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Transient nucleation effects in glass formation
Journal of Non-Crystalline Solids 79 (1986) 295-309 North-Holland, Amsterdam
295
T R A N S I E N T N U C L E A T I O N EFFECTS IN GLASS F O R M A T I O N K.F. KELTON * Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
A.L. G R E E R Department of Metallurgy and Materials Science, University of Cambridge, CB2 3QZ, UK Received 18 December 1984 Revised manuscript received 1 April 1985
We extend to the non-isothermal case a numerical technique that was developed to treat transient homogeneous nucleation in a one-component system by modeling directly the reaction by which clusters are produced. Calculations are presented for the nucleation frequency during the quench and for the number of nuclei produced and the volume fraction transformed at the end of quench for different rates of cooling from the melt. Three model systems are considered: an alkali silicate which is a relatively good glass former, and two metallic glasses. These show a wide range of critical cooling rates for glass formation. In some systems transient effects are predicted to be critical for glass formation. A simple technique is presented for determining when transient effects are important based on a calculation using steady state nucleation frequencies and macroscopic growth velocities.
1. Introduction
Glass formation occurs when crystallization is avoided upon cooling a liquid. An understanding of the variation of glass formability from system to system is therefore possible by considering the kinetics of crystallization. For the particular case of metallic glasses, such kinetic analyses have proved to be more widely applicable than treatments of glass formability based on structural, electronic or thermodynamic criteria [1]. In this paper we present a more thorough kinetic analysis than has been attempted to date, including both transient nucleation and size-dependent growth effects. Turnbull [2] proposed the avoidance of nucleation in a given volume of liquid during the quench as a kinetic criterion for glass formation. Uhlmann [3,4] suggested an operational but less idealized criterion that the crystalline volume fraction at the end of the quench should be less than some arbitrary value, typically chosen to be 10 -6 . Both approaches include the implication that any material can be made glassy given a sufficient cooling rate. Uhlmann's criterion is the more useful, especially since it is known that in some demonstrably glassy materials there are crystal nuclei [5]. * Present address: Department of Physics, Washington University, St. Louis, Missouri 63130, U.S.A. 0022-3093/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
296
Previous kinetic analyses have considered only materials that crystallize polymorphically (i.e. without compositional change). Since it is desired to calculate a critical cooling rate that is dependent on the materials parameters only, heterogeneous nucleation has not usually been considered. On the basis of classical nucleation theory it is expected that in an undercooled liquid that is initially devoid of crystalline clusters the nucleation frequency will rise sigmoidally with time from zero to the steady state value. Hillig [6] pointed out that such transient effects could become significant in rapid quenches. Despite this, most kinetic analyses have assumed steady state conditions throughout the quench; the temperature-dependent nucleation frequencies are calculated directly from the classical theory. Uhlmann [3] justified this approach, at least for metallic systems, by pointing out that typical nucleation transient times are less than 1 ~s at the onset temperature for homogeneous nucleation. For rapid quenches of the order of 10 6 K s -1, however, even such short transient times might be significant. In silicate systems the transient times can be as large as one day at the glass transition temperature, but the critical cooling rates are very low [3]. The growth rates of nuclei are reduced from their macroscopic values because of the curvature of the crystal-liquid interface. However, previous calculations of critical cooling rates have used the macroscopic, size-independent values. Several approaches that use the above assumptions have been employed to calculate the critical cooling rates for glass formation. Initially, Uhlmann [4] constructed an isothermal transformation curve for a volume fraction of 10 -6, and took the critical cooling rate to be the undercooling at the nose of the curve divided by the transformation time at that temperature. Onorato and Uhlmann [7] improved this analysis by using the method of Grange and Kiefer [8] to calculate a continuous cooling curve. The method of Grange and Kiefer, however, is valid only for an isokinetic reaction [9]. Hopper, Scherer and Uhlmann [10] presented a computer calculation of the volume fraction transformed at a given quench rate, using a "crystal distribution function". They showed that, in principle, the thermal history of the sample can be deduced from the crystal size distribution. Vreeswijk et al. [11] were the first to attempt to model the effects of transient nucleation during the quench. They used an approximate expression due to Zeldovich [12] for the time dependence of the nucleation frequency, I, at constant temperature:
I(T,
t) = / S ( T ) e x p [ - r ( T ) / t ] ,
(1)
where I s is the steady state nucleation frequency. They estimated the transient time r ( T ) using the expression of Hillig [6]: 'r(T) =
~rd2/D,
(2)
where d is the molecular diameter and D is the effective diffusion coefficient at the crystal-liquid interface. To obtain the number of nuclei formed during a quench, Vreeswijk et al. integrated I(T, t) given in eq. (1). This assumes that
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
297
the nucleation frequency at any instant in the quench is the same as it would have been after annealing at the instantaneous temperature for the elapsed time since the beginning of the quench. This is clearly unjustified. The most thorough treatment to date for modeling transient nucleation effects on glass formability is that of Yinnon and Uhtmann [13]. They used a numerical technique for determining the nucleation rate during the quench and calculated the volume fraction transformed using the method of Hopper et al. [10]. The quench was modeled as a series of short isothermal anneals. The nucleation frequency for any isothermal interval, for example at temperature T2, was given by: I ( T z , t) = I ( T 1 , f ) + [IS(T2) - I ( V l , f ) ] e x p [ - ' r ( T z ) / t ] ,
(3)
where T1 is the temperature of the previous isothermal interval, and I ( T 1, f ) is the nucleation frequency calculated at the end of that interval. Since the value of I(7"1, f ) was taken to be the initial nucleation frequency in the succeeding interval, this approach fails to properly include the temperature dependence of the nucleation frequency due to the change in the atomic mobility. As a result, the transient nucleation frequencies calculated from eq. (3) become larger than the steady state values at low temperature. In addition, Kelton et al. [14] have shown that eqs. (1) and (2) are seriously in error for the description of the isothermal transient nucleation behavior. A more fundamental flaw, however. is the assumption that any expression for the isothermal case can be used to describe non-isothermal behavior. During a quench the distribution of crystal cluster sizes in the liquid evolves in a complex way because the steady state distribution, toward which it tends, is a strong function of temperature. A proper calculation of the nucleation frequency during a quench must correctly account for the cluster dynamics. In this paper we present a numerical solution to this problem in which the cluster dynamics are modeled directly and a size-dependent growth velocity is taken into account. We calculate for a variety of model materials the total number of nuclei formed and the total volume fraction transformed as a function of quench rate. To illustrate the importance of transient effects, parallel calculations are performed using the same input parameters and assuming steady state behavior and planar growth kinetics.
