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Memory effects in glass transition
68
Journal of Non-Crystalline Solids 84 (1986) 68-85 North-Holland, Amsterdam
M E M O R Y EFFECTS IN GLASS TRANSITION
Simon M. REKHSON General Electric Company, Nela Park, Cleveland, OH 44112, USA
Behavior of glass in the glass transition region is reviewed with the emphasis on memory effects. Three examples of memory effects analyzed are the behavior of the fictive temperature in the cross-over, sub-Tg relaxation and simple reheating experiments. In all cases, the non-equilibrium system reveals a temporary neglect of the equilibrium state by moving away from it. A multiexponential model of structural relaxation is employed to model this phenomenon and to investigate the behavior of separate relaxation mechanisms contributing to a total effect. The model characterizes the glassy state by a broad distribution of partial fictive temperatures - low, produced by fast relaxation mechanisms, and high, produced by slow mechanisms. During reheating, each mechanism approaches the equilibrium line Tf = T in two stages: first from above (because Tfi > T) and then from below (when due to the overshoot Tfi < T). In the lower part of the glass transition region where slow mechanisms are still in the first stage of a very slow decrease of their Tfi's, the fast mechanisms are finishing their approach to equilibrium by a vigorous increase of their Tfi's. The Tf measured in the experiment is the weighted average of all Tfi's and therefore reflects the behavior of fast mechanisms by increasing its value although Tf is still greater than the actual temperature T. A complex thermal treatment designed as a combination of quenching and annealing steps will emphasize different groups of mechanisms and produce many intriguing effects. All these effects can be explained as a superposition of quenched and annealed mechanisms approaching the equilibrium state from opposite directions.
1. Introduction
This paper describes the performance of the multiexponential model of structural relaxation in predicting and interpreting memory effects in the glass transition region. The memory effects are best illustrated by cross-over experiments and by sub-Tg relaxation peaks. As shown below, in both cases the non-equilibrium system reveals a temporary neglect of the equilibrium state by moving away from it. This is a striking behavior because the very existence of a relaxation process is entirely due to the deviation of the system from equilibrium. The rate of the relaxation process is approximately proportional to the magnitude of this deviation, and the process ends when the system returns to equilibrium. The reason for the system to neglect its future (the equilibrium state) is generally seen in the memory of its past. The subject of memory effects is of general interest because they have been observed in inorganic [1], organic polymeric [2,3], and metallic [4,5] glasses. In addition, they represent a challenging material for testing theoretical models of glass transition. 0022-3093/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S.M. Rekhson / Memory effects in glass transition
69
H
TEMPERATURE
(c)
OEl ~g TEMPERATURE
Fig. 1. Evolution of the property (a) in isothermal condition following a sudden cooling (1) and heating (2) to the same hold temperature; evolution of the property (b) and the temperature coefficient of the property (c) during fast and slow cooling and reheating.
As shown in ref. [6], the multiexponential model is a special case of the Narayanaswamy model [7] where the structural retardation function is given by a sum of weighted exponentials. The model was also formulated by means of thermodynamics [8-10] to describe the evolution of glass properties in terms of order parameters. The first multiexponential model, containing only two terms, was proposed and successfully used for explanation of the cross-over effect by Macedo and Napolitano [11]. Kovacs and co-workers [8] as well as Hodge and Berens [12] showed that the multiparameter model does predict the sub-Tg relaxation peaks observed in glasses reheated following quenching and long stabilization at low temperatures. This work made it clear that the origin of these peaks is entirely in the complex thermal history employed. This is in contrast to the secondary relaxation peaks, extensively studied by Johari and Goldstein [13-15], whose origin is in corresponding peaks in the distribution of relaxation times. The secondary relaxation peaks are seen in simple cooling/ reheating experiments. The subject of memory effects requires a good understanding of the glass transition phenomenon. In the interest of those who do not work directly in the area of glass transition we shall build up our discussion gradually, starting with a review of the basic features of glass behavior in the glass transition region. This is shown schematically in fig. 1 using the glass volume, enthalpy and their temperature derivatives as examples. Following an abrupt cooling, both properties will decrease in time (fig. la, curve 1) and following heating to the same temperature they will increase (curve 2). In both cases, the property eventually arrives at the same value which is, therefore, an equilibrium value of the property. A cooled glass sample arrives at the temperature of the
70
S.M. Rekhson / Memory effects in glass transition
isothermal hold with lower density and higher mobility than the heated sample, and this results in a higher rate of retardation. As a first approximation, the duration of retardation is characterized by the structural retardation time defined as the time necessary for the system to cover about two-thirds of the way to equilibrium. The structural retardation time increases exponentially with decreasing temperature - about an order of magnitude for each 20°C in the case of silicate glasses. It is this single factor which determines the very essence of the phenomenon called liquid-glass, or simply glass transition. This phenomenon is illustrated in Fig. lb by the volume, or enthalpy vs temperature curves obtained during cooling and subsequent reheating. It is convenient to think of continuous cooling as a series of temperature jumps followed by isothermal holds. The duration of the isothermal hold z~t is determined by a cooling rate q and the magnitude of the temperature jump AT, i.e., ~t----AT/q. Let us assume some real numbers for clarity and consider cooling and reheating of the glass at a rate of 3 ° / m i n . Thus, envision temperature jumps of 1°C being followed by isothermal holds for 20 s. At high temperatures, 20 s is a very long time compared to the retardation times being on the order of ms. The system is definitely at equilibrium at the end of the hold. The retardation time quickly increases to minutes with decreasing temperature, and the duration of the hold will no longer allow the system to reach equilibrium. The cooling curve in fig. l b deviates from the equilibrium line. At even lower temperatures the retardation time becomes on the order of hours, days, years. Now the duration of the hold is negligible compared to the retardation time and therefore no changes will occur during the hold - the system is frozen. A reheating curve makes its way below the cooling curve. Again, this fact can be understood by considering reheating as a continuation of isothermal holds. Indeed, arriving at a certain temperature which we passed during cooling, we allow another 20 s for the system to proceed closer to the equilibrium - the reheating curve shifts downwards toward the equilibrium line. After crossing, below the equilibrium line the reheating curve vigorously directs upwards as retardation times again become similar in magnitude to the incremental isothermal hold. A complete convergence will not occur until the structural retardation time is a small portion of the holding time. A decrease in cooling/reheating rate is equivalent to extending the isothermal holds; this will shift all events toward longer retardation times, i.e., lower temperatures. Temperature plots for the derivative of property p with respect to temperature are shown in fig. lc. Both in the liquid and glassy states, the values of the derivatives are weak functions of temperature. The glass transition phenomenon in terms of the derivatives is seen then in the transition of dp/dT from the (dp/dT)~ to (dp/dT)g where the subscript 1 stands for "liquid" and g for "glassy". Figure 2 introduces the notion of fictive temperature defined as the intersection of extrapolations of liquid and glassy portions of the p = p ( T )
S.M. Rekhson / Memo~ effects in glass transition
GLASS TRANSITION
GLASS
71
SUPERCOOLED LIQUID
¢1
'if Tf(T)
T
,.
Fig. 2. Definition of fictive temperature in terms of measured property.
curve. This is the limiting fictive temperature, Tf. The value of the fictive temperature function, Tf(T), at any other temperature T is determined by drawing a glassy line from the cooling or heating curve at this temperature T toward the equilibrium line, as shown in fig. 2, Conversely, if the fictive temperature is determined by independent means, both the property p (T) and the derivative of the property, dp/dT, can be readily obtained using
p(T) -p(To) = JT~, %' d r ' +
f v,.,,
f w~apg d T '
(1)
and
ap(T)=apg(T)+[ap,(Tf)-C~pg(Tf)]
dT~(r) dT
(2)
and measured data for p(To), (dp/dT)l - apl and (dp/dT)g - OLpg.
