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Paper VII(ii) Some comments on the “glassy state” of lubricants in an EHD contact
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Paper Vll(ii)
Some comments on the "glassy state" of lubricants in an EHD contact P. Bezot, C. Hesse-Bezot and G.Rouill6
G e n e r a l i t i e s on t h e " g l a s s t r a n s i t i o n " i n molecular l i q u i d s a r e f i r s t b r i e f l y r e c a l l e d and some remarks concerning t h e v i s c o e l a s t i c t r a n s i t i o n s t u d i e d by means o f u l t r a s o n i c and hypersonic techniques a r e made. A L i g h t S c a t t e r i n g study o f t h e 5P4E i n t h e v i c i n i t y o f t h e " g l a s s t r a n s i t i o n " i n s t a t i c c o n d i t i o n s and immediately a f t e r a pressure s t e p i s a l s o presented. L a s t l y , r e s u l t s o f numerical s i m u l a t i o n s obtained i n a f l u i d element a f t e r a pressure s t e p near the "glass t r a n s i t i o n " pressure o r i n an E.H.D. c o n t a c t a r e given. 1 INTRODUCTION The question o f the g l a s s t r a n s i t i o n i n an E.H.D. experiment has been l a r g e l y discussed i n the l a s t two decades. The main elements o f t h i s problem have been c l e a r l y i n t r o d u c e d f o r i n s t a n ce i n the very good works o f Winer e t a l ( 1 ) and Johnson e t a1 ( 2 ) . For example, the i n f l u e n c e o f pressure and cooling speed on t h e " g l a s s t r a n s i t i o n " (G.T.) temperature has l a r g e l y been commented by the f i r s t authors. However the answer t o the q u e s t i o n o f t h e glass t r a n s i t i o n i n an E.H.D. c o n t a c t remains i n our o p i n i o n more q u a l i t a t i v e than q u a n t i t a t i v e . The reasons f o r t h i s a r e m a n i f o l d : one o f t h e most important i s t h a t t h e g l a s s t r a n s i t i o n i s generally measured by means o f experiments whose time scale ( t o ) i s t y p i c a l l y o f t h e order o f t e n minutes. kt what happens i n an E.H.D. c o n t a c t i n which the t r a n s i t time i s many o r d e r s shorter ? I n t h i s work, we s h a l l t r y t o c o n t r i b u t e t o the s o l u t i o n o f t h i s problem step by s t e p : A t f i r s t , i n the next p a r t ( 2 ) , some general considerations on t h e G.T. w i l l be reminded together w i t h general comments u s e f u l t o understand the f o l l o w i n g . I n the t h i r d p a r t ( 3 1 , experimental s t u d i e s of a polyphenylether ( t h e 5P4E) w i l l be p r e sented, from 0.1 MPa t o 0.35 GPa a t 17OC by means o f L i g h t S c a t t e r i n g technique, i n v a r i o u s dynamic s i t u a t i o n s . F i n a l l y , the f o u r t h p a r t ( 4 ) w i l l be devoted t o numerical s i m u l a t i o n s i n r e l a t i o n t o some of the previous experimental r e s u l t s and i n t h e case of E.H.D. experiments.
2 GENERAL CONSIDERATIONS ON THE GLASS TRANSITION Before discussing t h e problem o f t h e g l a s s t r a n s i t i o n i n an E.H.D. c o n t a c t , i t i s worth while r e c a l l i n g some g e n e r a l i t i e s on t h i s subj e c t ( 3 ) . I t i s w e l l known t h a t , by c o o l i n g a molecular l i q u i d a t c o n s t a n t pressure, s u f f i c i e n t l y below i t s m e l t i n g p o i n t , we can o b t a i n the formation o f a glass. I t i s a l s o known (perhaps l e s s ) t h a t , a t a g i v e n temperature, t h e
compression o f a l i q u i d above i t s m e l t i n g pressure can a l s o l e a d t o the f o r m a t i o n o f a glass. I n a l l these s i t u a t i o n s , t h e common f e a t u r e i s t h a t t h e r e e x i s t s a t l e a s t one s t r u c t u r a l r e l a x a t i o n process whose c h a r a c t e r i s t i c time increases d r a m a t i c a l l y as t h e pressure i s increased (or t h e temperature decreased). As t h i s c h a r a c t e r i s t i c time becomes much l a r g e r than t h e observat i o n time t ,then t h e l i q u i d i s "frozen" i n a non-equilibr%m s t a t e . These a r e t h e f i r s t qual i t a t i v e descriptions of a glass transition. To be a l i t t l e more q u a n t i t a t i v e , l e t us now g i v e some complementary d e f i n i t i o n s . A l i q u i d or a g l a s s i s d e f i n e d by a s e t o f thermodynamic f o r c e s (temperature T, pressure P, ...I t o g e t h e r w i t h , a t l e a s t , one order parameter which c h a r a c t e r i z e s t h e s t a t e of t h e l o c a l s t r u c t u r e i n a f l u i d . I f now, s t a r t i n g from the l i q u i d s t a t e above i t s m e l t i n g p o i n t , we i n c r e a se t h e pressure, keeping t h e o t h e r thermodynamic f o r c e s constant, then any e x t e n s i v e p r o p e r t y (volume V, .) o r r e l a t e d p h y s i c a l q u a n t i t y , w i l l vary along a curve corresponding t o the s t a t e o f thermodynamic e q u i l i b r i u m o f t h e l i q u i d . I f now t h e pressure i s increased above the m e l t i n g pressure (P,) and i f c r y s t a l l i z a t i o n can be prevented , then t h e l i q u i d i s u n s t a b l e ( r e l a t i v e t o t h e c r y s t a l l i z a t i o n ) but may s t i l l be i n thermodynamic e q u i l i b r i u m , i f t h e pressure i s increased s u f f i c i e n t l y s l o w l y . The range o f pressure above P over which t h e l i q u i d remains i n thermodynami2' e q u i l i b r i u m , depends on the r a t e o f compression q = dP/dt. F o r a g i v e n r a t e q, t h e l i q u i d w i l l s t a y i n a metastable thermodynamic e q u i l i b r i u m up t o a v a l u e of t h e p r e s sure c a l l e d t h e g l a s s pressure (PG, pressure corresponding t o the " g l a s s t r a n s i t i o n " ) a t which t h e l i q u i d departs from i t s e q u i l i b r i u m behavior More p r e c i s e l y , t h i s departure from e q u i l i b r i u m which i s e s s e n t i a l l y dynamic, begins t o occur when the mean r e l a x a t i o n time T, f o r t h e s t r u c t u r a l rearrangement i s l a r g e r than the pressure increment time. Consequently t h e f a s t e r t h e pressure i s a p p l i e d , t h e lower the PG value w i l l be. O f course, t h e o b s e r v a t i o n time
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( t i m e necessary t o measure an e x t e n s i v e p r o p e r t y o f t h e l i q u i d ) must a l s o be s h o r t e r than T L a s t l y l e t us make some remarks about t # e " v i s c o e l a s t i c t r a n s i t i o n " , o f t e n encountered i n
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t h e l i t e r a t u r e ( 4 ) , and which must n o t be confused w i t h a g l a s s t r a n s i t i o n i n a classical experiment : C o n s i d e r , f o r e x a m p l e , t h e case of a h y p e r s o n i c wave ( s t u d i e d by B r i l l o u i n S c a t t e r i n g ) o f frequency w t h a t propagates i n a viscous liquid, c h a - a c t e r i z e d by a mean s t r u c t u r a l r e l a x a t i o n time -rM. I f , b y c o m p r e s s i n g s l o w l y t h e s a m p l e a t constant temperature, t h e T values increase yT > > I , a "transition f r o m wT M < < I t o r e g i o n " is o b s e r v e d i n which t h e l o n g i t u d i n a l m o d u l u s , r e l a t e d t o t h e s o u n d v e l o c i t y , shorvs a marked v a r i a t i o n b e t w e e n r e s p e c t i v e l y a low a n d a h i g h f r e q u e n c y l i m i t . The p r e s s u r e r e g i o n i n which t h i s " v i s c o e l a s t i c t r a n s i t i o n " is observed, must n o t b e c o n f u s e d w i t h t h e g l a s s p r e s s u re PG w h i c h is much l a r g e r . However, t h i s " v i s c o e l a s t i c p r e s s u r e " w o u l d c e r t a i n l y correspond t o t h e g l a s s t r a n s i t i o n i n a n e x p e r i m e n t w h e r e a p r e s s u r e i n c r e m e n t would b e a p p l i e d -1 d u r i n g a time w W e h o p e now t h a t t h e p r e v i o u s comments w i l l j u s t i f y o u r f o l l o w i n g a p p r o a c h i n a n E.H.D. contact. A t f i r s t , i t seemed n e c e s s a r y t o m e a s u r e some e x t e n s i v e p r o p e r t i e s ( o r p h y s i c a l q u a n t i t i e s r e l a t e d t o t h e m ) of a p a r t i c u l a r l u b r i c a n t ( h e r e t h e 5P4E) a l o n g t h e c u r v e of t h e r m o d y n a m i c e q u i l i b r i u m ( s t a b l e or n o t , d e p e n d i n g on t h e p r e s s u r e value r e l a t i v e t o PM). I n o r d e r t o follow t h i s c u r v e , on t h e w i d e s t a c c e s s i b l e rang': of p r e s s u r e , t h e l a t t e r was i n c r e a s e d v e r y s l o w l y ( w a i t i n g time of two h o u r s a f t e r e a c h 1 0 MPa p r e s s u r e i n c r e m e n t ) . T h e n , w i t h t h e a i m of a p p r o a c h i n g t h e e x p e r i m e n t a l s i t u a t i o n e n c o u n t e r e d i n a n E.H.D. c o n t a c t , we i m a g i n e d a n e x p e r i m e n t a l a r r a n g e m e n t t o f o l l o w t h e b e h a v i o r of t h e i n f i n i t e f r e q u e n c y l o n g i t u d i n a l modulus immediately after a p r e s s u r e i n c r e m e n t . The a d v a i t a g e of t h i s p r o c e d u r e i s t o f o l l o w s t e p by s t e p t h e f l u i d b e h a v i o r i n t h e whole p r e s s u r e range i n c l u d i n g P G' Finally, since a t presently, d i r e c t experim e n t a l s t u d i e s of t h e f l u i d d y n a m i c i n a n E.H.D. c o n t a c t are n o t f e a s i b l e , we a r e o n l y l e f t w i t h t h e numerical s i m u l a t i o n for its behavior. Using a model d u e t o M o n t r o s e e t a l . ( 5 ) , we f i r s t compared o u r e x p e r i m e n t a l r e s u l t s o b t a i n e d a f t e r a p r e s s u r e s t e p with numerical simulations i n t h e same e x p e r i m e n t a l c o n d i t i o n s . Then u s i n g t h e same m o d e l , we s t u d i e d t h e d y n a m i c a l b e h a v i o r of a f l u i d a f t e r a s u c c e s s i o n of s m a l l p r e s s u r e jumps s u c h a s i n a n E . H . D . c o n t a c t . W e hope t h a t by t h i s a p p r o a c h , t h e q u e s t i o n of t h e g l a s s t r a n s i t i o n i n a n E.H.D. c o n t a c t w i l l b e made clearer.