2. Numerical method
Assuming classical nucleation theory, in which clusters are formed by a series of bimolecular reactions, we employ a numerical technique that directly simulates the dynamics of cluster formation by dividing the time into small intervals and calculating the change in the cluster size distribution for each interval. We have demonstrated the validity of this method for the description of steady state and transient nucleation in the isothermal case [14]. We now extend the technique to non-isothermal transformations by dividing the con-
298
K.F. Kelton, A.L Greer / Transient nucleation effects in glass formation
tinuous cooling into small isothermal intervals; the numerical simulation then proceeds as in the isothermal case over these intervals. Following our earlier treatment the algorithm chooses a very small calculational time increment at the start of the quench, typically 1 0 - 8 / y , where 7 is the unbiased molecular (or atomic)jump frequency: y = 6D/• 2,
(4)
where D is the diffusion coefficient at the crystal-liquid interface and A is the average jump distance. As the calculation proceeds, the algorithm changes this increment to reduce the CPU time requirements. In accordance with the quench rate and the temperature increment (typically 1 K), the algorithm periodically changes the temperature and recalculates the rate constants and the critical size (n*). At each temperature the calculation then continues using the new rate constants, starting with the distribution of clusters inherited from the previous temperature. By using this method the transient nucleation behavior during the quench is directly simulated without the artificial introduction of approximate expressions developed for the isothermal case. The free energy of formation of a spherical cluster of n molecules is taken to be: AG,, = n A G ' + 4on 2/3,
(5)
where ~G' is the Gibbs free energy difference per molecule between the initial and final phases, and o is the interfacial free energy per molecular site at the interface. AG' and o are taken to have macroscopic values independent of n. The L o t h e - P o u n d correction [15] and the contribution of stresses are ignored; these assumptions should be appropriate for liquid-to-crystal or glass-to-crystal transformations [14]. The rate constants, that is the average frequency with which a cluster of n molecules gains a molecule and the average frequency of the reverse reaction, are obtained from reaction rate theory and are expressed as"
k + = O.y e x p ( - A g . / 2 k B T )
(6a)
k.-+l = 0.7 e x p ( + A g J 2 k , T ) ,
(6b)
where On is the number of sites at the surface of a cluster of n molecules (taken to be 4n2/3), A g . is (AG.+ a --AGn) , k B is Boltzmann's constant, and T is the absolute temperature. The number of molecules in the initial phase is taken to be Avogadro's number, NA, so the nucleation rates reported are per mole. The upper limit of the cluster distribution is chosen always to exceed the critical cluster size, n*. It is normally set at 1.5n* for the highest temperature (largest n*) considered in the simulation; at the temperature of the maximum nucleation frequency this limit is typically greater than 4n*. The backward flux for clusters at the upper limit is taken to be zero. A 9-molecule cluster is chosen as the lower limit based on arguments presented elsewhere [14]. The steady state nucleation rates are
K.I~ Kelton, A.L. Greer / Transient nucleation effects in glass formation
299
obtained from [16]: i~ =
1
u N~k.
(7)
,
where v and u are the upper and lower limits of cluster size, and N.~ is the equilibrium number of clusters of size n. For our numerical treatment, Uhlmann's criterion was adopted to determine the critical cooling rate; to obtain a glass, the volume fraction transformed must be below the detectable limit of 1 0 - 6 . Although it is possible, in principle, to calculate the total volume transformed directly, using the numerical simulation to obtain the cluster size distribution at the end of the quench, it is not computationally practicable to use an upper limit of sufficient size. Fortunately, the evolution of the cluster size distribution at large sizes can be described analytically. As shown elsewhere [17], the behavior can be modeled to a very good approximation by calculating the average growth rate for a post-critical cluster and assuming that every cluster behaves in exactly the average manner. Using transition rate theory to calculate the net rate at which atoms are added to a cluster of a given size, and assuming spherical clusters, the average growth rate of a cluster (rate of increase in radius) is found to be both temperature and radius dependent, and is given by: dr dt
16D ~2
3V ~
20A
sinh[ 2-~BT [AG,' - - -r
(8)
where V is the molecular volume, AGv is the free energy decrease per unit volume on transformation and oA is the interracial energy per unit area. The expression is least accurate for growth from cluster sizes near the critical size, but is applied in our simulation only to large cluster sizes. For clusters with n > 1.5n* the error from using eq. (8) is at most 3%. The use of an artificial upper limit, v, in the numerical simulation has a negligible effect on the dynamics of cluster formation for n < (v - 10). During a simulated quench the number of clusters of size ( v - 10) generated in each temperature interval is stored. In subsequent intervals the growth of these clusters is calculated using eq. (8). In this way we obtain the cluster size distribution at the end of the quench. The transformed volume is found by summing the individual volumes of all the clusters (for n > 10). The number of nuclei generated during the quench is taken to be the number of clusters of size greater than (v - 11). The choice of (v - 10) as the minimum nucleus size, though arbitrary, ensures that the nuclei are post-critical at all temperatures in the simulation. Ideally the simulation of a quench would start at the melting temperature, but this is not possible because the critical cluster size is infinite at that temperature and remains too large for the numerical simulation until a substantial undercooling. An appropriate initial temperature was determined in each case by starting the simulation at successively higher temperature until the nucleation frequency (net forward flux between cluster sizes ( v - 11) and
300
K.F Kelton, A.L. Greer / Transient nucleation effects in glass formation
(v - 10)) at the end of the first temperature increment was at least 95% of the steady state frequency at that temperature. At higher temperatures steady state nucleation was assumed (see below). The cluster size distribution at the start of the simulation was taken to be the steady state distribution at that temperature. By using this technique no assumption was made for determining the point at which transient effects should become important, within the range of cluster sizes considered. The starting temperature for the simulation is somewhat arbitrary, being determined by the range of cluster sizes that is practical, but the results are independent of this starting point provided the steady state nucleation frequencies at higher temperatures are negligible. The total number of nuclei and the total volume transformed were also calculated by assuming that the nucleation frequency maintained its (temperature-dependent) steady state value (from eq. (7)) throughout the quench. For a valid comparison the nucleation frequency was defined in the same way as for the transient nucleation calculation. In effect, this procedure assumes that the steady state applies only up to cluster size (v - 10). In ideal steady state the forward flux is the same at all cluster sizes, but if this applied up to infinite size the rate of transformation would be infinite; presumably the steady state can apply only in a limited range of cluster sizes. In the steady state calculation we assumed that the macroscopic growth rate applied for all clusters larger than (v - 10). It should be emphasised that the steady state assumption, when applied to large cluster sizes, is seriously flawed. This is so even when transient effects are not expected to dominate. Calculations Of steady state nucleation do not represent physically realizable conditions, but are presented for comparison to illustrate the possible errors in estimating glass formability when simple models of nucleation and growth are used.
3. Results and discussion
3.1. Input parameters For our calculations we chose parameters which are appropriate for an approximate description of three real materials: lithium disilicate, an oxide glass former; and two metallic glass formers, (AussCu15)v7Si9Ge14 and AuslSi19 (compositions in at.%). These choices were made because there were sufficient data to estimate the material parameters and because the crystallization is polymorphic in each case. Our numerical techniques are valid only for polymorphic transformations, though the transformed volume fractions of interest in glass formation are so small ( - 1 0 -6) that the assumptions of a constant matrix composition may not be so restrictive. Lithium disilicate (Li 2° . 2SiO 2) is by far the best glass-former of the three materials, though it is a comparatively poor glass-former for a silicate system. It is known to exhibit transient nucleation effects in devitrification [18]. (AussCUls)77Si9Ge14 is a
k oA V
oA/T
-
a
Ionic d i a m e t e r
J u m p distance Interfacial e n e r g y 9.96 × 10
4.6 ~, 0.15 J m
-
Molecular v o l u m e
I K-1
29 m ~
2
2 X 10 9 m 2 s- 1 440 kJ tool. - 1
To
D0 E r/o
Pre-exponential diffusivity A c t i v a t i o n e n e r g y for diffusion Vogel-Fulcher parameters
1300 K 4 0 J mol.
-
T~ ASf
Melting temperature E n t r o p y o f fusion
Li 2 0 . 2SiO 2
~j
Symbol
Parameter
Table 1 T e m p e r a t u r e - i n d e p e n d e n t p a r a m e t e r s used in the c a l c u l a t i o n s
i K-l
4jm
1.80 × 10 29 m 3
1.95×10
2.88
1.70 ]~
150 K
2 K
1.12 × 10 s N s m - 2 6546 K
_
_
623 K 9.39 J mol.
( A u 85 Cu 15 ) 77Si 9Ge 14
i
A 2.19×10 1.80×10
2.88 ,~
1.70
169 K
4057 K
1.78x10
i K
i
4Jm 2 K 29 m 3
5 Nsm-2
631 K 12.29 J mol.
A u 81Si 19
1
e~
e~
,2
302
K.F Kelton, A.L. Greer / Transient nucleation effects in glass formation
good metallic glass-former also exhibiting transient effects in devitrification [19]. Au81Si19 is the worst glass-former of the three [20], with a critical cooling rate of - 1 0 6 K s 1. The choice of these three materials permits the investigation of transient nucleation effects over a wide range of critical cooling rates. Although we have chosen parameters that approximate those for real materials it is not the purpose of this work to make a detailed comparison with experimental results. The parameters chosen for the calculations are listed in table 1. Two techniques were used to calculate the free energy of crystallization. For Li 2° . 2SiO 2, the free energy per molecule was taken to be proportional to the undercooling: ASr AG t = ' ~ A ( 1 - - Tm) , (9) where Tm is the melting temperature, ASf is the molar entropy of fusion and N a is A v o g a d r o ' s n u m b e r . T h e free energy for A u 8 1 8 1 9 and (AussCUls)v7Si9Ge14 was calculated using the expression of Thompson and Spaepen [21]: AS~ ( T 2T AG' = -~A -- Tm ) ( T-----~-~m) .