2. Model of glass transition
The fictive temperature can be calculated using the definition given by Narayanaswamy [7] in terms of the kinetic parameters of the glass:
r~(To)M ,. dT d ' Tfg=V+J~(T ) p(~-~ ) ~ 7 ~,
(3)
where Mp(~) is the structural retardation function, To the initial temperature at t = 0, and ~ is the reduced time variable given by
~(t) = (tdp[T(t'), ao
Tf(t')] dt'.
(4)
S.M. Rekhson / Memory effects in glass transition
72
In eq. (4), q~(T, re) is the shift function given by e~(r, r e ) = X ( r r ) / h ( T , rf)
(5) x
= exp
T~
T
Tf
'
(6)
where X(re) and h(T, re) are the structural retardation time at the reference temperature Tr and the current structural retardation time, respectively, Z~h is the activation enthalpy, x is a constant (0 < x ~< 1), and R is the universal gas constant. The structural retardation function Mp(~) can be represented by a sum of weighted exponentials
Mp(~) = ~ u i e x p ( - ~ / h , ) ,
(7)
i=1
where u i and h i are weighting factor and partial retardation time at the reference temperature, Tr, respectively. Coefficients u~ and h i comprise the discrete spectrum of retardation times discussed in sect. 3 and given in Appendix 1. Substituting eq. (7) into eq. (3) and defining the partial fictive temperature by ' h dT , Tfi = T + fl~(r°)exp( - (~ - ~ ) / ,)-r77, d~ , J~(T)
(8)
"
we obtain
/1 Tf = E uire,.
(9)
i=1
Evaluation of the partial fictive temperatures is described by Tool's equation [161 dTfi
dt
T - Tfi hi ,
i=l...n,
(10)
as seen from differentiation of eq. (8) with respect to time. In eq. (10) the partial retardation time is given by eqs. (5) and (6), i.e.,
[ah{x
1-x
hi (T' re) = ~'"r exp[-R-- ~T + ~7-r
1)1
(11)
7"r 1.1 "
Eq. (10) was evaluated numerically using the equation derived by Scherer (see Appendix 2 in ref. [17]) * re i m) = V `m) -- ( r ( m ) -
Tf~m-l)) exp[-A/(m)//a
(m)]
i = 1, n.
(12)
In eq. (12) the superscript refers to the number of time steps taken, and T ( " ) = T O,
m=l.
(13)
* George Scherer also kindly provided his computer program, a substantial part of which I included in my own.
73
S.M. Rekhson / Memory effects in glass transition
3. Parameters for calculation 3.1. Spectrum of retardation times
Figure 3 shows the continuous spectrum L ( ~ ) of structural retardation times at the reference temperature, Tr, used in computations. This spectrum was obtained by shifting the shear retardation spectrum computed in ref. [18] to the longer retardation times. The shift factor equal to 6.3 was experimentally found in ref. [19]. Thus, in effect, the assumption is made that the two retardation spectra have a similar shape. The assumption is acceptable given the qualitative nature of the present study. A continuous spectrum is unique, i.e., it does not depend upon the method of its computation and should be considered as a material property. It is convenient for material characterization and comparison purposes. However, in computations, a discrete retardation spectrum is used, which is obtained from the continuous spectrum as follows. The retardation times, ~,, are chosen arbitrarily with a small spacing A In ~ and corresponding weighting factors, u i, are computed using the formula ui(Xi) = L ( X ) A In X,
~ = Xi i
(14)
Since the retardation times are chosen arbitrarily, the values of h i do not
0.3
0.2
/
0.1
0.0 10-6
18-4 18-2 Retardot Ion T i meG;, ~;eeond~;
Fig. 3. Spectrum of retardation times.
102
S.M. Rekhson / Memory effects in glass transition
74
indicate some specific retardation mechanisms existing in reality. The concept, rather, is that there exists a continuous spectrum of retardation times L(?,), and a discrete set of mechanisms is chosen to represent it. The discrete spectrum of retardation times used in computations is given in Appendix 1. Details of computing discrete and continuous spectra are described by Rekhson and Rekhson [18]. In particular, it was found that in order to avoid artifictual undulations in spectra or in response functions the A In X should be not greater than about 0.2. Practically, it leads to having as many pairs in the discrete spectrum as there are experimental points in the retardation function - a starting point of computing spectra. There are 54 pairs of numbers in the discrete spectrum of Appendix 1 which implies that 54 equations (12) should be solved. Fortunately, all the work is done by computer. The numbers of Appendix 1 were found, operated upon, and even typed into the final manuscript by a computer *
3.2. Other parameters Other parameters in eq. (11) are Ah/R=67811K, x = 0 . 5 , Tr = 8 5 2 K ,
(15)
9Og
A ~g
850 IAI II,
In IM
I-Id n L
8go
Ila
!
75{}
J
Sgg
,
,
I
i
J
,
,
,
,
6gg
,
,
,
s
J
,
,
,
,
•
,
,
,
,
700
,
,
i
i
BOg
,
t
i
,
,
,
,
,
ggg
Temtoor at~ur Q, OK
Fig. 4. Tf vs T during cooling/reheating at 3 K / m i n .
* Solving 54 equations (12) in the calculation involving cooling, a one-year long isothermal hold and reheating took only 39 s of CPU time on the VAX 780.
S.M. Rekhson / Memory effects in glass transition
75
/
950
A
'g 900 Ima at
P-
F"
850
/ / 500
800
700
808
Temperoture,
980
1000
OK
Fig. 5. ~ vs T ~ r quenching at 100 K / s and reheating at 3 K / m i n .
v 0
/
900 w I=¢
..=. L
=E 850 u u.
i t !
7 5 0
'
500
'
'
'
'
'
'
'
'
' ' ' ' '
600
.
.
.
.
.
708
'
'
'
'
'
'
.
.
.
.
.
800
.
.
.
.
.
.
''
908
Temperoture,°K Fig. 6. ~ vs T ~ r c o o l i n g a t 3 K / m i n and reheating a t l 0 0 K/s.
.
.
.
.
.
.
.
1888
76
S.M. Rekhson / Memory effects in glass transition
and the initial temperature T(0) was varied from 810 to 950 K, depending on rate of cooling.
4. Behavior of quenched and annealed glasses Figures 4-6 show the evolution of the fictive temperature of the glass during cooling and reheating at the same rate, during rapid quenching and slow reheating, and during slow cooling and rapid reheating. In fig. 4, the fictive temperature follows the actual temperature in the beginning of cooling and asymptotically approaches the constant value of 784°K in the end. During reheating, the fictive temperature crosses the equilibrium line, and then converges with it at higher temperatures. Rapid cooling (fig. 5) freezes in a higher limiting fictive temperature of 859 K. Slow reheating provides enough time for the system to approach the equilibrium line using a short route, closely following the equilibrium line afterwards. In contrast, rapid reheating of the annealed glass (see fig. 6) produces a large overshoot and vigorous approach toward the equilibrium line from below.
