DECONVOLUTION ANALYSIS TO DETERMINE RELAXATION TIME SPECTRA OF INTERNAL FRICTION PEAKS

HAL Id: jpa-00225294

https://hal.archives-ouvertes.fr/jpa-00225294

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

DECONVOLUTION ANALYSIS TO DETERMINE

RELAXATION TIME SPECTRA OF INTERNAL

FRICTION PEAKS

J. Cost

To cite this version:

JOURNAL

DE

PHYSIQUE

Colloque C10, suppl6ment au n012, Tome 46, dgcembre 1985

page

C10-31

DECONVOLUTION ANALYSIS TO DETERMINE RELAXATION TIME SPECTRA OF

INTERNAL FRICTION PEAKS

J.R. COST

Los Alamos National Laboratory, Los Alamos, N.M. 87545,

U.S.A.

RCsumh- O n d t c r i t u n e n o u v e l l e m t t h o d e d'analyse d e s pics d e f r o t t e m e n t i n t i r i e u r e n f o n c t i o n d e l a t e m p e r a t u r e . Elle p e r m e t d ' o b t e n i r u n e a p p r o x i m a t i o n d u s p e c t r e d e s t e m p s d e r e l a x a t i o n responsables d u pic. O n m o n t r e q u e c e t t e m i t h o d e , a p p e l i e analyse d i r e c t e d u s p e c t r e (DSA), s'applique I d i f f i r e n t s t y p e s d e spectres: (i) e l l e f o u r n i t d e s a p p r o x i m a t i o n s d e c e r t a i n e s f o r m e s d e s p e c t r e q u i r e p r o d u i s e n t l a p o s i t i o n , l ' a m p l i t u d e , l a l a r g e u r e t l a f o r m e a v e c u n e b o n n e p r i c i s i o n , q u i est d e l'ordre d e 10% (ii) elle n e c o n d u i t p a s I d e s a p p r o x i m a t i o n s a l i a t o i r e s d e l a f o r m e s p e c t r a l e d e s pics.

Abstract

-

A new m e t h o d f o r a n a l y s i s o f a n i n t e r n a l f r i c t i o n v s t e m p e r a t u r e peak t o o b t a i n a n a p p r o x i m a t i o n o f t h e s p e c t r u m o f r e l a x a t i o n t i m e s r e s p o n s i b l e f o r t h e peak i s described. T h i s m e t h o d , referred t o a s d i r e c t s p e c t r u m a n a l y s i s (DSA), i s s h o w n t o p r o v i d e a n a c c u r a t e e s t i m a t e o f t h e d i s t r i b u t i o n o f r e l a x a t i o n t i m e s . T h e m e t h o d i s v a l i d a t e d f o r v a r i o u s s p e c t r a , a n d i t i s s h o w n t h a t : (i) I t p r o v i d e s a p p r o x i m a t i o n s t o k n o w n i n p u t s p e c t r a which r e p l i c a t e t h e p o s i t i o n , a m p l i t u d e , w i d t h a n d s h a p e w i t h good a c c u r a c y (typically 10%). (ii) I t d o e s n o t yield a p p r o x i m a t i o n s which h a v e false s p e c t r a l peaks.

