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Verification of an analytical method for measuring crystal nucleation rates in glasses from DTA data
Journal of Non-Crystalline Solids 337 (2004) 261–267 www.elsevier.com/locate/jnoncrysol
Verification of an analytical method for measuring crystal nucleation rates in glasses from DTA data K.S. Ranasinghe a
d
a,*
, P.F. Wei b, K.F. Kelton c, C.S. Ray d, D.E. Day
e
Department of Physics and Materials Research Center, University of Missouri – Rolla, MO 45609-1140, USA b Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA c Department of Physics, Washington University, St Louis, MO 63130, USA Marshall Space and Flight Center, National Aeronautics and Space Administration, Huntsville, AL 35812, USA e Graduate Center for Materials Research, University of Missouri – Rolla, MO 45609-1140, USA Received 29 September 2003
Abstract A recently proposed analytical (DTA) method for estimating the nucleation rates in glasses has been evaluated by comparing experimental data with numerically computed nucleation rates for a model lithium disilicate glass. The time and temperature dependent nucleation rates were predicted using the model and compared with those values extracted from an analysis of numerically calculated DTA curves. The validity of the numerical approach was demonstrated earlier by a comparison with experimental data. The excellent agreement between the nucleation rates from the model calculations and the computer generated DTA data demonstrates the validity of the proposed analytical DTA method. 2004 Elsevier B.V. All rights reserved.
1. Introduction Detailed knowledge of the kinetics of nucleation and crystal growth rate is useful for materials design, since these kinetics control the phase formation and microstructure of the phases. Recently we developed an experimental method that uses differential thermal analysis (DTA) [1,2] to measure the nucleation and crystal growth rates as well as the number of quenchedin nuclei in glasses. The differential thermal analysis experiments, which characterize the transformation over a range in temperature, are particularly important since they are easy to conduct, require little sample preparation, and are independent of the sample geometry. In this DTA method, a series of isothermal and non-isothermal steps as shown schematically in Fig. 1 are required to obtain the numerical values for the rates for nucleation and crystal growth [1,2].
*
Corresponding author. Address: 3055 Marshal Ave., Apt. 13, Cincinnati, OH 45220, USA. Tel.: +1-513 751 5647. E-mail address: [email protected] (K.S. Ranasinghe). 0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.04.010
The goal of this paper is to demonstrate the validity of the analytical DTA method for measuring crystal nucleation rates. The crystal nucleation rate was calculated from computer generated DTA data that were produced using a numerical model developed and used by Kelton and co-workers [6–16]. The same Li2 O Æ 2SiO2 (LS2 ) glass that was originally used in the DTA experiments [1] for determining the nucleation and crystal growth rates is used in the present simulations. For many reasons LS2 is considered a model glass for studying of nucleation and crystallization processes. A good experimental data exists for the steady state and time dependent nucleation rates and growth rates [3–5] for this LS2 glass. Further, the free energy difference between the glass and crystal phases are known for this glass as a function of temperature, and these data have been used extensively to test the classical theory of nucleation [6,7]. Numerical solutions of the classical theory of nucleation have been used to simulate crystallization by homogeneous nucleation in this LS2 glass for isothermal and non-isothermal conditions [8–14]. The model lithium disilicate (Li2 O Æ 2SiO2 or LS2 ) glass was generated by numerically simulating a quench
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constant, the total number of nuclei, Ntot , present in the glass after nucleation at TN can be calculated from the following equation [1,2]:
Melt Crystal Growth Temperature
Quenching
Ntot ¼ ðIN tN þ Nq Þ ¼
Nucleation
Glass
Time
Fig. 1. The heat treatment schedule (schematic) used to simulate the nucleation experiments in the LS2 glass.
of the LS2 melt. The cluster distribution obtained after heat treatment that mimicked the DTA heat treatment cycles shown schematically in Fig. 1 was used in the numerical calculations. These calculations produced an exothermic crystallization peak, which is similar to the experimental DTA peak. The data from this computersimulated DTA peak are analyzed by an analytical equation (Eq. (1)) to calculate the nucleation rate of the computer-generated glass. This nucleation rate is then compared with the nucleation rate calculated directly from the model. It should be emphasized that it is not the purpose of this paper to make exact comparisons with experimental data, but to compare the nucleation data obtained by an analytical treatment of the simulated DTA data with that calculated directly within the numerical model.
2. Calculations of nucleation rate 2.1. The DTA method In the analytical DTA method [1,2], a small amount of glass, typically 40–60 mg, having a relatively large particle size (>400 lm), is first heated isothermally in the DTA apparatus at a temperature, TN , (nucleation temperature) for a time, tN , and then heated for a short time, tG , at a higher temperature, TG , (crystal growth temperature). After the nucleation and crystal growth heat treatments, the glass is heated non-isothermally in the DTA at a relatively high heating rate, typically 10– 25 C/min. An exothermic DTA peak is observed for the crystallization of the glass that remains untransformed after the crystal growth heat treatment at TG . The area of this DTA peak is proportional to the number of nuclei present in the untransformed glass. From the area of two such DTA peaks, where all the experimental parameters except the heat treatment time at TG are
3ðA1 A2 Þ ; 3 3 pðA1 tG2 A2 tG1 ÞU 3
ð1Þ