(10)
The interfacial energy, oA, for lithium disilicate was assumed to be temperature-independent, and its value was estimated from experimentally determined steady state nucleation rates. For the metallic glasses o A was assumed to be proportional to the temperature and was estimated using the expression of Spaepen [22]. The molecular jump frequency, ~,, was calculated from the diffusion coefficient D and the jump distance X (taken to be the molecular diameter in lithium disilicate and the atomic diameter of gold in the metallic glasses). An Arrhenius temperature dependence of D was assumed for lithium disilicate: D = D Oe x p ( - E / k s T
),
(11)
where E is the activation energy for diffusion. The viscosity of metallic glasses is approximately described by the Vogel-Fulcher expression:
where T0, ( and To are estimated from experimental data [19]. The diffusivity is obtained from the viscosity using the Stokes-Einstein relation: D = kBT/3~ra~,
(13)
where a is a characteristic distance taken to be the ionic diameter of Au 3÷ [19]. 3.2. R e s u l t s
Figs. 1-3 show the nucleation frequency as a function of temperature and quench rate for Li 20.2SIO2, (Au 85Cul 5) 77Si 9Ge14 and Au 81Si~9 respectively.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
303
10s ~'~
S,,feody ~
~E ~ 103f
~
~
/
LITHIUM ~
/
~
/
Di S',L/CATE
~
\ z ~
5 600
\
~
680
760
840
920
1000
TEMPERATURE (K)
1020
i
~,~
I
'
]
Steady State~
I
~
I
'
Fig. 1. The homogeneous nucleation frequency for lithium disilicate for a cluster of 138 molecules, calculated as a function of temperature and quench rate. The critical size at the peak in the steady state nucleation frequency, occurring at 750 K, is 23 molecules.
[
(Au85Cu15)77SigGe14
'a~ 1016 ,,~- 1012
n.z 10s
=, z
lo 4
1.0 300
I
J 330
I
J 360
I
J 390
I
420
~\
4,50
TEMPERATURE (K)
Fig. 2. The homogeneous nucleation frequency for (AussCuls)vv SigGel4 for a cluster of 297 atoms, calculated as a function of temperature and quench rate. The critical size at the peak in the steady state nucleation frequency, occurring at 332 K, is 31 atoms.
2G 10
~
"" 500
1
r
340
380
,
AusISi
420
TEMPERATURE (K)
l,9
460
,
500
Fig. 3. The homogeneous nucleation frequency for Au81Sil9 for a cluster of 330 atoms, calculated as a function of temperature and quench rate, The critical size at the peak in the steady state nucleation frequency, occurring at 330 K, is 17 atoms.
304
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
I 0 8 ~ Lithium Oisilicote 106I ~ : ~ - . ~ ' - ~ - o ~ ,; Stead),Stote ~ ~ - ~Nucleation 104I ~ '~"""'--~. Tronsient Nucleotion
z I
I
I
164( 16e
- SteadyState "~cleotion Transient J
10-1
1
l
I
101 102 QUENCH RATE (K/s)
103
Fig. 4. The number of nuclei (a) and transformed volume fraction (b) calculated for quenching lithium disilicate at variousrates, assumingeither steady state or transient nucleation.
The behavior is qualitatively similar in the three cases: an increasing quench rate lowers the nucleation frequencies below the steady state values and displaces the peaks to higher temperatures. For each system the nucleation frequencies converge to the steady state frequencies at high temperature. The increasing deviation from steady state as the temperature is lowered reflects the decreasing atomic mobility. A comparison of the systems shows that, for a given quench rate, the deviation from steady state is more pronounced in the better glass formers. Figs. 4 - 6 a show the influence of transient effects on the total number of nuclei produced during the quench. In all cases the number of nuclei deviates from the steady state value at an accelerating rate as the quench rate is increased. Transients in the nucleation frequencies are therefore very important in determining the total number of nuclei produced during the quench. Based on the criterion for glass formation suggested by Turnbull [2], transient effects are shown to be extremely important in glass formability. The more common criterion for glass formation, however, is based on the volume fraction transformed, as suggested by Uhlmann [3,4]. Figs. 4 b - 6 b show the volume fraction transformed as a result of nucleation and growth during the quench for the three materials considered. In agreement with the trends shown in figs. 4a-6a, the volume fraction drops precipitously below the values calculated assuming steady state as the quench rate is increased. A surprising difference, however, is the intersection with the steady state result for a quench rate at which the number of nuclei is still several orders of magnitude below the steady state result. This reflects the importance of the large growth velocity
305
K.F Kelton, A.L. Greer/ Transient nucleation effects in glass formation 102o A
'°
1
0
~
A
u
S
1
Si19
1012
~ 108
~
10!
Transient Nucleation - ~
10 4
,n4
\ Transient
'i I
l i
I
i
i
"o --
164
N ]~s
16 s
1012
.
.
.
.
.
.
.
.
.
.
.
.
1612 1016
~ 1016
=~ 1~2o 102
~ucleotion
I
~ 1()20
105 104 105 QUENCH RATE (K/s)
Fig. 5, The number of nuclei (a) and transformed volume fraction (b) calculated for quenching (Au85CUls)77SigGe14 at various rates, assuming either steady state or transient nucleation.
106
t04
~ I
,
Nucleatiol J\
105
106 l0 T QUENCH RATE (K/s)
Fig. 6. The number of nuclei (a) and transformed volume fraction (b) calculated for
quenching Au81Si19 at various rates, assuming either steady state or transient nucleation.
at temperatures where the steady state and transient nucleation frequencies are similar. At very high quench rates the transient nucleation frequencies are decreased to a point where the number of nuclei produced is so small that the volume transformation is insignificant, even with high growth velocities. Based on the Uhlmann criterion the critical cooling rate for L i 2 0 . 2 S i O 2 is 3.8 K s - t from both the steady state and transient calculations. For (AussCuls)vvSi9Ge]4 steady state calculations predict a critical cooling rate of 4.5 × 10 3 K s 1 and the transient calculations predict a value of 1.3 × 10 4 K s-1. For the Aus]Si]9, alloy, however, the two calculations give quite different results. Transient nucleation calculations predict a critical cooling rate of 1.0 × 10 5 K s 1 and steady state calculations predict a critical cooling rate of 2.4 × 10 7 K s ], a value not easily obtainable by disk quenching. In some systems, therefore, transient effects are significant and must be considered in glass formation. 3.3. Discussion To understand why transient effects are important in some cases and not in others, we examine the transition from steady state to transient nucleation during the quench. The ratio of transient to steady state frequencies as a function of temperature is plotted for lithium disilicate at various quench rates
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
306
,0 L,TH,0M # / / F:
y f
1o-K/ /K/t071o2/1o3/
/: / /i /i /
0.4
~
,
j
i
I
I
I
I
I
I
/ I /I
o.2
l
/
I I,I
/!
