On mechanisms of structural relaxation in a Pd48Ni32P20 glass

Journal of Non-Crystalline Solids 46 (1981) 289-305 North-Holland Publishing Company

289

ON MECHANISMS OF STRUCTURAL RELAXATION IN A Pd4sNi32 P20 G L A S S H.S. C H E N Bell Laboratories, Murray Hill, New Jersey 07974, USA

Received 20 August 1981

Structural relaxation processes are investigated calorimetrically for a pre-conditioned Pd4sNi32P20 glass over a wide temperature range from well below to just above the glass transition. The low temperature anneals further stabilize the glassy structure. Upon heating, the annealed sample shows an excess endothermic specific heat ACe above the annealing temperature and completely recovers the initial enthalpy before any manifestation of glass transition, Ts. Significantly the ACp peak evolves in a continuous manner with annealing time. A physically reasonable activation energy spectrum N~(Q) is obtained with the proper choice of coupling constants which are dependent on annealing temperature. Results suggest the existence of localized relaxation modes which do not contribute to macroscopic flow. A concept of distribution in glass transition temperatures H(Ts.m) is conceived to account for the reversible relaxation with temperature. A model glass transition based on percolation theory is proposed and is found to reproduce the calorimetric relaxation phenomena well.

I. Introduction In previous papers, we investigate calorimetrically the structural relaxation processes in the temperature range well below the glass transition Tg in two (Fe, Ni)-based metallic glasses [l] and in a glassy polystyrene [2]. The low temperature anneals stabilized the structure and led to volume contraction. U p o n heating, the annealed samples showed, with respect to the preconditioned sample by slow cooling through T~, an excess endothermic peak and a gradual expansion associated with structural recovery. Most significantly, the samples recovered the initial enthalpy and volume without heating through Tg. Consequently, the samples can be annealed cyclically between two temperatures well below Ts without change in the glass transition process. This relaxation behavior accounted for [ 1] the "reversivity" in some properties with annealing temperature and cross-over p h e n o m e n o n [3,4]. In this paper, we report a m o r e detailed study of the calorimetric behavior of a thermally m o r e stable metallic Pd48Ni32P20 glass. This enables us to investigate the relaxation behavior over a wider temperature range from well below to just above the glass transition, and to yield clearer pictures on the origins of relaxation processes. The activation energy spectrum is obtained with a p r o p e r coupling constant. F r o m the experimental results, the concept of 0 0 2 2 - 3 0 9 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1981 N o r t h - H o l l a n d

290

H.S. Chen / On mechanisms of structural relaxation

localized relaxation and the distribution of glass transition temperatures are conceived. A model glass transition in the framework of percolation theory is proposed to account for the experimental observation.

2. Experimental Glassy samples of Pd48Ni32P20 in ribbon form about 25 # m thick and 1 mm wide were prepared using a centrifugal spinning method. The apparent specific heat, Cp, was measured with a differential scanning calorimeter (perkin-Elmer DSC-2). Care was taken to reduce the thermal drift by pre-warming the calorimeter for at least 5 h in the temperature range of interest. The accuracy of the data was about 0.2 cal/mol K for the absolute Ce values, but was better than 0.1 c a l / m o l K for the relative Cp or ACp measurements. Prior to each test, the sample was thermally conditioned inside the calorimeter at 6 1 0 K for 2 min and then cooled to room temperature. Heating and cooling rates for the thermal cycling tests were always fixed at 20 K / m i n . The glass transition temperature Tg = 583 K was taken at the point of inflection in the C p - T curve. The thermally conditioned samples were then subjected to annealing treatments at temperatures T~ < Tg. Short-duration anneals (t a 30 h) were performed directly inside the calorimeter while long-duration anneals (up to 500 h) were performed in a well-controlled furnace after placing the encapsulated samples inside a vacuum-sealed Pyrex tube. Following the annealing treatment, the sample was thermally scanned at 20 K.min from 320 to 610 K to determine the Cp of the annealed sample. It was then cooled to 320 K, and reheated immediately to obtain the Cp data of the "reference" sample (i.e. the pre-conditioned sample without further low tem.perature annealing). This test procedure is essential in order to eliminate any possibly error that might result from the drift in the calorimeter due to the prolonged annealing time between :the measurements. The change in the calorimeter behavior with annealing was used to monitor the structural relaxation processes. The anneals were conducted in two temperature regions: a sub-sub Ts region e.g. Ta < Tg -- 100 K, and a sub-Tg region. The division being rather arbitrary is based on distinct relaxation kinetics in the, two temperature ranges as shown in the following. 2.1. Sub-sub Ts region, Ta < Ts -- l O O K