where, A1 and A2 are the area of the two DTA peaks after the crystal growth heat treatment at TG for times tG1 and tG2 , respectively. IN is the steady state nucleation rate at TN and U is the crystal growth rate at TG . The quantity (IN tN þ Nq ) is the total number of nuclei per unit volume of the glass, where (IN tN ) is the number of nuclei developed per unit volume due to the nucleation heat treatment only and Nq is the concentration of quenched-in nuclei that formed during quenching the melt to glass. A plot of (IN tN þ Nq ) as a function of tN yields a straight line at higher values of tN , whose slope gives the value of IN . By repeating the experiments at different TN , the complete nucleation rate curve as a function of temperature can be determined. The concentration of quenched-in nuclei, Nq , can be calculated from Eq. (1) when the experiments are performed using an as-quenched glass, i.e., without giving any nucleation treatment (IN tN ¼ 0). Also, using the Eq. (1), the crystal growth rate (U ) at different temperatures can be determined from two similar DTA runs at each temperature using an as-quenched glass, see Ref. [1,2] for further experimental details. 2.2. Numerical calculations The basic numerical model for an infinite sample is only briefly discussed in this section, since the model has been discussed extensively elsewhere [6–16]. The classical theory of nucleation is assumed [11,16], taking the time dependent nucleation rates into account. Within the classical theory, clusters of atoms in the configuration of the transformation product arise spontaneously in the original material. Assuming spherical clusters and negligible stress effects, the reversible work for forming a cluster of n molecules, Wn , at any temperature, TN , can be written as Wn ¼ nDGn þ ð36pÞ
1=3 2=3 2=3
n v rls ;
ð2Þ
where, DGn is the Gibbs free energy difference per molecule in the cluster (solid) and the liquid (glass), v is the molecular volume of the liquid and rls is the interfacial energy per unit area of the liquid–cluster interface. Since DGn is negative for a thermodynamically favorable transition and the energy required to create an interface is positive, a maximum in Wn is predicted at a critical cluster size, n , n ¼
ð32pÞr3ls 3vjDGv j
3
;
ð3Þ
K.S. Ranasinghe et al. / Journal of Non-Crystalline Solids 337 (2004) 261–267
where DGv is the free energy difference per unit volume of the cluster (final) and glass (initial) phases (¼ DGn =v). Clusters smaller than n will normally shrink, while those larger than n tend to grow. Clusters are assumed to evolve slowly in size by a series of bimolecular reactions: knþ
En þ E1 $ Enþ1 ; kn1
ð4Þ
where En represents a cluster of n molecules, and knþ and kn represent the rate of monomer addition and detachment, respectively, to a cluster of size n [10]. The time dependent cluster distribution is given by a system of coupled differential equations. dNn;t þ ¼ Nn1;t kn1 Nn;t kn þ Nn;t knþ Nnþ1;t knþ1 ; ð5Þ dt where Nn;t is the number of clusters of size n at a time t. The net forward rate of the reaction In;t , at a cluster size n, is the time-dependent flux of clusters past that size and is given by : In;t ¼ Nn;t knþ Nnþ1;t knþ1
ð6Þ
When a steady state distribution of clusters is established, the net forward rate, which is independent of time and n, is the steady state nucleation rate I s given by s knþ1 ; I s ¼ Nns knþ Nnþ1
ð7Þ
s
I at TN is comparable to IN described in Section 2.1 and Nns is the number of clusters of n molecules in the steady state at TN . In the strict sense, this is not a true steady state, because the number of molecules in the initial phase decreases as the clusters are formed, although this will have negligible impact on this study. Assuming detailed balance of the forward flux is equal to the backward flux and the nucleation rate is zero, i.e. e Nne knþ ¼ Nnþ1 knþ1 , where Nne is the equilibrium cluster distribution, allowing knþ to be expressed in terms of knþ1 . Eqs. (5) and (6) are solved using a finite difference method [6]. The time is divided into a large number of small intervals, dt, and the number of clusters of size n at the end of the interval, Nn;tþdt , is calculated from dNn;t : ð8Þ dt The behavior of clusters of size n < u (where u 2n , calculating n for the highest temperature considered in the numerical simulation) is calculated directly using these methods. The growth of clusters larger than this size is computed using an expression for the growth rate, uðrÞ which is the rate of increase of cluster radius dr=dt, as given by [8] 1 16D 3v 3 v 2r uðrÞ ¼ 2 DGv sinh ; ð9Þ 4p kT r k Nn;tþdt ¼ Nn;t þ dt
D is the diffusion coefficient in the parent phase, k is the jump distance, T is the temperature, k is the Boltzmann
263
constant and c is a constant that depends on the growth mode [13]. To simulate the non-isothermal DTA heating and cooling curves (scans), the rate of volume fraction transformed is computed by dividing the time into a series of isothermal scans of duration dt ¼ dT =/, where / is the heating/cooling scan rate per second and dT is the temperature step size, allowing the nucleation rate to evolve as in Eq. (6). Keeping track of number of clusters of a given size and assuming no overlap between clusters, the total extended volume fraction transformed is calculated at the end of each interval by m 1 X 4p 3 ; ð10Þ Ni ri;t xe ¼ V0 i¼1 3 where V0 is the sample volume, Ni is the number of nuclei generated in the interval i, and ri;t are the time-dependent radii of those nuclei. Cluster overlap is taken into account statistically using the Johnson–Mehl–Avrami equation [17] to relate the actual volume fraction transformed to the extended volume fraction transformed: xðtÞ ¼ 1 expðxe Þ:
ð11Þ
To simulate the DTA peak for continuous scans, it is necessary to mimic the DTA signal, which scales with the rate of volume fraction transformed, DTA signal / ½ xðTi þ dTi Þ xðTi Þ =dt:
ð12Þ
2.3. Input parameters In the present work a ‘molecule’ of lithium disilicate composition is taken to be one formula unit and the free energy is set accordingly. A third-order polynomial is used to fit the measured data [7,10] as DG ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 ;
ð13Þ
where T is the temperature and ai are given in Table 1. The diffusion coefficient was calculated from the Stokes– Einstein relation, D ¼ kB T =3pag;
ð14Þ
where, g is the viscosity and a is a characteristic distance of the order of an atomic diameter. The viscosity of the glass is assumed to obey the Fulcher–Vogel relation, g ¼ g0 expðn=ðT T0 ÞÞ;
ð15Þ
where g0 and T0 are estimated from experimental data. The values for the parameters used in Eqs. (13)–(15) are listed in Table 1. 2.4. Glass formation and nucleation In the present work, the non-isothermal simulation was used to calculate the cluster distribution produced (quenched in clusters) as the liquid was cooled to form a
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Table 1 Physical parameters used for numerical calculations [10]