III
y il rl ll
I
00B00"
080
/
i
?B0
840
I
920
TEMPERATURE (K)
I000
Fig. 7. The ratio of the transient nucleation frequency to the steady state nucleation frequency, I(T, Q/IS(T), for lithium disilicate as a function of t e m p e r a t u r e a n d quench rate. The dotted lines indicate the temperatures, TD, for which the ratio is 0.5.
in fig. 7. At each quench rate the transition from the steady state is quite sharp and occurs at a temperature To, defined to be the point at which the ratio of the nucleation frequencies is 0.5. This temperature can be estimated analytically by considering the transient time, ~-, for the isothermal generation of critical nuclei. As shown in an earlier paper [14], T is given to a good approximation by the expression of Kashchiev [23]: T= 24kBTn*/~Zk+,AG '. (14) -
In fig. 8 log T, calculated using eq. (14) for each of the values of T O in fig. 7, is plotted against the logarithm of the quench rate appropriate for each T O. Data calculated in a similar manner for the two metallic glasses are plotted in the same figure. The data show that at the transition temperature the transient time ~" and the quench rate Q are inversely proportional. For each system the product TQl't = 1.7 K. For our purposes we use the approximation sQ = 1 K;
102
~,~
o LITHIUMDISlLICATE o (AuBsCUls)7/Si90ei4 10
L,i
~E i-Z uJ
1(~2 104
I°!io-2
L
L
1
10 2
i I
~
104
QUENCH RATE (K/s)
10fi
108
Fig. 8. The transient time, ~-, at temperature T O (see fig. 7) as a function of the quench rate, Q, for the three systems considered.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation 1001
'(
307
'LITHIUI~ DISILII~ATE'
,~ 901 D I-- 800 rr"
lO(] bJ
t--
60C 500
I
I
I
I
103
I
I
1
I
106 109 TIME (s)
I
1012
1015
Fig. 9. The time required for isothermal transformation to 10 6 volume fraction for lithium disilicate as a function of temperature. Steady state nucleation frequencies and macroscopic growth velocities were assumed. The temperature T~) at which Q * ' r ( T ) = I K is shown, where Q* is the critical cooling rate calculated using the above assumptions.
then at any quench rate T D can be estimated by finding the temperature at which Q'r(TD)= 1 K. If Q is set to be the critical cooling rate calculated from steady state considerations, T D is the temperature below which transient nucleation effects are likely to dominate in a glass-forming quench. Figs. 9-11 show this temperature, T~, on isothermal transformation plots for the three materials. These plots show the time to 10 -6 volume fraction transformed assuming steady state nucleation and macroscopic growth rates. For lithium disilicate (fig. 9) the nose of the transformation curve lies well above T~, that is, the maximum rate of transformation occurs at a temperature where the nucleation frequency is in steady state. This is consistent with the result in fig. 4b that transient nucleation effects on the volume fraction transformed, and therefore on the critical cooling rate, are negligible. For Aus]Si]9 (fig. 11) on the other hand, the nose of the transformation curve lies well below T~), and fig. 6b shows that transient effects are significant. ( A u 8 5 C u ] 5 ) 7 7 S i g G e ] 4 is an intermediate case. Thus, by comparing T~ with the temperature for which the
500
E
I
i
i
I
I
i
I
I
( Au 85Cu 15) 77 Si9Ge14 450 LLI D~
'~ rr 400 - T D =395K LLI O_
t- 350
3°00 -1 r
1'
'
101
TIME (s)
102
105
104
Fig. 10. The isothermal transformation curve and temperature T~) for (Aus5Culs)77Si9Ge14. Other details as for fig. 9.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
308
500
'
U.I rr' I-L~J
~m~ =546K . . . . ~.~ - - - - ~ - - ' ~
Au81Sil9
45O
400
W ~- 350
300
D-5
10-~
I 101
I
TIME (s)
l 10 3
I
I 10 $
t 10 /
Fig. 11. T h e i s o t h e r m a l t r a n s formation curve and temperature T~ for Au81Sil9. O t h e r d e t a i l s as
for fig. 9.
transformation rate is maximum, as shown in figs. 9-11, one can readily determine whether transient effects will be important. A test of the treatments presented in this paper is not possible by making a detailed comparison of calculated and experimental critical cooling rates, because even small uncertainties in the material parameters can introduce uncertainties of several orders of magnitude in the calculated nucleation frequencies. However, there are manifestations of transient nucleation that can be used to test the theory. These experimental manifestations, both in devitrification and in glass formation, have been reviewed recently [17]. In glass formation, the dependence of the number of quenched-in nuclei on the cooling rate gives important information. For the metallic glass Fe80B20 it has been found that the number of quenched-in nuclei is not inversely proportional to the quench rate Q, as expected for steady state nucleation, but is proportional to Q-(2 to 4). This is clear evidence of transient nucleation in this system and confirms qualitatively the behavior predicted in figs. 4a-6a.
4. Conclusions
We have presented calculations of nucleation frequencies as a function of temperature, total numbers of nuclei produced, and total volume fractions transformed during quenches from the melt for three materials with thermodynamic and kinetic parameters that approximate those of L i 2 0 . 2 S i O 2, (Au85Cu]5)77Si9Ge]4 and Au81Si19. Transients in the nucleation frequency have been introduced in a natural way by directly simulating the cluster evolution during the non-isothermal treatments. These results are compared with those calculated by assuming a steady state nucleation frequency for the same thermal histories. In all systems, the nucleation frequency and number of nuclei produced are dramatically affected by the inclusion of transient effects.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
309
T o d e m o n s t r a t e the possible effect o n the glass formability of some alloys d u r i n g a rapid q u e n c h from the liquid, we have calculated the v o l u m e fraction t r a n s f o r m e d a n d c o m p a r e d it with the volume fraction t r a n s f o r m e d a s s u m i n g steady steady conditions. I n AusaSi~9 t r a n s i e n t n u c l e a t i o n effects are d e m o n strated to significantly i m p r o v e the prospect of o b t a i n i n g a glass at a n a t t a i n a b l e q u e n c h rate. W e have presented a simple analysis to evaluate the i m p o r t a n c e of transient effects in glass formability that involves only steady state considerations. This work was supported b y O N R u n d e r contract N00014-77-C-0002, by N A S A u n d e r contract NAS8-35416 a n d b y N S F u n d e r contract DMR80-20247.
References [1] [2] [3] [4] [5] [6]
H.A. Davies, Phys. Chem. Glasses 17 (1976) 159. D. Turnbull, Contemp. Phys. 10 (1969) 473. D.R. Uhlmann, in: Materials Science Research, Vol. 4 (Plenum, New York, 1969) p. 172. D.R. Uhlmann, J. Non-Cryst. Solids 7 (1972) 337. A.L. Greer, Acta Metall. 30 (1982) 171. W.B. Hillig, Syrup. on Nucleation and Crystallization of Glass (Amer. Ceram. Soc., Columbus, Ohio, 1962) p. 77. [7] P.I.K. Onorato and D.R. Uhlmann, J. Non-Cryst. Solids 22 (1976) 367. [8] R.A. Grange and J.M. Kiefer, Trans. ASM 29 (1941) 85. [9] J.W. Cabn, Acta Metall. 4 (1956) 572. [10] R.W. Hopper, G. Scherer and D.R. Uhlmann, J. Non-Cryst. Solids 15 (1974) 45+ [11] J.C.A. Vreeswijk, R.G. Gossink and J.M. Stevels, J. Non-Cryst. Solids 16 (1974) 15. [12] J.B. Zeldovich, Acta Physicochim. URSS 18 (1943) 1. [13] H. Yinnon and D.R. Uhlmann, J. Non-Cryst. Solids 50 (1982) 189. [14] K.F. Kelton, A.L. Greer and C.V. Thompson, J. Chem. Phys. 79 (1983) 6261. [15] J. Lothe and G.M. Pound, J. Chem. Phys. 36 (1962) 2080. [16] J.W. Christian, The Theory of Transformations in Metals and Alloys, 2nd ed. (Pergamon, Oxford, 1975) Chp. 10. [17] K.F. Kelton and A.L. Greet, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals Wi~rzburg, 3-7 September 1984, eds. S. Steeb, H. Warlimont (North-Holland, Amsterdam, 1985) p. 223. [18] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [19] C.V. Thompson, A.L. Greer and F. Spaepen, Acta Metall. 31 (1983) 1883. [20] C.V. Thompson, A.L. Greet and A.J. Drehman, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals, Vol. 1, eds., T. Masumoto and K. Susuki (Japan Inst. of Metals, Sendai, 1982) p. 743 [21] C.V. Thompson and F. Spaepen, Acta Metall. 27 (1979) 1855. [22] F. Spaepen, Acta Metall. 23 (1975) 729. [23] D. Kashchiev, Surface Sci. 14 (1969) 209.
