

at

59

middle temperatures that the parameters could be evaluated using data at a single temperature. The experimental temperature range was longer but the parameters could not be evaluated because the data was to the high or low frequency side of the relaxation time. In other words the data set was not balanced above or below these temperatures. A more complete comparison is given in Reference 14. A plot of the Zttf”with reciprocal temperature AbsoIute is given in Figure 3. The data in this graph are represented as W and + 2~ bars from the expectation value. 1.6 ANALYSIS

C

CONSTANT

TEMPERATURE

RESIDUALS

A comparison of the experimental and calcuIated values is given in Figure 2 for E’(O). These overlay plots are at best a qualitative assessment of the fit. The deviations (residuals) for these mixtures are barely discernable and #aretoo small to be evaluated qualitatively. A better estimation of the fit will be discussed in this Section. A statistical summary of the fit is given in Table 2 for all the temperatures of 33 m% isonmyl bromide that couId be analyzed. 6 In that Table is the scaled determinant, defined by Equation 6, at convergence. VAR REAL is the rea1 variance and is defined by :

VAR REAL In this expression

=i

i

DUF

the summation

=

DOF

is over the subscript i or all the experimental

quantities, e represents the experimental quantities while c represents the calcu-

latedquantities. A similar expression defines VAR IMAG. Taking the square root of VAR REAL yields the STD REAL or the real part of the model standard error of estimate. A similar quantity defines STD IMAG. This quantity is similar in concept to the standard deviation of a replic;?ted experiment and represents the model standard error of estimate. MEAN is simply the average of the experimental quanti:ies. COV REAL is lOG;,~TD REAUMEAN and represents in % the deviations from the predicted values. The COV for the real part is about 0.3 % and for the less about an order of magnitude larger, i.e. about 3.3%. This disparity comes about because the real and imaginary means are different by an order of magnitude while the STD for the fitting processes are similar. The real residuals are plotted with frequency for the different temperatures in Figure 4. It is apparent from these plots that there are no systematic deviations that depend on temperature, there by ruling out the possibility of a model brenkdown. A plot of the loss residuals with temperature and conclusions are similar to the real residuals. inspection of the plots in Figures 4 does suggests that the real residuals may have a linear dependence on log(frequency). A similar analysis suggests that the loss residuals may have a quadratic dependence on log&equency). Plots of the residuals with temperature, magnitude of the experiment31

observable such as E’(W) or E*+(co) suggest that other correlations may exist. These correlations can be estimated in the following way. First we assume that the reai residual, r’ can be sep,mted into four components, i.e.:

r ‘9

-

r’;-f-

r”,+

r"c.

TABLE zTATISTIC~L

+ r”c..

29.

2

PARAISllXERS SUMMARIXINC. -X-LIE:FIT BKIW’EEN EXPERIMENTAL AND CALCULATED VALUE!+ Ok- E*(O) BEHAVXOR OF 33 hIOLE % IABKT,~IW AT VAKIt)US TEhII’ERATURE - 1678 a . c

-166.6 “1 c

-1653 “C

-164.5 0 . c

-161.0 ‘C

S

7x111”

5X II)”

‘JxXC*

HXIWT

SXW

VAR REAL i..iblAG.

0.0003 O.(MMI

o.o(lo1 0.m

O.ono 1 o.I)(w3

::izi3

SYTL) REAL LRIAC

0.01‘) WH’t

0.083 MXW

0.010 0.018

nIlCAN REAL IMAr..

3.5x 0.5 1

3.7x 0.52

REAL In1 AC.

0.5

03 3.9

0.3

2.3

3.5

0.4 3.1

DOF

20

22

21

20

PARAhIE~~EH

3.!J4

0.52

cov

2

0.015 0.015

O.OmI o.m3 0.012 0.018

4.14

0.52

z7

In Equations 28 and 29 single and double primes represent real or imaginary terms. The subscripts f, T,E’, or E” designate whether the dependence is Log (frequency), Temperature, or magnitude of the relwation prccess E’ or E”. We can %r&er assume that each of these has a dependence of the following form. r;

with analogous

= u ;+

equations

b ‘#ogfi

+ c ‘#ogn’

for the other terms. i.e.:

30.