glass. It is essential to consider the possible influence of such pre-existing clusters when studying the effects of nucleation. When the modeled glass was obtained by quenching the melt at 0.1 K/s, the number of pre-existing clusters is comparable with the experimental value for the lithium disilicate glass [11]. Therefore, all simulations in the present work were conducted for a glass that was generated by quenching the melt at 0.1 K/s, unless stated otherwise. The numerical model was extended to simulate the DTA experiments, using the following isothermal and non-isothermal steps, which are also shown schematically in Fig. 1. These steps are the same as those used in the original DTA experiments [1,2]: Step 1: heat the glass particles in the DTA furnace to a temperature TN , the nucleation temperature (typically from 400 to 500 C), at 20 C/min; Step 2 (nucleation heat treatment): hold at TN for a time tN , typically from 30 min to 15 h; Step 3: heat from TN to a temperature TG , the crystal growth temperature, at 20 C/min; Step 4 (crystal growth heat treatment): hold at TG for a time tG1 ; Step 5: cool at a rate of 20 C/min to a lower temperature (about 350 C) where there is insignificant nucleation and crystal growth; Step 6: heat at 10 C/min until crystallization is complete, which will produce an exothermic peak of area A1 ; Step 7: using a new glass sample, repeat all of the steps above, except for a different heat treatment time, tG2; at TG , which will produce a different DTA peak area of A2 . The crystal growth temperature, TG , was maintained constant at 600 C for all the DTA runs, and the times used for tG1 and tG2 (at TG ) were 10 and 20 min, respectively. From the measured values of A1 and A2 of the computer generated exothermic DTA peak areas, the total number of nuclei was calculated using Eq. (1). Steps 1–7 mentioned above were simulated using the numerical model already discussed (Section 2.2) to develop quenched-in and new clusters in the crystal phase. Each step produced a cluster distribution that used as an input to calculate the number of clusters in the next step. Likewise, the final step gives the total number of nuclei developed after all the heat treatment cycles. In this
a0 ¼ 48 045, a1 ¼ 36:81, a2 ¼ 56:07 104 , a4 ¼ 4:31 106 (0:094 þ 7 105 T ) g0 ¼ 0:0363, n ¼ 7761, T0 ¼ 460 v ¼ 61:2 106 k ¼ 4:6
numerical model, a lower and an upper size limit for the cluster population was assumed. The lower cluster size limit does not have a significant effect on the results of the simulation, provided it is well below the critical size of a cluster. It was noted by Kelton et al. [6] that the smallest cluster with a demonstrable crystalline structure consists of 10 formula units and consequently the lower limit, l, of the cluster distribution is considered as n ¼ 9. It is also assumed that clusters of 9 molecules always have their equilibrium population. As observed by Kelton and co-workers [6–10], the choice of the upper limit, u, also has little effect on the results provided that n is well above the critical size (n > 2n ) for the temperatures used in the simulation. The highest temperature used in this work is 750 C where n is 205, hence the upper limit, u, was taken as 400 which is considered sufficiently large. The backward flux from clusters larger than the upper limit is taken to be zero. Since the critical cluster size increases with increasing temperature (see Fig. 2) many of the cluster sizes that are post-critical at a lower temperature become subcritical at higher temperatures, causing the fluxes for those clusters to reverse direction, thereby, decreasing their population. The temperature dependence of the critical size and its effect on the post critical cluster population are implicitly taken into account in the numerical model and the nucleation rate is always calculated at n ¼ 1:5n for the highest temperature con-
800
Critical Cluster Size, n*"
Constants in Gibb’s free energy (J/mol) Eq. (13) Interfacial energy (J/m2 ) Viscosity (Poise) Eq. (15) Atomic volume (m3 /mol): Jump distance (k, A)
T=750 ° C,n*=210
600
400 u=400 n=310 200
0
l=9
400
500
600
700
800
900
Temperature(C) Fig. 2. Critical cluster size of modeled LS2 glass as a function of temperature.
K.S. Ranasinghe et al. / Journal of Non-Crystalline Solids 337 (2004) 261–267
3. Results and discussion It is useful to first examine the behavior of the cluster distribution that underlies the nucleation and growth features. Fig. 3 compares the computed cluster distribution of the simulated glass after a 2 h nucleation treatment at 450 C with that of the as-quenched glass. As mentioned before, the cluster distribution at the end of each heat treatment step was used as input for the next heat treatment step. Fig. 4 shows the exothermic DTA type crystallization peaks that the numerical model produced for an as-quenched LS2 glass (no nucleation heat treatment) after crystal growth heat treatment at 600 C for 10 or 20 min. As expected, and also as observed experimentally [1,2], the DTA peak area for the glass after 20 min heat treatment is smaller
30
log(no of cluster/mol)
20 n=310 10 as-quenched (0.1K/sec) 2h/450 °C 0
0
50
100
150
200
250
300
350
400
Cluster size
Fig. 3. The cluster density as a function of cluster size after annealing at 450 C for 2 h compared with the cluster density of the as-quenched glass.
After 10 min (tG1) at 600°C After 20 min (tG2) at 600°C
0.005
Temperature Differential,°C/g
sidered in the simulation. In this model, therefore, the nucleation rate was calculated at n ¼ 310 (n ¼ 1:5n 1:5 210 310, Fig. 2) to ensure that nuclei of this size are post-critical at all the temperatures used in the simulation. Clusters of size, n > 310 will likely grow into crystals during the isothermal and non-isothermal steps of the simulated DTA experiments, contributing to the measured heat. During the isothermal heating step 4, almost all of the clusters of size n > n ðTg Þ that were generated during the nucleation step will grow into crystals, decreasing the volume of the glass in which subsequent nucleation can occur. The growth of clusters that become larger than the chosen upper limit for the cluster distribution is taken into account using an analytical expression for the size-dependent growth rate (see [8,10,11]). The distribution of all clusters (i.e. those smaller and larger than n ðTg Þ) is used as input to simulate the DTA scan (step 6), using Eq. (12). Step 5 was included to mimic the original DTA experiment [1,2] in which the DTA apparatus was stabilized before the non-isothermal scan. The DTA peak areas obtained from the numerical calculations following two different crystal growth times (tG1 and tG2 ) at TG are substituted into Eq. (1) to determine the total number of clusters (Ntot ) of n ¼ 310 developed due to the nucleation treatment at TN in step 2. Similar calculations for different tN at TN (step 2) allowed the determination of Ntot as a function of tN at TN . The steady state nucleation rate, IN at TN , was calculated from the slope of the straight line at larger values of tN for the Ntot vs tN plot. The steady state nucleation rate, I s , at any temperature (in this case, at TN ) was also calculated from the steady state cluster distribution (at TN ) evolved directly from the model and using Eq. (7). The values of I s calculated directly from the model were then compared with the values of IN , see above, to assess the validity of the DTA approach.