Results of typical Cp measurements for T~ = 420 K with t a = 16 h and 500 h are shown in fig. l a. The heating curve and the cooling curve for the "reference" sample show a hysteresis phenomenon in the glass transition region (between 530 K and 605 K) which has been commonly observed in many glasses, the observed Cp values are in good agreement with the previous data [5]. There are differences i n details however, e.g. slight dips in ACp(T) curves at T-~ 470 and 560 K were observed in the previous data. The dips

/ On mechanisms of structural relaxation

H.S. Chen

12 --

--

Cp'e

i~

(a)

8

291

Ir

"o =420K

,,,5 ( X ) _ h ~

I

Cp,~,

~

6 0

(b)

//

- IO0

<] b

z - 200

16h

, -300

,

t

,

450

,

J

,

I

t

500

,

,

T(K)

,

d

,

550

,

h

t

I

,

600

Fig. I. (a) The specific heats of the reference sample, Cp.s and the pre-conditioned PdNiP glass subjected to further anneals at 420 K. The annealing times (in hours) are shown in the figure. (b) The corresponding configurational enthalpy, A Ho(T), where A/-/o(620 K) is set to zero. A Hox is the A H o of the equilibrium liquid.

arose as a result of the fast cooling followed by the slower rate of heating. The heating curve for the annealed sample shows a Cp behavior which closely follows the Cp curve of the reference sample, Ce, s, up to about 420 K, then exhibits an excess endotherm relative to the reference sample before merging again with the reference at a temperature below T~. We may now state briefly three significant features revealed in fig. l a. Firstly, the annealed sample shows an excess endothermic specific heat beginning at 420 K which is about 110 K below the temperature (about 530 K) at which the hysteresis loop of the reference sample begins to materialise. This implies that the specific heat in the range about 420 K is dependent on the thermal history and consists of configurational contributions in addition to those arising from purely thermal vibrations. Therefore, if we denote the vibrational specific heat, by Cp,~, then the values of Cp,~ should be extrapolated from Cp values below 420 K and are described by Cp,~ = 6.87 + 2 × 1 0 - 3 ( T - 420) cal/mol K.

(1)

292

H.S. Chen / On mechanisms of structural relaxation

Similarly, the equilibrium specific heat, or the vibrational and configurational specific heat, Cp,e,'extrapolated from above 620 K is given by Cp,¢ = 11.34 + 2 × 10 -3(620 - T) cal/mol K.

(2)

Secondly, if the annealing is performed at temperatures well below Ts, it does not affect the glass transition process. This is indicated by the close overlap of Ce curves for the annealed and unannealed samples at temperatures above Tv Thirdly, regardless of the length of the annealing time, the excess endothermic curves always begin to rise at 420 K; furthermore, both the magnitude and the temperature of the ACp peak increase roughly in proportion to the logarithm of the time, in a continuous manner, (see figs. 2 and 3) indicative of the existence of a continuous relaxation spectrum. To further examine the significance of the present results, the configurational enthalpy, AHo(T), of the glassy sample was evaluated and is shown in fig. lb. In this case, the configurational enthalpy at 620 K was taken to be the reference with AHo(620 K ) = 0, and the relaxed configurational enthalpy was calculated from - c,.,)

O)