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3 EXPERIMENTAL APPROACH USING L I G H l SCATTERING TECHNIQUE I n t h e p r e s e n t w o r k , m e a s u r e m e n t s were p r i n c i p a l l y o b t a i n e d from t h e d y n a m i c L i g h t S c a t t e r i n g technique : When a l i q u i d s a m p l e i s i r r a d i a t e d by a n e l e c t r o m a g n e t i c wave ( f o r e x a m p l e a l a s e r b e a m ) , its molecules s c a t t e r l i g h t i n a l l d i r e c t i o n s . T h i s s c a t t e r e d l i g h t c o n t a i n s i n f o r m a t i o n on v a r i o u s m o l e c u l a r movements and c o n s e q u e n t l y o n some p h y s i c a l q u a n t i t i e s c h a r a c t e r i z i n g t h e l i q u i d . The a n a l y s i s s y s t e m t o b e u s e d d e p e n d s o n t h e c h a r a c t e r i s t i c time of t h e movement t o b e studied.
3 . 1 . Experimental set-up The p o l y p h e n y e t h e r (5P4E) s a m p l e s were s u p p l i e d by Monsanto Co a n d d i r e c t l y u s e d i n a s m a l l e o p t i c a l cell which w i l l be d e s c r i b e d later. W used a c o h e r e n t Innova 903 l a s e r a s e x c i t i n g s o u r c e e i t h e r i n a s i n g l e mode or multimode, depending on t h e a n a l y s i s t e c h n i q u e . To s t u d y t h e p r o p a g a t i v e waves, i . e . t o m e a s u r e t h e f r e q u e n c y of t h e p r o p a g a t i v e longi t u d i n a l and s h e a r w a v e s , t h e s c a t t e r e d l i g h t was a n a l y s e d w i t h a h i g h r e s o l u t i o n F a b r y - P e r o t s p e c t r o m e t e r . D e t a i l s of t h i s t e c h n i q u e have r e c e n t l y b e e n d e s c r i b e d e l s e w h e r e ( 6 ) . The nonp r o p a g a t i v e modes, i . e . t h e s t r u c t u r a l r e l a x a t i o n p r o c e s s e s , were s t u d i e d by t h e photon c o r r e l a t i o n technique using a d i g i t a l multitau m u l t i b i t L a n g l e y - F o r d c o r r e l a t o r . More d e t a i l s on t h i s e x p e r i m e n t a l arrangement t o g e t h e r with t h e a n a l y s i s p r o c e d u r e c a n b e f o u n d i n ( 7 ) . Note t h a t t h e a n a l y t i c a l c o r r e l a t i o n f u n c t i o n was chos e n t o b e t h e William-W tts f u n c t i o n : 4 ( t ) = exp( - ( t / T ) ) i n w h i c h 0 < B < 1 i s r e l a t e d t o t h e w i d t h of t h e d i s t r i b u t i o n and T c o r r e s p o n d s n e a r l y t o t h e time a t t h e maximum a m p l i t u d e of t h e d i s t r i b ution. Let u s now d e s c r i b e t h e h i g h p r e s s u r e o p t i c a l s y s t e m . Its g e n e r a l a r r a n g e m e n t h a s a l r e a d y b e e n d e s c r i b e d e l s e w h e r e ( 6 ) . However t h e p r e s s u r e c e l l i s now g r e a t l y m o d i f i e d . I n o u r p r e v i o u s works, t h e e n t i r e high pressure s y s t e m was f i l l e d w i t h t h e s t u d i e d l i q u i d used a l s o a s t h e p r e s s u r e t r a n s m i t t i n g medium. Conseq u e n t l y m e a s u r e m e n t s a s a f u n c t i o n of p r e s s u r e c o u l d n o t b e c a r r i e d o u t a b o v e v a l u e s f o r which t h e c o r r e s p o n d i n g v i s c o s i t y was too h i g h . Now t h e s a m p l e i s i s o l a t e d from t h e p r e s s u r e t r a n s m i t t i n g l i q u i d by a s m a l l g l a s s c e l l w h i c h i s p u t i n s i d e t h e h i g h p r e s s u r e c e l l and embedded i n t h e p r e s s u r e t r a n s m i t t i n g medium. Its volume i m m e d i a t e l y c h a n g e s , a f t e r a n e x t e r n a l p r e s s u r e v a r i a t i o n , so a s t o a d j u s t t h e pressure i n both liquids. C o n s e q u e n t l y , a s r e g a r d s t h e p r e s s u r e , it i s now p o s s i b l e t o s t u d y t h e l i q u i d b e h a v i o r following l a r g e compression (or depression) rat e s a n d a t p r e s s u r e s much l a r g e r t h a n PG. L a s t l y , b e f o r e presenting our experimental r e s u l t s , l e t u s d e s c r i b e some i m p r o v e m e n t s made i n o u r a n a l y s i s system. These improvements a r e p a r t i c u l a r i t y c r u c i a l i n t h e case of d y n a m i c a l studies. Let u s c o n s i d e r , i n f i g . 1 , a t y p i c a l s p e c t r u m o b t a i n e d a t t = 17OC a n d P = 0.22 GPa b y means of a F a b r y - P e r o t I n t e r f e r o m e t e r . One of t h e most i m p o r t a n t q u a n t i t i e s t o b e m e a s u r e d is t h e f r e q u e n c y s h i f t of t h e l o n g i t u d i n a l ( l a r g e s a t e l l i t e s A) or s h e a r ( s m a l l s a t e l l i t e s 8) waves. A s t h e time n e c e s s a r y t o r e c o r d t h i s t y p i c a l s p e c t r u m is a t least twenty minutes, d y n a m i c a l s t u d i e s s h o r t e r t h a n n e a r l y two h o u r s are d i f f i c u l t w i t h t h i s p r o c e d u r e . However when t h e large satellites s h i f t with t h e pressure, t h e i n t e n s i t y of t h e minimum b e t w e e n t h e two l a r g e s a t e l l i t e s (C) u n d e r g o e s a l a r g e v a r i a t i o n . C o n s e q u e n t l y i f we know a t h e o r e t i c a l r e l a t i o n b e t w e e n t h e minimum i n t e n s i t y and t h e f r e q u e n c y s h i f t , or i f we h a v e m e a s u r e d i t e x p e r i m e n t a l l y , it is p o s s i b l e , from a d i r e c t m e a s u r e m e n t of t h e i n t e n s i t y , t o d e d u c e t h e f r e q u e n c y s h i f t . The main a d v a n t a g e of t h i s p r o c e d u r e i s now t h e q u a s i - i n s t a n t a n e o u s r e s p o n se of t h e m e a s u r e m e n t s y s t e m .
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W e now p r e s e n t , i n t h e n e x t two p a r t s ( 3 . 2 and 3 . 3 ) , t h e v a r i o u s e x p e r i m e n t a l r e s u l t s . 3.2. L i q h t s c a t t e r i n g r e s u l t s experimental c o n d i t i o n s
i n quasi-static
The p r e s e n t s t u d y i s d e v o t e d t o t h e m e a s u r e m e n t of p h y s i c a l p a r a m e t e r s a s a f u n c t i o n of t h e p r e s s u r e f o r w a i t i n g times of t h e o r d e r o f two h o u r s a f t e r p r e s s u r e v a r i a t i o n s of n e a r l y 10 MPa These experimental conditions are generally encountered i n o t h e r techniques, such a s the c a l o r i m e t r i c one, for i n s t a n c e ( 8 ) . The e x p e r i m e n t a l v a l u e s of t h e l o n g i t u d i n a l wave f r e q u e n c i e s p l o t t e d a s a f u n c t i o n o f t h e p r e s s u r e are r e p o r t e d i n f i g . 2 . T h r e e domains c a n be d i s t i n g u i s h e d i n w h i c h t h e f r e q u e n c y varies linearly with t h e pressure : t h e first o n e , i n t h e l e f t p a r t of t h e c u r v e , c o r r e s p o n d s t o t h e v i s c o e l a s t i c (V.E.) r e g i o n . However, d u e t o t h e low t e m p e r a t u r e of t h e s a m p l e , e v e n a t P = 0.1 MPa, WT i s s t i l l l a r g e r t h a n o n e . I n t h e h i g h p r e s s u r e p a r t ( 0 . 1 0 GPa) of t h e V.E. r e g i o n , WT i s v e r y l a r g e a n d a n a n g u l a r p o i n t i s o b s e r v e d w h i c h c o r r e s p o n d s t o t h e c h a n g i n g of t h e c u r v e s l o p e . As p r e v i o u s l y o b s e r v e d , t h i s p o i n t must n o t b e e n c o n f u s e d w i t h PG. I n t h e m i d d l e r e g i o n where t h e v a l u e of s l o p e of t h e c u r v e i s lower, t h e l i q u i d i s s t i l l i n t h e r m o dynamic e q u i l i b r i u m b u t o u t s i d e t h e V.E. r e g i o n ( w T >> 1 ) . The e x t e n t of t h i s r e g i o n d e p e n d s o n t h e w a i t i n g time a f t e r e a c h p r e s s u r e i n c r e a s e : t h e l o n g e r t h e w a i t i n g time, t h e l a r g e r i t s l e n g t h . F o r e x a m p l e t h e crosses i n t h e c u r v e were o b t a i n e d f o r w a i t i n g times of a t l e a s t two d a y s ( o n e weak f o r t h e l a s t p o i n t ) w h e r e a s t h e o t h e r e x p e r i m e n t a l v a l u e s were m e a s u r e d two hours a f t e r t h e p r e s s u r e i n c r e a s e . I n t h e s e experimental c o n d i t i o n s , t h e g l a s s t r a n s i t i o n p r e s s u r e i s f o u n d t o b e 0 . 2 3 GPa. T h i s a g r e e s w i t h t h e r e s u l t s o b t a i n e d by Angel u s i n g a calorimetric technique ( 8 ) . F u r t h e r m o r e , o u r e x p e r i m e n t a l r e s u l t s for t h e s h e a r wave f r e q u e n c i e s , r e p o r t e d i n f i g . 3, e x h i b i t a g l a s s t r a n s i t i o n i n t h e same domain of pressure. F i n a l l y , c o n c e r n i n g t h e c e n t r a l component i n f i g . 1, its polarized p a r t is generally composed o f a l a r g e l i n e ( > 1 MHz) r e l a t e d t o t h e purely d i f f u s i v e t h e r m a l f l u c t u a t i o n and a narrow o n e , c a l l e d t h e M o u n t a i n l i n e , w h i c h i s related to the structural relaxation processes. The l a t t e r h a s b e e n a n a l y s e d by means of a c o r r e l a t o r . The mean r e l a x a t i o n times o b t a i n e d by a d j u s t m e n t w i t h t h e William-Watts f u n c t i o n (B 0.5) o n t h e e x p e r i m e n t a l c u r v e s are r e p o r t e d i n f i g . 4. Note t h a t , a t 0 . 2 3 GPa, t h e re a x a t i o n t i m e i s f o u n d t o b e of t h e o r d e r o f 10 s, i n agrement w i t h t h e o b s e r v a t i o n , i n t h i s p r e s s u r e r a n g e , of a g l a s s t r a n s i t i o n ( 9 ) .