265
0.004
A1
10 min
0.003
0.002 A2
20 min
0.001
0.000 500
550
600
650
700
750
Temperature, °C Fig. 4. Computed DTA curves at 10 C/min for the modeled LS2 glass after a 10 or 20 min growth treatment at 600 C.
than that for the glass after 10 min heat treatment. This is because the volume of glass (or the number of clusters) that remained untransformed and was available for further transformation during the rate-heating DTA scan is smaller for the glass that received a 20 min heat treatment than for the glass that received a heat treatment for 10 min. These two DTA peak areas in Fig. 4 were used in Eq. (1) as proposed in Refs. [1,2], to calculate the concentration of nuclei, Nq , in the as-quenched glass. The value of Nq determined by Eq. (1) for this computer generated lithium disilicate glass was about 1.96 · 106 mol1 . The value of Nq calculated directly from the numerical model for cluster distribution (Eq. (8)) was 2.06 · 106 mol1 . The excellent agreement between these two values for Nq justifies the DTA method, at least for the parameter space chosen for the numerical calculations. The total number of nuclei per unit volume, Ntot , calculated using Eq. (1) for the model glass after
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454 °C
2.5
8 7 Numerical Calculations (Eq. 7)
1.5
6
1.0
DTA-Type Equation (Eq. 1)
0.5
0.0 -2
0
2
4
6
8
10
12
14
16
Time at 450 °C, h
Fig. 5. Number of nuclei as a function of time at 450 C calculated using the numerical model (Eq. (7), solid curve) and the DTA-type equation (Eq. (1), broken curve and the data points).
nucleation at 450 C is shown in Fig. 5 as a function of nucleation time (broken line, solid squares). The solid line represents the number of nuclei calculated directly from the model (Eq. (8)) at 450 C for different times. The time-dependent nucleation rate is apparent in both calculations, and the nucleation rate saturates to the time-independent steady-state rate after an induction period. Although the induction time at 450 C calculated by these two methods differs by about 2 h, the slope of the linearly increasing portion (at longer times) is nearly the same for both curves, indicating that both methods yield nearly the same value for the steady-state nucleation rate, IN in Eq. (1) or I s in Eq. (7). Similar calculations were carried out using both Eqs. (1) and (7) to determine the temperature dependence of the nucleation rate for this computer generated LS2 glass and the results are shown in Fig. 6. As shown in Fig. 6, the nucleation rate calculated using the DTA approach (Eq. (1)) is generally in very good agreement with the nucleation rate computed directly from the numerical model (Eq. (7)) at all temperatures, again demonstrating the validity of the DTA approach. Since the steady state nucleation rates calculated using either Eq. (1) or Eq. (7) are very close to each other at any temperature, both sets of data show identical trends in their temperature-dependence (Fig. 6). The solid line in Fig. 6 is generated using the values for the nucleation rate calculated with the DTA equation (Eq. (1)) to show, as an example, the typical temperature-dependence profile for the calculated nucleation rate of the computer generated LS2 glass. As shown in Fig. 6, the onset temperature for nucleation is about 425 ± 3 C and the temperature for the maximum nucleation rate is about 454 ± 3 C, which are in excellent agreement with the experimental values [1,2], indicating that the parameters chosen for the numerical simulation are reasonably accurate.
I x 104(mol-1s-1)
N x 109(m-3s-1)
2.0
5 4 3 2 1 0
Onset=425 °C
-1 400
420
440
460
480
500
Temperature, °C Fig. 6. Temperature dependence of the steady state nucleation rate, I, calculated using Eq. (1) (s) and Eq. (7) ( ); the solid line represents the nucleation rate curve calculated with Eq. (1).
4. Conclusion A numerical model that has been shown previously to be valid for estimating nucleation rates in glasses was extended to model an experimental DTA method for measuring the concentration of quenched-in nuclei (Nq ) and the nucleation rate (I) for a lithium disilicate glass. When an identical heat treatment schedule is followed, the computer generated DTA curves are in good agreement with the experimental DTA curves, with regards to the area of the DTA peak and the temperature corresponding to peak maxima. The values of Nq and I calculated from the computer generated DTA peaks are in excellent agreement with those calculated directly from the cluster distribution predicted by the model. The results of this work establish the validity of the DTA method for a glass such as lithium disilicate, where the nucleation and growth regions are well separated in temperature. Future studies will examine the analytical DTA method in cases where there is significant overlap. Acknowledgements This work was supported by National Aeronautics and Space Administration, Contract NAG8-1465. The work at Washington University was partially supported by MEMC Electronic Materials SpA, NASA contract NAG5-908 and NSF grant DMR 00-72787. References [1] C.S. Ray, X. Fang, D.E. Day, J. Am. Ceram. Soc. 83 (2000) 865.
K.S. Ranasinghe et al. / Journal of Non-Crystalline Solids 337 (2004) 261–267 [2] K.S. Ranasinghe, C.S. Ray, D.E. Day, J. Mater. Sci. 37 (3) (2002) 547. [3] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [4] A.M. Kalinina, O.V. Potpov, V.N. Filipovich, V.M. Fokin, Glass Phys. Chem. 38&39 (1980) 723. [5] J. Deubener, R. Brukner, J. Non-Cryst. Solids 163 (1993) 1. [6] K.F. Kelton, A.L. Greer, C.V. Thompson, J. Chem. Phys. 79 (1983) 6261. [7] K.F. Kelton, A.L. Greer, Phys. Rev. B 33 (1988) 10089. [8] K.F. Kelton, A.L. Greer, J. Non-Cryst. Solids 79 (1986) 295. [9] A.L. Greer, K.F. Kelton, J. Am. Ceram. Soc. 74 (1991) 1015. [10] K.F. Kelton, J. Non-Cryst. Solids 163 (1993) 283.
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[11] K.F. Kelton, L.K. Narayan, L.E. Levine, T.C. Clull, C.S. Ray, J. Non-Cryst. Solids 204 (1996) 13. [12] C.S. Ray, D.E. Day, W. Haung, L.K. Narayan, T.C. Clull, K.F. Kelton, J. Non-Cryst. Solids 204 (1996) 1. [13] K.F. Kelton, M.C. Weinberg, J. Non-Cryst. Solids 180 (1994) 17. [14] L.K. Narayan, K.F. Kelton, C.S. Ray, J. Non-Cryst. Solids 195 (1996) 148. [15] K.F. Kelton, J. Am. Ceram. Soc. 75 (1992) 2449. [16] K.F. Kelton, in: Solid State Physics, vol. 45, Academic, NY, 1991, p. 75. [17] J.W. Christian, The Theory of Transformation of Metals and Alloys, Pergamon, Oxford, 1965.