Thus, if Alia, e is the AHo of the undercooled liquid, i.e. the relaxed glassy metal, then (AHo,e - A H o ) corresponds to the unrelaxed portion of the configurational enthalpy of the unrelaxed glass. As can be seen in fig. lb, the configurational enthalpy curve falls progressively with the annealing time, indicating that the low temperature anneals stabilize the glass structure. Despite the fact that the stabilization in the present cases is far from complete, however, upon heating, the --AHo of the annealed sample increases towards the reference value and merges with it before any manifestation of glass transition. Consequently, the -AHo(T) curves tend to stay to the left of the equilibrium line, --AHo,e(T). This implies that, as a result of the structural relaxation which occurred during the sub-sub T~ anneal, it is possible to recover the initial structure of the glass without reheating it through the glass transition. This feature differs significantly from the phenomenon commonly observed for the glassy materials in the usual sub- T~ anneals. To illustrate more details the evolution of the enhalpy relaxation, the differences in Cp between the annealed and reference sample, ACp(= Cp - Cp.s), are shown in fig. 2. We note that for each isothermal anneal, the raise in ACp begins at a critical temperature, Tcr (420 K) whenever the annealing temperature T~ ~ T~r (figs. 2a and b). For Ta > Tcr, the ACp curve begins to rise at Ta (fig. 2c, also figs. 4 and 5). In all cases where ACp disappears at temperatures below T~ (see also fig. 4), both the magnitude and temperature of the ACp peak increase with the annealing time, t~, in a continuous manner. As shown in fig. 3, the peak temperature, Tm, varies nearly as the logarithm of t~ while the

H.S. Chen / On mechanisms of structural relaxation

293

0.8

0.6I- (o) To=380K

Tm(40Oh)

0.4I 0

.

2

~

0L

0.6

oo.

A

_~ 0.4 u o.

<~ 0.2

0.6

~

(c)

Oh

0.4

0.2

I

420

450

~

i

t

I

t

500

T (K)

I

i\\\ 55O

i

6OO

Fig. 2. The differential specific heats, ACp, between the reference and annealed samples. The annealing temperatures, Ta, and duration of anneals, t a, are shown in the figure.

p e a k ACp, ACp. m change m o r e or less quadratically with In t a (not shown, ref. figs. 2 and 3). The Tm values for Ta = 500 K with t a <~ 3 h (ref. fig. 4) are also shown in fig. 3.

2.2. Sub-Tg anneal," Tg - 1 0 0 < To < Tg In the following, it will be shown that the same calorimetric behavior found in the sub-sub T~ anneal can also be seen for samples annealed at temperatures higher than 460 K. However, this phenomenon becomes less pronounced as Ta

294

H.S. Chen / On mechanisms of structural relaxation 105

o 103

K

102

i01

450

i

,

~

,

I

,

,

'

~

500

t

I

,

J

,

550 Tm

(K)

Fig. 3. Variation of the ACt, peak temperature, Tm, with annealing time l a.

is raised closer to Tv and depending on the length o f annealing time, a Cp peak characteristic of the sub-Tg anneal begins to appear at the glass transition. In the annealing study at 500 K as shown in fig. 4, if the annealing time was less than 3 h, e.g. for t a = 1 h the annealed sample exhibits the characteristic ACp peak appearing above Ta and diminished below Tg. The sample annealed for 3 h shows the similar ACp peak beginning at Ta(= 500 K), but extending further into glass transition and as a result, an additional but small ACp peak appears above Tg. For a longer annealing time of t a = 15 h, these two ACp peaks merge into a broad peak. For longest annealing (ta =210h), the annealed sample shows the whole Cp curve shifted slightly to higher temperatures relative to the reference one and exhibits a pronounced peak above Tg. As a consequence the relaxed configurational enthalpy, AHo(T ) exhibits a typical delayed recovery behavior, i.e. the occurence of enthalpy recovery on the right of the corresponding AHo.e(T) curve. The recovery proceeds rapidly once the temperature reaches Tg whereas the structural relaxation becomes comparable with the experimental scale. The delayed recovery behavior is known as a result of the decreased cooperative atomic mobility in glassy metals [6,7]. The observed Tv however, remains nearly the same at 583 K.

H.S. Chen / On mechanisms of structural relaxation

295

--~ .J~ -r ':3

\ \ A

I--O

kq

:°-~ /,

~ .-.,,.