TABLE SlJRlhfARY RESfDUALS EXPERIhfEhTAL VARIARLES T LWf+-l)

3

OF THE DEPENDEhCE OF REAL ON EXPERlhfEhTAL VARIARLFS R-spar+ ALL

R*lWC

CASE I

0.91 0.13

E-m)

0-W

E-‘(O)

0.01

0.00

In a11there are 130 real or imaginary residuals. Tfie residual equations have 4x3 -3 =9 parameters so that the ratio of experiments to parameters is about 15 making the quite analysis meaningful. A particulariy useful technique for thd analysis of these residuals is a stepwise procedure. The first step in this procedure is to test each of the variables on their effect of correlating with residuals using the model residual Equations 30 and 3 1. An important indicator is R-souare which cakulates l Lhe fraction of the range that can be accounted for by -the model. Regressing the residual Equation 30 yields: r; = -0.0008642

+ 0.0052937 + ~.00144S4T

32.

with an R-square value of 0.13, that is listed in Table 3 under R-square all. This procedure is repeated for the other three real residuals and the four loss residuals only the R-square values are listed in Table 4. The results of the re ression a& shown in Figure 4 and suggest that the strongest correlation of the ra residu(zlis with frequency.

62

The results of the regression suggest that I3 % of the variability in the msiduals can be represented by the residual model Equation 31. IF this dependence is removed by regression and the second order residuals analyzed with respect trJ the remaining variables no dependence is found. This means that 13% of the 0.3 % real COV exhibit systematic deviations due to frequency ant! the ren&ning 87 % of the 0.3 % show no correlation with the experimental variables. in other words any systematic deviations of the real residual with T, ~‘(6~1 or E”(O) is less than 0.9K&. TABLE

EXPERIMENTAL VARIARLFS

4

St’hIhJARY OF THE DE:PE!MXMX OF LOSS RESIDUALS ON E?WERI!tlENTAI, VARIABLES R-scpluore R-_squore CASE I ALL
T IwWRW)

C.42

afJd

0.35

O-09

E”(O)

053

021

0.01

The loss residuals can be analyzed in a similar fashion and are not listed here. For the loss residual, there 0.05% while the C3V for the loss is considerably higher, probably because both have been weighted evenly. In addition small frequency and loss calibration errors may exist.

I.7 VARIABLE

TEMTERATURE

ANALYSIS

AND MODEL

BUILDING

In the previous section we discussed the analysis of data taken at constant temperature. We can combine crPooL all the data from various temperatures into a single data set. In this pooled set we include those temperatures ‘lhat would not converge in the constant temperamre analysis. We assume model Equations 22 through 26 and ask what terms are neces.sary to represent the data. The simplest subset of the model Equations to represent the relaxation data of 33 m% IABRd2MEP solution is the five parameters (I,J5 ) and the temperature coefiicient for lnf, (Cs ). The 6 for this model is 0.00046. The effect on 8 by introducing the temperature dependences of the other four parameters into the

63

model Equations one a: a time is given in Table 5.0f the four 8 listed under I in Table 5 the temperature dppendence of p reduces 6 tile most. This parameter is now kept in the model equations and the other parameters introduced one at a time are used to evaLate 6 . Their effect on 6 is given in Table 5 under II. The next most effective telm in the model equations is the temperature coefficient for E, . The temperature dependences for p and E_ are then retained in the model equations and 6 determined for the remaining two parameters. This process is repeated systematically for all the terms in the model equations and the results are tabulated in Table 5. A plot of log(E) against t.ne number of +prs in the model is given in Figure 5. pa.rar,icW TABIJ3 5 SUMMAWY OF THE DEPENDENCE QF 6. IO ON LINEAR AND SQUAKE PAKAMIXERS

PARAI\lI3-EK

LINEAK Tl-cKhIs I

E”

3.x

E,

1.‘J

h!

_

u

2.1

B

I.8

LIh’EAK TERMS XI 1.7

LIXEAK

LiNEAK

TEKMS III

TERMS Iv

0.w

0.98

1.x

0.51

SCJUAKE TXKhIS V

a.42

rn’)

_

0.27

_

0.2X

026

0.23

0.40

026

0.24

036

c

1 E 2. g. 2

*

. -----------w-w--me_______

.:“1\

---~---._-_&___

M

It is cIear from an inspcitron of the data in Figure 5 that there is a stidden bre& in she dependence of log@) with number of parameten at 10. 3elow i0 parameters, there is a systematic and constant decrease in log@) with the number of parameters. This behavior suggests that the fractional decrease in 6 is about the same for each parameter. Above 10. log@) is approximately indeperkent of the number of parameters suggesting that any further inclusion of parameters into the model equations has no effect of reducing log@) and is therefore not justified.