Verification of an analytical method for measuring crystal nucleation rates in glasses from DTA data K.S. Ranasinghe a
d
a,*
, P.F. Wei b, K.F. Kelton c, C.S. Ray d, D.E. Day
e
Department of Physics and Materials Research Center, University of Missouri – Rolla, MO 45609-1140, USA b Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA c Department of Physics, Washington University, St Louis, MO 63130, USA Marshall Space and Flight Center, National Aeronautics and Space Administration, Huntsville, AL 35812, USA e Graduate Center for Materials Research, University of Missouri – Rolla, MO 45609-1140, USA Received 29 September 2003
Abstract A recently proposed analytical (DTA) method for estimating the nucleation rates in glasses has been evaluated by comparing experimental data with numerically computed nucleation rates for a model lithium disilicate glass. The time and temperature dependent nucleation rates were predicted using the model and compared with those values extracted from an analysis of numerically calculated DTA curves. The validity of the numerical approach was demonstrated earlier by a comparison with experimental data. The excellent agreement between the nucleation rates from the model calculations and the computer generated DTA data demonstrates the validity of the proposed analytical DTA method. 2004 Elsevier B.V. All rights reserved.
1. Introduction Detailed knowledge of the kinetics of nucleation and crystal growth rate is useful for materials design, since these kinetics control the phase formation and microstructure of the phases. Recently we developed an experimental method that uses differential thermal analysis (DTA) [1,2] to measure the nucleation and crystal growth rates as well as the number of quenchedin nuclei in glasses. The differential thermal analysis experiments, which characterize the transformation over a range in temperature, are particularly important since they are easy to conduct, require little sample preparation, and are independent of the sample geometry. In this DTA method, a series of isothermal and non-isothermal steps as shown schematically in Fig. 1 are required to obtain the numerical values for the rates for nucleation and crystal growth [1,2].
*
Corresponding author. Address: 3055 Marshal Ave., Apt. 13, Cincinnati, OH 45220, USA. Tel.: +1-513 751 5647. E-mail address: [email protected] (K.S. Ranasinghe). 0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.04.010
The goal of this paper is to demonstrate the validity of the analytical DTA method for measuring crystal nucleation rates. The crystal nucleation rate was calculated from computer generated DTA data that were produced using a numerical model developed and used by Kelton and co-workers [6–16]. The same Li2 O Æ 2SiO2 (LS2 ) glass that was originally used in the DTA experiments [1] for determining the nucleation and crystal growth rates is used in the present simulations. For many reasons LS2 is considered a model glass for studying of nucleation and crystallization processes. A good experimental data exists for the steady state and time dependent nucleation rates and growth rates [3–5] for this LS2 glass. Further, the free energy difference between the glass and crystal phases are known for this glass as a function of temperature, and these data have been used extensively to test the classical theory of nucleation [6,7]. Numerical solutions of the classical theory of nucleation have been used to simulate crystallization by homogeneous nucleation in this LS2 glass for isothermal and non-isothermal conditions [8–14]. The model lithium disilicate (Li2 O Æ 2SiO2 or LS2 ) glass was generated by numerically simulating a quench
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constant, the total number of nuclei, Ntot , present in the glass after nucleation at TN can be calculated from the following equation [1,2]:
Melt Crystal Growth Temperature
Quenching
Ntot ¼ ðIN tN þ Nq Þ ¼
Nucleation
Glass
Time
Fig. 1. The heat treatment schedule (schematic) used to simulate the nucleation experiments in the LS2 glass.
of the LS2 melt. The cluster distribution obtained after heat treatment that mimicked the DTA heat treatment cycles shown schematically in Fig. 1 was used in the numerical calculations. These calculations produced an exothermic crystallization peak, which is similar to the experimental DTA peak. The data from this computersimulated DTA peak are analyzed by an analytical equation (Eq. (1)) to calculate the nucleation rate of the computer-generated glass. This nucleation rate is then compared with the nucleation rate calculated directly from the model. It should be emphasized that it is not the purpose of this paper to make exact comparisons with experimental data, but to compare the nucleation data obtained by an analytical treatment of the simulated DTA data with that calculated directly within the numerical model.
2. Calculations of nucleation rate 2.1. The DTA method In the analytical DTA method [1,2], a small amount of glass, typically 40–60 mg, having a relatively large particle size (>400 lm), is first heated isothermally in the DTA apparatus at a temperature, TN , (nucleation temperature) for a time, tN , and then heated for a short time, tG , at a higher temperature, TG , (crystal growth temperature). After the nucleation and crystal growth heat treatments, the glass is heated non-isothermally in the DTA at a relatively high heating rate, typically 10– 25 C/min. An exothermic DTA peak is observed for the crystallization of the glass that remains untransformed after the crystal growth heat treatment at TG . The area of this DTA peak is proportional to the number of nuclei present in the untransformed glass. From the area of two such DTA peaks, where all the experimental parameters except the heat treatment time at TG are
3ðA1 A2 Þ ; 3 3 pðA1 tG2 A2 tG1 ÞU 3
ð1Þ
where, A1 and A2 are the area of the two DTA peaks after the crystal growth heat treatment at TG for times tG1 and tG2 , respectively. IN is the steady state nucleation rate at TN and U is the crystal growth rate at TG . The quantity (IN tN þ Nq ) is the total number of nuclei per unit volume of the glass, where (IN tN ) is the number of nuclei developed per unit volume due to the nucleation heat treatment only and Nq is the concentration of quenched-in nuclei that formed during quenching the melt to glass. A plot of (IN tN þ Nq ) as a function of tN yields a straight line at higher values of tN , whose slope gives the value of IN . By repeating the experiments at different TN , the complete nucleation rate curve as a function of temperature can be determined. The concentration of quenched-in nuclei, Nq , can be calculated from Eq. (1) when the experiments are performed using an as-quenched glass, i.e., without giving any nucleation treatment (IN tN ¼ 0). Also, using the Eq. (1), the crystal growth rate (U ) at different temperatures can be determined from two similar DTA runs at each temperature using an as-quenched glass, see Ref. [1,2] for further experimental details. 2.2. Numerical calculations The basic numerical model for an infinite sample is only briefly discussed in this section, since the model has been discussed extensively elsewhere [6–16]. The classical theory of nucleation is assumed [11,16], taking the time dependent nucleation rates into account. Within the classical theory, clusters of atoms in the configuration of the transformation product arise spontaneously in the original material. Assuming spherical clusters and negligible stress effects, the reversible work for forming a cluster of n molecules, Wn , at any temperature, TN , can be written as Wn ¼ nDGn þ ð36pÞ