.
3
T h i s s u d d e n v a r i a t i o n is m a i n l y d u e t o r a p i d p r o c e s s e s s u c h a s l a t t i c e v i b r a t i o n s . Then f o l l o w s a more g r a d u a l e v o l u t i o n d u e t o t h e structural relaxation. A t h i g h e r a n d lower p r e s s u r e s , c o n t r a r y t o w h a t i s s e e n i n f i g . 5 a n d 6, n o time e v o l u t i o n of t h e i n t e n s i t y , e x c e p t t h a t of t h e i n s t a n t a n e o u s o n e , is o b s e r v e d d u r i n g t h e v a r i o u s p o s s i b l e r e c o r d i n g times. A t low p r e s s u r e , a l l t h e r e l a x a t i o n times are t o o s h o r t t o b e r e s o l v e d and c o n t r i b u t e i n d i s t i n c t l y t o t h e i n s t a n t aneous response. A t t h e o p p o s i t e , for t h e high p r e s s u r e v a l u e s , t h e s t r u c t u r a l r e l a x a t i o n times a r e becoming t o o l a r g e ( s e v e r a l d a y s or more) t o be observed and t h e o n l y r a p i d p r o c e s s e s c o n t r i b u t e t o t h e instantaneous response. From c u r v e s similar t o f i g . 5 a n d 6, we d e d u c e d t h e time e v o l u t i o n of t h e c o m p r e s s i o n a l modulus a t i n f i n i t e frequency. T h i s kind of t e m p o r a l b e h a v i o r w i l l b e f o u n d i n p a r t 4.1 t o a g r e e w i t h t h e r e s u l t s of n u m e r i c a l s i m u l a t i o n s . L a s t l y , t h e v a l u e s of t h e mean s t r u c t u r a l r e l a x a t i o n times o b t a i n e d by t h i s method are p l o t t e d on f i g . 4. They are i n r o u g h a g r e e m e n t w i t h t h e v a l u e s m e a s u r e d by t h e p h o t o n correlat i o n technique. 4 NUMERICAL SIMULATION EXPERIMENTS
To a p p r o a c h more c l o s e l y t h e e x p e r i m e n t a l c o n d i t i o n s e n c o u n t e r e d i n a n E . H . D . c o n t a c t , o n e way would b e , f o r i n s t a n c e , t o a p p l y a time depende n t p r e s s u r e assuming a H e r t z shape and t o m e a s u r e s i m u l t a n e o u s l y t h e f r e q u e n c y s h i f t of the Brillouin lines. I n t h i s p a p e r we s h a l l r a t h e r p r e s e n t t h e r e s u l t s of n u m e r i c a l s i m u l a t i o n s o b t a i n e d f r o m a model r e c e n t l y d e v e l o p e d by M o n t r o s e e t a1 ( 5 ) . I n t h i s m o d e l , i t i s assumed t h a t , a f t e r a p r e s s u r e jump, a r e p r e s e n t a t i v e p r o p e r t y of t h e f l u i d w i l l experience an instantaneous response ( g l a s s l i k e b e h a v i o r ) f o l l o w e d by a r e t a r d e d c h a n g e ( l i q u i d l i k e b e h a v i o r ) . T h i s time d e p e n d e n c e i s e x p r e s s e d i n t h e case of t h e f r e e volume V f ( t ) by t h e f o l l o w i n g e q u a t i o n :
-
The free volume i s d e f i n e d by V f ( t ) = V ( t ) - V c
w h e r e V ( t ) a n d Vc a r e r e s p e c t i v e l y t h e t o t a l volume a t i n s t a n t t a n d t h e c l o s e - p a c k e d volume.
-
~ ( t i) s t h e s t r u c t u r a l r e l a x a t i o n time g i v e n
3.3 L i g h t s c a t t e r i n g r e s u l t s i n d y n a m i c experimental c o n d i t i o n s
by
Using t h e p r o c e d u r e d e s c r i b e d i n 3.1, we m e a s u r ed a f t e r e a c h p r e s s u r e jump t h e time e v o l u t i o n of t h e i n t e n s i t y minimum ( a r r o w C o n f i g . 1 ) between t h e two l o n g i t u d i n a l l i n e s . I n a r a n g e of p r e s s u r e e x t e n d i n g from 0 . 2 0 GPa t o 0 . 2 4 GPa, t h e i n t e n s i t y i s o b s e r v e d t o change w i t h time i n t h e manner shown i n F i g . 5 and 6 A t f i r s t t h e c h a n g e i s r a p i d compared t o t h e time r e s o l u t i o n of t h e m e a s u r i n g a p p a r a t u s .
- y ( t ) i s e q u a l t o Ko(t)/K, ( t ) , t h e r a t i o of t h e bulk moduli a t z e r o and i n f i n i t e frequency.
.
T (t)
=
T(O)
e x p [ vc
(l/Vf(t)-
l/vf(o))1
- K ( t ) c a n b e e Z p r e s s e d by 0 V ( t ) vc/ c r V f ( t ) I n a l l t h e following numerical simulations,
172
k ( o ) v a l u e s were d e d u c e d from o u r L i g h t Scatte-
r i n g m e a s u r e m e n t s ( 3 . 2 ) a n d t h e y v a l u e from M o n t r o s e e t a 1 ( 1 0 ) . The a v a l u e was c h o s e n e q u a l t o t h e s l o p e of t h e T = f(P) curve. 4.1 S t u d y of t h e f l u i d b e h a v i o r a f t e r a p r e s s u r e jump I n t h e s e n u m e r i c a l s i m u l a t i o n s , we t r i e d t o r e p r o d u c e a s c l o s e l y as p o s s i b l e t h e experimenta l c o n d i t i o n s o f p a r t 3.3. The p r e s s u r e 'ljump" was c h o s e n , a s i n t h e e x p e r i m e n t a l c o n d i t i o n s , t o v a r y l i n e a r l y from t h e i n i t i a l t o t h e f i n a l p r e s s u r e d u r i n g time t = 2 0 s. The s t r u c t u r a l r e l a x a t i o n time, T ( 0 ), a? t h e b e g i n n i n g of t h e p r e s s u r e jump was t a k e n e q u a l t o t h e v a l u e m e a s u r e d by t h e L i g h t Scatt e r i n g t e c h n i q u e i n t h e same t e m p e r a t u r e a n d pressure conditions. F i g . 7 a n d 8 show two t y p i c a l c u r v e s obtained near the g l a s s t r a n s i t i o n pressure a t 17OC. I n t h e s e c u r v e s , t h e time e v o l u t i o n of t h e i n f i n i t e c o m p r e s s i o n a l m o d u l u s K, i s composed o f a s u d d e n c h a n g e f o l l o w e d by a more g r a d u a l behavior, i n q u a l i t a t i v e agreement with t h e p r e v i o u s experimental o b s e r v a t i o n s (3.3). Furt h e r m o r e , t h e s t r u c t u r a l r e l a x a t i o n times o b t a i n e d i n t h e l o n g time p a r t of t h e c u r v e s a r e i n a g r e e m e n t , w i t h i n e x p e r i m e n t a l errors, w i t h t h e experimental r e s u l t s . 4 . 2 S t u d y of t h e f l u i d b e h a v i o r i n o u r E.H.D. contact The p r e v i o u s s t u d i e s h a v e shown t h a t much w i l l b e g a i n e d i n t h e u n d e r s t a n d i n g of t h e g l a s s t r a n s i t i o n i n a n E.H.D. c o n t a c t , i f we c a n d e t e r m i n e t h e t h e r m o d y n a m i c s t a t e of a t h i n s l i c e of l u b r i c a n t a s i t p a s s e s t h r o u g h t h e contact. The s t r u c t u r a l r e l a x a t i o n time h a s b e e n n u m e r i c a l l y computed a l o n g t h e c o n t a c t f o r d i f f e r e n t t r a n s i t times t These computations h a v e b e e n made a s s u m f n g i s o t h e r m a l a n d p u r e r o l l i n g c o n d i t i o n s . The Hertz p r o f i l e is i n f a c t r e p l a c e d by a s u c c e s s i o n of N p r e s s u r e s t e p s of d u r a t i o n A t d e f i n e d by t = N A t . The r e l e v a n t q u a n t i t y t o b e c o n s i d e r e d gere i s D(t) = T(t)/At ( i n s t a n t a n e o u s Deborah number) w h e r e T ( t ) i s t h e s t r u c t u r a l r e l a x a t i o n time a t the instant t i n the contact. Note h e r e t h a t i n t h e l i t e r a t u r e i t i s g e n e r a l l y t h e mean r e l a x a t i o n time :
.
t h a t is c o n s i d e r e d and c o n s e q u e n t l y o n l y q u a l i t a t i v e i n f o r m a t i o n i n t h e g l a s s t r a n s i t i o n c a n be deduced. I n o u r c o m p u t a t i o n s we h a v e d i s t i n g u i s h e d a t f i r s t two l i m i t i n g cases c o r r e s p o n d i n g t o v e r y l a r g e and v e r y small v a l u e s of D ( t ) : I n t h e f i r s t s i t u a t i o n , ( D ( t ) >> I ) , o n l y t h e i n s t a n t a n e o u s r e s p o n s e of t h e f l u i d , corresp o n d i n g t o a p u r e l y e l a s t i c b e h a v i o r , is o b s e r v e d . The f l u i d may b e c o n s i d e r e d a s a g l a s s a l o n g t h e w h o l e c o n t a c t . T h e s e r e s u l t s are r e p o r t e d i n f i g . 9 (A). Note a l s o t h a t a l l t h e r e s u l t s r e p o r t e d i n _ f i g . 9 were o b t a i n e d w i t h t h e mean p r e s s u r e P = 0.16 GPa a n d w i t h t h e same p a r a m e t e r s a s i n p a r t ( 4 . 1 ) .
