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INTERNAL FRICTION AND RHEOLOGICAL BEHAVIOUR OF GLASSES NEAR Tg
J. Perez
To cite this version:
J. Perez. INTERNAL FRICTION AND RHEOLOGICAL BEHAVIOUR OF GLASSES NEAR Tg.
Journal de Physique Colloques, 1985, 46 (C10), pp.C10-427-C10-437. �10.1051/jphyscol:19851096�.
�jpa-00225481�
JOURNAL DE PHYSIQUE
Colloque C10, supplbment au n012, Tome 46, dkcembre 1985 page C10-427
INTERNAL FRICTION AND RHEOLOGICAL BEHAVIOUR OF GLASSES NEAR Tg
J. PEREZ
Groupes d'Etudes d e MBtallurgie Physique et d e Physique des Matkriaux, LA CNRS 341, INSA de Lyon, B s t . 502,
69621 Villeurbanne Cedex, France
RBsum6 - Les principales caract6ristiques exp6rimentales concernant le frottement int6rieur et le comportement rh6ologique des solides vitreux vers la transition vitreuse, sont rappel6es ; une dist inction est faite entre les donn6es maintenant bien 6tablies et celles plus r6centes. L1interpr6ta- tion de l'ensemble exige l'intervention de concepts nouveaux : celui de
"d6fauttt oic serait localis6 le cisaillement et celui de mouvement atomique (mol6culaire) corr616s. De nouvelles expressions sont 6tablies et la loi de
~ohlrauschexp[ -(t/~)b] se trouve introduite i partir des concepts physiques pr6c6dents ; ces expressions sont confront4es aux donn6es exp6rimentales.
Abstract - The main experimental features about internal friction and rheo- logical behavior of glasses near the glass transition temperature Tg are recalled. Some are now well established, others have been obtained recently.
In order to interprete all these new ideas have to be developped : hence, the concepts of "defect" on whid: local shear is obtained and of correlated atomic (molecular) movements, are applied to non-crystal line solids. New Eq.
including the Kohlrausch law exp [ - (t/~)~] emerge naturally from such physical basis and are discussed by reference to experimental data.
I - INTRODUCTION
There have been numerous investigations on the homogeneous flow and anelastic- plastic deformation of vitreous solids. Most attention has been given to rheologi- cal behaviour of glasses near glass transition temperature Tg and several review articles have been published in this field /1,2,3/.
The main interest of internal friction studies done with glasses is connected to the informations thus obtained about atomic (molecular) mobi 1 i ty in this state of the mattes as internal friction is a very sensitive and effective tool to detect inter- nal atomic rearrangements. Atomic movements are usually associated with lattice defects. Since vitreous solids could be considered as extremely defective solids it is expected that they may exhibit a considerable amount of internal friction espe- cially when the temperature is increased approaching the glass transition tempera- ture Tg. Actually it is generally admitted that long range atomic (molecular) move- ments are only possible at temperature equal or higher than Tg although more local movements could exist at lower temperature /4/.
In the work reported in this paper, attempts were made to bring some new contribu- tion in the analysis of internal friction of glasses near Tg. In this aim the main experimental and theoretical features will be firstly recalled. In a second part, new results or results recently'obtained will be presented in order to focus our attention on the peculiar points which must be taken into account to improve the theories previously proposed. A new approach about the interpretation of atomic (molecular) mobility in glasses will be discussed in the third part which will be followed by the interpretation of anelasticity and viscous glow of glasses in the glass temperature region. As a conclusion, the vew thus presented will be tested with the available literature data.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851096
I 1 - INTERNAL FRICTION AND RHEOLOGICAL PROPERTIES OF GLASSES NEAR Tg : THE WELL ESTABLISHED FEATURES
As numerous experimental r e s u l t s a r e reviewed i n references /1,2,3/, o n l y t h e main p o i n t s w i l l be summarised i n t h e f o l l o w i n g .
a) The i n t e r n a l f r i c t i o n t g y increases and t h e modulus decreases as t h e temperature i s increased. I n t h e g l a s s t r a n s i t i o n temperature region, t g increases f u r t h e r i n t h e case o f oxides o r m e t a l l i c glasses b u t a maximum i s observed i n t h e case o f ma- cromolecular glasses due t o t h e r u b b e r y e f f e c t wich i s otherwise followed, a t h i g h e r temperature, by t h e d i f f u s i o n o f entangled chains.
b ) From t h e r e o l o g i c a l p o i n t o f vew t h e former and t h e l a t t e r case correspond t o a Maxwellian and a B u r g e r ' s behaviour r e s p e c t i v e l y .
c ) A c t u a l l y , t h e r e i s a broad d i s t r i b u t i o n o f r e l a x a t i o n times and t h e t h e o r y o f l i n e a r v i s c o e l a s t i c i t y gives r e l a t i o n s d e s c r i b i n g t h e s t o r a g e and l o s s modulus G and compliance J :
tgcQ = G"/G1 = J " / J '
w i t h s u b s c r i p t s R and U corresponding t o r e l a x e d and unrelaxed values r e s p e c t i v e l y ; H(Z) i s t h e r e l a x a t i o n t i m e spectrum and L ( t ) t h e r e t a r d a t i o n t i m e spectrum.