I

-

Introduction

T h e p r o b l e m o f a n a l y z i n g i n t e r n a l f r i c t i o n v s t e m p e r a t u r e p e a k s t o d e t e r m i n e t h e d i s t r i b u t i o n o f r e l a x a t i o n t i m e s r e s p o n s i b l e f o r t h e peak h a s been a d i f f i c u l t o n e f o r m a n y years. T h e t r a d i t i o n a l m e t h o d of a n a l y s i s h a s b e e n t o a s s u m e v a r i o u s f o r m s for t h e r e l a x a t i o n t i m e s p e c t r u m a n d t h e n c h o o s e t h e best f o r m b a s e d u p o n goodness-of-fit t o t h e d a t a . T h e p r o b l e m w i t h t h i s m e t h o d i s t h a t n o n e of t h e t r i a l f o r m s f o r t h e s p e c t r u m m a y be close t o t h e t r u e d i s t r i b u t i o n . I t would be p r e f e r r a b l e t o directly a n a l y z e t h e d a t a t o o b t a i n a n a p p r o x i m a t i o n of t h e s p e c t r u m , t h u s a v o i d i n g unnecessary a s s u m p t i o n s . Such a m e t h o d , referred t o a s d i r e c t s p e c t r u m a n a l y s i s (DSA), h a s been d e v e l o p e d a n d v a l i d a t e d f o r a p p r o x i m a t i n g r e l a x a t i o n t i m e s p e c t r a f o r processes w i t h f i r s t - o r d e r k i n e t i c s / l / . W i t h t h i s m e t h o d , n o n l i n e a r regression least-squares i s used t o u n f o l d t h e integral e q u a t i o n f o r e x p o n e n - tial decay w i t h a d i s t r i b u t i o n o f r e l a x a t i o n t i m e s a s t h e t i m e c o n s t a n t s w i t h i n t h e integral. I n a l a t e r p a p e r 121, t h e s a m e m e t h o d was a p p l i e d t o t h e D e b y e f u n c t i o n a s t h e kernel. T h i s allowed a n a l y s i s o f i n t e r n a l f r i c t i o n ( o r d i e l e c t r i c r e l a x a t i o n ) p e a k s when m e a s u r e d a s a f u n c t i o n o f frequency. I n t h e p r e s e n t p a p e r t h e s e m e t h o d s a r e e x t e n d e d t o t h e m e a s u r e m e n t o f i n t e r n a l f r i c t i o n v s t e m p e r a t u r e peaks. I1

-

Method T h i s m e t h o d h a s s o m e m i n i m a l r e q u i r e m e n t s of t h e d a t a . F i r s t , i t i s necessary t h a t t h e i n t e r n a l f r i c t i o n v s t e m p e r a t u r e d a t a h a v e had b a c k g r o u n d s u b t r a c t e d . I t i s i m p o r t a n t t h a t t h i s s u b t r a c t i o n be d o n e with t h e highest possible accuracy because t h e DSA m e t h o d t e n d s t o f i n d c o n t r i b u t i o n s t o t h e relaxation t i m e s p e c t r u m f o r a n y i n t e r n a l f r i c t i o n which m a y be p r e s e n t o v e r a n d a b o v e t h a t for a given peak. S e c o n d , i t i s necessary t h a t t h e r e b e roughly 2 0 o r m o r e d a t a points; a s h a s previously been s h o w n /1,2/, t h e r e s o l u t i o n c a p a b i l i t i e s o f t h e m e t h o d increase w i t h b o t h n u m b e r a n d accuracy of t h e d a t a points.

T o discuss t h e m e t h o d we will u s e t h e n o t a t i o n a n d d e v e l o p m e n t o f Nowick a n d Berry / 3 / . We c o n s i d e r d y n a m i c e x p e r i m e n t s i n which s t r e s s i s a p p l i e d periodically a t frequency o s o t h a t for a n a n e l a s t i c s o l i d t h e r e will be a p h a s e lag o f t h e s t r a i n b e h i n d t h e stress. T h e a n g l e o f t h i s phase lag i s cp,

t h e loss angle, a n d t h e c o m p l i a n c e i s d e s c r i b e d by t w o d y n a m i c r e s p o n s e f u n c t i o n s , a real p a r t , J , ,

(210-32

JOURNAL

DE

PHYSIQUE

and an imaginary part, J2. The internal friction i s given by tan ( P = J ~ / J I

.

For a standard anelastic solid /3/ responding t o periodic stress, these two dynamic response functions are given by

where GJ=J,-J., A=6J/J, is the strength of t h e anelastic relaxation and J, and J. are the relaxed a n d the unrelaxed compliances, respectively.

Because the internal friction is measured as a function of temperature, certain corrections for temperature-dependent parameters are required. Nowick and Berry again provide an excellent discussion of these temperature corrections 131. First, since we want our approximation o f the relaxation time spectrum a t some constant temperature, Tdi*,, we must correct w a n d A t o this temperature. For implementation of these two corrections, the reader is referred t o Weller, et a1 /4/.