C,

o

O

_L._~ 8

8

8

I

(M-elOU~/i03 ) d O

04

i

( alOW/lOO ) 'PHi7-

z ;.=

8

J

T ,~,,

o

<]

\

.=

0

2

=

°

8

a

2

\\ A v

~n

I..--

u

Q

O

O I

(~-elOWllO~) d3

O i

(elOm/IO3) "~HV-

8

<

296

H.S. Chen / On mechanisms of structural relaxation

When the annealing temperature increases closer to Tg, e.g. Ta = 540 K, as shown in fig. 5, a pronounced delayed relaxation phenomenon can be observed for anneals as short as ½ h (not shown). The calorimetric behavior regarding the Ce peak and A H o ( T ) curve at the glass transition for ta -- 1 h (fig. 5) are similar to that for t a = 2 1 0 h at Ta = 5 0 0 K (fig.4). The ACe also begins to appear at T~(= 540 K) in the present case. Annealing at 540 K for 50 h, the sample structurally relaxes completely to the equilibrium state, i.e. A H o ( T a ) = AHo.e(T~). The annealed sample then shows very pronounced delayed relaxation, and whole Cp curve shifted to higher temperatures by ~ 10 K so does Tg (fig. 5). It should be noted that the excess specific heat ACe(-- Cr - Cp.s) now becomes negative for temperatures below T~. Fig. 6 shows the isothermal enthalpy relaxations of the PdNiP glass. The total enthalpy relaxed AHs(T~, t~) is given by AHs(ta) = AHo(ta) -- AHo(0) or by

AH (r ,

= f 2,, ACe(r) dr'.

(4)

For the high temperature (T~ = 540 K) of anneal, AH~ varies linearly with the logarithm of t a for t a < 300 min, then tends to saturate for longer annealing time. This linear lnt a dependence has been reported widely for sub-Tg anneals [8,9] and is thought to be associated with a cooperative relaxation phenomenon 120

,oo

/

8o A ~D

/

6o b "r

<~ I

4O

• 0 I01

i

102

103 ta

JJ:l 104

105

(min)

Fig. 6. The isothermal enthalpy relaxation AH~AHo(t~)-AHo(O) of the glass. Annealing temperatures, T~, are shown in the figure.

H.S. Chen / On mechanisms of structural relaxation

297

in which the relaxation time depends on either the free volume, the configurational enthalpy or entropy. By contrast, AH s at sub-sub Tg anneals, e.g. Ta = 380K, 420K, varies with (lnta)" with n ranging from 2 to 3. This quadratic to cubic lnt a dependence of AH s is expected as both the magnitude and temperature of the ACe peak scale nearly as lnta (cf. figs. 2 and 3). For the intermediate annealing temperature (Ta = 500 K) AH s varies as (lnta) 2 initially as in the case of sub-sub Ta anneals but it is followed by a linear lnt~ dependence at longer annealing time at which a pronounced ACp peak emerges above T~. In foregoing discussion, it is shown that the glass transition process is not affected by low temperature (sub-sub-Tg) annealing. It implies that the effect of low temperature anneals can be superimposed on that of high temperature ones. Accordingly, successive low temperature annealings are possible provided that each annealing is followed successively by a lower temperature one. Fig. 7(a) shows the ACp curve of the pre-conditioned sample subjected to heat treatments at T~,~ = 460 K, for 2 h followed by Ta,2 = 380 K for 165 h. The ACe curve can be divided into two parts, I and II, as shown by the dotted lines in fig. 7(a) with each curve corresponding to each independent isothermal annealing. Fig. 7(b) illustrates ACp(T) for the glass annealed at Ta,l = 550 K for 3 h followed by Ta,2 = 420 K for 216 h. In this case the annealing tempera-

(a) Ta6 ' 2 H

Ta,2=380 K ,165H 0.60.4

f

l

IT

02

.

(b)

/

Ta,2 216H

0

450

/

"/--

500

550

I

t

600

T (K)

Fig. 7. The differentialspecificheat Acp, of samples subjected to successiveheat treatments.

298

H.S. Chen / On mechanisms of structural relaxation

tures are sufficiently far apart that the contribution to the ACp associated with each anneal is well separated. If the glass is then followed by a brief anneal at T~.3 = 480 K for 2 m in, a large portion of the ACp peak associated with Ta,2 = 420 K anneal disappears. Annealing at T~,4 = 520 K for 2 min eliminates most of the ACp peak below 550 K, but the ACp peak above 550 K remains unaffected. It should be mentioned that the measurements were performed on three samples subjected initially to the same Ta,~ = 550 K, and T~.2 = 420 K anneal.