5. Memory effects 5.1. Crossover experiment
Ritland [1] performed a classical experiment which in effect probed into the very definition of the fictive temperature as the actual temperature at which the glass is in equilibrium. It follows from this definition that glass quickly reheated to the temperature equal to its fictive temperature should be in equilibrium and no activities should be observed. Instead, Ritland discovered a time-dependent behavior described by a curve with the minimum shown in fig. 3 of ref. [1]. Ritland monitored the refractive index which has a negative temperature coefficient a,~ (n = index of refraction). Therefore, as seen from eq. (1) for T = const, the minimum on Ritland's curve indicates a maximum on the Tf (t) dependence. This maximum as predicted by the multi-exponential model is shown in fig. 7, curve Tf. The multiexponential model also provides the following explanation of this behavior. Although the average limiting fictive temperature of the glass cooled to room temperature is 784.3 K, there is a broad distribution of partial fictive temperatures for the 54 mechanisms as given in table 1. The time-dependent behavior of one moderate (No. 34) and one slow (No. 53) mechanism following very fast reheating of glass to T = 784.5 K is shown in fig. 7. The moderate mechanism frozen with Tt34 = 748 K arrives at the hold temperature severely overheated and demonstrates a vigorous approach to the equilibrium from below. (It is difficult to overheat rapid mechanisms - they equilibrate before the chosen hold temperature is
S.M. Rekhson / Memory effects in glass transition
77
848
vv
o
o
L
L
-
~53
820
780
/
760
I
AU
J 740 _ ' ............... 18 - z 10 -1
lg
1
102
103
104
Timo, minu~o~ Fig. 7. Cross-over effect in glass cooled at 3 K / m i n and reheated at 100 K / s to the temperature 784.5 K equal to its limiting fictive temperature, Tf; Tf34 and Tf53 are partial fictive temperatures of mechanisms Nos. 34 and 53, respectively.
Table 1 Partial fictive temperatures in glass cooled at 3 K / m i n (the weighted average fictive temperature is 784.30 K) Mech. no.
Tf,
Mech. no.
Tfi
Mech. no.
Tf,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
626.57 629.81 633.08 636.38 639.71 643.07 646.46 649.88 653.34 656.82 660.34 663.88 667.45 671.06 674.69 678.35 682.04 685.76
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
689.51 693.28 697.08 700.91 704.76 708.64 712.54 716.46 720.40 724.36 728.33 732.32 736.32 740.33 744.35 748.37 752.39 756.40
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
760.41 764.41 768.40 772.37 776.31 780.23 784.12 787.98 791.80 795.58 799.32 803.02 806.67 810.28 813.84 817.36 820.83 824.27
78
S.M. Rekhson / Memory effects in glass transition 828
-
.:
818
i L
,•53
708 T
~..--..._._.._
Tf34 ,
788
0
,
;08
,
,
,
I
,
,
,
,
.
.
1080
.
.
.
.
.
.
.
.
.
.
.
.
.
1580
.
.
.
.
.
.
.
.
.
.
.
.
2008
2580
Time, minu~e~ Fig. 8. No cross-over effect is seen in the glass cooled and reheated at 3 K / m i n to 784.5 K.
reached.) The slow mechanism is in a quenched state with Tf53= 821 K which is greater than the hold temperature, and therefore it approaches equilibrium from above. The maximum is seen to be produced by a superposition of moderate processes which determine the ascending part of the curve and slow processes controlling the descending part. This explanation was originally offered by Macedo and Napolitano [11], who used their two-exponential model. If the glass with the average fictive temperature 784.3 K is reheated at a rate 3 K / m i n , the cross-over effect disappears: the curve Tf(t) is a descending curve with no maximum seen (fig. 8). Apparently, in this case the majority of processes with Tr, lower than the temperature of the hold had enough time to equilibrate during reheating before they arrived at 784.5 K.
5.2. Sub-Tg relaxation The multiexponential model does produce a sub-Tg peak reported in refs. [3,4] (for additional references see ref. [3]), The peak is shown in fig. 9, curve 1, accentuated by comparison with the reheating curve for the same glass which was not subjected to stabilization at 630 K, curve 2. Figure 10 depicts this behavior in terms of the average fictive temperature. In the temperature interval 700-790 K the reheating curve seems to be kept off the equilibrium line by some intriguing forces. Again, this behavior is seen especially clearly when compared to the reheating curve for the rate cooled glass not subjected
S.M. Rekhson / Memory effects in glass transition
79
1.S
1.0
"o
/
0.5
J 0.0 600
650
700
.o , . . . , . , , . i , , , , , . , . . 7S0 800
Temloer ot;¢ar e,
BBO
,.. 000
OK
Fig. 9. d T f / d T vs T during reheating at 3 K / m i n of the glass cooled at 100 K / s and held for 1 y at 630 K, curve 1, and cooled at 3 K / m i n with no isothermal hold, curve 2.
to long stabilization, curve 2. The explanation of the phenomenon is given in figs. l l and 12. During a rapid cooling at 100 K / s , the majority of mechanisms were severely quenched and arrived at the hold temperature of 630 K with the following fictive temperatures given in K: Yf
Yfl 8
Tf2 5
Yf3 5
Tf4 0
Yf4 5
Yf5 3
857
743
774
821
845
868
894
During a one-year long isothermal hold these mechanisms equilibrate with very different rates as shown in fig. 11. This difference in kinetics can be better appreciated if the retardation times are approximately evaluated using eq. (11) along with ?'r, from Appendix 1, A h / R , x and Tr from eq. (15) and Tf from the table above at t = 0 and from fig. 11 at t > 1 min. According to this estimate the retardation times of mechanisms 18 and 25 are 17 and 129 min, respectively. This is at the very beginning of the hold. With time elapsing, 2~, will increase due to decrease in the average fictive temperature, T r, but the rapid mechanisms 18 and 25 equilibrate in a matter of hours before considerable changes in ?~, occur. The situation is very different with slow mechanisms, e.g., mechanism 53; its initial retardation time is 300 days, in a week it becomes close to 5 y, in 10 weeks it is on the order of 16 y, etc. Therefore, 1 y is too short a time for mechanism 53, it remains frozen in the quenched state.
S.M. Rekhson / Memory effects in glass transition
80 ~0
0 +
8.5O .=.. L
P. ta m.
~, o
B00
.=..
J
500
t300
700
880
900
Temper ot:ur e , oK Fig. 10. Same as fig. 9 but in terms of Tf vs T. Cooling curves are also shown.
9~
- - - ~
o~ o+ ~,..+
'-,~5
~-----~- ~
B8~
-l.lv~
x,|
ii.
n u.
7~
\ 10
\\+ +102
|B3
104
10S
106
Time+ mI nu~.~-, Fig. 11. Evolution of the average.fictive temperature, Tr, and partial fictive temperatures, Tr+ during the 1 year long isothermal hold at 630 K.
81
S,M. Rekhson / Memory effects in glass transition g~
"-
"-- 7 0 0
61~
650
7~
750
8~
850
g~0
TupQrokure, °K
Fig. 12. Behavior of the average fictive temperature, Tf, and partial fictive temperatures, Tf,, during reheating at 3 K/rain followingthe 1 y long isothermal hold at 630 K. The 1-y time scale is just right for mechanisms 35 and 40, they become deeply stabilized at the end of the hold. During subsequent reheating at a rate 100 K / s all annealed mechanisms are easily overheated and demonstrate a vigorous approach to equilibrium from below, as shown by curves Tfl8, Tf25, and Tf3 5 in fig. 12. It is these mechanisms that via eq. (9) contribute strongly to the increase in the average fictive temperature, curve Tf, in the temperature interval of 650-780 K. At these temperatures the slow mechanisms do not contribute to the slope of curve Tf as seen from the negligible slope of the curve Tr53. At temperatures above 780 K the slow, quenched processes contribute a negative slope, thus decreasing the slope of the average Tf curve. At temperatures above approximately 860 K all processes equilibrate and the slope d T f / d T = 1. Thus, the slope of d T f / d T is maximal at about 780 K, which is the position of the maximum in fig. 9. 5.3. Cooling~reheating curve
It turns out that no dramatic or complex thermal treatments are needed to demonstrate the memory type behavoir. The reheating curve for the average fictive temperature, Tf, shown in fig. 3, reveals a similar disregard of the equilibrium state. This is clear when compared to the behavior of separate
S.M. Rekhson / Memory effects in glass transition
82
91~8
°.
i!o
% X
Tf32 / / / i .........