I n t h e s e c o n d case, ( D ( t ) << I ) , t h e f l u i d i s , a t e a c h i n s t a n t , i n thermodynamic e q u i l i b r ium w i t h t h e e x t e r n a l p r e s s u r e and may be c o n s i d e r e d as a l i q u i d below its g l a s s t r a n s i t i o n p r e s s u r e . The r e s u l t s a r e r e p o r t e d i n f i g . 9 (e).
I n a d d i t i o n t o t h e two p r e v i o u s l i m i t i n g cases, i t i s i n t e r e s t i n g t o c o n s i d e r now more r e a l i s t i c s i t u a t i o n s e n c o u n t e r e d i n a n E.H.D. c o n t a c t i n w h i c h , f o r a g i v e n t r a n s i t time to, D ( t ) c a n v a r y from v e r y small v a l u e s a t t = 0 t o by many o r d e r l a r g e r v a l u e s a t t = t 12. F i g . 9 shows two c u r v e s o b t a i n e d f o r two d i f ? e r e n t t r a n s i t times t = 0.01 m s and 1 ms ( and o r e s p e c t i c e l y ) . I n b o t h cases, a l t h o u g h t h e mean H e r t z p r e s s u r e i s smaller t h a n t h e c o r r e s p o n d i n g s t a t ic glass transition pressure, the lubricant b e h a v e s l i k e a g l a s s o v e r a l a r g e domain i n t h e c o n t a c t . The w i d t h of t h i s domain d e p e n d s on t h e t r a n s i t time v a l u e . I n a l a s t n u m e r i c a l s i m u l a t i o n , we have followed t h e f l u i d behavior i n t h e contact after i n s t a n t t=t / 2 , k e e p i n g t h e p r e s s u r e c o n s t a n t a n d e q u a l Po t h e maximum, Pm, of t h e H e r t z p r e s s u r e . The a d v a n t a g e o f t h i s a p p a r e n t l y a c a d e m i c s t u d y is t o g i v e a v a l u e of t h e time n e c e s s a r y t o r e a c h t h e thermodynamic e q u i l i b r i u m . C o n s e q u e n t l y i t i s a l s o a measurement of t h e r m o d y n a m i c s t a t e of t h e f l u i d a t instant t = t /2. W e h a v e r e p o r t e d i n F i g . 10, t h e f r e e volumt? b e h a v i o r f o r v a r i o u s t r a n s i t times and f o r a maximum H e r t z p r e s s u r e P = 0 . 2 GPa. The m time n e c e s s a r y t o r e a c h t h e t h e r m o d y n a m i c e q u i l i b r i u m i s a l l t h e more l o n g a s t h e t r a n s i t time is s h o r t . As an e x a m p l e , f o r t o = 0.01 m s , t h e c h a r a c t e r i s t i c time i s f o u n d t o b e e q u a l t o 0.5 s. C o n s e q u e n t l y , f o r t h e s e p a r t i c u l a r c o n d i t i o n s , t h e l u b r i c a n t is i n a glassy state i n the m i d d l e of t h e c o n t a c t w h e r e t h e a p p l i e d p r e s s u r e i s 0 . 2 GPa. Note t h a t t h i s i s n o t t h e case i n t h e q u a s i s t a t i c e x p e r i m e n t s ( p a r t 3 ) : i n d e e d a t P = 0.2 GPa, i t i s f o u n d t h a t t h e l i q u i d i s s t i l l i n thermodynamic e q u i l i b r i u m .
*
5 CONCLUDING REMARKS I n t h i s p a p e r we h a v e f i r s t r e c a l l e d some g e n e r a l i t i e s on t h e g l a s s t r a n s i t i o n , t r y i n g to f o c u s a t t e n t i o n on its dynamic a s p e c t . I t h a s a l s o b e e n shown t h a t w i t h some i m p r o v e m e n t on a L i g h t S c a t t e r i n g a r r a n g e m e n t , i t is p o s s i b l e t o o b s e r v e t h e d y n a m i c b e h a v i o r of a f l u i d j u s t a f t e r a s h o r t p r e s s u r e s t e p . For e x a m p l e , t h e l o n g i t u d i n a l m o d u l u s h a s shown t o e x h i b i t a t first an instantaneous response f o l l o w e d by a more g r a d u a l b e h a v i o r . T h i s is i n a g r e e m e n t w i t h t h e model p r e v i o u s l y d e v e l o p e d by Montrose et al.. U s i n g t h i s m o d e l , we h a v e a l s o s i m u l a t e d t h e b e h a v i o r of t h e t h i n s l i c e of l u b r i c a n t a s it p a s s e s t h r o u g h t h e c o n t a c t . W e h a v e shown, i n p a r t i c u l a r , t h a t , e v e n when t h e maximum Hertz p r e s s u r e i s smaller t h a n t h e g l a s s t r a n s i t i o n p r e s s u r e measured i n q u a s i h y d r o s t a t i c condit i o n s , i t is p o s s i b l e t o o b s e r v e a g l a s s t r a n s i t i o n i n some p a r t of t h e c o n t a c t . C o n t r a r y t o t h e more g l o b a l s t u d i e s g e n e r a l l y e n c o u n t e r e d i n t h e l i t e r a t u r e , we h a v e t r i e d t o follow s t e p by s t e p t h e s t a t e of t h e f l u i d i n t h e c o n t a c t .
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6 ACKNOWLEDGEMENT
The a u t h o r s w i s h t o thank D r . C. P a p a r o d i t i s f o r h e l p f u l discussions. References ALSAAD, M.A., WINER, W.O., MEDINA, F.D., O'SHEA, D.C. ' L i g h t - s c a t t e r i n g s t u d y o f t h e glass t r a n s i t i o n i n lubricants',3. L u b r i c a t i o n Techn. 1978, 100. 418-422. ALSAAD, M., BAIR, S., SANBORN, D.M., WINER, W.O., 'Glass t r a n s i t i o n s i n l u b r i c a n t s : i t s r e l a t i o n t o elastohydrodynamic l u b r i c a t i o n (EHD)' 3. L u b r i c a t i o n Techn. 1978, 100, 404-417. EVANS, C.R. and JOHNSON, K.L. 'Regimes o f t r a c t i o n i n elastohydrodynamic l u b r i c a t i o n ' t o b e p u b l i s h e d i n Proc. I n s t . Mech. Engr. See f o r example: BRAWER, S.A. 'Theory o f r e l a x a t i o n i n v i s c o u s l i q u i d s and g l a s s e s ' . 3. Chem. Phys. 1984, 81, 954-975. HARRISON, G. 'The dynamic p r o p e r t i e s o f s u p e r c o o l e d l i q u i d s ' , Academic P r e s s , New York, 1976. HEYES, D.M., MONTROSE, C.3. ' The use o f l i n e and p o i n t c o n t a c t s i n d e t e r m i n i n g l u b r i c a n t r h e o l o g y under low s l i p elastohydrodynamic c o n d i t i o n s ' , 3 . L u b r i c a t i o n Techn. 1983, 105, 280-287. BEZOT, P., HESSE-BEZOT, C., BERTHE, D., DALMAZ, G., VERGNE, P. ' V i s c o e l a s t i c parameters o f 5P4E a s a f u n c t i o n o f p r e s s u r e and t e m p e r a t u r e by l i g h t s c a t t e r i n g t e c h n i q u e ' , 3. T r i b o l o g y 1986, 108, 579-583. BEZOT, P., HESSE-BEZOT, C., PETITET, 3.P. and ROYNARD, D. ' S t r u c t u r a l r e l a x a t i o n b e h a v i o r o f a 30% mole f r a c t i o n m i x t u r e o f DMSO and w a t e r ' 3. Mol. L i q u i d s 1987, 34, 317-328. ATAKE, T., ANGELL, C.A. ' P r e s s u r e dependence o f t h e g l a s s t r a n s i t i o n t e m p e r a t u r e i n m o l e c u l a r l i q u i d s and p l a s t i c c r y s t a l s ' 3. Phys. Chem. 1979, 83, 3218-3223. DILL, 3.F., DRAKE, P.W., L I T O V I T Z , T.A. 'The s t u d y o f v i s c o e l a s t i c p r o p e r t i e s o f l u b r i c a n t s using high pressure o p t i c a l t e c h n i q u e s ' A.S.L.E. T r a n s a c t i o n s 1974, 18. 202-210. G J~-~ (10) HEYES, D.M. and MONTROSE, C.3. ' A v i s c o e l a s t i c f r e e volume t h e o r y o f t r a c t i o n i n elastohydrodynamic l u b r i c a t i o n ' Fundamentals o f T r i b o l o g y Proceeding o f t h e I n t e r n a t i o n a l Conferences o f T r i b o l o g y 1978,1149-1168. ~
~
174
I I
A
I ~
\
I J
Fig. 1 Typical Light-Scattering spectrum a t t = 1 7 O C and P = 0 . 2 2 GPa : P o l a r i z e d ( A ) and d e p o l a r i z e d (B) s a t e l l i t e s .
F i g . 2 L o n g i t u d i n a l f r e q u e n c y 3 L as a f u n c t i o n of t h e p r e s s u r e a t 1 7 O C .
1E+03 1Et02 1E+01 1E+00 1E-0 1
1E-$f
4
Fig. 3 Transverse frequency g s f u n c t i o n of t h e p r e s s u r e a t 17OC.
P [GPa)
as a
3
0.17
0.21
Fig. 4 S t r u c t u r a l r e l a x a t i o n time Z a s a function of t h e pressure a t 1 7 O C . : photon c o r r e l a t i o n technique
+:
dynamic M,
behavior.
I
0.25
175
. I
Fig. 5
Time d e p e n d a n c e o f a B r i l l o u i n s a t e l l i t e
I
a f t e r a s t e p p r e s s u r e f r o m 0 . 2 2 t o 0.23 GPa.
8.
8.
w 0 Fig. 6 Idem f i g u r e 5 f o r a s t e p p r e s s u r e from 0.22 t o 0.21 GPa.
8
I
t El 460
8dO
12'00
Fig. 7 T i m e d e p e n d a n c e o f computed K, a s t e p p r e s s u r e f r o m 0 . 2 2 t o 0 . 2 3 GPa.
id00 for
176
8.
I
0
W
7. 7. 1E-06
7. 7.z I n
100
200
300
Fig. 8 Idem f i g u r e 7 f o r a s t e p p r e s s u r e f r o m 0.22 t o 0.21 GPa.