1=3 2=3 2=3
n v rls ;
ð2Þ
where, DGn is the Gibbs free energy difference per molecule in the cluster (solid) and the liquid (glass), v is the molecular volume of the liquid and rls is the interfacial energy per unit area of the liquid–cluster interface. Since DGn is negative for a thermodynamically favorable transition and the energy required to create an interface is positive, a maximum in Wn is predicted at a critical cluster size, n , n ¼
ð32pÞr3ls 3vjDGv j
3
;
ð3Þ
K.S. Ranasinghe et al. / Journal of Non-Crystalline Solids 337 (2004) 261–267
where DGv is the free energy difference per unit volume of the cluster (final) and glass (initial) phases (¼ DGn =v). Clusters smaller than n will normally shrink, while those larger than n tend to grow. Clusters are assumed to evolve slowly in size by a series of bimolecular reactions: knþ
En þ E1 $ Enþ1 ; kn1
ð4Þ
where En represents a cluster of n molecules, and knþ and kn represent the rate of monomer addition and detachment, respectively, to a cluster of size n [10]. The time dependent cluster distribution is given by a system of coupled differential equations. dNn;t þ ¼ Nn1;t kn1 Nn;t kn þ Nn;t knþ Nnþ1;t knþ1 ; ð5Þ dt where Nn;t is the number of clusters of size n at a time t. The net forward rate of the reaction In;t , at a cluster size n, is the time-dependent flux of clusters past that size and is given by : In;t ¼ Nn;t knþ Nnþ1;t knþ1
ð6Þ
When a steady state distribution of clusters is established, the net forward rate, which is independent of time and n, is the steady state nucleation rate I s given by s knþ1 ; I s ¼ Nns knþ Nnþ1
ð7Þ
s
I at TN is comparable to IN described in Section 2.1 and Nns is the number of clusters of n molecules in the steady state at TN . In the strict sense, this is not a true steady state, because the number of molecules in the initial phase decreases as the clusters are formed, although this will have negligible impact on this study. Assuming detailed balance of the forward flux is equal to the backward flux and the nucleation rate is zero, i.e. e Nne knþ ¼ Nnþ1 knþ1 , where Nne is the equilibrium cluster distribution, allowing knþ to be expressed in terms of knþ1 . Eqs. (5) and (6) are solved using a finite difference method [6]. The time is divided into a large number of small intervals, dt, and the number of clusters of size n at the end of the interval, Nn;tþdt , is calculated from dNn;t : ð8Þ dt The behavior of clusters of size n < u (where u 2n , calculating n for the highest temperature considered in the numerical simulation) is calculated directly using these methods. The growth of clusters larger than this size is computed using an expression for the growth rate, uðrÞ which is the rate of increase of cluster radius dr=dt, as given by [8] 1 16D 3v 3 v 2r uðrÞ ¼ 2 DGv sinh ; ð9Þ 4p kT r k Nn;tþdt ¼ Nn;t þ dt
D is the diffusion coefficient in the parent phase, k is the jump distance, T is the temperature, k is the Boltzmann
263
constant and c is a constant that depends on the growth mode [13]. To simulate the non-isothermal DTA heating and cooling curves (scans), the rate of volume fraction transformed is computed by dividing the time into a series of isothermal scans of duration dt ¼ dT =/, where / is the heating/cooling scan rate per second and dT is the temperature step size, allowing the nucleation rate to evolve as in Eq. (6). Keeping track of number of clusters of a given size and assuming no overlap between clusters, the total extended volume fraction transformed is calculated at the end of each interval by m 1 X 4p 3 ; ð10Þ Ni ri;t xe ¼ V0 i¼1 3 where V0 is the sample volume, Ni is the number of nuclei generated in the interval i, and ri;t are the time-dependent radii of those nuclei. Cluster overlap is taken into account statistically using the Johnson–Mehl–Avrami equation [17] to relate the actual volume fraction transformed to the extended volume fraction transformed: xðtÞ ¼ 1 expðxe Þ:
ð11Þ
To simulate the DTA peak for continuous scans, it is necessary to mimic the DTA signal, which scales with the rate of volume fraction transformed, DTA signal / ½ xðTi þ dTi Þ xðTi Þ =dt:
ð12Þ
2.3. Input parameters In the present work a ‘molecule’ of lithium disilicate composition is taken to be one formula unit and the free energy is set accordingly. A third-order polynomial is used to fit the measured data [7,10] as DG ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 ;
ð13Þ
where T is the temperature and ai are given in Table 1. The diffusion coefficient was calculated from the Stokes– Einstein relation, D ¼ kB T =3pag;
ð14Þ
where, g is the viscosity and a is a characteristic distance of the order of an atomic diameter. The viscosity of the glass is assumed to obey the Fulcher–Vogel relation, g ¼ g0 expðn=ðT T0 ÞÞ;
ð15Þ
where g0 and T0 are estimated from experimental data. The values for the parameters used in Eqs. (13)–(15) are listed in Table 1. 2.4. Glass formation and nucleation In the present work, the non-isothermal simulation was used to calculate the cluster distribution produced (quenched in clusters) as the liquid was cooled to form a
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Table 1 Physical parameters used for numerical calculations [10]