788 S~
60~
. . . . . . . . . . . . . . . . . . . 880 708
9~
Temperaluro, °K
Fig. 13. Behavior of the average fictive temperature, Tf, and partial fictive temperatures, Tfi, during reheating at 3 K/min following cooling at the same rate.
mechanisms - Nos. 32 and 53 as shown in fig. 13. The fictive temperature of the single exponential mechanism - the partial fictive temperature - decreases slightly during reheating until it crosses the equilibrium line. This behavior can be clarified by resorting again to the representation of continuous heating as a series of temperature jumps followed by isothermal holds. The equilibrium condition for each mechanism is Tfi = T, which is the equation of the straight line in fig. 13. Thus, during the hold at any temperature, the Tf, should draw closer to the equilibrium line. They certainly do so as illustrated in fig. 13. In contrast, the average fictive temperature increases in the temperature interval where Tf > T, i.e., the system as a whole moves away from equilibrium. The cause of this behavior is those fast mechanisms (e.g. Tf32 which at these temperatures proceed through the final stages of their approach to equilibrium. As seen in fig. 13, this stage is characterized by a vigorous rise in the partial fictive temperature. Again, via Tf = ~uiTf~, these mechanisms will i
contribute to the increase in the average fictive temperature. A positive slope of Tf vs T curves to the left of the equilibrium line was noticed and correctly identified as a memory effect by Moynihan, Macedo and co-workers [20].
S.M. Rekhson / Memory effects in glass transition
83
6. Summary The multi-exponential model offers an interpretation of the frozen system as an ensemble of states characterized by a distribution of fictive temperatures. This distribution is a direct result of the distribution of structural retardation times: fast mechanisms will retain equilibrium states during cooling to lower temperatures and will produce lower fictive temperatures, while slower mechanisms will freeze at higher temperatures resulting in higher fictive temperatures. Whereas only the average fictive temperature is measured in the experiment, its behavior appears meaningful only if analyzed as a result of superposition of separate mechanisms. In this work we primarily examined the behavior of glass during reheating. The latter can be considered as a process of re-establishing the equilibrium state. We observed three memory effects whose essence can be outlined as follows. A non-equilibrium and partially relaxed system, when forced back to equilibrium, gradually or abruptly, may exhibit some resistance by moving temporarily away from equilibrium. This seemingly paradoxical behavior can be explained by recognizing that we force the system into the state which we inadequately define as "equilibrium" by using a single parameter - the average fictive temperature, Tf. The enforced condition Tf = T can produce nothing but a mixture of non-equilibrium states at any time shorter than the longest retardation time. As was shown above, when during reheating the entire system is still in a supercooled state characterized by Tf > T, there are some overheated states with Tfi < T. These will proceed at this time through the final stages of equilibration characterized by a vigorous increase in Tf,. With their weighting coefficients they will contribute to the "non-intuitive" increase in the average Tf. In the cross-over experiment, these mechanisms are forced into an overheated state by abrupt reheating. In the sub-Tg experiments both slow and intermediate mechanisms are forced although in different directioxls: slow mechanisms are frozen in a severely quenched state by rapid cooling, and intermediate mechanisms are deeply annealed during long stabilization at low temperature. Thus, by forcing certain mechanisms we can produce many intriguing effects. They all, however, can be understood if the system is properly Characterized by a distribution of fictive temperatures. Although the material properties used in the above computations were those of soda-lime-silicate glass, in this work we limited ourselves to a qualitative modeling of glass behavior. This is because the model used is based on the principle of thermorheological simplicity which assumes the same activation energy for all mechanisms. In reality, there must be a distribution of activation energies with slow mechanisms having higher and fast mechanisms lower activation energy. Therefore the present model may produce some numerical orders when applied to experiments emphasizing the role of one group of mechanisms possibly having an activation energy substantially different from the average. None of the conclusions of this work - which are purely qualitative should be affected by the simplification employed.
S.M. Rekhson / Memory effects in glass transition
84
Continuing guidance in computer science by Michael Rekhson of the Massachusetts Institute of Technology, Cambridge, Mass., is deeply appreciated.
Appendix 1 Discrete spectrum of retardation times Mechanism no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Retardation time, )~
Weighting factor,
(s)
u~
7.5858E- 06 1.0136E-05 1.3543E-05 1.8095E - 05 2.4177E- 05 3.2303E - 05 4.3161E-05 5.7669E-05 7.7054E - 05 1.0295E-04 1.3756E-04 1.8380E-04 2.4558E-04 3.2812E-04 4.3842E-04 5.8578E - 04 7.8268E-04 1.0458E - 03 1.3973E - 03 1.8669E - 03 2.4945E-03 3.3330E-03 4.4533E- 03 5.9501E-03 7.9502E-03 1.0622E - 02 1.4193E-02
0.0018 0.0023 0.0027 0.0022 0.0022 0.0022 0.0024 0.0028 0.0034 0.0039 0.0042 0.0042 0.0039 0.0037 0.0036 0.0039 0.0044 0.0048 0.0048 0.0041 0.0029 0.0015 0.0006 0.0009 0.0026 0.0057 0.0095
Mechanism no. 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Retardation time, ),~
Weighting factor
(s)
u~
1.8964E - 02 2.5338E-02 3.3855E-02 4.5235E - 02 6.0439E- 02 8.0755E- 02 1.0790E-01 1.4417E-01 1.9263E - 01 2.5737E-01 3.4388E-01 4.5947E-01 6.1392E-01 8.2027E-01 1.0960E+00 1.4644E + 00 1.9566E+00 2.6143E + 00 3.4930E + 00 4.6672E + 00 6.2359E+00 8.3320E+00 1.1133E + 01 1.4875E+01 1.9875E+01 2.6555E + 01 3.5481E+01
0.0130 0.0153 0.0157 0.0143 0.0116 0.0088 0.0071 0.0071 0.0091 0.0127 0.0168 0.0205 0.0231 0.0246 0.0257 0.0277 0.0320 0.0392 0.0491 0.0600 0.0694 0.0743 0.0724 0.0631 0.0480 0.0302 0.1181
References [1] [2] [3] [4] [5] [6]
H.N. Ritland, J. Am. Ceram. Soc. 39 (1956) 403. A.J. Kovacs, Fortschr. Hochpolym. Forsch. 3 (1963) 394. A.R. Berens and I.M. Hodge, Macromolecules 15 (1982) 756. H.S. Chen, J. Appl. Phys. 52 (1981) 1868. A.L. Greer and J.A. Leake, J. Non-Cryst. Solids 33 (1979) 291. S.M. Rekhson, Glass: Science and Technology, Vol. 3, eds. D.R. Uhlmann and N.J. Kreidl (Academic Press, New York) in press.