Correction of A t o T d i r t involves a Curie-Weiss temperature dependence. Thus, i t is required t h a t a value be chosen for the critical temperature, T,. Since the value may not be known, i t is important t o d o the analysis using upper and lower limiting estimates of this parameter.

Analysis involves making two approximations both of which are valid for the usual experimental condition t h a t tan cp<<l. First, we consider J l

-

J. so t h a t tan cp*J2/J.. Second, we consider t h a t the T

which i s measured (the relaxation time a t constant stress) is equal t o the average T (the one i n the above equations) /3/. We now proceed to obtain our approximation of N ( l n ~ ) , the spectrum of relaxation times a t Tdlrt. We first speci'fy the upper and lower T limits for the spectrum. For the first attempt a wide range i s chosen; with later tries this is adjusted keeping the range sufficient to show the tail regions of the spectrum. We next d i v i d e the range of 1og10.r in t o n bins of equal width a n d designate the relaxation time of the ith bin as T , , th e midpoint value (10g~o.r scale) of the bin. The number of bins chosen depends upon the resolution desired. I t i s typically 10<n<100, h u t with the constraint n<=m, where m is the number of data points.

The crux of the DSA method is to make a sum approximation for the integral (see Ref. 2) which contains the function N(ln T). This gives

where A, is the relaxation strength i n the ith bin. Eq. (3) i s t h e same as given by Nowick and Berry 131 for multiple relaxations with discrete relaxation times. Thus, using t h i s equation is the same a s considering the relaxation t o be made up of n different discrete relaxations all of which are equally spaced in log T.

The .ri in Eq. 3 will exhibit a temperature dependence according to the Arrhenius relation, T=ro exp(Q/kT), where 70 i s the pre-exponential factor, Q i s the activation energy a n d k i s Boltzmann's constant. For a given measurement of the internal friction a t some temperature other t h a n Tdlot, the .ri need t o be corrected t o their values a t Tdist. This i s done using either of two models for the distribution of relaxation times as discussed by Nowick a n d Berry 131. The first model considers t h a t the distribution is only due to a distribution of activation energies and that T O i s constant. The second model is just the opposite of the first, i.e., there i s only distribution of To's and Q i s a constant. We will use the first of these models in the discussion which follows, although the second has been used a n d found t o provide good spectral estimates.

After choosing n and the 7 limits for the spectrum, we continue as follows: (i) Choose Tdirt; typically this will be the temperature of the peak. (ii) I n p u t the experimental value for w. (iii) Provide a n estimate of T o . (iv) Estimate the total relaxation strength, A. Typically, this i s estimated a s twice the peak height, Q;:,

.

T o start the iteration process we set the Ai, the relaxation strength i n the ith bin, all t o the same value, Ai=A/n. The nonlinear regression least-squares method is now ready t o be used. I t will iterate adjusting the Ai t o minimize the least-square differences between the calculated and the experimental points. We terminate the calculation by either choosing a desired number of iterations or by a statistical evaluation of convergence.

111.

-

VALIDATION

c o m b i n a t i o n of several l o g n o r m a l ( G a u s s i a n in log,,r) d i s t r i b u t i o n s .

Fig. 1 shows a n e x a m p l e o f s u c h a c o m p u t e r - g e n e r a t e d i n t e r n a l f r i c t i o n peak p l o t t e d v s reciprocal t e m p e r a t u r e . T h e i n t e r n a l f r i c t i o n i s n o r m a l i z e d a s t a n q / A . T h e d a t a a t t h e v a r i o u s t e m p e r a t u r e s h a v e been c a l c u l a t e d f r o m t h e A r r h e n i u s r e l a t i o n with a n a c t i v a t i o n energy, Q=1.4 eV a n d pre- e x p o n e n t i a l , ~ ~ = 1 0 - ' ~ S. T h e 5 0 d a t a p o i n t s h a v e been g e n e r a t e d for w=1.0 s f r o m a single l o g n o r m a l

d i s t r i b u t i o n c e n t e r e d a t r,=1.0 s a n d w i t h w i d t h P=1.0. R a n d o m f r a c t i o n a l e r r o r w i t h s t a n d a r d d e v i a t i o n o=0.01 h a s been a d d e d t o t h e d a t a .