But these r e l a t i o n s l e a d o n l y t o a phenomenological d e s c r i p t i o n o f r e s u l t s and t h e p h y s i c a l b a s i s o f such a d e s c r i p t i o n i s g e n e r a l l y unclear.
d) I n o r d e r t o use more convenient r e l a t i o n s having a s i m p l e r a n a l y t i c form, o t h e r expressions have been proposed l e t us mention :
- t h e Davidson-Cole equation : ~ ' ( i w ) = l/Gu [l + ( i u c M ) - l t s(l t i w 5 ) - b } which has been shown t o be n e a r l y i d e n t i c a l t o Zener's p a r a b o l i c model /3/.
- t h e B.E.L. equation /5/ ~ * ( i w ) = l/Gu \! + ( i u r M ) - l t 2 K (iwrM)0.5} Although e m p i r i c a l i n t h e i r o r i g i n , s u c h equa i o n s f i t g e n e r a l l y f a i r l y w e l l w i t h experimental d a t a i n t h e case o f m o l e c u l a r glasses /5/ and o x i d e o r m e t a l l i c glasses /3/.
I n some cases, authors have made attempts t o g i v e a p h y s i c a l basis t o such equation.
Thus, P h i l l i p s e t a l . /6/, extending t h e work o f Glarum /7/ found an expression e q u i v a l e n t t o be BEL equation ; th e model proposed by Glarum i n v o l v e s t h e concept o f a d e f e c t i n t h e l i q u i d s t r u c t u r e , moving by a process o f d i f f u s i o n through t h e l i q u i d : i n t h e presence o f such a d e f e c t , i n s t a n t a n n e o u s r e o r i e n t a t i o n o f a molecule i s p o s s i b l e ; two c h a r a c t e r i s t i c t i m e s a r e i n t r o d u c e d t 1 ( m o l e c u l a r r e o r i e n t a t i o n i n t h e absence o f d e f e c t s ) and ZD ( d i f f u s i o n ) . Another approach has been p u t f o r w a r d ( 8 ) i n terms o f thermomechanical a c t i v a t i o n (mean t i m e Z 1 ) o f some s i t e s l e a d i n g t o t h e f o r m a t i o n o f sheared microdomains (smd) ; such smd c o u l d be extended w i t h t h e assistance o f d i f f u s i o n n a l process, t h u s i n d u c i n g t h e l o s t o f t h e l o c a l memory o f s t r e s s (mean t i m e r2) ; such a concept have l e a d t o an expression v e r y s i m i l a r t o t h e Davidson-Cole equation and a p p l i e d t o experimental r e s u l t s obtained w i t h o x i d e glasses /3,8/.
I n t h e case o f macromolecular glasses, t h e s i t u a t i o n i s more complicated as t h e rub- b e r y e f f e c t and d i f f u s i o n o f i n t a n g l e d chains must be taken i n t o account /9/.
e ) The r e l a x a t i o n t i m e obtained by e i t h e r expression p r e c e d i n g l y r e c a l l e d i s ob- v i o u s l y temperature dependant : an Arhenius law i s observed a t T<Tg and a Tamann- Vogel-Fulcher (TVF) v a r i a t i o n ( ' t o < exp(cte/(T-To)) i s g e n e r a l l y v e r i f i e d a t T>Tg
/lo/.
It i s w e l l know, now /11/, t h a t v i t r e o u s s o l i d s , m e t a l l i c as w e l l as non m e t a l l i c , are i n s t a t e s which a r e c o n f i g u r a t i o n a l l y frozen. Upon annealing, these s o l i d s may r e l a x c o n f i g u r a t i o n a l l y w i t h o u t c r y s t a l 1 iz a t i o n . Thus two t y p e s o f atomic t r a n s p o r t behaviour may be d i s t i n g h i s h e d . Indeed, t h e r a t e constant f o r atomic movement v a r i e s
roughly as the probability of some critical excitation of the configuration required for the movement) multiplied by the frequency of rearrangement of the system in its critical state. Thus at T7Tg the former factor is temperature dependant since the material is in a metastable equilibrium leading to the TVF equation (equilibrium measurements). On the contrary, at T<Tg,only the latter factor depends on tempera- ture leading to the Arrhenius law (iso-configurationnal measurements).
d) Vitreous solids are well known as structural relaxation sensitive materials. Thus internal friction of glasses is decreased during aging at temperature lower but not too different than Tg. During the same aging treatement,the storage modulus increases. This is a very general behaviour observed with oxide glasses /12/, metal 1 ic glasses /12/, selenium glass /14/, organic glasses /15/.
To state more clearly the interpretation of internal friction and other rheological properties of glassy solids near the glass transition temperature, other experimental informations are needed. As new results have recently been obtained, we suggest to emphasize these in the following part.
111 - EXPERIMENTAL : NEW RESULTS OR PARTICULAR CHARACTERISTICS 3.1 - Frequency dependance
The temperature dependance of internal friction just mentionned above is connected to frequency dependance ; then, internal friction and modulus measurements made by frequency scanning in isothermal conditions bring us very useful1 data. For instance the figure 1 shows results obtained in the case of an oxide glass /a/. It was ob- served that t g y varies as w - ~ . Thus, b was determined in a large temperature range between T ( Tg and T > T g : it appears that b increases with the temperature from 0.3 to 0.94. Furthermore, it seems, at least at lowest temperature, that b, which is the slope of the curves log tgy - log P (P = 2r /\r> is the period of the cyclic stress applied on the material during internal friction measurements), increases with P.