glass. It is essential to consider the possible influence of such pre-existing clusters when studying the effects of nucleation. When the modeled glass was obtained by quenching the melt at 0.1 K/s, the number of pre-existing clusters is comparable with the experimental value for the lithium disilicate glass [11]. Therefore, all simulations in the present work were conducted for a glass that was generated by quenching the melt at 0.1 K/s, unless stated otherwise. The numerical model was extended to simulate the DTA experiments, using the following isothermal and non-isothermal steps, which are also shown schematically in Fig. 1. These steps are the same as those used in the original DTA experiments [1,2]: Step 1: heat the glass particles in the DTA furnace to a temperature TN , the nucleation temperature (typically from 400 to 500 C), at 20 C/min; Step 2 (nucleation heat treatment): hold at TN for a time tN , typically from 30 min to 15 h; Step 3: heat from TN to a temperature TG , the crystal growth temperature, at 20 C/min; Step 4 (crystal growth heat treatment): hold at TG for a time tG1 ; Step 5: cool at a rate of 20 C/min to a lower temperature (about 350 C) where there is insignificant nucleation and crystal growth; Step 6: heat at 10 C/min until crystallization is complete, which will produce an exothermic peak of area A1 ; Step 7: using a new glass sample, repeat all of the steps above, except for a different heat treatment time, tG2; at TG , which will produce a different DTA peak area of A2 . The crystal growth temperature, TG , was maintained constant at 600 C for all the DTA runs, and the times used for tG1 and tG2 (at TG ) were 10 and 20 min, respectively. From the measured values of A1 and A2 of the computer generated exothermic DTA peak areas, the total number of nuclei was calculated using Eq. (1). Steps 1–7 mentioned above were simulated using the numerical model already discussed (Section 2.2) to develop quenched-in and new clusters in the crystal phase. Each step produced a cluster distribution that used as an input to calculate the number of clusters in the next step. Likewise, the final step gives the total number of nuclei developed after all the heat treatment cycles. In this
a0 ¼ 48 045, a1 ¼ 36:81, a2 ¼ 56:07 104 , a4 ¼ 4:31 106 (0:094 þ 7 105 T ) g0 ¼ 0:0363, n ¼ 7761, T0 ¼ 460 v ¼ 61:2 106 k ¼ 4:6
numerical model, a lower and an upper size limit for the cluster population was assumed. The lower cluster size limit does not have a significant effect on the results of the simulation, provided it is well below the critical size of a cluster. It was noted by Kelton et al. [6] that the smallest cluster with a demonstrable crystalline structure consists of 10 formula units and consequently the lower limit, l, of the cluster distribution is considered as n ¼ 9. It is also assumed that clusters of 9 molecules always have their equilibrium population. As observed by Kelton and co-workers [6–10], the choice of the upper limit, u, also has little effect on the results provided that n is well above the critical size (n > 2n ) for the temperatures used in the simulation. The highest temperature used in this work is 750 C where n is 205, hence the upper limit, u, was taken as 400 which is considered sufficiently large. The backward flux from clusters larger than the upper limit is taken to be zero. Since the critical cluster size increases with increasing temperature (see Fig. 2) many of the cluster sizes that are post-critical at a lower temperature become subcritical at higher temperatures, causing the fluxes for those clusters to reverse direction, thereby, decreasing their population. The temperature dependence of the critical size and its effect on the post critical cluster population are implicitly taken into account in the numerical model and the nucleation rate is always calculated at n ¼ 1:5n for the highest temperature con-
800
Critical Cluster Size, n*"
Constants in Gibb’s free energy (J/mol) Eq. (13) Interfacial energy (J/m2 ) Viscosity (Poise) Eq. (15) Atomic volume (m3 /mol): Jump distance (k, A)
T=750 ° C,n*=210
600
400 u=400 n=310 200
0
l=9
400
500
600
700
800
900
Temperature(C) Fig. 2. Critical cluster size of modeled LS2 glass as a function of temperature.
K.S. Ranasinghe et al. / Journal of Non-Crystalline Solids 337 (2004) 261–267
3. Results and discussion It is useful to first examine the behavior of the cluster distribution that underlies the nucleation and growth features. Fig. 3 compares the computed cluster distribution of the simulated glass after a 2 h nucleation treatment at 450 C with that of the as-quenched glass. As mentioned before, the cluster distribution at the end of each heat treatment step was used as input for the next heat treatment step. Fig. 4 shows the exothermic DTA type crystallization peaks that the numerical model produced for an as-quenched LS2 glass (no nucleation heat treatment) after crystal growth heat treatment at 600 C for 10 or 20 min. As expected, and also as observed experimentally [1,2], the DTA peak area for the glass after 20 min heat treatment is smaller
30
log(no of cluster/mol)
20 n=310 10 as-quenched (0.1K/sec) 2h/450 °C 0
0
50
100
150
200
250
300
350
400
Cluster size
Fig. 3. The cluster density as a function of cluster size after annealing at 450 C for 2 h compared with the cluster density of the as-quenched glass.
After 10 min (tG1) at 600°C After 20 min (tG2) at 600°C
0.005
Temperature Differential,°C/g
sidered in the simulation. In this model, therefore, the nucleation rate was calculated at n ¼ 310 (n ¼ 1:5n 1:5 210 310, Fig. 2) to ensure that nuclei of this size are post-critical at all the temperatures used in the simulation. Clusters of size, n > 310 will likely grow into crystals during the isothermal and non-isothermal steps of the simulated DTA experiments, contributing to the measured heat. During the isothermal heating step 4, almost all of the clusters of size n > n ðTg Þ that were generated during the nucleation step will grow into crystals, decreasing the volume of the glass in which subsequent nucleation can occur. The growth of clusters that become larger than the chosen upper limit for the cluster distribution is taken into account using an analytical expression for the size-dependent growth rate (see [8,10,11]). The distribution of all clusters (i.e. those smaller and larger than n ðTg Þ) is used as input to simulate the DTA scan (step 6), using Eq. (12). Step 5 was included to mimic the original DTA experiment [1,2] in which the DTA apparatus was stabilized before the non-isothermal scan. The DTA peak areas obtained from the numerical calculations following two different crystal growth times (tG1 and tG2 ) at TG are substituted into Eq. (1) to determine the total number of clusters (Ntot ) of n ¼ 310 developed due to the nucleation treatment at TN in step 2. Similar calculations for different tN at TN (step 2) allowed the determination of Ntot as a function of tN at TN . The steady state nucleation rate, IN at TN , was calculated from the slope of the straight line at larger values of tN for the Ntot vs tN plot. The steady state nucleation rate, I s , at any temperature (in this case, at TN ) was also calculated from the steady state cluster distribution (at TN ) evolved directly from the model and using Eq. (7). The values of I s calculated directly from the model were then compared with the values of IN , see above, to assess the validity of the DTA approach.