S.M. Rekhson / Memory effects in glass transition
85
[7] O.S. Narayanaswamy, J. Am. Ceram. Soc. 54 (1971) 691. [8] A.J. Kovacs, J.J. Aklonis, J.M. Hutchinson and A.R. Ramos, J. Polym. Sci., Polym. Phys. Ed. 17 (1979) 1097. [9] C.T. Moynihan, A.J. Easteal, M.A. DeBolt and J. Tucker, J. Am. Ceram. Soc. 59 (1976) 12. [10] C.T. Moynihan and P.K. Gupta, J. Non-Crystl. Solids 29 (1978) 143. [11] P.B. Macedo and A. Napolitano, J. Res. Nat. Bur. Stand., Sect. A, 71 (1967) 231. [121 I.M. Hodge and A.R. Berens, Macromolecules 15 (1982) 762. [13] G.P. Johari and M. Goldstein, J. Chem. Phys. 53 (1970) 2372; 55 (1971) 4245. [14] G.P. Johari, Ann. N.Y. Acad. Sci. 279 (1976) 117. [15] G.P. Johari, Phil. Mag. B46 (1982) 549. [16] A.Q. Tool, J. Am. Ceram. Soc. 29 (1946) 240. [17] G.W. Scherer, Relexation in Glass and Composites (Wiley, New York, 1986). [18] M.S. Rekhson and S.M. Rekhson, J. Am. Ceram. Soc., in press. [19] S.M. Rekhson and V.A. Ginzburg, Sov. J. Glass Phys. Chem. 2 (1976) 422. [20] C.T. Moynihan, P.B. Macedo, C.J. Montrose, P.K. Gupta, M.A. DeBolt, J.F. Dill, B.E. Dom, P.W. Drake, A.J. Easteal, P.B. Elterman, R.P. Moeller, H Sasabe and J.A. Wilder, Ann. N.Y. Acad. Sci. 279 (1976) 15.
Journal of Non-Crystalline Solids 84 (1986) 68-85 North-Holland, Amsterdam
M E M O R Y EFFECTS IN GLASS TRANSITION
Simon M. REKHSON General Electric Company, Nela Park, Cleveland, OH 44112, USA
Behavior of glass in the glass transition region is reviewed with the emphasis on memory effects. Three examples of memory effects analyzed are the behavior of the fictive temperature in the cross-over, sub-Tg relaxation and simple reheating experiments. In all cases, the non-equilibrium system reveals a temporary neglect of the equilibrium state by moving away from it. A multiexponential model of structural relaxation is employed to model this phenomenon and to investigate the behavior of separate relaxation mechanisms contributing to a total effect. The model characterizes the glassy state by a broad distribution of partial fictive temperatures - low, produced by fast relaxation mechanisms, and high, produced by slow mechanisms. During reheating, each mechanism approaches the equilibrium line Tf = T in two stages: first from above (because Tfi > T) and then from below (when due to the overshoot Tfi < T). In the lower part of the glass transition region where slow mechanisms are still in the first stage of a very slow decrease of their Tfi's, the fast mechanisms are finishing their approach to equilibrium by a vigorous increase of their Tfi's. The Tf measured in the experiment is the weighted average of all Tfi's and therefore reflects the behavior of fast mechanisms by increasing its value although Tf is still greater than the actual temperature T. A complex thermal treatment designed as a combination of quenching and annealing steps will emphasize different groups of mechanisms and produce many intriguing effects. All these effects can be explained as a superposition of quenched and annealed mechanisms approaching the equilibrium state from opposite directions.
1. Introduction
This paper describes the performance of the multiexponential model of structural relaxation in predicting and interpreting memory effects in the glass transition region. The memory effects are best illustrated by cross-over experiments and by sub-Tg relaxation peaks. As shown below, in both cases the non-equilibrium system reveals a temporary neglect of the equilibrium state by moving away from it. This is a striking behavior because the very existence of a relaxation process is entirely due to the deviation of the system from equilibrium. The rate of the relaxation process is approximately proportional to the magnitude of this deviation, and the process ends when the system returns to equilibrium. The reason for the system to neglect its future (the equilibrium state) is generally seen in the memory of its past. The subject of memory effects is of general interest because they have been observed in inorganic [1], organic polymeric [2,3], and metallic [4,5] glasses. In addition, they represent a challenging material for testing theoretical models of glass transition. 0022-3093/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S.M. Rekhson / Memory effects in glass transition
69
H
TEMPERATURE
(c)
OEl ~g TEMPERATURE
Fig. 1. Evolution of the property (a) in isothermal condition following a sudden cooling (1) and heating (2) to the same hold temperature; evolution of the property (b) and the temperature coefficient of the property (c) during fast and slow cooling and reheating.
As shown in ref. [6], the multiexponential model is a special case of the Narayanaswamy model [7] where the structural retardation function is given by a sum of weighted exponentials. The model was also formulated by means of thermodynamics [8-10] to describe the evolution of glass properties in terms of order parameters. The first multiexponential model, containing only two terms, was proposed and successfully used for explanation of the cross-over effect by Macedo and Napolitano [11]. Kovacs and co-workers [8] as well as Hodge and Berens [12] showed that the multiparameter model does predict the sub-Tg relaxation peaks observed in glasses reheated following quenching and long stabilization at low temperatures. This work made it clear that the origin of these peaks is entirely in the complex thermal history employed. This is in contrast to the secondary relaxation peaks, extensively studied by Johari and Goldstein [13-15], whose origin is in corresponding peaks in the distribution of relaxation times. The secondary relaxation peaks are seen in simple cooling/ reheating experiments. The subject of memory effects requires a good understanding of the glass transition phenomenon. In the interest of those who do not work directly in the area of glass transition we shall build up our discussion gradually, starting with a review of the basic features of glass behavior in the glass transition region. This is shown schematically in fig. 1 using the glass volume, enthalpy and their temperature derivatives as examples. Following an abrupt cooling, both properties will decrease in time (fig. la, curve 1) and following heating to the same temperature they will increase (curve 2). In both cases, the property eventually arrives at the same value which is, therefore, an equilibrium value of the property. A cooled glass sample arrives at the temperature of the
70
S.M. Rekhson / Memory effects in glass transition
isothermal hold with lower density and higher mobility than the heated sample, and this results in a higher rate of retardation. As a first approximation, the duration of retardation is characterized by the structural retardation time defined as the time necessary for the system to cover about two-thirds of the way to equilibrium. The structural retardation time increases exponentially with decreasing temperature - about an order of magnitude for each 20°C in the case of silicate glasses. It is this single factor which determines the very essence of the phenomenon called liquid-glass, or simply glass transition. This phenomenon is illustrated in Fig. lb by the volume, or enthalpy vs temperature curves obtained during cooling and subsequent reheating. It is convenient to think of continuous cooling as a series of temperature jumps followed by isothermal holds. The duration of the isothermal hold z~t is determined by a cooling rate q and the magnitude of the temperature jump AT, i.e., ~t----AT/q. Let us assume some real numbers for clarity and consider cooling and reheating of the glass at a rate of 3 ° / m i n . Thus, envision temperature jumps of 1°C being followed by isothermal holds for 20 s. At high temperatures, 20 s is a very long time compared to the retardation times being on the order of ms. The system is definitely at equilibrium at the end of the hold. The retardation time quickly increases to minutes with decreasing temperature, and the duration of the hold will no longer allow the system to reach equilibrium. The cooling curve in fig. l b deviates from the equilibrium line. At even lower temperatures the retardation time becomes on the order of hours, days, years. Now the duration of the hold is negligible compared to the retardation time and therefore no changes will occur during the hold - the system is frozen. A reheating curve makes its way below the cooling curve. Again, this fact can be understood by considering reheating as a continuation of isothermal holds. Indeed, arriving at a certain temperature which we passed during cooling, we allow another 20 s for the system to proceed closer to the equilibrium - the reheating curve shifts downwards toward the equilibrium line. After crossing, below the equilibrium line the reheating curve vigorously directs upwards as retardation times again become similar in magnitude to the incremental isothermal hold. A complete convergence will not occur until the structural retardation time is a small portion of the holding time. A decrease in cooling/reheating rate is equivalent to extending the isothermal holds; this will shift all events toward longer retardation times, i.e., lower temperatures. Temperature plots for the derivative of property p with respect to temperature are shown in fig. lc. Both in the liquid and glassy states, the values of the derivatives are weak functions of temperature. The glass transition phenomenon in terms of the derivatives is seen then in the transition of dp/dT from the (dp/dT)~ to (dp/dT)g where the subscript 1 stands for "liquid" and g for "glassy". Figure 2 introduces the notion of fictive temperature defined as the intersection of extrapolations of liquid and glassy portions of the p = p ( T )
S.M. Rekhson / Memo~ effects in glass transition
GLASS TRANSITION
GLASS
71
SUPERCOOLED LIQUID
¢1
'if Tf(T)
T
,.