The non-linear frequency dependance has to be compared to the non exponential time dependance of strain (stress) observed during a creep (stress relaxation) test : the Kohlrausch's Eq. implying an exponential factor such as exp [-(t~)~] with Otb<l is generally accepted to obtain a good fit between experimental data and calculated curves..
Such a behaviour is also observed when the dielectric response of materials is stu- died ; this point has been revieved by Jonscher /16/ who concluded to the existence of a remarkable "universality" of frequency and time responses suggesting the dom- inant role of many-body interactions.
3.2 - Relaxation time and activation energy
The results of figure 1 were analysed using differents methods /a/ but all have lead to determine a characteristic time which was plotted against 1/T : an apparent ac- tivation energy (4,3 eV) and a frequency factor (103~ S-1) were found but these values were considered to have no physical meaning. In order to distinguish equilibrium from iso-configurationnal measurements as explained in the preceding part, creep and recovery deformation measurements were made with an other oxide glass /17/. Assuming the rules of linear viscoelasticity, the experimental data were analysed as explained in the figure 2 and the slope l/n of the curves Jvisco 1. (t) are shown as a function of 1/T in the figure 3. Each point has been obtained agter a thermal treatment of the material allowing us to separate results corresponding to the material in equilibrium or isoconfigurational states. In a first approximation, a characteristic time 77= r) /Gu can be considered and we obtain :
- Z = To e x p u/kT w i t h u = (2.8 + 0.1) eV and To = 5.10(-15+1) f o r iso-configurational conditions.
- T d e x p A/T-To with T o + 790 K for metastable equilibrium state.
Such a result is in agreement with the general pattern about glass behaviour near Tg. Although, it can be noticed that even in iso-configurationnal state, the ap- parent activation energy and the frequency factor seem rather large by comparison to values expected from the assumptions of simple atomic (molecular) jump and "lattice"
frequency (1013 s). Such large values are also obtained as well with organic macromolecular glasses /la/ as with amorphous Se /19/.
to'-
to?.
lo-'-
Fig. 1 - Internal friction and storage modulus of an oxide glass /8/.
Fig. 2 - Compliance as obtained from a creep test ; separation of components : Janel (t) = Janel (t-tl) and Jviscopl (t)
= Jtot(t)-J anel (t)
To conclude, let us recall that Berry /13/ nearly ten years ago, did mention similar features in the case of metal- lic glasses, implying a frequency fac- tor 1025 s-1, which was considered as an "apparently ridiculous result".
Is this large value of the apparent activation energy observed at lower temperature ? The figure 4 concerning results obtained with the same material as for results of the figure 3, shows that this is not the case since the application of the time superposition principle, although not strictly veri- fied, need a shift factor varying with the temperature with an apparent ac- tivation energy equal to (0.9 + 0.l)eV.
Fig. 3 - Oxide glass : Variation of the slope d Jviscop(t)/dt with temperature either in isoconfigurational condition
( X ) or in metastable equilibrium ( a ) .
3.3 - Aging e f f e c t
Many p a p e r s have been p u b l i s h e d on aging e f f e c t observed w i t h glasses. A l a r g e l y developped r e v i e w o f t h e s i t u a - t i o n c o n c e r n i n g p o l y m e r i c o r g a n i c glasses can be found i n S t r u i k ' s book / 2 0 / a l t h o u g h some f e a t u r e s may be argued. The main p o i n t p u t forward by s t r u i k i s t h a t aging induces a s h i f t o f t h e whole spectrum o f r e l a x a t i o n ( r e - t a r d a t i o n ) times w i t h o u t deformation o f t h i s spectrum. Such a p o i n t o f view i s questionnable as r e s u l t s o b t a i n e d w i t h g l a s s y selenium r a t h e r support t h e idea o f a m o d i f i c a t i o n o f t h e f o r m o f F i g . 4 - I n t e r n a l f r i c t i o n as a f u n c t i o n t h e s p e c t r u m d u r i n g a g i n g . More frequency : experimental ( f i l l l i n e ) and r e c e n t l y , CHAI and Mc CRUM /21/ have c a l c u l a t e d ( d o t t e d l i n e ) curves (same s u b j e c t e d t h e s t r u i k i m o d e l t o an m a t e r i a l as f o r Fig. 3 ) . e x p e r i m e n t a l v e r i f i c a t i o n . F o r i s o t a c t i c p o l y p r o p y l e n e w h i c h was quenched f r o m 353 K t o 313 K, t h e y found t h a t t h e d e r i v a t i v e d l o g J/d l o g t v a r i e d smoothly f r o m - 0.749 t o - 0.858 as t h e aging t i m e v a r i e d f r o m 180 s t o 1.91 105 s, t h u s i m p l y i n g a m o d i f i c a t i o n o f t h e f o r m o f t h e r e t a r d a t i o n spectrum. Hence, a t l e a s t f o r t h i s m a t e r i a l , i t appears t h a t t h e StruiWmodel i s n o t c o n s i s t a n t w i t h t h e experimental f i n d i n g o f CHAI and Mc CRUM. T h i s p o i n t may be connected t o t h e f a c t t h a t t h e r a t i o Jviscopl./Janel i s decreased a f t e r aging /22, 23/.