TABLE4 E+(O) MODEL PARAhllZTTERS AND hIIXTUREC!j USL.iC TEhIl’ERATtiHF PARA MI!XZR 11 Q

THEIW LIMITS AS A VARIABLE

FOR iABZ’!!MEP OF REGRWION

-1653 “C

LSF

333 h! r, XAl3K

50 hi % IARR

75 I;1 % IABR

I00M % IABR

5.51 0.01

554 0.02

5.!xi 0.01

7.19 0.01

:0.19 0.01

13.51 0.02

-0.02-l 0.001

-0.03fl O.M)f

-fl.o(~ 0JX-E

-0.03ii o.cK)-I

G u

-0m7 0.OC-U

1, a

_.Y 3 S? 0.0 1

2.20 0.03

2.2 1 0.01

2.2s 0.02

2x 0.01

2.65 0.02

cz Q

_

-0.030 o.co7

-0.!113 O.oCl3

-0.01s 0.003

-0.004 mm

-0.021 O.MIS

I, 0

0.24 ?oYi

9.5 0.3

8.62 0.02

9.59 0.01

11.31 0.0:

-6.3 0.s

4.e

-8.12 0.0 1

-5.00

0.74 0.0 1

0.733 WKlS

0.023 0.002

0,013 O.o()!

I).CiS O.c)l

o.(is O.OI

4.033 O.(K)2

41.mI9 0.IXl.t

c, u

1, u

0.74 OS)!

0.64 0.02

0.07

10.121 0.011.5

0.W 1.0 CLcli

0.CfM-l

o.ows 0.60

0.55

0.01

0.01

TABLE

7

STATISITCAL PARAMETERS SURIblARIZlXG T.CIE FIT FETWEEN CX’ERIMENTAL CALZULATED V 4LUES OF E*(O) FOR 33 M ‘5 IABWZ?dEP SOLVTIOKS USING TEMPERATURE AS A REGRESSION VARIABLE

SCAL. DET-

4.2xxIG5

cov REAL IMAG.

0.8 3.0

0.7 -1.0

0.8 3.2

1.8 5.6

DOF

21K;

L-i

245

206

T,

-1ti6.S

-160

-1%

-1329

I.8 XNTF,RDEkENDEblCE

6x HY

ZOxlW

OF REAL

AND IMAGINARY

AiWD

49x loz

PARTS

Anycne who has balanced a capacitance - conductance bridge knows that starting from a seriously unbalanced condition, the capacitance dial (whose magnitude determines the real dielectric constant) and the conductance switches and dials (whose magnitude determines the dielectric loss) appear to be independent of ea,h other. As bridge balance is approached the dials appear to interact. This interaction may of course be due to an artificial coupling of the components through the detector: on the other hand this coupling may suggest that -&;ereal and imaginary parts are not imldependentof each other. This confusion causes a dilemma in how to set up the regression conditions for the determination of rhe parameters from the relairation equation. Arguments base on Bcltzmann’s Superposition integral [ 151yield the result that the real and imagina_T harts of the dielectric constant arise from the time dependence of the process and :he nature of the periodic field_ Rese arguments, which are general to a wide variety of relaxation processes, are cast in general terms and show that the real and imaginary parts are not independent of each other but are funL -,x-tallyrelated. GaII :hev derived the theoretical equntions of Kronig’s [ 16j relationship between tlx index of refraction and the linear absorption coefficient and aIso Kramers [ 171relationship between the real and imaginary components of the dielectric constant. The resuI% of this shldy are: E’(q_)and

1+

DD E”(C9) _._ -__ Jx K _-. J_ r

33.

34.

