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ELSEVIER S0032-3861 (96)00274-1

PolymerVol. 37 No. 18, pp. 4107 4110, 1996Copyright 1996 Elsevier Science LtdPrinted in Great Britain. All rights reserved0032-3861/96/$15.00 + 0.00

C o m p a r is o n o f t h e H a v r i l ia k - N e g a m i a nds t r e t c h e d e x p o n e n t i a l f u n c t i o n sS . H a v r i l i a k , J r *Rohm and H aas Research. Br is to l Research Park. PO Box 219, B r is to l PA 19007 . USAa n d S . J . H a v r i l i a kHavr il iak Software D evelopment Co., Hu nting don Val ley, PA 190 06, USA(Received 23 December 1994; revised 16 Janua ry 1996)

A lv a re z , A lg e r i a a n d C o lm e n e ro c o m p a re d th e s t r e t c h e d e x p o n e n t i a l (K WW) a n d H a v r i l i a k -N e g a m i(H -N ) fu n c tio n s b y c o m p a r in g th e i r d i s tr ib u t io n s o f r e la x a t io n t im e s . In th e c as e o f t he K W W fu n c tio n , t h ed is t r ibu t ion o f re laxa t ion t imes was ca lcu la ted numerica l ly . The numerica l re su l ts were then represen ted int e rm s o f t h e c lo se d fo rm d i s t r ib u t io n o f r e l a x a tio n t im e s a v a il a b le fo r t h e H - N fu n c tio n . T h e y fo u n d th a t aspec ific KW W k pa ram ete r corresponds to a spec ific H - N a , /3 pa ir . In th is pape r the time-dependen td ie lec tr ic cons tan t , ca lcu la ted f rom the s t re tched exponen t ia l fo r va r ious ks and over a t ime rangesufficiently long to define the t = 0 or cc limits , was tra nsfo rm ed to the com plex dielectric con stan t using theex tended Schwarz l m e thod . T he complex d ielectric con s tan t was f i t ted d irect ly to the H - N func t ion . Theresu l ts o f th is p rocedure a re com pare d with those o f Alvarez e t a l . The two d iffe ren t me thods g ive the sameresults and the consequences o f this agreem ent are discussed. C opy righ t 1996 Elsevier Science Ltd.(Keywords: t ime to frequency transforms; s tretched exponential; H-N funct ion)

I N T R O D U C T I O NA n e x p r e s s i o n o f t e n u s e d t o r e p r e s e n t t h e t i m e d e c a y ( o rr i s e ) o f p h y s i c a l p r o c e s s i s t h e s o - c a l l e d s t r e t c h e de x p o n e n t i a l o r K W W 1 3 f u n c t i o n , g i v e n i n t h e n o r m a l -i z e d f o r m f o r t h e d i e l e c t r i c c a s e b y e q u a t i o n ( 1 ) :

e . ( t ) = ( e ( , )_ - - e o ~ = 1 _ e x p ( _ ( t / r o ) k ) ( 1)\ ~o - ~ /I n t h i s e x p r e s s i o n c , ( t ) i s t h e t i m e - d e p e n d e n t d i e l e c t r i cc o n s t a n t a n d k i s t h e K W W p a r a m e t e r , T o i s t h e r e la x -a t i o n t i m e , w h i l e e0 a n d E o~ r e p r e s e n t t h e e q u i l i b r i u m a n di n s t a n t a n e o u s d i e l e c t r i c c o n s t a n t s , r e s p e c t i v e l y .A n o t h e r f u n c t i o n u s e d t o r e p r e s e n t d i e l e c t r i c r e l a x a -t i o n d a t a i n t h e f r e q u e n c y d o m a i n i s t h e H - N f u n c t i o n 4'5d e f i n e d b y

e~,(w) = (g* (w ) -_~oc, .'~ = {1 + ( i w r o ) ~ } = J (2 )\ S o - e ~ /I n t h i s e x p r e s s i o n , e * ( w ) i s t h e n o r m a l i z e d c o m p l e xd i e l e c t r i c c o n s t a n t ( o r c o m p l e x r e l a t i v e p e r m i t t i v i t y )m e a s u r e d a t r a d i a n f r e q u e n c y w = 2 7 r f , f i s i n H z a n d i i sv / S T . T h e p a r a m e t e r s a a n d / 3 r e p r es e n t t h e w i d t h a n ds k e w n e s s o f t h e d i e l e c t ri c l o s s ( e n ( w ) ) w h e n v i e w e d i n al o g ( w ) p l o t . T h e s e p a r a m e t e r s a l s o d e s c r i b 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 w h i c h c a n b e o b t a i n e d i n a c lo s e df o r m f r o m e q u a t i o n ( 2 ) . T h e r e l a x a t i o n t i m e i s r e p r e -s e n te d b y 7 -

T h e d i s t r i b u t i o n o f r e l ax a t i o n t i m e s c a n b e o b t a i n e d* To whom correspondence should be addressed

f r o m e q u a t i o n ( 2) a n d t h e c o r r e s p o n d i n g i n t e g r ale q u a t i o n r e l a t i n g it t o a d i s t r i b u t i o n o f r e l a x a t i o nt i m e s . T h e d i s t r i b u t i o n f u n c t i o n , F ( y ) , i s g i v e n b y

F ( y ) = ( 1 ) y ~ ( s i n / 3 0 ) ( y Z ' ~ + 2 y ~ c o s T r a + l ) - / 3 /2( 3 )

I n t h i s e x p r e s s i o n , y = 7 - / r0 a n d0 = a r c t a n ( 4 )y~ + cos 7 ra J

R E S U L T S A N D D I S C U S S I O NA l v a r e z e t a l . 6 c a l c u l a te d n u m e r i c a l l y t he d i s t r i b u t i o n o fr e l a x a t i o n t i m e s f r o m e q u a t i o n ( 1) f o r g i v e n va l u e s o f ki n t h e r a n g e o f 0 .1 t o 1 .0 a n d t h e n f i t t e d t h e n u m e r i c a lr e s u l t s t o e q u a t i o n ( 3 ) u s i n g a n o n - l i n e a r r e g r e s s i o nr o u t i n e . P l o t s o f th e i r r e s u l t s a r e g i v e n in F i g u r e s 1 , 2 a n d3 . N o s t a t i s t i c a l i n f o r m a t i o n w a s s u p p l i e d b y t h e s ea u t h o r s t o d e s c r ib e p a r a m e t e r c o n f i d e n c e i n te r v a l s o r t hem o d e l s t a n d a r d e r r o r o f e s t i m a t e .K o i z u m i a n d K i t a 7 t r a n s f o r m e d e , ( t ) t o e ~ ( w ),c a l c u l at e d o v e r a l o g ( t im e ) r a n g e o f - 3 t o + 3 f r o m t h es t r e t ch e d e x p o n e n t i a l K W W f u n c t i o n f o r g iv e n v a l u e s o ft h e k p a r a m e t e r f r o m 0 .2 9 t o 1 .0 i n c r e m e n t e d b y 0 . 0 1.D i s h o n e t a l . 8 a l s o t r e a t e d t h e s a m e p r o b l e m b u t d i d n o tm e n t i o n t h e r e su l t s o f K o i z u m i a n d K i t a . T h e i rp a r a m e t e r r a n g e w a s f r o m 0 . 02 t o 1 a n d t h e y u s e d t h es a m e t i m e r a n g e . T h e i r p r o c e d u r e w a s s i m i l a r a n d t h er e s u l t s w e r e r e p o r t e d t o s i x p l a c e s .

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