On another hand a r e v e r s i b l e s t r u c t u r a l r e l a x a t i o n i s sometimes mentionned f r o m de- t a i l e d s t u d y o f t h e e f f e c t o f a n n e a l i n g on t h e i n t e r n a l f r i c t i o n o f m e t a l l i c glasses : i t was found /24, 25/ t h a t t h e glass reaches, a f t e r an extended p e r i o d o f annealing an i n t e r n a l pseudo-equilibrium s t a t e , which i s f u n c t i o n o f t h e annealing temperature. A change i n t h e annealing temperature r e s u l t s i n a r e v e r s i b l e change f r o m one such s t a t e t o another.
To sum up, a success f u l l t h e o r y o f atomic (molecular) m o b i l i t y i n glasses near Tg must be c o n s i s t e n t w i t h those c h a r a c t e r i s t i c s o f aging phenomena and t h i s has t o be
i n c l u d e d i n whatever k i n d o f model o f i n t e r n a l f r i c t i o n .
I V - ANELASTICITY AND VISCOPLASTICITY OF GLASSES NEAR Tg - PHYSICAL BASIS FOR A THEORY
4.1 - General grounds
I n a r e c e n t paper /22/, we proposed t h a t t h e b a s i c deformation mechanism i s t h e n u c l e a t i o n o f shear microdomains. The n u c l e a t i o n occurs under t h e e f f e c t o f t h e ap- p l i e d s t r e s s and i s t h e r m a l l y assisted. A general case o f shear microdomains was i l l u s t r a t e d as f o l l o w /22/ : t h e shear i s along a s u r f a c e S and t h e c o o p e r a t i v e atomic rearrangement occurs i n s i d e t h e volume o f m a t t e r l i m i t e d by a s u r f a c e z . The
curve Cn d e f i n e d by t h e i n t e r s e c t i o n o f Z and S, separates t h e area S1 where shear has occured f r o m t h e non sheared p a r t o f S. I n mechanics o f continuous media, t h e l i n e Cn i s a d i s l o c a t i o n loop ; i n amorphous s o l i d s , d i s l o c a t i o n s , as f a r as t h i s concept i s v a l i d , would be o f t h e Somigliana type, as L I /26/ has p o i n t e d out. The n u c l e a t i o n r a t e has been c a l c u l a t e d by several authors b u t r e s u l t s appear t o be u n r e a l i s t i c as f a r as such n u c l e a t i o n occurs every where i n t h e amorphous s o l i d . Therefore, i t was proposed /22/ t h a t n u c l e a t i o n can occur o n l y i n those r e g i o n s where r e s i s t a n c e t o shear i s a p p r e c i a b l y weaker t h a n i n t h e r e s t o f t h e m a t e r i a l . Such s o f t s i t e s may be regarded as "defect< The thermomechanical a c t i v a t i o n o f a d e f e c t (mean t i m e 2 1 ) may l e a d t o t h e f o r m a t i o n o f a s.m.d.. When t h e s t r e s s i s removed, t h e s o l i d r e c o v e r s i t s p r e v i o u s c o n f i g u r a t i o n t h u s c o r r e s p o n d i n g t o a n e l a s t i c behaviour. I n o r d e r t o o b t a i n p l a s t i c deformation t h e growth o f t h e s a d . must be n e c e s s a r i l y invoked, b u t t h e l i n e Cn b e i n g a Somigliana d i s l o c a t i o n , i s 3 s e s s i l e defect. Nevertheless, such a growth can be obtained through d i f f u s i o n n a l mechanism : t h i s growth covers a d i s t a n c e (mean t i m e r2) a t which t h e l i n e Cn
losses its identity by combination with other similar lines formed from the neigh- bouring defects, and become ineffective. This leads to a viscoplastic behaviour.
Following these hypothesis, a quantitative description of the non-elastic deforma- tion of glassy solids was given /22/ in terms of the following Eq. :
with A = No vl va f A?$ /kT ; No : number of defect (volume vl) per unit volume ; va
: activation volume ; A 8 : local shear ; f : schmid factor ; 1 / Z = l/Cl + l/ q.
By assuming a distribution of values of 2 1 (depending on distribution of either activation energy or frequency factor of both) we, recently obtained from Eq. (1) approximated Eq. in order to describe rheological behaviour of glasses near Tg /3/ :
J(t) 2 (l/GU) +
g$2\B
- exp,(t/z)9) + 2 nt/(cl,rz) and Jqio) 2 (l/~~){l + (iwrM)-l + s (1 + i u'c )-b)with CM = (el +T$/ZAG, and S = G ~ A (22-21)73(21+5)2
These Eq. can thus be "explained" by a suitable choice of the weight distribution of Cl. However, this approach is microscopically arbitrary and does not explain the universality of Kohlrausch's law e~~[-(t/~)~]. So, instead of the picture of parallel relaxation, in which each degree of freedom i relaxes independantly with characteristic time Zi, Palmer et al. /27/ considered a series interpretation, involving a distribution having a microscopic source in the correlations between different degrees of freedom. In other words, a hierarchy of degrees of freedom, from fast to slow is involved : the fastest might correspond to single-atom motion ; other atoms, or groups of atoms, might only be able to move appreciably when several of the fastest happen to be placed in just the right way, leaving "free volume" or weakening a bond.