4d0
1E-08 Fig. 9
Is+
d
*A t/tO 0.2 0.4
0.6 0.8
i
R e l a x a t i o n t i m e 2 as a f u n c t i o n of
t / t o f o r P = 0.20 GPa rn
A %
D(t)7> 1 to = 0.01 ms
D(t)Cc 1
o
to = 1 ms
0.042
0.036 0.034
0m03 0.03
F i g . 10
+
Time dependance o f V f / V o P = P
for
= 0.2 GPa
m to = 0.01 ms
t
0
=
Is
Paper Vll(ii)
Some comments on the "glassy state" of lubricants in an EHD contact P. Bezot, C. Hesse-Bezot and G.Rouill6
G e n e r a l i t i e s on t h e " g l a s s t r a n s i t i o n " i n molecular l i q u i d s a r e f i r s t b r i e f l y r e c a l l e d and some remarks concerning t h e v i s c o e l a s t i c t r a n s i t i o n s t u d i e d by means o f u l t r a s o n i c and hypersonic techniques a r e made. A L i g h t S c a t t e r i n g study o f t h e 5P4E i n t h e v i c i n i t y o f t h e " g l a s s t r a n s i t i o n " i n s t a t i c c o n d i t i o n s and immediately a f t e r a pressure s t e p i s a l s o presented. L a s t l y , r e s u l t s o f numerical s i m u l a t i o n s obtained i n a f l u i d element a f t e r a pressure s t e p near the "glass t r a n s i t i o n " pressure o r i n an E.H.D. c o n t a c t a r e given. 1 INTRODUCTION The question o f the g l a s s t r a n s i t i o n i n an E.H.D. experiment has been l a r g e l y discussed i n the l a s t two decades. The main elements o f t h i s problem have been c l e a r l y i n t r o d u c e d f o r i n s t a n ce i n the very good works o f Winer e t a l ( 1 ) and Johnson e t a1 ( 2 ) . For example, the i n f l u e n c e o f pressure and cooling speed on t h e " g l a s s t r a n s i t i o n " (G.T.) temperature has l a r g e l y been commented by the f i r s t authors. However the answer t o the q u e s t i o n o f t h e glass t r a n s i t i o n i n an E.H.D. c o n t a c t remains i n our o p i n i o n more q u a l i t a t i v e than q u a n t i t a t i v e . The reasons f o r t h i s a r e m a n i f o l d : one o f t h e most important i s t h a t t h e g l a s s t r a n s i t i o n i s generally measured by means o f experiments whose time scale ( t o ) i s t y p i c a l l y o f t h e order o f t e n minutes. kt what happens i n an E.H.D. c o n t a c t i n which the t r a n s i t time i s many o r d e r s shorter ? I n t h i s work, we s h a l l t r y t o c o n t r i b u t e t o the s o l u t i o n o f t h i s problem step by s t e p : A t f i r s t , i n the next p a r t ( 2 ) , some general considerations on t h e G.T. w i l l be reminded together w i t h general comments u s e f u l t o understand the f o l l o w i n g . I n the t h i r d p a r t ( 3 1 , experimental s t u d i e s of a polyphenylether ( t h e 5P4E) w i l l be p r e sented, from 0.1 MPa t o 0.35 GPa a t 17OC by means o f L i g h t S c a t t e r i n g technique, i n v a r i o u s dynamic s i t u a t i o n s . F i n a l l y , the f o u r t h p a r t ( 4 ) w i l l be devoted t o numerical s i m u l a t i o n s i n r e l a t i o n t o some of the previous experimental r e s u l t s and i n t h e case of E.H.D. experiments.
2 GENERAL CONSIDERATIONS ON THE GLASS TRANSITION Before discussing t h e problem o f t h e g l a s s t r a n s i t i o n i n an E.H.D. c o n t a c t , i t i s worth while r e c a l l i n g some g e n e r a l i t i e s on t h i s subj e c t ( 3 ) . I t i s w e l l known t h a t , by c o o l i n g a molecular l i q u i d a t c o n s t a n t pressure, s u f f i c i e n t l y below i t s m e l t i n g p o i n t , we can o b t a i n the formation o f a glass. I t i s a l s o known (perhaps l e s s ) t h a t , a t a g i v e n temperature, t h e
compression o f a l i q u i d above i t s m e l t i n g pressure can a l s o l e a d t o the f o r m a t i o n o f a glass. I n a l l these s i t u a t i o n s , t h e common f e a t u r e i s t h a t t h e r e e x i s t s a t l e a s t one s t r u c t u r a l r e l a x a t i o n process whose c h a r a c t e r i s t i c time increases d r a m a t i c a l l y as t h e pressure i s increased (or t h e temperature decreased). As t h i s c h a r a c t e r i s t i c time becomes much l a r g e r than t h e observat i o n time t ,then t h e l i q u i d i s "frozen" i n a non-equilibr%m s t a t e . These a r e t h e f i r s t qual i t a t i v e descriptions of a glass transition. To be a l i t t l e more q u a n t i t a t i v e , l e t us now g i v e some complementary d e f i n i t i o n s . A l i q u i d or a g l a s s i s d e f i n e d by a s e t o f thermodynamic f o r c e s (temperature T, pressure P, ...I t o g e t h e r w i t h , a t l e a s t , one order parameter which c h a r a c t e r i z e s t h e s t a t e of t h e l o c a l s t r u c t u r e i n a f l u i d . I f now, s t a r t i n g from the l i q u i d s t a t e above i t s m e l t i n g p o i n t , we i n c r e a se t h e pressure, keeping t h e o t h e r thermodynamic f o r c e s constant, then any e x t e n s i v e p r o p e r t y (volume V, .) o r r e l a t e d p h y s i c a l q u a n t i t y , w i l l vary along a curve corresponding t o the s t a t e o f thermodynamic e q u i l i b r i u m o f t h e l i q u i d . I f now t h e pressure i s increased above the m e l t i n g pressure (P,) and i f c r y s t a l l i z a t i o n can be prevented , then t h e l i q u i d i s u n s t a b l e ( r e l a t i v e t o t h e c r y s t a l l i z a t i o n ) but may s t i l l be i n thermodynamic e q u i l i b r i u m , i f t h e pressure i s increased s u f f i c i e n t l y s l o w l y . The range o f pressure above P over which t h e l i q u i d remains i n thermodynami2' e q u i l i b r i u m , depends on the r a t e o f compression q = dP/dt. F o r a g i v e n r a t e q, t h e l i q u i d w i l l s t a y i n a metastable thermodynamic e q u i l i b r i u m up t o a v a l u e of t h e p r e s sure c a l l e d t h e g l a s s pressure (PG, pressure corresponding t o the " g l a s s t r a n s i t i o n " ) a t which t h e l i q u i d departs from i t s e q u i l i b r i u m behavior More p r e c i s e l y , t h i s departure from e q u i l i b r i u m which i s e s s e n t i a l l y dynamic, begins t o occur when the mean r e l a x a t i o n time T, f o r t h e s t r u c t u r a l rearrangement i s l a r g e r than the pressure increment time. Consequently t h e f a s t e r t h e pressure i s a p p l i e d , t h e lower the PG value w i l l be. O f course, t h e o b s e r v a t i o n time
..
.
( t i m e necessary t o measure an e x t e n s i v e p r o p e r t y o f t h e l i q u i d ) must a l s o be s h o r t e r than T L a s t l y l e t us make some remarks about t # e " v i s c o e l a s t i c t r a n s i t i o n " , o f t e n encountered i n
.
170
t h e l i t e r a t u r e ( 4 ) , and which must n o t be confused w i t h a g l a s s t r a n s i t i o n i n a classical experiment : C o n s i d e r , f o r e x a m p l e , t h e case of a h y p e r s o n i c wave ( s t u d i e d by B r i l l o u i n S c a t t e r i n g ) o f frequency w t h a t propagates i n a viscous liquid, c h a - a c t e r i z e d by a mean s t r u c t u r a l r e l a x a t i o n time -rM. I f , b y c o m p r e s s i n g s l o w l y t h e s a m p l e a t constant temperature, t h e T values increase yT > > I , a "transition f r o m wT M < < I t o r e g i o n " is o b s e r v e d i n which t h e l o n g i t u d i n a l m o d u l u s , r e l a t e d t o t h e s o u n d v e l o c i t y , shorvs a marked v a r i a t i o n b e t w e e n r e s p e c t i v e l y a low a n d a h i g h f r e q u e n c y l i m i t . The p r e s s u r e r e g i o n i n which t h i s " v i s c o e l a s t i c t r a n s i t i o n " is observed, must n o t b e c o n f u s e d w i t h t h e g l a s s p r e s s u re PG w h i c h is much l a r g e r . However, t h i s " v i s c o e l a s t i c p r e s s u r e " w o u l d c e r t a i n l y correspond t o t h e g l a s s t r a n s i t i o n i n a n e x p e r i m e n t w h e r e a p r e s s u r e i n c r e m e n t would b e a p p l i e d -1 d u r i n g a time w W e h o p e now t h a t t h e p r e v i o u s comments w i l l j u s t i f y o u r f o l l o w i n g a p p r o a c h i n a n E.H.D. contact. A t f i r s t , i t seemed n e c e s s a r y t o m e a s u r e some e x t e n s i v e p r o p e r t i e s ( o r p h y s i c a l q u a n t i t i e s r e l a t e d t o t h e m ) of a p a r t i c u l a r l u b r i c a n t ( h e r e t h e 5P4E) a l o n g t h e c u r v e of t h e r m o d y n a m i c e q u i l i b r i u m ( s t a b l e or n o t , d e p e n d i n g on t h e p r e s s u r e value r e l a t i v e t o PM). I n o r d e r t o follow t h i s c u r v e , on t h e w i d e s t a c c e s s i b l e rang': of p r e s s u r e , t h e l a t t e r was i n c r e a s e d v e r y s l o w l y ( w a i t i n g time of two h o u r s a f t e r e a c h 1 0 MPa p r e s s u r e i n c r e m e n t ) . T h e n , w i t h t h e a i m of a p p r o a c h i n g t h e e x p e r i m e n t a l s i t u a t i o n e n c o u n t e r e d i n a n E.H.D. c o n t a c t , we i m a g i n e d a n e x p e r i m e n t a l a r r a n g e m e n t t o f o l l o w t h e b e h a v i o r of t h e i n f i n i t e f r e q u e n c y l o n g i t u d i n a l modulus immediately after a p r e s s u r e i n c r e m e n t . The a d v a i t a g e of t h i s p r o c e d u r e i s t o f o l l o w s t e p by s t e p t h e f l u i d b e h a v i o r i n t h e whole p r e s s u r e range i n c l u d i n g P G' Finally, since a t presently, d i r e c t experim e n t a l s t u d i e s of t h e f l u i d d y n a m i c i n a n E.H.D. c o n t a c t are n o t f e a s i b l e , we a r e o n l y l e f t w i t h t h e numerical s i m u l a t i o n for its behavior. Using a model d u e t o M o n t r o s e e t a l . ( 5 ) , we f i r s t compared o u r e x p e r i m e n t a l r e s u l t s o b t a i n e d a f t e r a p r e s s u r e s t e p with numerical simulations i n t h e same e x p e r i m e n t a l c o n d i t i o n s . Then u s i n g t h e same m o d e l , we s t u d i e d t h e d y n a m i c a l b e h a v i o r of a f l u i d a f t e r a s u c c e s s i o n of s m a l l p r e s s u r e jumps s u c h a s i n a n E . H . D . c o n t a c t . W e hope t h a t by t h i s a p p r o a c h , t h e q u e s t i o n of t h e g l a s s t r a n s i t i o n i n a n E.H.D. c o n t a c t w i l l b e made clearer.
