Tabular compilation of data on exponential relations

Materials Science and Engineering, 56 (1982) 247 - 251

247

Tabular Compilation of Data on Exponential Relations WERNER PRANTL

Erich-Schmid-lnstitut fiir Festk6rperphysik, Osterreichische Akademie der Wissenschaften, und Institut fiir Metallphysik der Montanuniversit~t, A-8 700 Leoben (Austria) (Received January 21, 1982; in revised form April 30, 1982)

SUMMARY

Measured data o f relations which can be described by an Arrhenius equation are usually handled by applying a linear regression analysis to their logarithms. In this paper, after a short s u m m a r y o f the results o f a linear regression analysis the requirements o f data collection are briefly discussed. A modified set o f constants, which enables the user to draw m a x i m u m information from a m i n i m u m n u m b e r o f data, is proposed.

straight line because every measurement is subject to error. To find the straight line which best fits all the measured points or to be able to derive the error limits for the diffusion coefficient, a linear regression analysis using the least-squares m e t h o d is usually applied. The most important results obtained from this type of analysis will be summarized briefly [ 1]. For a linear relation between the measured values x and y the regression analysis yields the regression line

y = a + bx The dependence on temperature of numerous physical and chemical quantities (e.g. the rate of chemical reactions, the solubility of small amounts of impurity atoms in solids, activities, the density of vacancies in solids, the thermionic emission current, the diffusion coefficient, leakage currents in semiconductors) can be described by the so-called Arrhenius equation. For the diffusion coefficient, for example, this equation has the following form:

D = D o e x p ( - - R--~)

(1)

where Q is the apparent activation energy, Do is a constant factor, R is the gas constant and T is the absolute temperature. If the diffusion coefficient has been measured at different temperatures and if these values plotted on semilogarithmic paper, i.e. In D versus 1/T, prove to lie along a straight line, at least within a certain temperature range, then this means t h a t the temperature dependence o f the measured diffusion coefficient can be described by eqn. (1) and is therefore characterized by the constants Q and D 0. In general, of course, the points on the semilogarithmic plot will n o t exactly follow a 0025-5416/82/0000-0000/$02.75

(2)

with the regression coefficient b and the ordinate intercept a. The confidence interval for the regression coefficient b, i.e. the slope of the regression line, is found to be b + Ab

(3)

with

cp l j2 Ab =

sl{(n -- 1)(n -- 2))1/2

(4)

where p is the sum of the squares of the vertical distances of the sample points (x,y) from the regression line, s l is the standard deviation of the abscissa values of the sample points and n is the number of sample points. c is the solution of the equation 1 F(c) = ~ (1 + 7)

(5)

to be determined from a table of the t distribution with n -- 2 degrees of freedom. The so-called confidence level 7 (95%, 99% or some similar value) means t h a t the u n k n o w n true value of the slope lies within the confidence interval b + Ab with a probability 7. For the value of y belonging to a given value © Elsevier Sequoia/Printed in The Netherlands

248

of x the regression analysis yields a confidence interval

y = a + bx + l(x)

(6)

with

By analogy with eqn. (6) the confidence interval for the logarithm of the diffusion coefficient can be obtained as Q lnD=lnD0--- +l(T)

RT

l(x) = A b t s 2 -n -- - 1 + ( x - - ~ ) : 1~/2

(7)

n

where ~ is the mean value of all measured abscissa values. As can be seen, the confidence interval increases with x departing from its mean value 2. A graphical representation of these relations is given in Fig. 1. To transfer these results to the Arrhenius equation, eqn. (1), logarithms of both sides of eqn. (1) have to be taken: Q lnD=lnD0-RT (8) A comparison of eqn. (2) and eqn. (8) suggests the following assignments: y-~ln D a -~ In Do

Q b

--->

--

--

R 1 X-->

__

T From the confidence interval (eqn. (3)) for the regression coefficient a confidence interval for Q can be derived

Q+AQ where

Q = --Rb

AQ = R Ab

(9)

Y a+Zib2 , 0

%, %. '"~%~

O y-]= (b÷4b) (x-2) ~ y=a÷b.x (~ y-p= {b-Llb) {x-R) ,~ v- a÷bx÷l{x) (~ y= a÷b x - l l x )

~..~,,,~

.......... i':_C-:_~.~~.~. ~..~ q~ •.~.~,~

- ~.~.~. "%%~'~,,,@

X

D X

Fig. 1. The results of linear regression analysis.

and the diffusion coefficient itself is given by

D = D o e x p ( - - R ~ ) K±'