3. Analysis of data: activation energy spectrum 3.1. Activation energy Qm(Tm) The activation energy for structural relaxation, Qm, can be obtained either by tempering or isothermal annealing with some modifications as follows: (a) Isothermal annealing. We may associate the ACp peak with a relaxation entity with a characteristic Qm which has a relaxation time r* at the peak temperature Tm. Here ~'* is related to the scanning rate a = ½ K s - ~ such that [10],

~'* = kBT2m//Qma ~ 30s

(5)

and the activation energy

Qm(Tm)/kB = d In t * / d ( 1 / T ~ ) ,

(6)

where l* is the annealing time for the appearance of ACp,m at Tm and k B is Boltzmann's constant. Based on the data of fig. 3, log t versus (1/T~) relations are replotted in fig. 8. As t* -, ~*(= 30 s), Tm -, Ta, indeed the fact can be seen from the extrapolation in the figure. Qm(Tm) is found to vary from 1.60 eV at Tm= 480 K to 2.60 eV at T~ = 560 K as shown in fig. 9. The activation energy viscous flow in the relaxed state, Q~ [10], is also shown in fig. 9. As Tg represents the temperature at which the relaxation time T = r* -~ 30 s, Qm(Tm) would connect Qn at Tg. The observed Qm would increase rapidly from 2.6 eV at 560 K to a high value of 4.8 eV at 583 K. The rapid increase in Qm(Tm) above Tm = 560 K in which a ACp peak appears above Tg reflects the occurrence of cooperative structural relaxation (figs. 4 and 5). (b) Tempering. Alternatively, Qm(Tm) can be evaluated from the shift in the ACp spectrum with scanning rate, ot such that

Qm(Tm)/kB ~

d In ( Tm2 / a ) d(1/Tm ) -

T

d ln a d(1/Tm ) ~

d ln a d(1/Tm ) ,

(7)

where T<< Qm/ka. Tins were measured at scanning rates of 10, 20, 40 and 80 K / m i n . The Qm(A) thus obtained agree fairly well with the isothermal results as shown in fig. 9. If we assume the first order rate reaction process for the enthalpy relaxa-

H.S. Chen / On mechanisms of structural relaxation

299

i05

i04

103 E

E o

102

I01

I

Y

2.00

I

I

[.

2.20

-I

240

I

I

2.60

103/T0 (K -I) Fig. 8. The annealing time, t*, for the appearance of the ACp peak at Tm as a function of annealing temperature Ta.

tion, the frequency factor 1'o is given by log 1,oz" -~ Q m / 2 . 3 k B T m.

(8)

The calculated vo(Tm) is shown in fig. 9. uo increases with Tm from vo = I015 s - i at 480 K to a high value of 1022 s - J at 560 K. It attains a ridiculous value of 104°s - I at T~(= 583 K). The calculated values of i)o(= 1015-1022s - I ) for T ~ 560 K are much higher than the characteristic Debye frequency v D ~ 1013 S --I

300

H.S. C h e n /

O n m e c h a n i s m s o f structural relaxation

,./

\

2O

b~

~3

:£

!

_8' 2

0

10

I

I

I

500

I

I

I

I

I

600

I

I

I

I

700

Tm(K)

Fig. 9. The activationenergyspectrum Qra(Tm)and frequencyfactor Poof the glass and activation energies of the undercooled liquid, Q,r The solid triangles are those obtained by the timetemperature shift method.