Hence as Palmer et al. /27/ let us consider a discrete series of levels n = 0,1,2 ...
with the degrees of freedom in level n represented by Nn Ising spins. Each spin in level n is only free to change if a condition on some spins in le el n is satisfied, e.g. pn-1 spins in level n-1 attain a particular state of their 2'"-1 possible ones.
The average relaxation time 2; will be rebated by : Zn = 2 "-lZn-l
giving n-4
tn = 1 exp ( Z Ln 2. Pk) (4)
One of the possibilities discussed by Palmerset a1 was Ln 2. ( 1 ~ = k-P so that
Eq. (4) becomes : 9-?
2,= elp ~1 k-P (5)
In order to have a clearer connection with the microstructure of the material, we propose to modify the approach of Palmer et a1 in the two following ways :
a) If to is the mean time necessary for the system to go from levels k to k + 1, the ratio k = tc/tn may be introduced (tc is the time after which the system is at level - ..
k). ~ q . (5)"cai be-re laced by - u
~ ( t ) ; Tl exp ( p1 2 (tk/to)-;l) = 2 1 eXP ( P1 u - ~ du) with = tlt0
giving C,
z(t) = % expi Pl (1-ul-p)/(p-l)] (6) 0 1 may be regarded as a structural parameter characterizing the correlation between the different atomic movements in the glass which, as a solid, is a strongly interacting system. For = 0 the matter surrounding the atom moving firstly, is not sensitive to such a movement and T(t) reduces to TI. Forpi = 1, the primary movement induces numerous correlated atomic movements and Z(t) can reach the maximum value ZmaX.. If experimental time texp is shorter than tmax., glassy systems clearly breaks ergodicity so that equi llbrium distributions in configuration space are not usable. On the contrary, if texp. 7 Tmax., the system is ergodic, so that statistics equilibrium laws becomes correct and relaxation phenomena require pure exponential relations.
We furthermore expect such features to be relevant to glassy materials when (C rmax (a.
This is the case whith p = 1 + & (&(l) and Eq. (6) can be written
Zmax = r 1 exp(Pl/e) ( 7 4
Finally, Z;nax must increase when the temperature is decreased, more rapidly than pl ; this is bbtained with & = €1 T/T giving withti = Q expCul/kT)
rmax. = P- expt (ul + K TI/ E~)/xT) (7b) b) An equivalent description can be given on the basis of the formation of a s.m.d.
from the activation of a defect (mean time TI) as discussed above. But, instead of atomic diffusion assisted growth of s.m.d., let us assume a repeated nucleation at and behind the slip front of the Somigliana dislocation Cn /26,28/ ; such nucleated micro-loops might be structurally equivalent to the pseudo-spins considered by Palmer et a1 /27/. In such conditions,
- the activation of the defect in order to form a s.m.d. need the time
- the preceding ,phenomenon followed by the nucleation of the micro-loops No 1 need the time r 2 =Q exp (hul/kT)
- with the micro loop No n : 2-6 = 2;-1 exp(&un-l/kT)
giving n - l
= T l exp( Z 4 ui/kT)
i.?
Similarly to the preceding paragraph ( A U ~ = Aul x i-'l ; P1 = 1 + E l ; E1(l) one can calculate
= rl exp ( A U .i;P" x 4 z -4-7
and, for n-+m , Emax. = ca exp((u1 + hul/ ~i)/k'i) which is equivalent to Eq. (7b) with Aul = P I k Ti.
In fact *the t h e kmax. thus introduced might be similar to the time r 2 /22/ as
rmax corresponds to the formation of a s.m.d whose size has just the value above which the growth of the Smd is irreversible due to annihilation with neighbouring dislocation loops like Cn ; besides, the micro-loops nucleated at the front of two Somigliana dislocations being annhihilated must be of reverse character and this leads to a diffusion transfer of matter.
The present description may be compared to the work of Ngai and White /28/ who pro- posed an unified theory of low frequency dynamic response of condensed matter : the relaxation of a primary species (side group, chain segment and, more generally, a dipole ...) is followed by an interaction with correlated states presenting low ener- gy excitations. To underline this comparison, let us calculate Z(t) from Eq. (6) by developping it in series around & = 0 : at the first order (which corresponds to p=l), we have :
r(t) = (tito) pl which can be identified to the result of Ngai and White :
C(t) = (t/tc)"
with t'l = rl exp nib ; = 0.577 ; n 4 1 ; tc = h/Ec and Ec is the upper "cutoff"
of the correlated state excitation. The expansion to the second order is more interesting : e(t) = r1 (t/t )a
with a = r1 91 - ~ / 2 loqu) (8) (9)
Finally, by taking into account of the condition Z(t) = rmaX for t = rmax we obtain q t ) = rma,l-a ta.
As said above, the system is ergodic when t 3 ZTmax. : relaxation phenomena then require pure exponential relations implying the characteristic time Cmax.
to sum up, as far as Z(t) C Zmax, the elemental shear process resulting from hierarchically constrained atomic movements, is reversible (anelasticity) ; for higher temperature, or longer time, Z(t) = Zmax and shear is irreversible (viscoplasticity).