Fig. 2. Definition of fictive temperature in terms of measured property.
curve. This is the limiting fictive temperature, Tf. The value of the fictive temperature function, Tf(T), at any other temperature T is determined by drawing a glassy line from the cooling or heating curve at this temperature T toward the equilibrium line, as shown in fig. 2, Conversely, if the fictive temperature is determined by independent means, both the property p (T) and the derivative of the property, dp/dT, can be readily obtained using
p(T) -p(To) = JT~, %' d r ' +
f v,.,,
f w~apg d T '
(1)
and
ap(T)=apg(T)+[ap,(Tf)-C~pg(Tf)]
dT~(r) dT
(2)
and measured data for p(To), (dp/dT)l - apl and (dp/dT)g - OLpg.
2. Model of glass transition
The fictive temperature can be calculated using the definition given by Narayanaswamy [7] in terms of the kinetic parameters of the glass:
r~(To)M ,. dT d ' Tfg=V+J~(T ) p(~-~ ) ~ 7 ~,
(3)
where Mp(~) is the structural retardation function, To the initial temperature at t = 0, and ~ is the reduced time variable given by
~(t) = (tdp[T(t'), ao
Tf(t')] dt'.
(4)
S.M. Rekhson / Memory effects in glass transition
72
In eq. (4), q~(T, re) is the shift function given by e~(r, r e ) = X ( r r ) / h ( T , rf)
(5) x
= exp
T~
T
Tf
'
(6)
where X(re) and h(T, re) are the structural retardation time at the reference temperature Tr and the current structural retardation time, respectively, Z~h is the activation enthalpy, x is a constant (0 < x ~< 1), and R is the universal gas constant. The structural retardation function Mp(~) can be represented by a sum of weighted exponentials
Mp(~) = ~ u i e x p ( - ~ / h , ) ,
(7)
i=1
where u i and h i are weighting factor and partial retardation time at the reference temperature, Tr, respectively. Coefficients u~ and h i comprise the discrete spectrum of retardation times discussed in sect. 3 and given in Appendix 1. Substituting eq. (7) into eq. (3) and defining the partial fictive temperature by ' h dT , Tfi = T + fl~(r°)exp( - (~ - ~ ) / ,)-r77, d~ , J~(T)
(8)
"
we obtain
/1 Tf = E uire,.
(9)
i=1
Evaluation of the partial fictive temperatures is described by Tool's equation [161 dTfi
dt
T - Tfi hi ,
i=l...n,
(10)
as seen from differentiation of eq. (8) with respect to time. In eq. (10) the partial retardation time is given by eqs. (5) and (6), i.e.,
[ah{x
1-x
hi (T' re) = ~'"r exp[-R-- ~T + ~7-r
1)1
(11)
7"r 1.1 "
Eq. (10) was evaluated numerically using the equation derived by Scherer (see Appendix 2 in ref. [17]) * re i m) = V `m) -- ( r ( m ) -
Tf~m-l)) exp[-A/(m)//a
(m)]
i = 1, n.
(12)
In eq. (12) the superscript refers to the number of time steps taken, and T ( " ) = T O,
m=l.
(13)
* George Scherer also kindly provided his computer program, a substantial part of which I included in my own.
73
S.M. Rekhson / Memory effects in glass transition
3. Parameters for calculation 3.1. Spectrum of retardation times
Figure 3 shows the continuous spectrum L ( ~ ) of structural retardation times at the reference temperature, Tr, used in computations. This spectrum was obtained by shifting the shear retardation spectrum computed in ref. [18] to the longer retardation times. The shift factor equal to 6.3 was experimentally found in ref. [19]. Thus, in effect, the assumption is made that the two retardation spectra have a similar shape. The assumption is acceptable given the qualitative nature of the present study. A continuous spectrum is unique, i.e., it does not depend upon the method of its computation and should be considered as a material property. It is convenient for material characterization and comparison purposes. However, in computations, a discrete retardation spectrum is used, which is obtained from the continuous spectrum as follows. The retardation times, ~,, are chosen arbitrarily with a small spacing A In ~ and corresponding weighting factors, u i, are computed using the formula ui(Xi) = L ( X ) A In X,
~ = Xi i
(14)
Since the retardation times are chosen arbitrarily, the values of h i do not
0.3
0.2
/
0.1
0.0 10-6
18-4 18-2 Retardot Ion T i meG;, ~;eeond~;
Fig. 3. Spectrum of retardation times.
102
S.M. Rekhson / Memory effects in glass transition
74
indicate some specific retardation mechanisms existing in reality. The concept, rather, is that there exists a continuous spectrum of retardation times L(?,), and a discrete set of mechanisms is chosen to represent it. The discrete spectrum of retardation times used in computations is given in Appendix 1. Details of computing discrete and continuous spectra are described by Rekhson and Rekhson [18]. In particular, it was found that in order to avoid artifictual undulations in spectra or in response functions the A In X should be not greater than about 0.2. Practically, it leads to having as many pairs in the discrete spectrum as there are experimental points in the retardation function - a starting point of computing spectra. There are 54 pairs of numbers in the discrete spectrum of Appendix 1 which implies that 54 equations (12) should be solved. Fortunately, all the work is done by computer. The numbers of Appendix 1 were found, operated upon, and even typed into the final manuscript by a computer *
3.2. Other parameters Other parameters in eq. (11) are Ah/R=67811K, x = 0 . 5 , Tr = 8 5 2 K ,
(15)
9Og
A ~g
850 IAI II,
In IM
I-Id n L
8go
Ila
!
75{}
J
Sgg
,
,
I
i
J
,
,
,
,
6gg
,
,
,
s
J
,
,
,
,
•
,
,
,
,
700
,
,
i
i
BOg
,
t
i
,
,
,
,
,
ggg
Temtoor at~ur Q, OK
Fig. 4. Tf vs T during cooling/reheating at 3 K / m i n .
* Solving 54 equations (12) in the calculation involving cooling, a one-year long isothermal hold and reheating took only 39 s of CPU time on the VAX 780.
S.M. Rekhson / Memory effects in glass transition
75
/
950
A
'g 900 Ima at
P-
F"
850
/ / 500
800
700
808
Temperoture,
980
1000
OK
Fig. 5. ~ vs T ~ r quenching at 100 K / s and reheating at 3 K / m i n .
v 0
/
900 w I=¢
..=. L
=E 850 u u.
i t !
7 5 0
'
500
'
'
'
'
'
'
'
'
' ' ' ' '
600
.
.
.
.
.
708
'
'
'
'
'
'
.
.
.
.
.
800
.
.
.
.
.
.
''
908
Temperoture,°K Fig. 6. ~ vs T ~ r c o o l i n g a t 3 K / m i n and reheating a t l 0 0 K/s.
.
.
.
.
.
.
.
1888
76
S.M. Rekhson / Memory effects in glass transition
and the initial temperature T(0) was varied from 810 to 950 K, depending on rate of cooling.