Fig. 2 shows t h e histogram f o r t h e s t a r t o f t h e n o n l i n e a r regression a n a l y s i s o f t h e d a t a i n Fig. 1. Each of t h e 40 b i n s h a s e q u a l s p e c t r a l a m p l i t u d e such t h a t t h e a r e a o f t h e h i s t o g r a m i s u n i t y ( o n a In r scale). T h e s p e c t r a l l i m i t s f o r t h e c a l c u l a t i o n h a v e been c h o s e n t o c o v e r t h r e e d e c a d e s with t h e c e n t e r a t roughly r = I / o . Fig. 3 shows o n t h e s a m e a x e s t h e h i s t o g r a m a p p r o x i m a t i o n of t h e i n p u t s p e c t r u m for Tdlst=500 K a f t e r 500 i t e r a t i o n s using DSA a n d a s s u m i n g t o be 10-l4 S. At t h e t o p of t h e figure

the scale i s f o r t h e e q u i v a l e n t a c t i v a t i o n energy. I t m a y be n o t e d t h a t t h e histogram a p p r o x i m a t i o n agrees well w i t h t h e original i n p u t d i s t r i b u t i o n which i s s h o w n a s a solid c u r v e i n t h e figure. A lognormal f i t t o t h e histogram gives r,=1.02 s a n d P=1.03 i n good agreement with t h e s e p a r a m e t e r s for t h e i n p u t s p e c t r u m .

V a l i d a t i o n was a l s o d o n e f o r i n p u t s p e c t r a w i t h a single l o g n o r m a l d i s t r i b u t i o n , b u t for values of P ranging f r o m 0.25 t o 3.0 a n d r a n d o m f r a c t i o n a l e r r o r w i t h o=0.01. As with t h e result i n Fig. 3, t h e DSA m e t h o d was f o u n d t o p r o v i d e a good a p p r o x i m a t i o n o f t h e i n p u t s p e c t r a o v e r t h i s range o f

p.

T h e capability o f t h e m e t h o d t o give good s p e c t r a l a p p r o x i m a t i o n s for s y m m e t r i c a l d i s t r i b u t i o n s with w i d t h s ranging f r o m t h a t o f a nearly d i s c r e t e r e l a x a t i o n (P=0.25) t o d i s t r i b u t i o n s c o v e r i n g u p t o f i v e decades i n r i s encouraging.

T o e x a m i n e t h e recovery o f a s y m m e t r i c a l spectra, d a t a were g e n e r a t e d from m u l t i p l e , b u t o v e r l a p - ping, l o g n o r m a l s p e c t r a o f varying w i d t h s . T h e s e s p e c t r a g a v e o v e r l a p p i n g i n t e r n a l f r i c t i o n p e a k s which would be expected t o be v e r y difficult t o d e c o n v o l u t e . An e x a m p l e o f s u c h a c o m p l e x peak i s shown in Fig. 4 a s a b r o a d i n t e r n a l f r i c t i o n peak m a d e u p a s t h e s u m o f f o u r s e p a r a t e peaks s h o w n a s d a s h e d curves. Again, 5 0 d a t a p o i n t s h a v e been used f o r o=1.0. F r a c t i o n a l e r r o r with s t a n d a r d d e v i a t i o n a=0.01 h a s b e e n a d d e d t o s i m u l a t e e x p e r i m e n t . T h e f o u r s e p a r a t e peaks a r e e a c h g e n e r a t e d f r o m a l o g n o r m a l d i s t r i b u t i o n . T h e s e v a r y i n f r a c t i o n a l s t r e n g t h a n d width a n d a r e c e n t e r e d roughly a factor of f o u r a p a r t a s s h o w n i n Fig. 5. ' 0 5 ,