4.2 - Calculation of mechanical response
A general representation of the stress relaxation modulus (or any other rheological properties) must contain the relaxation time T(t) but also a h(r) distribution having a microscopic source in the correlations between different degrees of freedom. Palmer et al. 1271 considered the weights hn = ho ),-n ; as pointed out by these authors themselves, thereds many ways to choose the h(r) distribution ; but in all cases, the condition /h('~) d Z = A (A being the same constant as in Eq.
(1)) must be fulfilled; Thus, t'he following pattern can be proposed :
(i) In glassy solids, atomic movements induced by both the stress and thermal activation occur on special sites ("defects", the concentration of which is No per
u n i t volume,
( i i ) each movement need a d e l a y between 5 and q t ) ,
( i i i ) t h e number o f p o s s i b l e values o f t h i s d e l a y i s dependant on t h e c h a r a c t e r i s - t i c s o f t h e defects so t h a t t h e c o n s t a n t A o f t h e Eq. ( 1 ) must be r e p l a c e d by t h e product AIP(Z), P(z) r e p r e s e n t i n g t h e f r a c t i o n o f t h e t o t a l number o f d e f e c t s , being a c t i v a t e d w i t h t h e t i m e t . We suggest t o t e s t t h e symnetric Gaussian r e l a t i o n P(z) = 1 / B F exp
6-
LO^ s / c ~ ~ ~ ) / B ] ~ ~ . Such a r e l a t i o n would t a k e i n t o account :- t h e proba i l ~ t y t o have a move e n t n e c e s s i t a t i n g a g i v e n delay,
- t h e s t r u c t u r a l source o f t h e c h a r a c t e r i s t i c s o f t h e movement,
- t h e c o u p l i n g s t r e n g t h o f t h e processus.
By c o n s i d e r i n g Eq.(7), i t appears t h a t any v a r i a t i o n o f t h e s t r u c t u r a l f a c t o r P.1 induces a v a r i a t i o n o f b o t h apparent a c t i v a t i o n energy and frequency f a c t o r ; th i s suggest t o use a f a c t o r B o f t h e f o r m B = Bo + Bl/T.
Thus, i n o r d e r t o t r y some q u a n t i t a t i v e d e s c r i p t i o n o f t h e mechanical response o f g l a s s y s o l i d s , two extreme cases may be considered :
i ) f o r low temperatures (texp. 4 . Cmax), t h e mechanical response i s m a i n l y s e n s i t i v e t o t h e d i s t r i b u t i o n r e l a t i o n P ( r ) . Using tgcQ = &(TI * W ( l + ~ ~ c ~ ) # o . s A ( T ) t h e r e l a x a t i o n s t r e n g t h can be taken from Eq. (3), r e c a l r i n g than i n t e r n a l f r i c t i o n comes o n l y f r o m t h e a n e l a s t i c behaviour i n t h i s temperature range : t h u s one has :
t g (q = 0.5 A Gu P(Z) (10)
i i ) f o r h i g h temperatures ( t e x p & t m a X ) a l l values o f C l < Z ( t ) 4 Tmax may i n t e r v e n e and a l l t h e d e f e c t s may be concerned. The mechanical response i s , then, m a i n l y s e n s i t i v e t o t h e form o f athe expression g i v i n g C ( t ) . The c a l c u l a t i o n o f t h e compliance i n g l a s s y m a t e r i a l s was made e a r l i e r /22/ b u t t h e t i m e Z1 must be r e p l a c e d b y 2 ( t ) and some steps o f t h e c a l c u l a t i o n t h u s a r e m o d i f i e d : t h e most i m p o r t a n t m o d i f i c a t i o n i s i n i n t e g r a t i n g Eq. (6) ( r e f . 22) as t h e c h a r a c t e r i s t i c t i m e i s now t i m e dependant. A good approximation can be found as f o l l o w : on one hand, t h e a n e l a s t i c behaviour i s r u l e d o u t by t h e Eq. d N ( t ) / d t = - N ( t ) / T ( t ) N ( t ) : number of non a c t i v e d d e f e c t s p e r u n i t volume g i v i n g , w i t h Eq. ( 8 ) -
N ( t ) = No exp [ - l / b ( t / zm,,)b] w i t h b = 1-a = 1- p1 + rl 1 / 2 Lag"
The s t r a i n r a t e i s t h e n ranel = rl ( N ( t ) / Z ( t ) ) ( f Q va/kT) ( 1 1 j
and s i m i l a r l y t v i s c o = g2 (No/rmax) ( f 6 va'/kT)
B y a d m i t t i n g t h a t r l = A g v l / l and va a r e n o t t o o d i f f e r e n t o f x2 and v a 8 r e s p e c t i v e l y , we have t h e compliance
J ( t ) = l / G u + A {l - exp [ - l / b ( t / T ~ ~ ~ ) ~ ] ) + A t / C m a x (12) w i t h may be compared t o Eq. (2). S j m i l a r l y t o Palmer e t a1 /27/ and t o Ngai /29/, we have obtained a Kohlrausch e x p o n e n t i a l i s based upon p h y s i c a l basis. The same Eq. ( 3 ) can be used t o c a l c u l a t e i n t e r n a l f r i c t i o n b u t we have now :
cM = Tmax/A Gu and 'C = b l / b '=ma,. (13)
V - COMPARISON WITH EXPERIMENTAL RESULTS AND DISCUSSION
We f i r s t l y emphasize t h a t t h e r e s u l t s fo F i g . 1 now can be c l e a r l y understood : on one hand, as t h e temperature i s increased, t h e v a l u e o f b increases, approaching t h e u n i t when T 2 1.2 Tg ; t h i s appears t o be induced by a decrease o f t h e f a c t o r P l (see Eq. ( 9 ) and ( 1 1 ) ) as t h e c o r r e l a t i o n between t h e p r e l i m i n a r y and subsequent atomic movements becomes l e s s and l e s s e f f i c i e n t . On t h e o t h e r hand, Eq. (11) shows us t h a t b m i g h t increase w i t h t h e t i m e and t h i s c o u l d e x p l a i n t h a t b i s h i g h e r when t h e p e r i o d o f measurement i s l a r g e r . To conclude about t h e r e s u l t s o f F i g . 1 we o n l y mention t h a t i t i s a l s o p o s s i b l e t o understand t h e i r p r e s e n t a t i o n i n t h e Cole-Cole diagram /8/ showing t h a t ( i ) b i s constant a t T 4 Tg ( i i ) b increases w i t h T a t T >
Tg and ( i i i ) t h e f o r m o f t h e c o l e - c o l e diagram /3/ duggests = 5 (Eq. ( 3 ) ) and i t happens t h a t t h i s i s j u s t t h e l i m i t corresponding t o b = 1 (see Eq. (12)).