4. Behavior of quenched and annealed glasses Figures 4-6 show the evolution of the fictive temperature of the glass during cooling and reheating at the same rate, during rapid quenching and slow reheating, and during slow cooling and rapid reheating. In fig. 4, the fictive temperature follows the actual temperature in the beginning of cooling and asymptotically approaches the constant value of 784°K in the end. During reheating, the fictive temperature crosses the equilibrium line, and then converges with it at higher temperatures. Rapid cooling (fig. 5) freezes in a higher limiting fictive temperature of 859 K. Slow reheating provides enough time for the system to approach the equilibrium line using a short route, closely following the equilibrium line afterwards. In contrast, rapid reheating of the annealed glass (see fig. 6) produces a large overshoot and vigorous approach toward the equilibrium line from below.
5. Memory effects 5.1. Crossover experiment
Ritland [1] performed a classical experiment which in effect probed into the very definition of the fictive temperature as the actual temperature at which the glass is in equilibrium. It follows from this definition that glass quickly reheated to the temperature equal to its fictive temperature should be in equilibrium and no activities should be observed. Instead, Ritland discovered a time-dependent behavior described by a curve with the minimum shown in fig. 3 of ref. [1]. Ritland monitored the refractive index which has a negative temperature coefficient a,~ (n = index of refraction). Therefore, as seen from eq. (1) for T = const, the minimum on Ritland's curve indicates a maximum on the Tf (t) dependence. This maximum as predicted by the multi-exponential model is shown in fig. 7, curve Tf. The multiexponential model also provides the following explanation of this behavior. Although the average limiting fictive temperature of the glass cooled to room temperature is 784.3 K, there is a broad distribution of partial fictive temperatures for the 54 mechanisms as given in table 1. The time-dependent behavior of one moderate (No. 34) and one slow (No. 53) mechanism following very fast reheating of glass to T = 784.5 K is shown in fig. 7. The moderate mechanism frozen with Tt34 = 748 K arrives at the hold temperature severely overheated and demonstrates a vigorous approach to the equilibrium from below. (It is difficult to overheat rapid mechanisms - they equilibrate before the chosen hold temperature is
S.M. Rekhson / Memory effects in glass transition
77
848
vv
o
o
L
L
-
~53
820
780
/
760
I
AU
J 740 _ ' ............... 18 - z 10 -1
lg
1
102
103
104
Timo, minu~o~ Fig. 7. Cross-over effect in glass cooled at 3 K / m i n and reheated at 100 K / s to the temperature 784.5 K equal to its limiting fictive temperature, Tf; Tf34 and Tf53 are partial fictive temperatures of mechanisms Nos. 34 and 53, respectively.
Table 1 Partial fictive temperatures in glass cooled at 3 K / m i n (the weighted average fictive temperature is 784.30 K) Mech. no.
Tf,
Mech. no.
Tfi
Mech. no.
Tf,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
626.57 629.81 633.08 636.38 639.71 643.07 646.46 649.88 653.34 656.82 660.34 663.88 667.45 671.06 674.69 678.35 682.04 685.76
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
689.51 693.28 697.08 700.91 704.76 708.64 712.54 716.46 720.40 724.36 728.33 732.32 736.32 740.33 744.35 748.37 752.39 756.40
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
760.41 764.41 768.40 772.37 776.31 780.23 784.12 787.98 791.80 795.58 799.32 803.02 806.67 810.28 813.84 817.36 820.83 824.27
78
S.M. Rekhson / Memory effects in glass transition 828
-
.:
818
i L
,•53
708 T
~..--..._._.._
Tf34 ,
788
0
,
;08
,
,
,
I
,
,
,
,
.
.
1080
.
.
.
.
.
.
.
.
.
.
.
.
.
1580
.
.
.
.
.
.
.
.
.
.
.
.
2008
2580
Time, minu~e~ Fig. 8. No cross-over effect is seen in the glass cooled and reheated at 3 K / m i n to 784.5 K.
reached.) The slow mechanism is in a quenched state with Tf53= 821 K which is greater than the hold temperature, and therefore it approaches equilibrium from above. The maximum is seen to be produced by a superposition of moderate processes which determine the ascending part of the curve and slow processes controlling the descending part. This explanation was originally offered by Macedo and Napolitano [11], who used their two-exponential model. If the glass with the average fictive temperature 784.3 K is reheated at a rate 3 K / m i n , the cross-over effect disappears: the curve Tf(t) is a descending curve with no maximum seen (fig. 8). Apparently, in this case the majority of processes with Tr, lower than the temperature of the hold had enough time to equilibrate during reheating before they arrived at 784.5 K.
5.2. Sub-Tg relaxation The multiexponential model does produce a sub-Tg peak reported in refs. [3,4] (for additional references see ref. [3]), The peak is shown in fig. 9, curve 1, accentuated by comparison with the reheating curve for the same glass which was not subjected to stabilization at 630 K, curve 2. Figure 10 depicts this behavior in terms of the average fictive temperature. In the temperature interval 700-790 K the reheating curve seems to be kept off the equilibrium line by some intriguing forces. Again, this behavior is seen especially clearly when compared to the reheating curve for the rate cooled glass not subjected
S.M. Rekhson / Memory effects in glass transition
79
1.S
1.0
"o
/
0.5
J 0.0 600
650
700
.o , . . . , . , , . i , , , , , . , . . 7S0 800
Temloer ot;¢ar e,
BBO
,.. 000
OK
Fig. 9. d T f / d T vs T during reheating at 3 K / m i n of the glass cooled at 100 K / s and held for 1 y at 630 K, curve 1, and cooled at 3 K / m i n with no isothermal hold, curve 2.
to long stabilization, curve 2. The explanation of the phenomenon is given in figs. l l and 12. During a rapid cooling at 100 K / s , the majority of mechanisms were severely quenched and arrived at the hold temperature of 630 K with the following fictive temperatures given in K: Yf
Yfl 8
Tf2 5
Yf3 5
Tf4 0
Yf4 5
Yf5 3
857
743
774
821
845
868
894
During a one-year long isothermal hold these mechanisms equilibrate with very different rates as shown in fig. 11. This difference in kinetics can be better appreciated if the retardation times are approximately evaluated using eq. (11) along with ?'r, from Appendix 1, A h / R , x and Tr from eq. (15) and Tf from the table above at t = 0 and from fig. 11 at t > 1 min. According to this estimate the retardation times of mechanisms 18 and 25 are 17 and 129 min, respectively. This is at the very beginning of the hold. With time elapsing, 2~, will increase due to decrease in the average fictive temperature, T r, but the rapid mechanisms 18 and 25 equilibrate in a matter of hours before considerable changes in ?~, occur. The situation is very different with slow mechanisms, e.g., mechanism 53; its initial retardation time is 300 days, in a week it becomes close to 5 y, in 10 weeks it is on the order of 16 y, etc. Therefore, 1 y is too short a time for mechanism 53, it remains frozen in the quenched state.
S.M. Rekhson / Memory effects in glass transition
80 ~0
0 +
8.5O .=.. L
P. ta m.
~, o
B00
.=..
J
500
t300
700
880
900
Temper ot:ur e , oK Fig. 10. Same as fig. 9 but in terms of Tf vs T. Cooling curves are also shown.
9~
- - - ~
o~ o+ ~,..+
'-,~5
~-----~- ~
B8~
-l.lv~
x,|
ii.
n u.
7~
\ 10
\\+ +102
|B3
104
10S
106
Time+ mI nu~.~-, Fig. 11. Evolution of the average.fictive temperature, Tr, and partial fictive temperatures, Tr+ during the 1 year long isothermal hold at 630 K.