In these equations P represents the principle value of the integral. These results show that if either E’(O) or E”(O) are known over the entire frequency range then the other component is also uniquely determined. In other words the real and imaginary parts of the complex dielectric constant are not independent functions but tie related to each other. Havriiiak and Watts suggested writing the relaxation equations in the following form: E’(W) - &, =

E”(O)

=

l=‘E(&,,-

r-y&,,

-E,)

&,) cos(

PO) + e ’

35.

sin@@) + r ”

These equations include a real error (e’) and a loss error (e”) that may or may not be auto-correlated. The real and loss errors though related are not the real and loss residuals. The residuais may contain model breakdown as well as bridge calibration errors The errors are those that obey all the rules of random noise. If these errors are set to zero, the two equations are not independent ones since the same set of parameters fit both equations. On the other hand, in the presence of error these equations may bc sufficiently decoupled so that !hey may be considered to be independent equations. In this paper both assumptions will be studied. 1.9 TWO ISR/p

PERTURBATION

ONE RESPONSE

REGRESSION

FORMAL-

X9.1 Introduction In the previous section we developed and applied two perturbation two response regression formalism to the analysis of complex dielectric data. In that section we treated the complex dielectric constant as thougb it consisted of two statistically independent responses. In this section we extend that analysis to the two perturbation one response formalism, the case where the complex dielectric constant is viewed statistically as a single complex response. This analysis proceeds in a similar manner to the one in the previous section, the only exception being the way in which the residual matrrx, i.e R(L)=

E -P(L)

37.

67

is defined. Here we collect the responses (complex numbers) for N combined values of frequency and temperature into a Nx 1 matrix E. The Nxl matrix P(L) is similarly defined, containing the predicted values corresponding to the elements The elements of P(L) are defined as A!Zi[~,~~.j

.

P;,,.& 1= Km where tion.

i,&,,(T),

1.9.2 Dielectric

&,(T),q,(T),

Constant

f a(T)

38.

Gf-) - c.U) (1 + (ior I,(T)Y”)O(T) and P(T)are

defined

as in the previous

sec-

as a Chmplex Number

If Equation 37 is written in complex notation. then the dimensions of the matrices are Nxi because each element is a complex number. In other words the experimental values, predicted values and their residuals zre complex quantities. If we re-write Equation 10 in complex notation we have:

w>=&

detemlinant

(R*(I)R(I)}

39.

In this expression R*(I) is the adjoint matrix of R (I). An adjoint matrix is similar to a transpose matrix bzcause not oniy have the terms been interchanged in the same way but each element in the adjoin: matrix is the complex conjugate of the corresponding element in the original matrix. The determinant on the LHS of Equation 39 is a 1x1 instead of a 2x2 determinant used in the preceding sections. It is assumed that the best fit parameters ;!re those that minimize the this 1x1 determinant. These parameters were determined using a commercial software package developed by SAS“” and known frs PROC NLIN. This nonlinear regression procedure yields best fit parameters for a nonlinea: model. The convergence criteria is defined by: 40.

In this expression d has been defined by Equation 39, and the subscript i-l represents the previous (relative to i) d. The default value for c is l@’ . In other words, a minimum in d has been determined to 1 in 8 * place. In addition to the parameter estimates, confidence intervals for their estimates and the parameter correlations are part of the output. This procedure provides the analyst with all the necessary statistical information to form an unbiased estimation of the parameters. To use this procedure three statements must be made. The first one is the parameter statement or trial values which is a list of the fitting (regression)

parameters. These are obtained from the graphieaI procedures describedearlier. The second is the model statement which is dktated by the form (number of columns) in the residuai matrix. Thirdly, the fir.% derivatives must be specified and these have been listed in the Appendix. The advantages of SAS@” are evident, it has eIiminatsd the tedium of writing code for looping, making conditional statements, or evaluating second lerivativcs and provide the scientist with a11 the information necessary to make an unbiased statement about his results. It should be pointed out that th w confidence interval for the parameter estimate at convergence is a product of this procedure_ The corrfidence interval for the parameters is essential when making comparisons. The utihty of *this information will be illustrated throughout the rest of this work. The results of ana’ryzing the data as a complex number is given in Table 8 under II. 1.93

Absofute

Value of the Complex

Dielectric

Constant

Another quantity associated with the complex dielectric constant is its absoIute value given by, (see the Appendix at the end of this paper): (ABSOLUTE

VALUE)’

The results for this regression 1.9.4 Real Dictcctrk

= ~*(a, T)‘+

are given in Table

E-(co, T)”

41.

8 under Ill.