.
3 EXPERIMENTAL APPROACH USING L I G H l SCATTERING TECHNIQUE I n t h e p r e s e n t w o r k , m e a s u r e m e n t s were p r i n c i p a l l y o b t a i n e d from t h e d y n a m i c L i g h t S c a t t e r i n g technique : When a l i q u i d s a m p l e i s i r r a d i a t e d by a n e l e c t r o m a g n e t i c wave ( f o r e x a m p l e a l a s e r b e a m ) , its molecules s c a t t e r l i g h t i n a l l d i r e c t i o n s . T h i s s c a t t e r e d l i g h t c o n t a i n s i n f o r m a t i o n on v a r i o u s m o l e c u l a r movements and c o n s e q u e n t l y o n some p h y s i c a l q u a n t i t i e s c h a r a c t e r i z i n g t h e l i q u i d . The a n a l y s i s s y s t e m t o b e u s e d d e p e n d s o n t h e c h a r a c t e r i s t i c time of t h e movement t o b e studied.
3 . 1 . Experimental set-up The p o l y p h e n y e t h e r (5P4E) s a m p l e s were s u p p l i e d by Monsanto Co a n d d i r e c t l y u s e d i n a s m a l l e o p t i c a l cell which w i l l be d e s c r i b e d later. W used a c o h e r e n t Innova 903 l a s e r a s e x c i t i n g s o u r c e e i t h e r i n a s i n g l e mode or multimode, depending on t h e a n a l y s i s t e c h n i q u e . To s t u d y t h e p r o p a g a t i v e waves, i . e . t o m e a s u r e t h e f r e q u e n c y of t h e p r o p a g a t i v e longi t u d i n a l and s h e a r w a v e s , t h e s c a t t e r e d l i g h t was a n a l y s e d w i t h a h i g h r e s o l u t i o n F a b r y - P e r o t s p e c t r o m e t e r . D e t a i l s of t h i s t e c h n i q u e have r e c e n t l y b e e n d e s c r i b e d e l s e w h e r e ( 6 ) . The nonp r o p a g a t i v e modes, i . e . t h e s t r u c t u r a l r e l a x a t i o n p r o c e s s e s , were s t u d i e d by t h e photon c o r r e l a t i o n technique using a d i g i t a l multitau m u l t i b i t L a n g l e y - F o r d c o r r e l a t o r . More d e t a i l s on t h i s e x p e r i m e n t a l arrangement t o g e t h e r with t h e a n a l y s i s p r o c e d u r e c a n b e f o u n d i n ( 7 ) . Note t h a t t h e a n a l y t i c a l c o r r e l a t i o n f u n c t i o n was chos e n t o b e t h e William-W tts f u n c t i o n : 4 ( t ) = exp( - ( t / T ) ) i n w h i c h 0 < B < 1 i s r e l a t e d t o t h e w i d t h of t h e d i s t r i b u t i o n and T c o r r e s p o n d s n e a r l y t o t h e time a t t h e maximum a m p l i t u d e of t h e d i s t r i b ution. Let u s now d e s c r i b e t h e h i g h p r e s s u r e o p t i c a l s y s t e m . Its g e n e r a l a r r a n g e m e n t h a s a l r e a d y b e e n d e s c r i b e d e l s e w h e r e ( 6 ) . However t h e p r e s s u r e c e l l i s now g r e a t l y m o d i f i e d . I n o u r p r e v i o u s works, t h e e n t i r e high pressure s y s t e m was f i l l e d w i t h t h e s t u d i e d l i q u i d used a l s o a s t h e p r e s s u r e t r a n s m i t t i n g medium. Conseq u e n t l y m e a s u r e m e n t s a s a f u n c t i o n of p r e s s u r e c o u l d n o t b e c a r r i e d o u t a b o v e v a l u e s f o r which t h e c o r r e s p o n d i n g v i s c o s i t y was too h i g h . Now t h e s a m p l e i s i s o l a t e d from t h e p r e s s u r e t r a n s m i t t i n g l i q u i d by a s m a l l g l a s s c e l l w h i c h i s p u t i n s i d e t h e h i g h p r e s s u r e c e l l and embedded i n t h e p r e s s u r e t r a n s m i t t i n g medium. Its volume i m m e d i a t e l y c h a n g e s , a f t e r a n e x t e r n a l p r e s s u r e v a r i a t i o n , so a s t o a d j u s t t h e pressure i n both liquids. C o n s e q u e n t l y , a s r e g a r d s t h e p r e s s u r e , it i s now p o s s i b l e t o s t u d y t h e l i q u i d b e h a v i o r following l a r g e compression (or depression) rat e s a n d a t p r e s s u r e s much l a r g e r t h a n PG. L a s t l y , b e f o r e presenting our experimental r e s u l t s , l e t u s d e s c r i b e some i m p r o v e m e n t s made i n o u r a n a l y s i s system. These improvements a r e p a r t i c u l a r i t y c r u c i a l i n t h e case of d y n a m i c a l studies. Let u s c o n s i d e r , i n f i g . 1 , a t y p i c a l s p e c t r u m o b t a i n e d a t t = 17OC a n d P = 0.22 GPa b y means of a F a b r y - P e r o t I n t e r f e r o m e t e r . One of t h e most i m p o r t a n t q u a n t i t i e s t o b e m e a s u r e d is t h e f r e q u e n c y s h i f t of t h e l o n g i t u d i n a l ( l a r g e s a t e l l i t e s A) or s h e a r ( s m a l l s a t e l l i t e s 8) waves. A s t h e time n e c e s s a r y t o r e c o r d t h i s t y p i c a l s p e c t r u m is a t least twenty minutes, d y n a m i c a l s t u d i e s s h o r t e r t h a n n e a r l y two h o u r s are d i f f i c u l t w i t h t h i s p r o c e d u r e . However when t h e large satellites s h i f t with t h e pressure, t h e i n t e n s i t y of t h e minimum b e t w e e n t h e two l a r g e s a t e l l i t e s (C) u n d e r g o e s a l a r g e v a r i a t i o n . C o n s e q u e n t l y i f we know a t h e o r e t i c a l r e l a t i o n b e t w e e n t h e minimum i n t e n s i t y and t h e f r e q u e n c y s h i f t , or i f we h a v e m e a s u r e d i t e x p e r i m e n t a l l y , it is p o s s i b l e , from a d i r e c t m e a s u r e m e n t of t h e i n t e n s i t y , t o d e d u c e t h e f r e q u e n c y s h i f t . The main a d v a n t a g e of t h i s p r o c e d u r e i s now t h e q u a s i - i n s t a n t a n e o u s r e s p o n se of t h e m e a s u r e m e n t s y s t e m .
B
171
W e now p r e s e n t , i n t h e n e x t two p a r t s ( 3 . 2 and 3 . 3 ) , t h e v a r i o u s e x p e r i m e n t a l r e s u l t s . 3.2. L i q h t s c a t t e r i n g r e s u l t s experimental c o n d i t i o n s
i n quasi-static
The p r e s e n t s t u d y i s d e v o t e d t o t h e m e a s u r e m e n t of p h y s i c a l p a r a m e t e r s a s a f u n c t i o n of t h e p r e s s u r e f o r w a i t i n g times of t h e o r d e r o f two h o u r s a f t e r p r e s s u r e v a r i a t i o n s of n e a r l y 10 MPa These experimental conditions are generally encountered i n o t h e r techniques, such a s the c a l o r i m e t r i c one, for i n s t a n c e ( 8 ) . The e x p e r i m e n t a l v a l u e s of t h e l o n g i t u d i n a l wave f r e q u e n c i e s p l o t t e d a s a f u n c t i o n o f t h e p r e s s u r e are r e p o r t e d i n f i g . 2 . T h r e e domains c a n be d i s t i n g u i s h e d i n w h i c h t h e f r e q u e n c y varies linearly with t h e pressure : t h e first o n e , i n t h e l e f t p a r t of t h e c u r v e , c o r r e s p o n d s t o t h e v i s c o e l a s t i c (V.E.) r e g i o n . However, d u e t o t h e low t e m p e r a t u r e of t h e s a m p l e , e v e n a t P = 0.1 MPa, WT i s s t i l l l a r g e r t h a n o n e . I n t h e h i g h p r e s s u r e p a r t ( 0 . 1 0 GPa) of t h e V.E. r e g i o n , WT i s v e r y l a r g e a n d a n a n g u l a r p o i n t i s o b s e r v e d w h i c h c o r r e s p o n d s t o t h e c h a n g i n g of t h e c u r v e s l o p e . As p r e v i o u s l y o b s e r v e d , t h i s p o i n t must n o t b e e n c o n f u s e d w i t h PG. I n t h e m i d d l e r e g i o n where t h e v a l u e of s l o p e of t h e c u r v e i s lower, t h e l i q u i d i s s t i l l i n t h e r m o dynamic e q u i l i b r i u m b u t o u t s i d e t h e V.E. r e g i o n ( w T >> 1 ) . The e x t e n t of t h i s r e g i o n d e p e n d s o n t h e w a i t i n g time a f t e r e a c h p r e s s u r e i n c r e a s e : t h e l o n g e r t h e w a i t i n g time, t h e l a r g e r i t s l e n g t h . F o r e x a m p l e t h e crosses i n t h e c u r v e were o b t a i n e d f o r w a i t i n g times of a t l e a s t two d a y s ( o n e weak f o r t h e l a s t p o i n t ) w h e r e a s t h e o t h e r e x p e r i m e n t a l v a l u e s were m e a s u r e d two hours a f t e r t h e p r e s s u r e i n c r e a s e . I n t h e s e experimental c o n d i t i o n s , t h e g l a s s t r a n s i t i o n p r e s s u r e i s f o u n d t o b e 0 . 2 3 GPa. T h i s a g r e e s w i t h t h e r e s u l t s o b t a i n e d by Angel u s i n g a calorimetric technique ( 8 ) . F u r t h e r m o r e , o u r e x p e r i m e n t a l r e s u l t s for t h e s h e a r wave f r e q u e n c i e s , r e p o r t e d i n f i g . 3, e x h i b i t a g l a s s t r a n s i t i o n i n t h e same domain of pressure. F i n a l l y , c o n c e r n i n g t h e c e n t r a l component i n f i g . 1, its polarized p a r t is generally composed o f a l a r g e l i n e ( > 1 MHz) r e l a t e d t o t h e purely d i f f u s i v e t h e r m a l f l u c t u a t i o n and a narrow o n e , c a l l e d t h e M o u n t a i n l i n e , w h i c h i s related to the structural relaxation processes. The l a t t e r h a s b e e n a n a l y s e d by means of a c o r r e l a t o r . The mean r e l a x a t i o n times o b t a i n e d by a d j u s t m e n t w i t h t h e William-Watts f u n c t i o n (B 0.5) o n t h e e x p e r i m e n t a l c u r v e s are r e p o r t e d i n f i g . 4. Note t h a t , a t 0 . 2 3 GPa, t h e re a x a t i o n t i m e i s f o u n d t o b e of t h e o r d e r o f 10 s, i n agrement w i t h t h e o b s e r v a t i o n , i n t h i s p r e s s u r e r a n g e , of a g l a s s t r a n s i t i o n ( 9 ) .