3.2. Relaxation spectra Po(Q=) and No(Q= ) When both activation energy and frequency factor exhibit a broad distribution, the deduction of the relaxation spectrum from the experimental data is very dificult. For a qualitative study of the phenomelogical behavior of relaxation processes, we deal at present with the case for a broad distribution only in activation energies, assuming Po to be constant. Following the approximation of Primak [11], the initial characteristic distribution P0(Qm) is given for an isothermal anneal at Ta after time t a, P0(Qm) = 7(Qm)No(Qm) ~- - (l/kBra)(d A H s / d In ta) ,

(9)

where N0(Qm ) is the distribution of the relaxation entity, 7(Qm) is the coupling strength contributing to the heat content AH~, and the activation energy Qm is k a T~ In pot* = kaT m In ~0~'* = Qm(Tm). Po(Qm) evaluated from the data of figs. 6 and 9 are shown in fig. 10(a). In fact, AH, can be approximated from fig. 2 as ACp,m(Tm-To), where To = Ta except for Ta < To in which TO = Tcr. AS ACv.ma(Tm-To)" with n = 1.-2, and d T m / d In t ~ 2 X 10 -2 T~ (see fig. 3), eq. (9) reduces to Po(Qm) -~ 2ACp'm d T m / d In t ks Ta ~- (4-6) × 10 -2 ACp.m/kn"

(10)

Indeed P0(Qm) reproduces fairly well the leading edge of the ACp curves. We do not know prior the coupling strength 7(Qm). However based on the results of sub-sub Tg annealing, No(Qm ) would be independent of the tempera-

H.S. Chen / On mechanisms of structural relaxation

301

ture of annealing for which Tm < 560 K, then from fig. 10(a), the best guess would be that Y(Qm) dt(Qm - Qa) or C ( T m -- Ta) with C ~ 1.2 × 10 -2 e V / K , where Qa is the activation energy corresponding to Ta. The ( Tm - T a ) dependence of the couphng constant "t has significant physical meaning which will be elaborated in the last section. The unnormalized distribution spectrum N~(Qm) =-Po(Qm)/(QmQa) are shown in fig. 10(b). The N~(Qm ) curves now are seen to nearly superimpose on each other. It shows a cut-off spectrum at Qm ~ 1.0 eV and increases with Qm-

T m (K)

4OO

450 i

i

i

]

5OO i

i

i

i

I

550 I

i

i

i

I

(b)

i

-"-'~v 15[

o

W~o Z

(o)

I0 8

6 Cl

a? 4

0

~

._L._.I

1.0

1.5 Qm (eV)

2.()

2.5

Fig. 10. (a) The enthalpyrelaxationspectraPo(Q,~) and (b) the unnormalizedrelaxationspectrum Ng(Qm)~l'o(Om)/(Om-Oa).

302

H.S. Chen / On mechanisms of structural relaxation

4. Summary and implications of the results Metallic glasses, even pre-condutioned by slow cooling through the glass transition temperature, Tg, undergo an appreciable structural relaxation at temperatures well below Tg. It is manifested by internal friction measurements [12-14] a decrease in enthalpy [1,2] an increase in density [2], and a raise in Curie temperature [1,3,15], Youngs modulus [16] and resistivity [17]. The changes are small of the order of - 10-3 in the enthalpy and density but are an order of magnitude larger in Curie temperature and resistivityl We observe here that the processes of the sub-sub Tg relaxation are quite unique and distinctively different from those of usual sub-Tg relaxation. The significances of the present results are summarized in the following. (1) Upon heating, the sub-sub Tg annealed sample shows an excess endo-

5

-

(o)

T~

TI

l

0 ~=T W

t--

(b)

>tY
..r

0

(c)

T2
r W

5-

LOCALIZED

0

-I0

!

i

i

i

-5

0

5

log "t- ((;)

Fig. 1 I. Schematic relaxation spectrum H(log 1-) of a typical metallic glass near Tg.