295
T R A N S I E N T N U C L E A T I O N EFFECTS IN GLASS F O R M A T I O N K.F. KELTON * Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
A.L. G R E E R Department of Metallurgy and Materials Science, University of Cambridge, CB2 3QZ, UK Received 18 December 1984 Revised manuscript received 1 April 1985
We extend to the non-isothermal case a numerical technique that was developed to treat transient homogeneous nucleation in a one-component system by modeling directly the reaction by which clusters are produced. Calculations are presented for the nucleation frequency during the quench and for the number of nuclei produced and the volume fraction transformed at the end of quench for different rates of cooling from the melt. Three model systems are considered: an alkali silicate which is a relatively good glass former, and two metallic glasses. These show a wide range of critical cooling rates for glass formation. In some systems transient effects are predicted to be critical for glass formation. A simple technique is presented for determining when transient effects are important based on a calculation using steady state nucleation frequencies and macroscopic growth velocities.
1. Introduction
Glass formation occurs when crystallization is avoided upon cooling a liquid. An understanding of the variation of glass formability from system to system is therefore possible by considering the kinetics of crystallization. For the particular case of metallic glasses, such kinetic analyses have proved to be more widely applicable than treatments of glass formability based on structural, electronic or thermodynamic criteria [1]. In this paper we present a more thorough kinetic analysis than has been attempted to date, including both transient nucleation and size-dependent growth effects. Turnbull [2] proposed the avoidance of nucleation in a given volume of liquid during the quench as a kinetic criterion for glass formation. Uhlmann [3,4] suggested an operational but less idealized criterion that the crystalline volume fraction at the end of the quench should be less than some arbitrary value, typically chosen to be 10 -6 . Both approaches include the implication that any material can be made glassy given a sufficient cooling rate. Uhlmann's criterion is the more useful, especially since it is known that in some demonstrably glassy materials there are crystal nuclei [5]. * Present address: Department of Physics, Washington University, St. Louis, Missouri 63130, U.S.A. 0022-3093/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
296
Previous kinetic analyses have considered only materials that crystallize polymorphically (i.e. without compositional change). Since it is desired to calculate a critical cooling rate that is dependent on the materials parameters only, heterogeneous nucleation has not usually been considered. On the basis of classical nucleation theory it is expected that in an undercooled liquid that is initially devoid of crystalline clusters the nucleation frequency will rise sigmoidally with time from zero to the steady state value. Hillig [6] pointed out that such transient effects could become significant in rapid quenches. Despite this, most kinetic analyses have assumed steady state conditions throughout the quench; the temperature-dependent nucleation frequencies are calculated directly from the classical theory. Uhlmann [3] justified this approach, at least for metallic systems, by pointing out that typical nucleation transient times are less than 1 ~s at the onset temperature for homogeneous nucleation. For rapid quenches of the order of 10 6 K s -1, however, even such short transient times might be significant. In silicate systems the transient times can be as large as one day at the glass transition temperature, but the critical cooling rates are very low [3]. The growth rates of nuclei are reduced from their macroscopic values because of the curvature of the crystal-liquid interface. However, previous calculations of critical cooling rates have used the macroscopic, size-independent values. Several approaches that use the above assumptions have been employed to calculate the critical cooling rates for glass formation. Initially, Uhlmann [4] constructed an isothermal transformation curve for a volume fraction of 10 -6, and took the critical cooling rate to be the undercooling at the nose of the curve divided by the transformation time at that temperature. Onorato and Uhlmann [7] improved this analysis by using the method of Grange and Kiefer [8] to calculate a continuous cooling curve. The method of Grange and Kiefer, however, is valid only for an isokinetic reaction [9]. Hopper, Scherer and Uhlmann [10] presented a computer calculation of the volume fraction transformed at a given quench rate, using a "crystal distribution function". They showed that, in principle, the thermal history of the sample can be deduced from the crystal size distribution. Vreeswijk et al. [11] were the first to attempt to model the effects of transient nucleation during the quench. They used an approximate expression due to Zeldovich [12] for the time dependence of the nucleation frequency, I, at constant temperature:
I(T,
t) = / S ( T ) e x p [ - r ( T ) / t ] ,
(1)
where I s is the steady state nucleation frequency. They estimated the transient time r ( T ) using the expression of Hillig [6]: 'r(T) =
~rd2/D,
(2)
where d is the molecular diameter and D is the effective diffusion coefficient at the crystal-liquid interface. To obtain the number of nuclei formed during a quench, Vreeswijk et al. integrated I(T, t) given in eq. (1). This assumes that
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
297
the nucleation frequency at any instant in the quench is the same as it would have been after annealing at the instantaneous temperature for the elapsed time since the beginning of the quench. This is clearly unjustified. The most thorough treatment to date for modeling transient nucleation effects on glass formability is that of Yinnon and Uhtmann [13]. They used a numerical technique for determining the nucleation rate during the quench and calculated the volume fraction transformed using the method of Hopper et al. [10]. The quench was modeled as a series of short isothermal anneals. The nucleation frequency for any isothermal interval, for example at temperature T2, was given by: I ( T z , t) = I ( T 1 , f ) + [IS(T2) - I ( V l , f ) ] e x p [ - ' r ( T z ) / t ] ,
(3)
where T1 is the temperature of the previous isothermal interval, and I ( T 1, f ) is the nucleation frequency calculated at the end of that interval. Since the value of I(7"1, f ) was taken to be the initial nucleation frequency in the succeeding interval, this approach fails to properly include the temperature dependence of the nucleation frequency due to the change in the atomic mobility. As a result, the transient nucleation frequencies calculated from eq. (3) become larger than the steady state values at low temperature. In addition, Kelton et al. [14] have shown that eqs. (1) and (2) are seriously in error for the description of the isothermal transient nucleation behavior. A more fundamental flaw, however. is the assumption that any expression for the isothermal case can be used to describe non-isothermal behavior. During a quench the distribution of crystal cluster sizes in the liquid evolves in a complex way because the steady state distribution, toward which it tends, is a strong function of temperature. A proper calculation of the nucleation frequency during a quench must correctly account for the cluster dynamics. In this paper we present a numerical solution to this problem in which the cluster dynamics are modeled directly and a size-dependent growth velocity is taken into account. We calculate for a variety of model materials the total number of nuclei formed and the total volume fraction transformed as a function of quench rate. To illustrate the importance of transient effects, parallel calculations are performed using the same input parameters and assuming steady state behavior and planar growth kinetics.
2. Numerical method
Assuming classical nucleation theory, in which clusters are formed by a series of bimolecular reactions, we employ a numerical technique that directly simulates the dynamics of cluster formation by dividing the time into small intervals and calculating the change in the cluster size distribution for each interval. We have demonstrated the validity of this method for the description of steady state and transient nucleation in the isothermal case [14]. We now extend the technique to non-isothermal transformations by dividing the con-
298
K.F. Kelton, A.L Greer / Transient nucleation effects in glass formation
tinuous cooling into small isothermal intervals; the numerical simulation then proceeds as in the isothermal case over these intervals. Following our earlier treatment the algorithm chooses a very small calculational time increment at the start of the quench, typically 1 0 - 8 / y , where 7 is the unbiased molecular (or atomic)jump frequency: y = 6D/• 2,
(4)
where D is the diffusion coefficient at the crystal-liquid interface and A is the average jump distance. As the calculation proceeds, the algorithm changes this increment to reduce the CPU time requirements. In accordance with the quench rate and the temperature increment (typically 1 K), the algorithm periodically changes the temperature and recalculates the rate constants and the critical size (n*). At each temperature the calculation then continues using the new rate constants, starting with the distribution of clusters inherited from the previous temperature. By using this method the transient nucleation behavior during the quench is directly simulated without the artificial introduction of approximate expressions developed for the isothermal case. The free energy of formation of a spherical cluster of n molecules is taken to be: AG,, = n A G ' + 4on 2/3,
(5)
where ~G' is the Gibbs free energy difference per molecule between the initial and final phases, and o is the interfacial free energy per molecular site at the interface. AG' and o are taken to have macroscopic values independent of n. The L o t h e - P o u n d correction [15] and the contribution of stresses are ignored; these assumptions should be appropriate for liquid-to-crystal or glass-to-crystal transformations [14]. The rate constants, that is the average frequency with which a cluster of n molecules gains a molecule and the average frequency of the reverse reaction, are obtained from reaction rate theory and are expressed as"
k + = O.y e x p ( - A g . / 2 k B T )
(6a)
k.-+l = 0.7 e x p ( + A g J 2 k , T ) ,
(6b)
where On is the number of sites at the surface of a cluster of n molecules (taken to be 4n2/3), A g . is (AG.+ a --AGn) , k B is Boltzmann's constant, and T is the absolute temperature. The number of molecules in the initial phase is taken to be Avogadro's number, NA, so the nucleation rates reported are per mole. The upper limit of the cluster distribution is chosen always to exceed the critical cluster size, n*. It is normally set at 1.5n* for the highest temperature (largest n*) considered in the simulation; at the temperature of the maximum nucleation frequency this limit is typically greater than 4n*. The backward flux for clusters at the upper limit is taken to be zero. A 9-molecule cluster is chosen as the lower limit based on arguments presented elsewhere [14]. The steady state nucleation rates are
K.I~ Kelton, A.L. Greer / Transient nucleation effects in glass formation
299
obtained from [16]: i~ =
1
u N~k.