265
0.004
A1
10 min
0.003
0.002 A2
20 min
0.001
0.000 500
550
600
650
700
750
Temperature, °C Fig. 4. Computed DTA curves at 10 C/min for the modeled LS2 glass after a 10 or 20 min growth treatment at 600 C.
than that for the glass after 10 min heat treatment. This is because the volume of glass (or the number of clusters) that remained untransformed and was available for further transformation during the rate-heating DTA scan is smaller for the glass that received a 20 min heat treatment than for the glass that received a heat treatment for 10 min. These two DTA peak areas in Fig. 4 were used in Eq. (1) as proposed in Refs. [1,2], to calculate the concentration of nuclei, Nq , in the as-quenched glass. The value of Nq determined by Eq. (1) for this computer generated lithium disilicate glass was about 1.96 · 106 mol1 . The value of Nq calculated directly from the numerical model for cluster distribution (Eq. (8)) was 2.06 · 106 mol1 . The excellent agreement between these two values for Nq justifies the DTA method, at least for the parameter space chosen for the numerical calculations. The total number of nuclei per unit volume, Ntot , calculated using Eq. (1) for the model glass after
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454 °C
2.5
8 7 Numerical Calculations (Eq. 7)
1.5
6
1.0
DTA-Type Equation (Eq. 1)
0.5
0.0 -2
0
2
4
6
8
10
12
14
16
Time at 450 °C, h
Fig. 5. Number of nuclei as a function of time at 450 C calculated using the numerical model (Eq. (7), solid curve) and the DTA-type equation (Eq. (1), broken curve and the data points).
nucleation at 450 C is shown in Fig. 5 as a function of nucleation time (broken line, solid squares). The solid line represents the number of nuclei calculated directly from the model (Eq. (8)) at 450 C for different times. The time-dependent nucleation rate is apparent in both calculations, and the nucleation rate saturates to the time-independent steady-state rate after an induction period. Although the induction time at 450 C calculated by these two methods differs by about 2 h, the slope of the linearly increasing portion (at longer times) is nearly the same for both curves, indicating that both methods yield nearly the same value for the steady-state nucleation rate, IN in Eq. (1) or I s in Eq. (7). Similar calculations were carried out using both Eqs. (1) and (7) to determine the temperature dependence of the nucleation rate for this computer generated LS2 glass and the results are shown in Fig. 6. As shown in Fig. 6, the nucleation rate calculated using the DTA approach (Eq. (1)) is generally in very good agreement with the nucleation rate computed directly from the numerical model (Eq. (7)) at all temperatures, again demonstrating the validity of the DTA approach. Since the steady state nucleation rates calculated using either Eq. (1) or Eq. (7) are very close to each other at any temperature, both sets of data show identical trends in their temperature-dependence (Fig. 6). The solid line in Fig. 6 is generated using the values for the nucleation rate calculated with the DTA equation (Eq. (1)) to show, as an example, the typical temperature-dependence profile for the calculated nucleation rate of the computer generated LS2 glass. As shown in Fig. 6, the onset temperature for nucleation is about 425 ± 3 C and the temperature for the maximum nucleation rate is about 454 ± 3 C, which are in excellent agreement with the experimental values [1,2], indicating that the parameters chosen for the numerical simulation are reasonably accurate.
I x 104(mol-1s-1)
N x 109(m-3s-1)
2.0
5 4 3 2 1 0
Onset=425 °C
-1 400
420
440
460
480
500
Temperature, °C Fig. 6. Temperature dependence of the steady state nucleation rate, I, calculated using Eq. (1) (s) and Eq. (7) ( ); the solid line represents the nucleation rate curve calculated with Eq. (1).
4. Conclusion A numerical model that has been shown previously to be valid for estimating nucleation rates in glasses was extended to model an experimental DTA method for measuring the concentration of quenched-in nuclei (Nq ) and the nucleation rate (I) for a lithium disilicate glass. When an identical heat treatment schedule is followed, the computer generated DTA curves are in good agreement with the experimental DTA curves, with regards to the area of the DTA peak and the temperature corresponding to peak maxima. The values of Nq and I calculated from the computer generated DTA peaks are in excellent agreement with those calculated directly from the cluster distribution predicted by the model. The results of this work establish the validity of the DTA method for a glass such as lithium disilicate, where the nucleation and growth regions are well separated in temperature. Future studies will examine the analytical DTA method in cases where there is significant overlap. Acknowledgements This work was supported by National Aeronautics and Space Administration, Contract NAG8-1465. The work at Washington University was partially supported by MEMC Electronic Materials SpA, NASA contract NAG5-908 and NSF grant DMR 00-72787. References [1] C.S. Ray, X. Fang, D.E. Day, J. Am. Ceram. Soc. 83 (2000) 865.
K.S. Ranasinghe et al. / Journal of Non-Crystalline Solids 337 (2004) 261–267 [2] K.S. Ranasinghe, C.S. Ray, D.E. Day, J. Mater. Sci. 37 (3) (2002) 547. [3] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [4] A.M. Kalinina, O.V. Potpov, V.N. Filipovich, V.M. Fokin, Glass Phys. Chem. 38&39 (1980) 723. [5] J. Deubener, R. Brukner, J. Non-Cryst. Solids 163 (1993) 1. [6] K.F. Kelton, A.L. Greer, C.V. Thompson, J. Chem. Phys. 79 (1983) 6261. [7] K.F. Kelton, A.L. Greer, Phys. Rev. B 33 (1988) 10089. [8] K.F. Kelton, A.L. Greer, J. Non-Cryst. Solids 79 (1986) 295. [9] A.L. Greer, K.F. Kelton, J. Am. Ceram. Soc. 74 (1991) 1015. [10] K.F. Kelton, J. Non-Cryst. Solids 163 (1993) 283.
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[11] K.F. Kelton, L.K. Narayan, L.E. Levine, T.C. Clull, C.S. Ray, J. Non-Cryst. Solids 204 (1996) 13. [12] C.S. Ray, D.E. Day, W. Haung, L.K. Narayan, T.C. Clull, K.F. Kelton, J. Non-Cryst. Solids 204 (1996) 1. [13] K.F. Kelton, M.C. Weinberg, J. Non-Cryst. Solids 180 (1994) 17. [14] L.K. Narayan, K.F. Kelton, C.S. Ray, J. Non-Cryst. Solids 195 (1996) 148. [15] K.F. Kelton, J. Am. Ceram. Soc. 75 (1992) 2449. [16] K.F. Kelton, in: Solid State Physics, vol. 45, Academic, NY, 1991, p. 75. [17] J.W. Christian, The Theory of Transformation of Metals and Alloys, Pergamon, Oxford, 1965.