H.S. Chen / On mechanisms of structural relaxation

303

thermic specific heat, ACp, above the annealing temperature and recovers the initial enthalpy without reheating through the glas s transition. Both the temperature and amplitude of the ACp peak evolve with annealing time, t a, such as ln t a in a continuous manner. These features suggest the existence of a continuous relaxation spectrum in which each relaxation mode involving a group of atoms is operating more or less independently from each other in a non-cooperative manner although the atomic rearrangement within the duster may be cooperative. (2) The relaxation spectrum H(ln ~) can be seen to have a wide distribution in both activation energy and frequency factors, u0. It has a time constant width of at least 7 decades (cf. fig. 11). Qm ranges from 1 eV to 2.5 eV with the corresponding values for r 0 varying from 1014 s - l to 10 22 s - 1. The unnormalized activation energy spectrum N0*(Qm) is approximated and shown to have a cut-off energy of - 1 eV and to increase nearly linearly with Qm. In deducing reasonable N*(Qm), the coupling constant Y(Qm) is found to be annealing temperature dependent such that "/(Qa, Om) °t(Om - Oa) °t(Tm - Ta)" this Ta dependence of the coupling constant and the relatively large frequency factor PO >> PD ( ~ 1013 S -- I ), observed are similar to the relaxation behavior commonly observed in the sub-Tg anneal. We therefore attribute the sub-sub Tg relaxation spectrum phenomenologically to a distribution of characteristic glass transitions Tg.m with an apparent activation energy Qm. (3) While the sub-Tg annealed sample (cf. fig. 6) show a strong delayed relaxation indicative of a cooperative non-linear relaxation process which contrasts strongly to the sub-sub Tg behavior mentioned above. It is tempting to refer these two relaxation processes to two distinctly different mechanisms, however, the fact that the low temperature process transforms continuously with the temperature and the duration of the anneal suggests that the two seemingly different relaxation processes have a common origin, although there are differences in details, and can be described by a single relaxation spectrum with a long tail extending towards the short time range.

5. A model glass transition Our. data indicate that the relaxation entities responsible for the short relaxation times operate more or less independently of each other while the long relaxation time process which occurs near Tg is cooperative in nature. To emerge these two processes arising from a single continuous relaxation spectrum, we propose a liquid structure near Ts being inhomogeneous and treat the glass transition through a percolation process. Similar models, but different in detail, have been conceived previously by Cohen and Grest [ 18] and Cyrot [ 19]. We visualize that a liquid consists of liquid-like regions of large free volume or high local free energy, and solid-like regions with small free volume or low local free energy. The inhomogeneity arises in monatomic materials from

304

H.S. Chen / On mechanisms of structural relaxation

fluctuations of density, and in alloys from fluctuations in concentration as well. In the potential energy surface picture of Goldstein [20] each region undergoes infrequent transitions, as compared with the Debye frequency ~D, between local potential minima separated by energy barriers, A~m, in configurational space. Each local minimum represents a different structure, the differences in structure being larger, the larger the energy bariers. A distribution in the heights of mcm is responsible for the relaxation times as "/'mOt exp[ -- Acm / k B T ]. The relaxation spectra near Tg are generally seen to be asymmetric with a log tail in the shorter times and have a half-width of 2-3 decades. With lowering temperature, the spectra become broader and the main relaxation time ~ increases [21]. The relaxation spectra are shown schematically in fig. 11. At the temperature T, > Tg, (fig. 1 la) the whole spectrum lies to the left of the time of measurement z*, so that the whole system undergoes frequent configurational transformation and is liquid-like. During cooling, small parts of regions with • > ~'*, behave like solid clusters in the sense that there are no local configuration transitions, while most of the regions corresponds to a liquid form. More precisely, the liquid consists of solid-like clusters embedded in the liquid structure. As the temperature decreases, the number of atoms or clusters which become solid-like increases. When the solid-like regions form an infinite cluster a drastic decrease in macroscopic flow occurs leading to the liquid-glass transition. The overall configuration is frozen-in at Tg (fig. 1lb). The glass transition is then related to percolation process. A study of this percolation process is very difficult and at present, there is no rigorous solution to the problem, however, the fraction of the solid-like, t, may be taken between ½ to ¼ at the percolation limit [22]. Although the remaining liquid-like regions occupy a larger fraction, 1 - t , of the atoms, they contribute little to macroscopic flow. At temperature, T2 slightly lower than Tg (fig. 1 lc), when solid-like regions grow to such an extent that the fraction of liquid-like regions decreases to the percolation limit, t ' ( ~ t), then most of liquid-like regions are isolated from each other embedded in the solid-like matrix. Each liquid-like region undergoes configurational change being localized and independent of each other, or non-cooperative in nature. It may be emphasized however, that the local structural relaxation itself in the cluster involving a number of atoms can be cooperative. The sizes of the clusters can be - 2 0 , ~ . By analogy, each liquid-like region, m, manifests a liquid-glass transition at Tg.~ when the local relaxation time 'l'm(Tg,m)[ ~ P0.me x p [ - mcm/kTg, m )] equals z*. The difference in specific heat, ACe.~, between the reference sample Cp.s and the vibrational contribution Cp., then would represent the distribution of the glass transition T~,m i.e. mCp,g(Tg,m) ota(Tg,m ). Annealing at temperatures well below Tg, the regions with characteristic relaxation times,