,

l : STARTING HISTOGRAY

-

G- - ; n~108111S ; r,.003t - 3 - ; t 2 0 5 -

~

3

A - : a AREA. 112 303 10" 100 10' R E L A X A T I O N T I M E . r ( s )

g

O P - 4 a W \ LL B LL Z 0 . 3 $ F -,... 1 . 0 ~ ~ : I 3

,

1.4 1.5 l 6 Fig. 1 - I n t e r n a l f r i c t i o n v s t e m p e r a t u r e d a t a

generated f r o m a l o g n o r m a l d i s t r i b u t i o n t o check

;

lWUT SPECIRIM.N<," 71 W P " T n l S T O G R A M . ~ - . l

j SINGLE LOGNORMAL

1

4081NS

t h e capability o f t h e DSA m e t h o d t o recover t h e

,

-i : B.IO r.:0031 INTERNAL FRICTION PEAK INPUT LOGNORMAL .

C10-34

JOURNAL

DE

PHYSIQUE

The results o f the calculation using t h e DSA method o n t h e d a t a i n Fig. 4 a r e also shown in Fig. 5. Here they may be compared with t h e i n p u t spectrum (solid curve) which defines t h e internal friction peak. T h e historgram a p p r o x i m a t i o n of t h e i n p u t spectrum is d o n e a t Td,,,=500 K f o r 40 bins of T. The only assumption used for t h e calculation was t h a t .ro=10-14 S. I t may be seen t h a t t h e DSA approximation shows very satisfactory agreement with t h e spectrum which was used t o generate t h e data. I t i s particularly satisfying t h a t the analysis i s able t o reproduce t h e four peaks which make up t h e i n p u t spectrum. Similar validations using i n p u t spectra of two a n d three peaks also showed t h i s good resolving power. Many different k i n d s of spectra have been validated using t h e m e t h o d described here. I m p o r t a n t l y , these v a l i d a t i o n s never gave false approximations t o t h e i n p u t spec- t r u m , i.e., i n n o case d i d t h e results of t h e DSA show extra peaks o r miss peaks which were o f reasonable magnitude ( - 10% of t h e t o t a l spectrum).

The validation described i n Fig. 5 d e m o n s t r a t e s t h a t t h e amplitude, shape, width a n d location o f t h e peaks a r e all reconstructed with acceptable accuracy. Only a m i n o r difficulty i s evident; t h i s i s f o r the location of t h e t h i r d peak. The histogram shows t h i s peak t o be a t a relaxation t i m e which i s roughly 40% too high. T h i s tendency was observed i n o t h e r analyses, particularly when t h e i n p u t peaks were close together. Typically, the m e t h o d resolves i n p u t peaks which a r e a factor of two o r more apart, b u t the peaks t e n d t o be displaced away from each o t h e r a s shown i n Fig. 5 .

Fig. 4 - I n t e r n a l friction d a t a generated from 4- Fig. 5 - I n p u t relaxation t i m e spectrum a n d out- peak i n p u t spectrum. put histogram a p p r o x i m a t i o n after DSA of t h e d a t a in Fig. 4. T h e a's a r e the fractional contribu- tion of each lognormal spectrum.

V

-

CONCLUSIONS

(a)

A new method i s described for analysis of an i n t e r n a l friction vs temperature peak t o o b t a i n a n approximation o f t h e spectrum of relaxation t i m e s responsible f o r t h e peak.

(b) I t i s shown t h a t t h i s m e t h o d provides a p p r o x i m a t i o n s to known i n p u t spectra which replicate t h e position, amplitude, width a n d shape with good accuracy (typically

-

10%).

(c) Validation of t h i s method f o r a wide variety of spectra shows t h a t i t does n o t yield approxima- tions which have false spectral peaks.

REFERENCES

/ l / Cost, J. R., J. Appl. Phys., 5 4 ( 1 9 8 3 ) 2137.

/2/ Cost, J. R., Nontraditional Methods in Diffusion, ed. b y Murch, G., TMS-AIME, New York (1984).

/3/ Nowick, A. S. a n d Berry, B. S., Anelastic Relaxations in CrystallineSolids, Academic Press, New York (1972).

/4/ Weller, M., Zhang, J. X., Li, G. Y., Ke, T. S. a n d Diehl, J., Acta Met., 29 (1981) 1055.