A chalenge more d i f f i c u l t i s t o e x p l a i n a t l e a s t s e m i - q u a n t i t a t i v e l y i n t h e scope o f t h e p r e s e n t t h e o r y r e s u l t s observed w i t h t h e same m a t e r i a l i n two d i f f e r e n t tempera- t u r e range ( f i g . 3 and 4). L e t us consider t h e r e s u l t s o f Fig. 3 : Eq. (12) i n d i - c a t e s t h a t t h e slope 1/q i s equal t o A/ rmax. I n i s o c o n f i g u r a t i o n a l c o n d i t i o n s A i s found t o be c l o s e l y independant o f t h e temperature T and equal t o (1.3+ 0.1) .lo-
'lo pa-1 /17/ ; hence Tmax v a r i e s as a f u n c t i o n o f T w i t h an apparent a c t i v a t i o n energy equal t o (2.8 + 0.l)eV and a frequency f a c t o r equal t o 1016+1 S-1, On t h e o t h e r hand, i n metastable c o n d i t i o n s t h e slope i s d i f f e r e n t and v a r i e s w i t h T due t o b o t h A and Tmax s t r u c t u r e dependance.
Internal friction measurements (Fig. 4) does correspond to isoconfigurationnal conditions ; so, it is worthwhile to verify if the preceding value of A and Cmax
(extrapolated towards the temperature range 473-673 K for the latter) can lead to calculated results in agreement with experiment : using Eq. (10) and the gaussian expression for P(P), theoritical curves are calculated (dotted line : Fig. 4);
Although the fit is very poor, the following features can be deduced : (i) the ap- parent activation energy is the same for both calculated and experimental curves (0.9 + 0.1 eV) ; (ii) B is the lower, the temperature is high : the value B .v 0 can be extrapolated for T #: 1.2 Tg meaning that all moving species behave in the same way without correlation effects ; this is in agreement with the conclusion deduced from observing b # 1 in the same temperature range. (iii) the Fig. 5 shows the variation of Tmax obtained from both sets of results (Fig. 2 and 4) ; consequently, the curves P(T) calculated for different temperatures are shown in the same Fig. 5.
Fig. 5 - Time relaxation spectra for 3 temperatures and variation of rmax (isocon- f igurationnal : a (creep test) and o (internal friction) ; metastable equilibrium :
+) and of B (x) with temperature.
Such curves although being not in agreement with the thermorheological simplicity assumption may be compared to relaxation spectra shown, by Chen /30/ : in both cases, the whole spectrum lies either to the left of texp, so that the whole system undergoes frequent configurational transformation and is liquid-like, or, during cooling, the number of solid-like regions ( T , texp) increases until1 they form an infinite cluster. In the latter case, the overall configuration is frozen-in at Tg and the glass transition is then related to percolation process /31/. It may be em- phasized however, that even at T < Tg, atomic (molecular) movements are possible in those defects for which t(t) 4 tex thus leading to anelastic phenomena.
On another hand in metastable con&'tion, the variation of rmax with T shown in Fig.
5 illustrates the decreasing correlation effect when the temperature is increased : for T # 1.2 Tg each movement could correspond to the characteristic time 2-1 which obeys to an Arrhenius law with an activation energy equal to that obtained from low temperature internal friction measurements (0.9 eV) thus justifizing that p relaxa- tion times generally merge in d relaxation times at high temperature /2/. Further- more, the viscosity is 4 = rmax/A so that the variation of A (i.e. the defect concentration, with the temperature results in a temperature dependance of viscosity (or diffusion) which obeys to the well known TVF Eq. /9/.