[ A,m(, 1)]

"rm- "r* e x p - - - ~ B

T.

r ,m

'

(11)

H.S. Chen / On mechanisms of structural relaxation

305

shorter than the duration of annealing, t a, would undergo local relaxations towards the local equilibrium states. Each local relaxation contributes to the enthalpy relaxation proportional to (T~,m - T a ) . Upon heating the anealed sample, each region, m, recovers the initial structure and contributes to an excess endothermic specific heat as the glass-liquid transition occurs at or slightly above Tg,m. Thus both the temperature and intensity of the ACe peak evolve in a continuous manner as -- In ta with the coupling constant " / a ( T - Ta). The experimental activation energy spectrum No*(Qm) or N ~ ( T m) shown in fig. 10b, indeed resembles fairly well the configurational specific distribution ACe.g(T ) = Ce,s(T ) -Cp,~(T) (cf. fig. 1). It should be kept in mind that the energy barriers A~m(Om) are dependent on the local configuration, or free volume, thus the observed apparent activation energy Qm(Tm) differs from Ac m, where Tm corresponds to Tg.m. It is quite encouraging that the present model reproduces many of the phenomelogical features observed for low temperature anneals of the PdNiP glass fairly well, although a quantitative comparison between the model and measurements has not been made. The existence of localized relaxation modes and the concept of the distribution of the glass transition for the short relaxation time are significant and are valuable for an understanding of the stability of the glasses at low temperature. The localized relaxation can be many orders of magnitude faster than the cooperative relaxation process responsible for the commonly oberved glass transition. Further exploration on the stability of different glassy materials is still in progress.

References [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [I 1] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

H.S. Chen, J. Appl. Phys. 52 (1981) 1668. H.S. Chen and T.T. Wang, Proc. APS March Meeting, 1981, J. Appl. Phys. 52 (1981) 5898. T. Egami, Mat. Res. Bull. 13 (1978) 557. C.T. Moynihan et al., Ann. New York Academy of Sciences 279 (1976) 15. H.S. Chen and E. Coleman, Appl. Phys. Lett. 28 (1976) 245. D. Turnbull and M.H. Cohen, J. chem. Phys. 34 (1961) 120. G. Adam and J.H. Gibbs, J. Chem. Phys. 43 (1963) 139. A.J. Kavacs, Fotchr. Hochpolym-Forsch 3 (1963) 394. H.S. Chen, J. Appl. Phys. 49 (1978) 4595. H.S. Chen, J. Non-Crystalline Solids 27 (1978) 257. W. Primak, Phys. Rev. 100 (1955) 1677. H.S. Chen, H.J. Leamy and M. Barmatz, J. Non-Crystalline Solids 5 (1971) 448. M. Barmatz and H.S. Chen, Phys. Rev. 139 (1974) 4073. B.S. Berry and W.C. Pritcher, J. Appl. Phys. 44 (1973) 3122. A.L. Greet and J.A. Leak, J. Non-Crystalline Solids 33 (1978) 291. A. Kursumovic, M.G. Scott, E. Girst and R.W. Cahn, Scrip. Met. 14 (1980) 1303. E. Balanzat, C. Maivy and J. Hillaret, J. Phys. C8 (1980) 871. M.H. Cohen and G.S. Gres, Phys. Rev. B20 (1979) 1077. M. Cyat, J. Phys. C8 (1980) 107. M. Goldstein, J. Chem. Phys. 51 (1969) 3728. D. Ulhmann, in Metallic glasses, eds., J.J. Gilman and H.J. Leamy, (ASM, Metals Park, 1978) p. 128. [22] K.K.S. Shante and S. Kirkpatrick, Adv. Phys. 20 (1971) 325.