Constant

Often times the dieIectric relaxation data is availabk only in a limited form, such as only th,p dielectric constant or loss is available as a function of temperature and frequency. Under these limited conditions it may be important to determine the relaxation parameters. The real dielectric constant can also be used in the model statement and a?1 5 parameters are determined, see Section 1.1 I_ 1. The results for this regression are given in Table 8 under IV. 1.9-5 Dielectric Loss The dielectric loss Equation A2 contains 4 of the 5 l-324 parameters so that d’(wj can also be used in the SAS” model statement_ A limitation to this approach is that &-and its temperature dependence will no longer be determined. The results for this regression are given in Tables 8 under V. The parameters 1, = & is redefined , while 1, and Cz are not determined. l-9-6 Lass Tangent The loss tangent is defined as:

42.

In this equation C, is the cell constant, g is the test specimens equivaknr electrical conductance, C, is its equivalent electrical capacitance and f is the frequency of measurement in Hz. It migh: appear that parameters such as the equilibrium and instantaneous dieIectric constants. i.e. E,, and E, respectively, have been deleted from Equation 42 and they can not be obtained by regressing tan 6(T, o) data. However if we take the ratio of the imaginary to real parts of Equ&ons A? and A2 we have: tan 6(o) =

43.

E*‘(O)

r+‘{E,, -E,) sin(Pa)

E’(N

= E, + rBw(E,,- E,) cos( f30)

We see from this equation that if &_ can be defined. then the remaining parameters are determined. It is well known irxx] that the instantaneous dielectric constant is related to the refractive index with some allowance made for atomic polarization. i.e. E_ = II’ 3 1 .OSn’ . Denney irxx] reported the room temperature values for the refractive index of the halide to be 1.444, its density 1.204 (g/cc), the density of rhe hydrocarbon to be 0.648 (g/cc) while its refractive index was not listed. He did extrapolate E,, far his mixtures to infinite dilutiun. i. e. pure ha?ide to obtain 1.87, at room temperature. If we assume that E,, = E, for the hydrocarbon. then the refractive index computes to be ? .37. A more recent value [ 181 for the refractive index is ?.37? and a density of 0.653. Both sets of values are in good agreement_ If we assume that the refractive indices are additive according to mole wt 55, the refractive index for the 33.3 m% tixture at room temperature is 1.39. A vallle for the density of 0.803 g/cc was obtained from Denney’s data by interpolation. This is sufficient information to ca?cu?ate the molar pokization at room temperature from the Lorenz-Lorentz equation 141. Denney also determined mixture densities at three temperature so that the refractive index could be estimated in the dielectric experimental range. The temperature dependence of E, was estimated from the temperature dependence of the refractive index. The result of this ca?cu?ation without the 5% contribution from the atomic polarization is given by Equation 44 and is listed in Table I under Case VI. E, = I2 + C2(T - T,,) = 2.49 - 0.0023

- (T

+

165.5)

44.

The estimated parameters Iz and C2, defined by Equation 44 are cons’mts in the regression of the loss tangent data. The results of the regression are given in Table 8 under Vi.

1.9.7 Results of Yarious Methods of Regrcssiorr

There is essentially no difference in the results using the first four methods listed in Table 8. Method V, which treats only the !oss data does not determine Ar, can be determined. The E,, and&_ . uniquely. aithough their difference difference. E,, -E, for cases I through iV is about 3.35 + 0.0 1 while for case V it is 3.42 k 0.01. Although the agreement between the differences is better than the agreement between- the absolute values, it is still outside the confidence interval. TA3LE

8

V

I

0.02

557 0.0 1

S.Sb O.OI

0.01

-0.024 o.oin

-0.Lr27 o.tm 1

-D.O27 O.!xi;ll

-0.W~ 0.0 1

7 73

2.40

S.Sh

S!b.

VI

2.20

773 _.w_

_.I

0.01

0.02

0.02

S.34

-0.01 t o.Iw1s

-0.W~ O.(WM

-0SKI3 o.cHM

X.A3 o.of#

X.63 0.06

K.59 0.06

K.7C 0.03

4.7 I 0.13

4i.S-r 0.13

-6.S‘J 0.13

-6.7 1 0.07

0.72 0.0 1

0.73 0.n i

0.73 0.0 1

0.71 0.01

0.721 0.w

11.011 O.(K)7

0.016 O.fW

0.01s O.Oo2

O.l)iO O.IK11

0.01s 0.001

0.M 0.03

0.63 0.03

0.66 Cl.02

0.67 0.01

-0.012 o.D!u

-o.OlG 0.On.F

M-I 0.02 -o.mci OSXLI

O.cK123

-0.016 0.003

Definition of column headings: I = the mahod CICI-Iavrili;lk and Watts; II = absolufc vduc; III = complex diclcctric corst;lnt; IV real pan of the diclcctic constant; V dicktic less; VJ Ioss t;lngcnt

x.72 ME

-0.017 0.0112

71

1

2

I-

-169.76

I

6

-ltS_56 -a--_

166.26

B.-.-w

-16.46

_.*-.