.
3
T h i s s u d d e n v a r i a t i o n is m a i n l y d u e t o r a p i d p r o c e s s e s s u c h a s l a t t i c e v i b r a t i o n s . Then f o l l o w s a more g r a d u a l e v o l u t i o n d u e t o t h e structural relaxation. A t h i g h e r a n d lower p r e s s u r e s , c o n t r a r y t o w h a t i s s e e n i n f i g . 5 a n d 6, n o time e v o l u t i o n of t h e i n t e n s i t y , e x c e p t t h a t of t h e i n s t a n t a n e o u s o n e , is o b s e r v e d d u r i n g t h e v a r i o u s p o s s i b l e r e c o r d i n g times. A t low p r e s s u r e , a l l t h e r e l a x a t i o n times are t o o s h o r t t o b e r e s o l v e d and c o n t r i b u t e i n d i s t i n c t l y t o t h e i n s t a n t aneous response. A t t h e o p p o s i t e , for t h e high p r e s s u r e v a l u e s , t h e s t r u c t u r a l r e l a x a t i o n times a r e becoming t o o l a r g e ( s e v e r a l d a y s or more) t o be observed and t h e o n l y r a p i d p r o c e s s e s c o n t r i b u t e t o t h e instantaneous response. From c u r v e s similar t o f i g . 5 a n d 6, we d e d u c e d t h e time e v o l u t i o n of t h e c o m p r e s s i o n a l modulus a t i n f i n i t e frequency. T h i s kind of t e m p o r a l b e h a v i o r w i l l b e f o u n d i n p a r t 4.1 t o a g r e e w i t h t h e r e s u l t s of n u m e r i c a l s i m u l a t i o n s . L a s t l y , t h e v a l u e s of t h e mean s t r u c t u r a l r e l a x a t i o n times o b t a i n e d by t h i s method are p l o t t e d on f i g . 4. They are i n r o u g h a g r e e m e n t w i t h t h e v a l u e s m e a s u r e d by t h e p h o t o n correlat i o n technique. 4 NUMERICAL SIMULATION EXPERIMENTS
To a p p r o a c h more c l o s e l y t h e e x p e r i m e n t a l c o n d i t i o n s e n c o u n t e r e d i n a n E . H . D . c o n t a c t , o n e way would b e , f o r i n s t a n c e , t o a p p l y a time depende n t p r e s s u r e assuming a H e r t z shape and t o m e a s u r e s i m u l t a n e o u s l y t h e f r e q u e n c y s h i f t of the Brillouin lines. I n t h i s p a p e r we s h a l l r a t h e r p r e s e n t t h e r e s u l t s of n u m e r i c a l s i m u l a t i o n s o b t a i n e d f r o m a model r e c e n t l y d e v e l o p e d by M o n t r o s e e t a1 ( 5 ) . I n t h i s m o d e l , i t i s assumed t h a t , a f t e r a p r e s s u r e jump, a r e p r e s e n t a t i v e p r o p e r t y of t h e f l u i d w i l l experience an instantaneous response ( g l a s s l i k e b e h a v i o r ) f o l l o w e d by a r e t a r d e d c h a n g e ( l i q u i d l i k e b e h a v i o r ) . T h i s time d e p e n d e n c e i s e x p r e s s e d i n t h e case of t h e f r e e volume V f ( t ) by t h e f o l l o w i n g e q u a t i o n :
-
The free volume i s d e f i n e d by V f ( t ) = V ( t ) - V c
w h e r e V ( t ) a n d Vc a r e r e s p e c t i v e l y t h e t o t a l volume a t i n s t a n t t a n d t h e c l o s e - p a c k e d volume.
-
~ ( t i) s t h e s t r u c t u r a l r e l a x a t i o n time g i v e n
3.3 L i g h t s c a t t e r i n g r e s u l t s i n d y n a m i c experimental c o n d i t i o n s
by
Using t h e p r o c e d u r e d e s c r i b e d i n 3.1, we m e a s u r ed a f t e r e a c h p r e s s u r e jump t h e time e v o l u t i o n of t h e i n t e n s i t y minimum ( a r r o w C o n f i g . 1 ) between t h e two l o n g i t u d i n a l l i n e s . I n a r a n g e of p r e s s u r e e x t e n d i n g from 0 . 2 0 GPa t o 0 . 2 4 GPa, t h e i n t e n s i t y i s o b s e r v e d t o change w i t h time i n t h e manner shown i n F i g . 5 and 6 A t f i r s t t h e c h a n g e i s r a p i d compared t o t h e time r e s o l u t i o n of t h e m e a s u r i n g a p p a r a t u s .
- y ( t ) i s e q u a l t o Ko(t)/K, ( t ) , t h e r a t i o of t h e bulk moduli a t z e r o and i n f i n i t e frequency.
.
T (t)
=
T(O)
e x p [ vc
(l/Vf(t)-
l/vf(o))1
- K ( t ) c a n b e e Z p r e s s e d by 0 V ( t ) vc/ c r V f ( t ) I n a l l t h e following numerical simulations,
172
k ( o ) v a l u e s were d e d u c e d from o u r L i g h t Scatte-
r i n g m e a s u r e m e n t s ( 3 . 2 ) a n d t h e y v a l u e from M o n t r o s e e t a 1 ( 1 0 ) . The a v a l u e was c h o s e n e q u a l t o t h e s l o p e of t h e T = f(P) curve. 4.1 S t u d y of t h e f l u i d b e h a v i o r a f t e r a p r e s s u r e jump I n t h e s e n u m e r i c a l s i m u l a t i o n s , we t r i e d t o r e p r o d u c e a s c l o s e l y as p o s s i b l e t h e experimenta l c o n d i t i o n s o f p a r t 3.3. The p r e s s u r e 'ljump" was c h o s e n , a s i n t h e e x p e r i m e n t a l c o n d i t i o n s , t o v a r y l i n e a r l y from t h e i n i t i a l t o t h e f i n a l p r e s s u r e d u r i n g time t = 2 0 s. The s t r u c t u r a l r e l a x a t i o n time, T ( 0 ), a? t h e b e g i n n i n g of t h e p r e s s u r e jump was t a k e n e q u a l t o t h e v a l u e m e a s u r e d by t h e L i g h t Scatt e r i n g t e c h n i q u e i n t h e same t e m p e r a t u r e a n d pressure conditions. F i g . 7 a n d 8 show two t y p i c a l c u r v e s obtained near the g l a s s t r a n s i t i o n pressure a t 17OC. I n t h e s e c u r v e s , t h e time e v o l u t i o n of t h e i n f i n i t e c o m p r e s s i o n a l m o d u l u s K, i s composed o f a s u d d e n c h a n g e f o l l o w e d by a more g r a d u a l behavior, i n q u a l i t a t i v e agreement with t h e p r e v i o u s experimental o b s e r v a t i o n s (3.3). Furt h e r m o r e , t h e s t r u c t u r a l r e l a x a t i o n times o b t a i n e d i n t h e l o n g time p a r t of t h e c u r v e s a r e i n a g r e e m e n t , w i t h i n e x p e r i m e n t a l errors, w i t h t h e experimental r e s u l t s . 4 . 2 S t u d y of t h e f l u i d b e h a v i o r i n o u r E.H.D. contact The p r e v i o u s s t u d i e s h a v e shown t h a t much w i l l b e g a i n e d i n t h e u n d e r s t a n d i n g of t h e g l a s s t r a n s i t i o n i n a n E.H.D. c o n t a c t , i f we c a n d e t e r m i n e t h e t h e r m o d y n a m i c s t a t e of a t h i n s l i c e of l u b r i c a n t a s i t p a s s e s t h r o u g h t h e contact. The s t r u c t u r a l r e l a x a t i o n time h a s b e e n n u m e r i c a l l y computed a l o n g t h e c o n t a c t f o r d i f f e r e n t t r a n s i t times t These computations h a v e b e e n made a s s u m f n g i s o t h e r m a l a n d p u r e r o l l i n g c o n d i t i o n s . The Hertz p r o f i l e is i n f a c t r e p l a c e d by a s u c c e s s i o n of N p r e s s u r e s t e p s of d u r a t i o n A t d e f i n e d by t = N A t . The r e l e v a n t q u a n t i t y t o b e c o n s i d e r e d gere i s D(t) = T(t)/At ( i n s t a n t a n e o u s Deborah number) w h e r e T ( t ) i s t h e s t r u c t u r a l r e l a x a t i o n time a t the instant t i n the contact. Note h e r e t h a t i n t h e l i t e r a t u r e i t i s g e n e r a l l y t h e mean r e l a x a t i o n time :
.
t h a t is c o n s i d e r e d and c o n s e q u e n t l y o n l y q u a l i t a t i v e i n f o r m a t i o n i n t h e g l a s s t r a n s i t i o n c a n be deduced. I n o u r c o m p u t a t i o n s we h a v e d i s t i n g u i s h e d a t f i r s t two l i m i t i n g cases c o r r e s p o n d i n g t o v e r y l a r g e and v e r y small v a l u e s of D ( t ) : I n t h e f i r s t s i t u a t i o n , ( D ( t ) >> I ) , o n l y t h e i n s t a n t a n e o u s r e s p o n s e of t h e f l u i d , corresp o n d i n g t o a p u r e l y e l a s t i c b e h a v i o r , is o b s e r v e d . The f l u i d may b e c o n s i d e r e d a s a g l a s s a l o n g t h e w h o l e c o n t a c t . T h e s e r e s u l t s are r e p o r t e d i n f i g . 9 (A). Note a l s o t h a t a l l t h e r e s u l t s r e p o r t e d i n _ f i g . 9 were o b t a i n e d w i t h t h e mean p r e s s u r e P = 0.16 GPa a n d w i t h t h e same p a r a m e t e r s a s i n p a r t ( 4 . 1 ) .