(7)
,
where v and u are the upper and lower limits of cluster size, and N.~ is the equilibrium number of clusters of size n. For our numerical treatment, Uhlmann's criterion was adopted to determine the critical cooling rate; to obtain a glass, the volume fraction transformed must be below the detectable limit of 1 0 - 6 . Although it is possible, in principle, to calculate the total volume transformed directly, using the numerical simulation to obtain the cluster size distribution at the end of the quench, it is not computationally practicable to use an upper limit of sufficient size. Fortunately, the evolution of the cluster size distribution at large sizes can be described analytically. As shown elsewhere [17], the behavior can be modeled to a very good approximation by calculating the average growth rate for a post-critical cluster and assuming that every cluster behaves in exactly the average manner. Using transition rate theory to calculate the net rate at which atoms are added to a cluster of a given size, and assuming spherical clusters, the average growth rate of a cluster (rate of increase in radius) is found to be both temperature and radius dependent, and is given by: dr dt
16D ~2
3V ~
20A
sinh[ 2-~BT [AG,' - - -r
(8)
where V is the molecular volume, AGv is the free energy decrease per unit volume on transformation and oA is the interracial energy per unit area. The expression is least accurate for growth from cluster sizes near the critical size, but is applied in our simulation only to large cluster sizes. For clusters with n > 1.5n* the error from using eq. (8) is at most 3%. The use of an artificial upper limit, v, in the numerical simulation has a negligible effect on the dynamics of cluster formation for n < (v - 10). During a simulated quench the number of clusters of size ( v - 10) generated in each temperature interval is stored. In subsequent intervals the growth of these clusters is calculated using eq. (8). In this way we obtain the cluster size distribution at the end of the quench. The transformed volume is found by summing the individual volumes of all the clusters (for n > 10). The number of nuclei generated during the quench is taken to be the number of clusters of size greater than (v - 11). The choice of (v - 10) as the minimum nucleus size, though arbitrary, ensures that the nuclei are post-critical at all temperatures in the simulation. Ideally the simulation of a quench would start at the melting temperature, but this is not possible because the critical cluster size is infinite at that temperature and remains too large for the numerical simulation until a substantial undercooling. An appropriate initial temperature was determined in each case by starting the simulation at successively higher temperature until the nucleation frequency (net forward flux between cluster sizes ( v - 11) and
300
K.F Kelton, A.L. Greer / Transient nucleation effects in glass formation
(v - 10)) at the end of the first temperature increment was at least 95% of the steady state frequency at that temperature. At higher temperatures steady state nucleation was assumed (see below). The cluster size distribution at the start of the simulation was taken to be the steady state distribution at that temperature. By using this technique no assumption was made for determining the point at which transient effects should become important, within the range of cluster sizes considered. The starting temperature for the simulation is somewhat arbitrary, being determined by the range of cluster sizes that is practical, but the results are independent of this starting point provided the steady state nucleation frequencies at higher temperatures are negligible. The total number of nuclei and the total volume transformed were also calculated by assuming that the nucleation frequency maintained its (temperature-dependent) steady state value (from eq. (7)) throughout the quench. For a valid comparison the nucleation frequency was defined in the same way as for the transient nucleation calculation. In effect, this procedure assumes that the steady state applies only up to cluster size (v - 10). In ideal steady state the forward flux is the same at all cluster sizes, but if this applied up to infinite size the rate of transformation would be infinite; presumably the steady state can apply only in a limited range of cluster sizes. In the steady state calculation we assumed that the macroscopic growth rate applied for all clusters larger than (v - 10). It should be emphasised that the steady state assumption, when applied to large cluster sizes, is seriously flawed. This is so even when transient effects are not expected to dominate. Calculations Of steady state nucleation do not represent physically realizable conditions, but are presented for comparison to illustrate the possible errors in estimating glass formability when simple models of nucleation and growth are used.
3. Results and discussion
3.1. Input parameters For our calculations we chose parameters which are appropriate for an approximate description of three real materials: lithium disilicate, an oxide glass former; and two metallic glass formers, (AussCu15)v7Si9Ge14 and AuslSi19 (compositions in at.%). These choices were made because there were sufficient data to estimate the material parameters and because the crystallization is polymorphic in each case. Our numerical techniques are valid only for polymorphic transformations, though the transformed volume fractions of interest in glass formation are so small ( - 1 0 -6) that the assumptions of a constant matrix composition may not be so restrictive. Lithium disilicate (Li 2° . 2SiO 2) is by far the best glass-former of the three materials, though it is a comparatively poor glass-former for a silicate system. It is known to exhibit transient nucleation effects in devitrification [18]. (AussCUls)77Si9Ge14 is a
k oA V
oA/T
-
a
Ionic d i a m e t e r
J u m p distance Interfacial e n e r g y 9.96 × 10
4.6 ~, 0.15 J m
-
Molecular v o l u m e
I K-1
29 m ~
2
2 X 10 9 m 2 s- 1 440 kJ tool. - 1
To
D0 E r/o
Pre-exponential diffusivity A c t i v a t i o n e n e r g y for diffusion Vogel-Fulcher parameters
1300 K 4 0 J mol.
-
T~ ASf
Melting temperature E n t r o p y o f fusion
Li 2 0 . 2SiO 2
~j
Symbol
Parameter
Table 1 T e m p e r a t u r e - i n d e p e n d e n t p a r a m e t e r s used in the c a l c u l a t i o n s
i K-l
4jm
1.80 × 10 29 m 3
1.95×10
2.88
1.70 ]~
150 K
2 K
1.12 × 10 s N s m - 2 6546 K
_
_
623 K 9.39 J mol.
( A u 85 Cu 15 ) 77Si 9Ge 14
i
A 2.19×10 1.80×10
2.88 ,~
1.70
169 K
4057 K
1.78x10
i K
i
4Jm 2 K 29 m 3
5 Nsm-2
631 K 12.29 J mol.
A u 81Si 19
1
e~
e~
,2
302
K.F Kelton, A.L. Greer / Transient nucleation effects in glass formation
good metallic glass-former also exhibiting transient effects in devitrification [19]. Au81Si19 is the worst glass-former of the three [20], with a critical cooling rate of - 1 0 6 K s 1. The choice of these three materials permits the investigation of transient nucleation effects over a wide range of critical cooling rates. Although we have chosen parameters that approximate those for real materials it is not the purpose of this work to make a detailed comparison with experimental results. The parameters chosen for the calculations are listed in table 1. Two techniques were used to calculate the free energy of crystallization. For Li 2° . 2SiO 2, the free energy per molecule was taken to be proportional to the undercooling: ASr AG t = ' ~ A ( 1 - - Tm) , (9) where Tm is the melting temperature, ASf is the molar entropy of fusion and N a is A v o g a d r o ' s n u m b e r . T h e free energy for A u 8 1 8 1 9 and (AussCUls)v7Si9Ge14 was calculated using the expression of Thompson and Spaepen [21]: AS~ ( T 2T AG' = -~A -- Tm ) ( T-----~-~m) .