(10)

where

K = K ( T ) = exp(l(T)}

(11)

with

l(T) = ~

s12--n

+

--

(12)

by analogy with eqn. (7). Here s l means the standard deviation and ( l / T ) t h e mean value of all measured 1/T values. As can be seen, in eqn. (10) the confidence interval is expressed as a factor K instead of an addend, which results from the exponential relation between ordinate and abscissa. These are the results of linear regression analysis as usually applied to Arrhenius relations. At this point we should briefly remember the assumptions a b o u t the variables y necessary for the regression analysis model to hold. The y variables (a) are independent normally distributed variables and (b) have c o m m o n variance for all x. Departures from these assumptions, as long as they are not t o o drastic, fortunately do n o t have a marked effect on the results of regression analysis [2]. If, however, assumption (b} is strongly violated, each value of y must be weighted inversely as its variance to achieve efficient results [ 3 ]. As already mentioned, the temperature dependence of the diffusion coefficient is characterized by the t w o constants Q and Do over the whole temperature range within which the results of measurements are properly described by the Arrhenius equation, eqn. (1). Therefore it is obviously expedient to collect these constants in a tabular form. From Q and Do it is possible to evaluate the diffusion coefficient for any given temperature. If, however, the limits of error need to be indicated, some more information about the experiments is necessary. In the literature there is usually also a confidence interval AQ for the activation energy, although sometimes

249

w i t h o u t notification of the pertinent confidence level 7. However, this value signifies only that the u n k n o w n real value of the activation energy lies within Q + AQ and Q -- AQ, with a probability 7. Also no statem e n t can be made about the error limits of the diffusion coefficient, nor is it possible to plot the two straight lines with the slopes Q + AQ and Q -- AQ, because the point of intersection ( ( l / T ) , (ln D)) of these lines with the regression line is u n k n o w n (cf. Fig. 1). Frequently also confidence intervals Do + A1Do to Do and Do to Do -- A2Do are given in the literature. These differences A1Do and A2Do must be different, because the corresponding distances in the semilogarithmic plot are equal according to linear regression analysis and this equality of course is lost, when going backwards from In Do to D 0. S y m m e t r y is only possible when the confidence intervals are expressed by means of a factor, i.e.

data from diffusion experiments will be developed. Q, Do and A Q (e.g. with 7 = 95%) are used as usual. Instead of the confidence intervals AIDo and A2D0 for Do, the mean value ( l / T ) and the standard deviation Sl of all measured l I T values and the number n of the measured points available are introduced. When these six constants are used, a typical data collection from diffusion experiments would be as in Table 1. The following example demonstrates the type of information about the diffusion of 14C in F e 17wt.%Cr-12wt.%Ni t h a t could be extracted from Table 1 (third row). (1) From Q and Do the diffusion coefficient for any temperature (e.g. 0 = 500 °C) can be evaluated in the usual way from eqn. (10) by setting K equal to unity:

o O0ex ( ) ( 1.719X10-s t = 9.43 × 10 -s exp -- 8.-~-9)~ ~ /

Do ~±1

where = 2.334 × 10 -16 m 2 s-1 5 = exp ~ If n o t only Q and Do but also confidence intervals for these two constants axe given, the straight lines with slopes Q -+ AQ can of course be plotted but, as before, it is n o t possible to evaluate a confidence interval in the sense of regression analysis for the diffusion coefficient for any given temperature. To attain this the original measurement data still have to be obtained and the regression analysis has to be carried out. In this case the utilization of data collection is rather d o u b t f u l and appears in this form to be somewhat unsatisfactory. Thus, as an alternative, a proposal for a modified set of constants for compilation of

(2) From Q and AQ a confidence interval for Q can be determined with a confidence level 7 = 95%; this means that the (unknown) true value of Q lies with a probability of 95% within Q + AQ = 1.719 X l 0 s + 4.27 × 103 = 1.762 × l 0 s J mo1-1 and Q--AQ=1.719X10

s-4.27X103

= 1.676 x l 0 s J mo1-1 (3) From Q, Do, AQ, sl, ( l / T ) and n using eqns. (10) - (12) for any temperature (e.g. 0 = 500 °C) a confidence interval for the related diffusion coefficient D can be determined

TABLE 1 Diffusion in Fe-17wt.%Cr-12wt.%Ni Diffusant

S9Fe 14C SlCr 63Ni

AQ(95%) ( x 1 0 - S m 2 s -1) ( × 1 0 3 J m o l - l )

Sl ( X 1 0 - 4 K -1)

(l/T) ( X l 0 - 3 K -1)

n

(×10SJmo1-1) 2.798 1.719 2.644 2.582

3.69 9.43 1.33 0.197

1.626 1.698 1.790 1.525

0.8360 1.025 0.8788 0.8254

13 [4] 18 [5] 17 [4] 18 [6]

Q

DO

14.3 4.27 15.5 12.9

Reference

250 w i t h a c o n f i d e n c e level 7 = 95%; this m e a n s t h a t t h e ( u n k n o w n ) t r u e value o f D lies w i t h a p r o b a b i l i t y o f 95% w i t h i n

and 1

F(c) = -~ (1

+ 0.90)