-I

ST.76

6

-166.56

___a.._

__*_

-16-S --c_

-1u --_&--_

Finally the loss tangent, i.e. case VI also gives results provided estimates of E_ and its tempxature dependence can be determined. Estimates for these parameters from refractive indes and density - temperature data are in poor agreement with those determined from analysis ofcm~plcx dielectric relaxation data. What is disturbing is that the estimated value for E, is higher that it is from the direct analvsis of the data, even though no contribution was allotted for atomic polar&ion. The remaining or dynamic parameters are in good agreement. The real and imaginary coefficients of variation (COV) listed in that TabIe for Case II are also poor. In fact the real COV for Case II is 20 times that of the original analysis. A plot of the predicted and experimental tan& is given in Figure 6 left,

72

where it can be seen that the agreem?nt is quite good. However, a complex plane plot of the calculated and experimental real and imaginary parts shown in Figure 6 right, indicates that the agreement is quite poor. A plot of the real residual with log(frequency) for a number of temperatures is given in Figure 7 and is an excellent illustration of a model breakdown. In this case the model breakdown is due to E, and its dependence on temperature. 1.10 CONCLUSIONS A review of many dieIectric papers in any journal or language reveals an In thissection the experimental procedures used to make experimental section. the &electric measurements are described. Even though the Iength may be 1 or 2 pxagmphs, the equipment is described and generally the reader obtains some idea about the value of the experimental results. Recently, many papers have appeared with the results from a least squares technique to determine the Cole-Cole plot parameters. AJmost always three importantpieces of informations arenot i ncluded. First, js a statement about the criteria USed for covergence, i. e. minimum to how nearly always the standard model error of many significant figures. Second, estimate is not reported. Overlay plots are not a substitute for a statement of model plot is better than an overlay plot. error of estimates. Furthermore a residual Finally fie parameter confidence intervals arc never stated even though interpretation and comparison of the parameters is the most important objective of the regression procedure. Estimation of these confidence intervals is straight forward In addition several commercial [S] from the transpose of the residual matrix. software packages are now av&l3ble tc estimate confidence intervals an convergence. Several important concepts have been discussed and applied in this work, First is the notion of an rrrtbiascd analysis of experimental data. Unbiused parameter estimation is a technical term used by Statisticians to indicate the nature of the assumptions used in the analysis and is related to the principle of maximum Iikclihood. The second important con.=@ developed in this work is the notion of apooled data set. The comparisons gwen in Table 8 show that there is a factor of two or more improvement in the parameter confidence intervals for pooled data sets over single te.mperature anaWlsThis c cmes about simply because the ratio of experiments to parameters i ncreaSeS about a factor of ten. Commensurate with this decrease is an increase of in CoV for the mnlysis. The confidence interval for a was fmmd to be COOS while the corresponding interval for p was 0.01. This means th ai OLis different from I by about 500 while p is different from ! by 350 - Such results cannot be ignored because they are significant. Furthermore, the same results are obtained from analysis of E*(W), E’(C) or E”(W) data as shown in Table 8. Finally the residuals &arefreeofsystematic and correlated tendencies. These results mean that Equation 3 represents the data for IABRDMEP mixtures to about 0.01%. Conversely the data for these solutions

73

cannot be represented by Equations 1 or 2. All the procedures discussed in this work were programmed in SASfi using their PROC NLIN. This procedure not on3y has vlzry stringent convergence conditions, i. e. Equation 40 but also reports parameter confidence intervals at convergence. Parameter confidence intervaIs i;sabsolutely necessary for malting comparisons -Nhich is one of the objectives of this work. The regression results discussed in this work was obtained on a PC of rnodect dimensions. Specifically, the unit used is a COMPAQ Model I486 operatint; at 22 MH;-.: Mont rtyresskns are completed in a few minutes. Occasionally, there are some tedious calculations that may take 30 n.#in.The point is that the requirements for ur:birrsed analysis no longer requires te.;ium or high cost. I.11 APPENDIX

OF THE FIRST DERIVATIVES

1.11.2 LISTING

In this nppcndix wc give tt ccrdilrg of the functi
CODE

$.