(10)
The interfacial energy, oA, for lithium disilicate was assumed to be temperature-independent, and its value was estimated from experimentally determined steady state nucleation rates. For the metallic glasses o A was assumed to be proportional to the temperature and was estimated using the expression of Spaepen [22]. The molecular jump frequency, ~,, was calculated from the diffusion coefficient D and the jump distance X (taken to be the molecular diameter in lithium disilicate and the atomic diameter of gold in the metallic glasses). An Arrhenius temperature dependence of D was assumed for lithium disilicate: D = D Oe x p ( - E / k s T
),
(11)
where E is the activation energy for diffusion. The viscosity of metallic glasses is approximately described by the Vogel-Fulcher expression:
where T0, ( and To are estimated from experimental data [19]. The diffusivity is obtained from the viscosity using the Stokes-Einstein relation: D = kBT/3~ra~,
(13)
where a is a characteristic distance taken to be the ionic diameter of Au 3÷ [19]. 3.2. R e s u l t s
Figs. 1-3 show the nucleation frequency as a function of temperature and quench rate for Li 20.2SIO2, (Au 85Cul 5) 77Si 9Ge14 and Au 81Si~9 respectively.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
303
10s ~'~
S,,feody ~
~E ~ 103f
~
~
/
LITHIUM ~
/
~
/
Di S',L/CATE
~
\ z ~
5 600
\
~
680
760
840
920
1000
TEMPERATURE (K)
1020
i
~,~
I
'
]
Steady State~
I
~
I
'
Fig. 1. The homogeneous nucleation frequency for lithium disilicate for a cluster of 138 molecules, calculated as a function of temperature and quench rate. The critical size at the peak in the steady state nucleation frequency, occurring at 750 K, is 23 molecules.
[
(Au85Cu15)77SigGe14
'a~ 1016 ,,~- 1012
n.z 10s
=, z
lo 4
1.0 300
I
J 330
I
J 360
I
J 390
I
420
~\
4,50
TEMPERATURE (K)
Fig. 2. The homogeneous nucleation frequency for (AussCuls)vv SigGel4 for a cluster of 297 atoms, calculated as a function of temperature and quench rate. The critical size at the peak in the steady state nucleation frequency, occurring at 332 K, is 31 atoms.
2G 10
~
"" 500
1
r
340
380
,
AusISi
420
TEMPERATURE (K)
l,9
460
,
500
Fig. 3. The homogeneous nucleation frequency for Au81Sil9 for a cluster of 330 atoms, calculated as a function of temperature and quench rate, The critical size at the peak in the steady state nucleation frequency, occurring at 330 K, is 17 atoms.
304
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
I 0 8 ~ Lithium Oisilicote 106I ~ : ~ - . ~ ' - ~ - o ~ ,; Stead),Stote ~ ~ - ~Nucleation 104I ~ '~"""'--~. Tronsient Nucleotion
z I
I
I
164( 16e
- SteadyState "~cleotion Transient J
10-1
1
l
I
101 102 QUENCH RATE (K/s)
103
Fig. 4. The number of nuclei (a) and transformed volume fraction (b) calculated for quenching lithium disilicate at variousrates, assumingeither steady state or transient nucleation.
The behavior is qualitatively similar in the three cases: an increasing quench rate lowers the nucleation frequencies below the steady state values and displaces the peaks to higher temperatures. For each system the nucleation frequencies converge to the steady state frequencies at high temperature. The increasing deviation from steady state as the temperature is lowered reflects the decreasing atomic mobility. A comparison of the systems shows that, for a given quench rate, the deviation from steady state is more pronounced in the better glass formers. Figs. 4 - 6 a show the influence of transient effects on the total number of nuclei produced during the quench. In all cases the number of nuclei deviates from the steady state value at an accelerating rate as the quench rate is increased. Transients in the nucleation frequencies are therefore very important in determining the total number of nuclei produced during the quench. Based on the criterion for glass formation suggested by Turnbull [2], transient effects are shown to be extremely important in glass formability. The more common criterion for glass formation, however, is based on the volume fraction transformed, as suggested by Uhlmann [3,4]. Figs. 4 b - 6 b show the volume fraction transformed as a result of nucleation and growth during the quench for the three materials considered. In agreement with the trends shown in figs. 4a-6a, the volume fraction drops precipitously below the values calculated assuming steady state as the quench rate is increased. A surprising difference, however, is the intersection with the steady state result for a quench rate at which the number of nuclei is still several orders of magnitude below the steady state result. This reflects the importance of the large growth velocity
305
K.F Kelton, A.L. Greer/ Transient nucleation effects in glass formation 102o A
'°
1
0
~
A
u
S
1
Si19
1012
~ 108
~
10!
Transient Nucleation - ~
10 4
,n4
\ Transient
'i I
l i
I
i
i
"o --
164
N ]~s
16 s
1012
.
.
.
.
.
.
.
.
.
.
.
.
1612 1016
~ 1016
=~ 1~2o 102
~ucleotion
I
~ 1()20
105 104 105 QUENCH RATE (K/s)
Fig. 5, The number of nuclei (a) and transformed volume fraction (b) calculated for quenching (Au85CUls)77SigGe14 at various rates, assuming either steady state or transient nucleation.
106
t04
~ I
,
Nucleatiol J\
105
106 l0 T QUENCH RATE (K/s)
Fig. 6. The number of nuclei (a) and transformed volume fraction (b) calculated for
quenching Au81Si19 at various rates, assuming either steady state or transient nucleation.
at temperatures where the steady state and transient nucleation frequencies are similar. At very high quench rates the transient nucleation frequencies are decreased to a point where the number of nuclei produced is so small that the volume transformation is insignificant, even with high growth velocities. Based on the Uhlmann criterion the critical cooling rate for L i 2 0 . 2 S i O 2 is 3.8 K s - t from both the steady state and transient calculations. For (AussCuls)vvSi9Ge]4 steady state calculations predict a critical cooling rate of 4.5 × 10 3 K s 1 and the transient calculations predict a value of 1.3 × 10 4 K s-1. For the Aus]Si]9, alloy, however, the two calculations give quite different results. Transient nucleation calculations predict a critical cooling rate of 1.0 × 10 5 K s 1 and steady state calculations predict a critical cooling rate of 2.4 × 10 7 K s ], a value not easily obtainable by disk quenching. In some systems, therefore, transient effects are significant and must be considered in glass formation. 3.3. Discussion To understand why transient effects are important in some cases and not in others, we examine the transition from steady state to transient nucleation during the quench. The ratio of transient to steady state frequencies as a function of temperature is plotted for lithium disilicate at various quench rates
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
306
,0 L,TH,0M # / / F:
y f
1o-K/ /K/t071o2/1o3/
/: / /i /i /
0.4
~
,
j
i
I
I
I
I
I
I
/ I /I
o.2
l
/
I I,I
/!