= 0.95

D=Doexp(--R-~)K

-+ c(90%) = 2 . 3 3 4 × 10 -16 × 1 . 1 7 6 = 2 . 7 4 2 × 10 -16 m 2 s-1 and

= 1.75 where n--2=16

D = Do e x p - -

K 1

9..334 x 10 -16 x ~

--

1.176

T h e c o n f i d e n c e interval f o r D w i t h this n e w c o n f i d e n c e level 7 = 90% is t h e n f o u n d t o range from

D= Doexp(-- R~) K(90%)

= 1 . 9 8 6 X 1 0 -16 m 2 S- 1 = 2 . 3 3 4 × 10 -16 × 1 . 1 4 3 with l

= 2 . 6 6 8 × 10 -16 m 2 s-1 R

4.27 X 10 3 -

to

n

8.319 +

i 17 × 1( 1 . 6 9 8 × 1 0 - 4 ) 2 × - - + 18

1 . 0 2 5 X 10 .3 773.2

D = Do e x p - -

K(90%)

1 = 2 . 3 3 4 X 10 -16 X - 1.143 = 2 . 0 4 2 × 10 -16 m 2 s-1

= 0.1617 and K = exp(l) = 1.175 (4) I f t h e c o n f i d e n c e intervals in (2) a n d (3) are r e q u i r e d to be e v a l u a t e d w i t h a n y o t h e r c o n f i d e n c e level (e.g. '~ = 90%), this c a n b e d o n e b y r e p l a c i n g A Q ( 9 5 % ) listed in Table 1 by c(90%) AQ(90%) = AQ(95%)---c(95%) 1.75 = 4 . 2 7 × 103 X - 2.12

(5) F r o m Do, AQ and ( l / T ) a c o n f i d e n c e interval f o r Do c a n be d e t e r m i n e d w i t h a c o n f i d e n c e level o f 7 = 95%; this m e a n s t h a t t h e ( u n k n o w n ) t r u e value o f Do lies w i t h a p r o b a b i l i t y o f 95% b e t w e e n D08 = 9.43 × 10 - s × 1.69 = 1.59 × 1 0 - 4 m 2 s-~ and 1 D o - ~ = 9.43 × 10 - s ×

1 1.6---9

= 5 . 5 8 × 10 - s m 2 s-~ with

= 3 . 5 2 × 10 a J mo1-1 w h e r e c(95%) a n d c(90%) are t o b e t a k e n from a table of the t distribution with n -- 2 d e g r e e s o f f r e e d o m as s o l u t i o n s o f e q n . (5): 1

F(c) = ~

(1 + 0.95)

= 0.975 -~ c(95%) =

2.12

--

exp(-/4.278_.3_19X103 × 1.025 × I0-

,)

1.69 A n a l o g o u s t o (4) a c o n f i d e n c e interval f o r Do can be determined with any other confidence level. (6) F r o m ( l / T ) a n d t h e s t a n d a r d d e v i a t i o n s l o f t h e abscissa values t h e r a n g e o f t e m p e r a =

251

ture within which the measurements were carried out can be roughly estimated, namely from 01 =

1

(1/T)+sl

273.2

1 1.025 × 10 -3 + 1.698 X 10 -4

-- 273.2

= 564 °C to 0u =

1 (l/T) --sl

necessary, because all this information can be derived from only the six constants Do, Q, AQ, sl, ( l / T ) and n. As can be seen, the information content of this set of data markedly exceeds that of usual data sets. However, since the task of data collection is to provide m a x i m u m information from the minimum number of compiled data, the data set developed here appears to be particularly appropriate for the purpose of collecting data from experiments which can be described by an Arrhenius equation.

273.2 1

1.025 X 10 - 3 - 1.698 X 10 -4

ACKNOWLEDGMENT

-- 273.2

= 896 °C For comparison, the measurements were really carried o u t between 500 and 996 °C. The example shows t h a t , with the proposed set of data, it is possible for any given temperature T to evaluate the corresponding diffusion coefficient D(T), to derive confidence intervals for the activation energy Q, for the diffusion coefficient D(T) and for the constant Do whereby the confidence level 7 can be chosen arbitrarily by the user, and moreover to give a rough estimate of the temperature range in which the measurements were carried out. Knowledge of the original measurement data is no longer

The author wishes to t h a n k Professor H. P. Sti~we for useful and stimulating discussions.

REFERENCES 1 E. Kreyszig, Statistische Methoden und ihre Anwendungen, Vandenhoeck und Ruprecht, GSttingen, 1973. 2 L. Breiman, Statistics, Houghton Mifflin, Boston, MA, 1973. 3 E. J. Williams, Regression Analysis, Wiley, New York, 1959. 4 R. A. Perkins, R. A. Padgett and N. K. Tunali, MetaIL Trans., 4 (1973) 2535. 5 R. A. Perkins and P. T. Carlson, MetalL Trans., 5 (1974) 1511. 6 R. A. Perkins, Metall. Trans., 4 (1973) 1665.