I n t h e s e c o n d case, ( D ( t ) << I ) , t h e f l u i d i s , a t e a c h i n s t a n t , i n thermodynamic e q u i l i b r ium w i t h t h e e x t e r n a l p r e s s u r e and may be c o n s i d e r e d as a l i q u i d below its g l a s s t r a n s i t i o n p r e s s u r e . The r e s u l t s a r e r e p o r t e d i n f i g . 9 (e).
I n a d d i t i o n t o t h e two p r e v i o u s l i m i t i n g cases, i t i s i n t e r e s t i n g t o c o n s i d e r now more r e a l i s t i c s i t u a t i o n s e n c o u n t e r e d i n a n E.H.D. c o n t a c t i n w h i c h , f o r a g i v e n t r a n s i t time to, D ( t ) c a n v a r y from v e r y small v a l u e s a t t = 0 t o by many o r d e r l a r g e r v a l u e s a t t = t 12. F i g . 9 shows two c u r v e s o b t a i n e d f o r two d i f ? e r e n t t r a n s i t times t = 0.01 m s and 1 ms ( and o r e s p e c t i c e l y ) . I n b o t h cases, a l t h o u g h t h e mean H e r t z p r e s s u r e i s smaller t h a n t h e c o r r e s p o n d i n g s t a t ic glass transition pressure, the lubricant b e h a v e s l i k e a g l a s s o v e r a l a r g e domain i n t h e c o n t a c t . The w i d t h of t h i s domain d e p e n d s on t h e t r a n s i t time v a l u e . I n a l a s t n u m e r i c a l s i m u l a t i o n , we have followed t h e f l u i d behavior i n t h e contact after i n s t a n t t=t / 2 , k e e p i n g t h e p r e s s u r e c o n s t a n t a n d e q u a l Po t h e maximum, Pm, of t h e H e r t z p r e s s u r e . The a d v a n t a g e o f t h i s a p p a r e n t l y a c a d e m i c s t u d y is t o g i v e a v a l u e of t h e time n e c e s s a r y t o r e a c h t h e thermodynamic e q u i l i b r i u m . C o n s e q u e n t l y i t i s a l s o a measurement of t h e r m o d y n a m i c s t a t e of t h e f l u i d a t instant t = t /2. W e h a v e r e p o r t e d i n F i g . 10, t h e f r e e volumt? b e h a v i o r f o r v a r i o u s t r a n s i t times and f o r a maximum H e r t z p r e s s u r e P = 0 . 2 GPa. The m time n e c e s s a r y t o r e a c h t h e t h e r m o d y n a m i c e q u i l i b r i u m i s a l l t h e more l o n g a s t h e t r a n s i t time is s h o r t . As an e x a m p l e , f o r t o = 0.01 m s , t h e c h a r a c t e r i s t i c time i s f o u n d t o b e e q u a l t o 0.5 s. C o n s e q u e n t l y , f o r t h e s e p a r t i c u l a r c o n d i t i o n s , t h e l u b r i c a n t is i n a glassy state i n the m i d d l e of t h e c o n t a c t w h e r e t h e a p p l i e d p r e s s u r e i s 0 . 2 GPa. Note t h a t t h i s i s n o t t h e case i n t h e q u a s i s t a t i c e x p e r i m e n t s ( p a r t 3 ) : i n d e e d a t P = 0.2 GPa, i t i s f o u n d t h a t t h e l i q u i d i s s t i l l i n thermodynamic e q u i l i b r i u m .
*
5 CONCLUDING REMARKS I n t h i s p a p e r we h a v e f i r s t r e c a l l e d some g e n e r a l i t i e s on t h e g l a s s t r a n s i t i o n , t r y i n g to f o c u s a t t e n t i o n on its dynamic a s p e c t . I t h a s a l s o b e e n shown t h a t w i t h some i m p r o v e m e n t on a L i g h t S c a t t e r i n g a r r a n g e m e n t , i t is p o s s i b l e t o o b s e r v e t h e d y n a m i c b e h a v i o r of a f l u i d j u s t a f t e r a s h o r t p r e s s u r e s t e p . For e x a m p l e , t h e l o n g i t u d i n a l m o d u l u s h a s shown t o e x h i b i t a t first an instantaneous response f o l l o w e d by a more g r a d u a l b e h a v i o r . T h i s is i n a g r e e m e n t w i t h t h e model p r e v i o u s l y d e v e l o p e d by Montrose et al.. U s i n g t h i s m o d e l , we h a v e a l s o s i m u l a t e d t h e b e h a v i o r of t h e t h i n s l i c e of l u b r i c a n t a s it p a s s e s t h r o u g h t h e c o n t a c t . W e h a v e shown, i n p a r t i c u l a r , t h a t , e v e n when t h e maximum Hertz p r e s s u r e i s smaller t h a n t h e g l a s s t r a n s i t i o n p r e s s u r e measured i n q u a s i h y d r o s t a t i c condit i o n s , i t is p o s s i b l e t o o b s e r v e a g l a s s t r a n s i t i o n i n some p a r t of t h e c o n t a c t . C o n t r a r y t o t h e more g l o b a l s t u d i e s g e n e r a l l y e n c o u n t e r e d i n t h e l i t e r a t u r e , we h a v e t r i e d t o follow s t e p by s t e p t h e s t a t e of t h e f l u i d i n t h e c o n t a c t .
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6 ACKNOWLEDGEMENT
The a u t h o r s w i s h t o thank D r . C. P a p a r o d i t i s f o r h e l p f u l discussions. References ALSAAD, M.A., WINER, W.O., MEDINA, F.D., O'SHEA, D.C. ' L i g h t - s c a t t e r i n g s t u d y o f t h e glass t r a n s i t i o n i n lubricants',3. L u b r i c a t i o n Techn. 1978, 100. 418-422. ALSAAD, M., BAIR, S., SANBORN, D.M., WINER, W.O., 'Glass t r a n s i t i o n s i n l u b r i c a n t s : i t s r e l a t i o n t o elastohydrodynamic l u b r i c a t i o n (EHD)' 3. L u b r i c a t i o n Techn. 1978, 100, 404-417. EVANS, C.R. and JOHNSON, K.L. 'Regimes o f t r a c t i o n i n elastohydrodynamic l u b r i c a t i o n ' t o b e p u b l i s h e d i n Proc. I n s t . Mech. Engr. See f o r example: BRAWER, S.A. 'Theory o f r e l a x a t i o n i n v i s c o u s l i q u i d s and g l a s s e s ' . 3. Chem. Phys. 1984, 81, 954-975. HARRISON, G. 'The dynamic p r o p e r t i e s o f s u p e r c o o l e d l i q u i d s ' , Academic P r e s s , New York, 1976. HEYES, D.M., MONTROSE, C.3. ' The use o f l i n e and p o i n t c o n t a c t s i n d e t e r m i n i n g l u b r i c a n t r h e o l o g y under low s l i p elastohydrodynamic c o n d i t i o n s ' , 3 . L u b r i c a t i o n Techn. 1983, 105, 280-287. BEZOT, P., HESSE-BEZOT, C., BERTHE, D., DALMAZ, G., VERGNE, P. ' V i s c o e l a s t i c parameters o f 5P4E a s a f u n c t i o n o f p r e s s u r e and t e m p e r a t u r e by l i g h t s c a t t e r i n g t e c h n i q u e ' , 3. T r i b o l o g y 1986, 108, 579-583. BEZOT, P., HESSE-BEZOT, C., PETITET, 3.P. and ROYNARD, D. ' S t r u c t u r a l r e l a x a t i o n b e h a v i o r o f a 30% mole f r a c t i o n m i x t u r e o f DMSO and w a t e r ' 3. Mol. L i q u i d s 1987, 34, 317-328. ATAKE, T., ANGELL, C.A. ' P r e s s u r e dependence o f t h e g l a s s t r a n s i t i o n t e m p e r a t u r e i n m o l e c u l a r l i q u i d s and p l a s t i c c r y s t a l s ' 3. Phys. Chem. 1979, 83, 3218-3223. DILL, 3.F., DRAKE, P.W., L I T O V I T Z , T.A. 'The s t u d y o f v i s c o e l a s t i c p r o p e r t i e s o f l u b r i c a n t s using high pressure o p t i c a l t e c h n i q u e s ' A.S.L.E. T r a n s a c t i o n s 1974, 18. 202-210. G J~-~ (10) HEYES, D.M. and MONTROSE, C.3. ' A v i s c o e l a s t i c f r e e volume t h e o r y o f t r a c t i o n i n elastohydrodynamic l u b r i c a t i o n ' Fundamentals o f T r i b o l o g y Proceeding o f t h e I n t e r n a t i o n a l Conferences o f T r i b o l o g y 1978,1149-1168. ~
~
174
I I
A
I ~
\
I J
Fig. 1 Typical Light-Scattering spectrum a t t = 1 7 O C and P = 0 . 2 2 GPa : P o l a r i z e d ( A ) and d e p o l a r i z e d (B) s a t e l l i t e s .
F i g . 2 L o n g i t u d i n a l f r e q u e n c y 3 L as a f u n c t i o n of t h e p r e s s u r e a t 1 7 O C .
1E+03 1Et02 1E+01 1E+00 1E-0 1
1E-$f
4
Fig. 3 Transverse frequency g s f u n c t i o n of t h e p r e s s u r e a t 17OC.
P [GPa)
as a
3
0.17
0.21
Fig. 4 S t r u c t u r a l r e l a x a t i o n time Z a s a function of t h e pressure a t 1 7 O C . : photon c o r r e l a t i o n technique
+:
dynamic M,
behavior.
I
0.25
175
. I
Fig. 5
Time d e p e n d a n c e o f a B r i l l o u i n s a t e l l i t e
I
a f t e r a s t e p p r e s s u r e f r o m 0 . 2 2 t o 0.23 GPa.
8.
8.
w 0 Fig. 6 Idem f i g u r e 5 f o r a s t e p p r e s s u r e from 0.22 t o 0.21 GPa.
8
I
t El 460
8dO
12'00
Fig. 7 T i m e d e p e n d a n c e o f computed K, a s t e p p r e s s u r e f r o m 0 . 2 2 t o 0 . 2 3 GPa.
id00 for
176
8.
I
0
W
7. 7. 1E-06
7. 7.z I n
100
200
300
Fig. 8 Idem f i g u r e 7 f o r a s t e p p r e s s u r e f r o m 0.22 t o 0.21 GPa.
4d0
1E-08 Fig. 9
Is+
d
*A t/tO 0.2 0.4
0.6 0.8
i
R e l a x a t i o n t i m e 2 as a f u n c t i o n of
t / t o f o r P = 0.20 GPa rn
A %
D(t)7> 1 to = 0.01 ms
D(t)Cc 1
o
to = 1 ms
0.042
0.036 0.034
0m03 0.03
F i g . 10
+
Time dependance o f V f / V o P = P
for
= 0.2 GPa
m to = 0.01 ms
t
0
=
Is