‘nIErA1

E,

‘1-lI E-l-,\2

WW

Tt tBl’h3

n

‘I7 IErA

P

‘I’11WhS

2W

< )XlEGA

?c e’ .I E

PI 1zm1. 1MhG

USC is

l-cd imaginary

rccl1

KEALI

imaginary

IMACiI

rlxll imqginxry Tcnrpwxfy hlln_‘13

=

<1RD3 =

ibnclions:

-TiIITI-AJ*hlIIl_-I’ M111.13*S1W

ABS3

=

X11 IIa-D*SIN-t

ANCii_I3

=

(OKl)3*ABS-ABS3+ORl))IKN)SU

SINE3 COSIhI3

= TIII-TrA.F*C<)SINi~~,~~C;iJ33 =

-nrI~As*sls~3~Ah’c;I1:~

RAIXQ

=

ABS+ABS.1+

KMULT3

=

-nIIn-As*Rhl~n,I-*KMSu~/RA~Sy

l

OKIY’OKI>3

Tcmpwary

limrtion~:

SINES = C<~SlNl-:+ANCiI.I~ C10S1K1~ = slNE+A.Ncil.ls KM f Jl_T.F=

-IdX;(RADSQ)*Rh41

II,T/’

lkriwtivcs

&l/dB:

‘l’lII~l’AI~+(~~l?l_-IY+COS15riI3 ‘I1iI:~AI~*(~IIll.‘l’.srSINI~

1.Coic.

+ RMIIl.-!yC’OSlh73.5)

+ KL4IJi_-I’*SC\;i’.C)

1.12 REFERENCES R. II.: Cdr.

2. Ihvidson.

D.W.:

K. S. J, C/wm

3. iIavri1iak.S. Jr.: Ncprni. $F.F.cCIU~L

19-%t, 0. p. _Ul

PIIw..

Cole. R. II. J/Phys. S. Polwnrr.

5. Babs. D. W.: W~~LS. D. G. Abnlint~uurRegression New York 6. SAS Inslimlc Inc. Users Guide.

8. IIavrili&.

S. Jr.: WILLS. D. G. Prrfwner. 19X6,27.

9. Havrihk.

S. Jr. CoUoidPolym

14. Iiavrihk.

Sci.. 19?0.268.

18. Aldrich

Z Annrg. Allgem

S. Jr. J. Phys. Chcm,

of Curlzberl

Daniels.

(IX

C. MJ-

p_ 426

Chenz.

3990,35.

Am.

Caluloguc.

1926,

WI3

156. p. 255

p_ JCitl

p. 43-U of Efwtric

Vol. II. Dielectrics

in 7bw

Inremuzir~nale dei Fisici a C&w.

Bologna.

Pohti~tion.

Depetient

1926. 12. p. 547

II. A. Esrrarrp gagi! Ani de1 Congresso Chemical

NC. UbA.

1921,22. p. MS

15. Boctchu. C. I. F.: Bordcwijk. P. 77~0~ .1978, Elscvia PuMicltion Co.. New York 17. B

1988. John Wiley & Sons,

p. 1509.

D. J.; Ring. J. W. J. ChEm. Ph_vs., 1966.

16. Kmnig. R. 1. Opt. Sk.

& I1.s Applicurion.

195%. 21(10). p. IX50

12. Tammann. G.: Hcssc. W. 13. Dcmcy.

uncl Anu:ysis

1967. John Wiley &

E$@xs in IWyners,

, Dufu. ord Anul_vsh. Ly some Frien&

S. Jr.: Warn.. D. G. Des& Wiley. New York.

11_ Vogci. M. PitysSz Z

andDieIecfric

Version 6.01. Guy.

7. I kwriIhk. lows). 19%

C~~TL Pl+.

IJXJ

p. 161

1967,8.

13. IL: Williams. ~3.Anclrtsbc

N. G.; Read.

IO. Bucchc.J.

10(12).p.

C/rem, 1951.

Aldrich Chcmicd

Co.. Milwaukee. Wi-nsin

1927.

Fields.