The aging experiments may be explained, too, in the light of the present theory : as an example the Fig. 6 shows experimental curves J(t) obtained with the same oxide
glass as that giving results shown in Fig. 3 and 4. A good fit is obtained with curves calculated with Eq. (12) and, using the method summarized in Fig. 2 the parameters A, Gax, b are obtained successively. Thus, during aging, (i)A decreases in agreement with the decreasing defect concentration /22/ (ii) b decreases corresponding to an increase of that is an increase of correlation phenomena ; such a feature similar to that discussed in ref. /21, 32/, results in an increase of
Gax and it can be verified through Eq. (7a) that both observed variation of Tmax and b are consistent. On another hand, reversible structural relaxation might be concerned with more local atomic rearrangementyhence should not be sensitive to such aging treatment, contrarily to lower values of ~7at)j ; this can be described by considering that the constant B of the gaussian distribution law decreases during a reversible structural relaxation.
Fig. 6 - Aging effect on the compliance (creep test and recovery of deformation made on the same material as for Fig. 3 and 4) ; experimental (full line) and calculated (dotted line) curves.
As a conclusion one may emphasize the main following points characterizing rheologi- cal properties of glasses near Tg : (a) atomic (molecular) movements in glasses occur in "defects" which are particular sites where entropy (disorder) is high and density is low ; (b) the main and preliminary movement involve the excitation of correlated states : this corresponds to a hierarchy of degrees of freedom, from fast to slow ; (c) when a stress is applied, the response of amorphous solids might be analysed in terms of (i) thermomechanical activation of defects leading to the for- mation of s.m. d. (main preliminary atomic movement) and (i i ) subsequent extension of the Somigliana dislocation boarding the smd through hierarchically correlated atomic movements ; (d) after the delay Gax, the extended loop is annihilated by combina- tion with neighbouring loops ; (e) the solid is mainly anelastic when texp .(. TmaX but viscous flow becomes important when texp approaches rmax ; (f) the value ofrmax is the higher the correlation effects are important, i.e. the temperature is low or the glass is structurally relaxed.
On the basis of such ideas a semiquantitative description of non-elastic deformation of glasses near Tg has been proposed : Eq. (lo), (12) and (13) are consistent with experimental features ; in these Eq. the Kohlrausch factor exp C-(t/ z)b]is intro- duced on physical arguments, the parameter b caracterizing the effectiveness of correlation effects.
REFERENCES
/1/ Perez J. and Gobin P.F., Rev. Phys. Ap., 12, (1977) 819.
/2/ Perez J. Cavaille J.Y., Etienne S., Fouquet F. and Guyot F., Ann. Phys., 8,
(1983) 417.
/3/ Perez J., J. de Phys., 2, (1983) C9-3.
/4/ Johari G., "Plast. Def. of Am. and Semi-cryst. mat.', Ed. Phys., (1982) 109.
/5/ Harrison G., "The dyn. prop. of supercoll. liq.", Ac. Press, (1976) 117.
/6/ Phillips M.C., Barlow A.J. and Lamb J., Proc. R. Soc. Land., A, (1972) 193.
/7/ Glarum S.M., The J. of Chem. Phys., 2, (1960) 639.
/8/ Perez J., Duperray B. and Lefevre D., J. of Non Cryst. Sol ., 44, (1981) 113.
/9/ Perez J., J. de Phys. Appl., acc. for public.
/lo/ Etienne S., Cavaille J.Y., Fouquet F. and Perez J., J. de Phys., 42, (1981)
C5-575.
/11/ Spaepen F. and Turnbull D., "Metal-Glasses", Ed. Am. Soc. Met., (1976) 673.
/12/ Borde C., Mai C. and Perez J., J. Non Cryst. Sol., 56, (1983) 399.
/13/ Berry B.S., "Metal-Glasses", Ed. Am. Soc. Met., (19z) 161.
/14/ Etienne S., Cavaill6 J.Y., Perez J., Bonjour E. and Calemczuk, J. de Phys., 43, (1982) C9-599.
/15/ Cavaille J.Y., Etienne S., Perez J., Monnerie L. and Johari G.P., accept. for publ. J. Chem. Phys..
/16/ Jonscher A.K., Nature, 267, (1977) 673.
/17/ Borde C., Thesis, LYON m 8 4 ) .
/18/ Cavaill6 J.Y., Etienne S., Perez J., Monnerie L. and Johari G.P., accept. for publ. Polymer.
/19/ Etienne S., Cavaill6 J.Y. and Perez J., J. de Phys., 44, (1983) C9-157.
/20/ Struik L.C.E., "Phys. Aging in Amorph. Pol. and others Mat.", Elsevier (1978).
/21/ Chai C.K. and Mc Crum N.C., Polymer, 21 (1980) 706.
/22/ Perez J., Acta Met., 32, (1984) 2133.-
/23/ Haimovich J. and ~gamrT., Mat. Sc and Engin., 61, (1983) 89.
/24/ Balanzat E., Thesis, GRENOBLE (1983).
/25/ Morito N. and Egami T., Acta Met., 32, (1984) 603.
/26/ LI J.C.M.. "Met. Glasses". Am. Soc. Met.. (1978) 224.
/27/ Palmer ~.t., Stein D.L., braha hams E. a& '~nderson P.N., Phys. Rev. Let., 2,
(1984) 958.
/28/ iscaig B., Symp. macromolecules, Cleveland (1983).
/29/ Ngai K.L. and White C.T., Phys. Rev. B, 20 (1979) 2475.
/30/ Chen H.S., J. Non Cryst. Sol., 46, (1981F289.
/31/ Cyrot M., J. de Phys., 41, (198m C8-107.
/32/ Ngai K.L. and Manning ,1 J. de Phys., 12 (1982) C9-607.