III
y il rl ll
I
00B00"
080
/
i
?B0
840
I
920
TEMPERATURE (K)
I000
Fig. 7. The ratio of the transient nucleation frequency to the steady state nucleation frequency, I(T, Q/IS(T), for lithium disilicate as a function of t e m p e r a t u r e a n d quench rate. The dotted lines indicate the temperatures, TD, for which the ratio is 0.5.
in fig. 7. At each quench rate the transition from the steady state is quite sharp and occurs at a temperature To, defined to be the point at which the ratio of the nucleation frequencies is 0.5. This temperature can be estimated analytically by considering the transient time, ~-, for the isothermal generation of critical nuclei. As shown in an earlier paper [14], T is given to a good approximation by the expression of Kashchiev [23]: T= 24kBTn*/~Zk+,AG '. (14) -
In fig. 8 log T, calculated using eq. (14) for each of the values of T O in fig. 7, is plotted against the logarithm of the quench rate appropriate for each T O. Data calculated in a similar manner for the two metallic glasses are plotted in the same figure. The data show that at the transition temperature the transient time ~" and the quench rate Q are inversely proportional. For each system the product TQl't = 1.7 K. For our purposes we use the approximation sQ = 1 K;
102
~,~
o LITHIUMDISlLICATE o (AuBsCUls)7/Si90ei4 10
L,i
~E i-Z uJ
1(~2 104
I°!io-2
L
L
1
10 2
i I
~
104
QUENCH RATE (K/s)
10fi
108
Fig. 8. The transient time, ~-, at temperature T O (see fig. 7) as a function of the quench rate, Q, for the three systems considered.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation 1001
'(
307
'LITHIUI~ DISILII~ATE'
,~ 901 D I-- 800 rr"
lO(] bJ
t--
60C 500
I
I
I
I
103
I
I
1
I
106 109 TIME (s)
I
1012
1015
Fig. 9. The time required for isothermal transformation to 10 6 volume fraction for lithium disilicate as a function of temperature. Steady state nucleation frequencies and macroscopic growth velocities were assumed. The temperature T~) at which Q * ' r ( T ) = I K is shown, where Q* is the critical cooling rate calculated using the above assumptions.
then at any quench rate T D can be estimated by finding the temperature at which Q'r(TD)= 1 K. If Q is set to be the critical cooling rate calculated from steady state considerations, T D is the temperature below which transient nucleation effects are likely to dominate in a glass-forming quench. Figs. 9-11 show this temperature, T~, on isothermal transformation plots for the three materials. These plots show the time to 10 -6 volume fraction transformed assuming steady state nucleation and macroscopic growth rates. For lithium disilicate (fig. 9) the nose of the transformation curve lies well above T~, that is, the maximum rate of transformation occurs at a temperature where the nucleation frequency is in steady state. This is consistent with the result in fig. 4b that transient nucleation effects on the volume fraction transformed, and therefore on the critical cooling rate, are negligible. For Aus]Si]9 (fig. 11) on the other hand, the nose of the transformation curve lies well below T~), and fig. 6b shows that transient effects are significant. ( A u 8 5 C u ] 5 ) 7 7 S i g G e ] 4 is an intermediate case. Thus, by comparing T~ with the temperature for which the
500
E
I
i
i
I
I
i
I
I
( Au 85Cu 15) 77 Si9Ge14 450 LLI D~
'~ rr 400 - T D =395K LLI O_
t- 350
3°00 -1 r
1'
'
101
TIME (s)
102
105
104
Fig. 10. The isothermal transformation curve and temperature T~) for (Aus5Culs)77Si9Ge14. Other details as for fig. 9.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
308
500
'
U.I rr' I-L~J
~m~ =546K . . . . ~.~ - - - - ~ - - ' ~
Au81Sil9
45O
400
W ~- 350
300
D-5
10-~
I 101
I
TIME (s)
l 10 3
I
I 10 $
t 10 /
Fig. 11. T h e i s o t h e r m a l t r a n s formation curve and temperature T~ for Au81Sil9. O t h e r d e t a i l s as
for fig. 9.
transformation rate is maximum, as shown in figs. 9-11, one can readily determine whether transient effects will be important. A test of the treatments presented in this paper is not possible by making a detailed comparison of calculated and experimental critical cooling rates, because even small uncertainties in the material parameters can introduce uncertainties of several orders of magnitude in the calculated nucleation frequencies. However, there are manifestations of transient nucleation that can be used to test the theory. These experimental manifestations, both in devitrification and in glass formation, have been reviewed recently [17]. In glass formation, the dependence of the number of quenched-in nuclei on the cooling rate gives important information. For the metallic glass Fe80B20 it has been found that the number of quenched-in nuclei is not inversely proportional to the quench rate Q, as expected for steady state nucleation, but is proportional to Q-(2 to 4). This is clear evidence of transient nucleation in this system and confirms qualitatively the behavior predicted in figs. 4a-6a.
4. Conclusions
We have presented calculations of nucleation frequencies as a function of temperature, total numbers of nuclei produced, and total volume fractions transformed during quenches from the melt for three materials with thermodynamic and kinetic parameters that approximate those of L i 2 0 . 2 S i O 2, (Au85Cu]5)77Si9Ge]4 and Au81Si19. Transients in the nucleation frequency have been introduced in a natural way by directly simulating the cluster evolution during the non-isothermal treatments. These results are compared with those calculated by assuming a steady state nucleation frequency for the same thermal histories. In all systems, the nucleation frequency and number of nuclei produced are dramatically affected by the inclusion of transient effects.
K.F. Kelton, A.L. Greer / Transient nucleation effects in glass formation
309
T o d e m o n s t r a t e the possible effect o n the glass formability of some alloys d u r i n g a rapid q u e n c h from the liquid, we have calculated the v o l u m e fraction t r a n s f o r m e d a n d c o m p a r e d it with the volume fraction t r a n s f o r m e d a s s u m i n g steady steady conditions. I n AusaSi~9 t r a n s i e n t n u c l e a t i o n effects are d e m o n strated to significantly i m p r o v e the prospect of o b t a i n i n g a glass at a n a t t a i n a b l e q u e n c h rate. W e have presented a simple analysis to evaluate the i m p o r t a n c e of transient effects in glass formability that involves only steady state considerations. This work was supported b y O N R u n d e r contract N00014-77-C-0002, by N A S A u n d e r contract NAS8-35416 a n d b y N S F u n d e r contract DMR80-20247.
References [1] [2] [3] [4] [5] [6]
H.A. Davies, Phys. Chem. Glasses 17 (1976) 159. D. Turnbull, Contemp. Phys. 10 (1969) 473. D.R. Uhlmann, in: Materials Science Research, Vol. 4 (Plenum, New York, 1969) p. 172. D.R. Uhlmann, J. Non-Cryst. Solids 7 (1972) 337. A.L. Greer, Acta Metall. 30 (1982) 171. W.B. Hillig, Syrup. on Nucleation and Crystallization of Glass (Amer. Ceram. Soc., Columbus, Ohio, 1962) p. 77. [7] P.I.K. Onorato and D.R. Uhlmann, J. Non-Cryst. Solids 22 (1976) 367. [8] R.A. Grange and J.M. Kiefer, Trans. ASM 29 (1941) 85. [9] J.W. Cabn, Acta Metall. 4 (1956) 572. [10] R.W. Hopper, G. Scherer and D.R. Uhlmann, J. Non-Cryst. Solids 15 (1974) 45+ [11] J.C.A. Vreeswijk, R.G. Gossink and J.M. Stevels, J. Non-Cryst. Solids 16 (1974) 15. [12] J.B. Zeldovich, Acta Physicochim. URSS 18 (1943) 1. [13] H. Yinnon and D.R. Uhlmann, J. Non-Cryst. Solids 50 (1982) 189. [14] K.F. Kelton, A.L. Greer and C.V. Thompson, J. Chem. Phys. 79 (1983) 6261. [15] J. Lothe and G.M. Pound, J. Chem. Phys. 36 (1962) 2080. [16] J.W. Christian, The Theory of Transformations in Metals and Alloys, 2nd ed. (Pergamon, Oxford, 1975) Chp. 10. [17] K.F. Kelton and A.L. Greet, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals Wi~rzburg, 3-7 September 1984, eds. S. Steeb, H. Warlimont (North-Holland, Amsterdam, 1985) p. 223. [18] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [19] C.V. Thompson, A.L. Greer and F. Spaepen, Acta Metall. 31 (1983) 1883. [20] C.V. Thompson, A.L. Greet and A.J. Drehman, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals, Vol. 1, eds., T. Masumoto and K. Susuki (Japan Inst. of Metals, Sendai, 1982) p. 743 [21] C.V. Thompson and F. Spaepen, Acta Metall. 27 (1979) 1855. [22] F. Spaepen, Acta Metall. 23 (1975) 729. [23] D. Kashchiev, Surface Sci. 14 (1969) 209.