81
S,M. Rekhson / Memory effects in glass transition g~
"-
"-- 7 0 0
61~
650
7~
750
8~
850
g~0
TupQrokure, °K
Fig. 12. Behavior of the average fictive temperature, Tf, and partial fictive temperatures, Tf,, during reheating at 3 K/rain followingthe 1 y long isothermal hold at 630 K. The 1-y time scale is just right for mechanisms 35 and 40, they become deeply stabilized at the end of the hold. During subsequent reheating at a rate 100 K / s all annealed mechanisms are easily overheated and demonstrate a vigorous approach to equilibrium from below, as shown by curves Tfl8, Tf25, and Tf3 5 in fig. 12. It is these mechanisms that via eq. (9) contribute strongly to the increase in the average fictive temperature, curve Tf, in the temperature interval of 650-780 K. At these temperatures the slow mechanisms do not contribute to the slope of curve Tf as seen from the negligible slope of the curve Tr53. At temperatures above 780 K the slow, quenched processes contribute a negative slope, thus decreasing the slope of the average Tf curve. At temperatures above approximately 860 K all processes equilibrate and the slope d T f / d T = 1. Thus, the slope of d T f / d T is maximal at about 780 K, which is the position of the maximum in fig. 9. 5.3. Cooling~reheating curve
It turns out that no dramatic or complex thermal treatments are needed to demonstrate the memory type behavoir. The reheating curve for the average fictive temperature, Tf, shown in fig. 3, reveals a similar disregard of the equilibrium state. This is clear when compared to the behavior of separate
S.M. Rekhson / Memory effects in glass transition
82
91~8
°.
i!o
% X
Tf32 / / / i .........
788 S~
60~
. . . . . . . . . . . . . . . . . . . 880 708
9~
Temperaluro, °K
Fig. 13. Behavior of the average fictive temperature, Tf, and partial fictive temperatures, Tfi, during reheating at 3 K/min following cooling at the same rate.
mechanisms - Nos. 32 and 53 as shown in fig. 13. The fictive temperature of the single exponential mechanism - the partial fictive temperature - decreases slightly during reheating until it crosses the equilibrium line. This behavior can be clarified by resorting again to the representation of continuous heating as a series of temperature jumps followed by isothermal holds. The equilibrium condition for each mechanism is Tfi = T, which is the equation of the straight line in fig. 13. Thus, during the hold at any temperature, the Tf, should draw closer to the equilibrium line. They certainly do so as illustrated in fig. 13. In contrast, the average fictive temperature increases in the temperature interval where Tf > T, i.e., the system as a whole moves away from equilibrium. The cause of this behavior is those fast mechanisms (e.g. Tf32 which at these temperatures proceed through the final stages of their approach to equilibrium. As seen in fig. 13, this stage is characterized by a vigorous rise in the partial fictive temperature. Again, via Tf = ~uiTf~, these mechanisms will i
contribute to the increase in the average fictive temperature. A positive slope of Tf vs T curves to the left of the equilibrium line was noticed and correctly identified as a memory effect by Moynihan, Macedo and co-workers [20].
S.M. Rekhson / Memory effects in glass transition
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6. Summary The multi-exponential model offers an interpretation of the frozen system as an ensemble of states characterized by a distribution of fictive temperatures. This distribution is a direct result of the distribution of structural retardation times: fast mechanisms will retain equilibrium states during cooling to lower temperatures and will produce lower fictive temperatures, while slower mechanisms will freeze at higher temperatures resulting in higher fictive temperatures. Whereas only the average fictive temperature is measured in the experiment, its behavior appears meaningful only if analyzed as a result of superposition of separate mechanisms. In this work we primarily examined the behavior of glass during reheating. The latter can be considered as a process of re-establishing the equilibrium state. We observed three memory effects whose essence can be outlined as follows. A non-equilibrium and partially relaxed system, when forced back to equilibrium, gradually or abruptly, may exhibit some resistance by moving temporarily away from equilibrium. This seemingly paradoxical behavior can be explained by recognizing that we force the system into the state which we inadequately define as "equilibrium" by using a single parameter - the average fictive temperature, Tf. The enforced condition Tf = T can produce nothing but a mixture of non-equilibrium states at any time shorter than the longest retardation time. As was shown above, when during reheating the entire system is still in a supercooled state characterized by Tf > T, there are some overheated states with Tfi < T. These will proceed at this time through the final stages of equilibration characterized by a vigorous increase in Tf,. With their weighting coefficients they will contribute to the "non-intuitive" increase in the average Tf. In the cross-over experiment, these mechanisms are forced into an overheated state by abrupt reheating. In the sub-Tg experiments both slow and intermediate mechanisms are forced although in different directioxls: slow mechanisms are frozen in a severely quenched state by rapid cooling, and intermediate mechanisms are deeply annealed during long stabilization at low temperature. Thus, by forcing certain mechanisms we can produce many intriguing effects. They all, however, can be understood if the system is properly Characterized by a distribution of fictive temperatures. Although the material properties used in the above computations were those of soda-lime-silicate glass, in this work we limited ourselves to a qualitative modeling of glass behavior. This is because the model used is based on the principle of thermorheological simplicity which assumes the same activation energy for all mechanisms. In reality, there must be a distribution of activation energies with slow mechanisms having higher and fast mechanisms lower activation energy. Therefore the present model may produce some numerical orders when applied to experiments emphasizing the role of one group of mechanisms possibly having an activation energy substantially different from the average. None of the conclusions of this work - which are purely qualitative should be affected by the simplification employed.
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Continuing guidance in computer science by Michael Rekhson of the Massachusetts Institute of Technology, Cambridge, Mass., is deeply appreciated.
Appendix 1 Discrete spectrum of retardation times Mechanism no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Retardation time, )~
Weighting factor,
(s)
u~
7.5858E- 06 1.0136E-05 1.3543E-05 1.8095E - 05 2.4177E- 05 3.2303E - 05 4.3161E-05 5.7669E-05 7.7054E - 05 1.0295E-04 1.3756E-04 1.8380E-04 2.4558E-04 3.2812E-04 4.3842E-04 5.8578E - 04 7.8268E-04 1.0458E - 03 1.3973E - 03 1.8669E - 03 2.4945E-03 3.3330E-03 4.4533E- 03 5.9501E-03 7.9502E-03 1.0622E - 02 1.4193E-02
0.0018 0.0023 0.0027 0.0022 0.0022 0.0022 0.0024 0.0028 0.0034 0.0039 0.0042 0.0042 0.0039 0.0037 0.0036 0.0039 0.0044 0.0048 0.0048 0.0041 0.0029 0.0015 0.0006 0.0009 0.0026 0.0057 0.0095
Mechanism no. 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Retardation time, ),~
Weighting factor
(s)
u~
1.8964E - 02 2.5338E-02 3.3855E-02 4.5235E - 02 6.0439E- 02 8.0755E- 02 1.0790E-01 1.4417E-01 1.9263E - 01 2.5737E-01 3.4388E-01 4.5947E-01 6.1392E-01 8.2027E-01 1.0960E+00 1.4644E + 00 1.9566E+00 2.6143E + 00 3.4930E + 00 4.6672E + 00 6.2359E+00 8.3320E+00 1.1133E + 01 1.4875E+01 1.9875E+01 2.6555E + 01 3.5481E+01
0.0130 0.0153 0.0157 0.0143 0.0116 0.0088 0.0071 0.0071 0.0091 0.0127 0.0168 0.0205 0.0231 0.0246 0.0257 0.0277 0.0320 0.0392 0.0491 0.0600 0.0694 0.0743 0.0724 0.0631 0.0480 0.0302 0.1181
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