Kinetics of crystal nucleation in silicate glasses

Journal of Non-Crystalline Solids 73 (1985) 517-540 North-Holland, A m s t e r d a m

517

KINETICS OF C R Y S T A L N U C L E A T I O N IN SILICATE G L A S S E S P.F. JAMES Department of Ceramics, Glasses and Polymers, University of Sheffield, Sheffield, England

The kinetics of volume nucleation are analysed in a number of "simple" systems in which the crystallising phase has the same composition as the parent glass. The compositions considered are Li20-2SiO 2, N a 2 0 . 2 C a O . 3 S i O 2, BaO.2SiO 2, 2 N a 2 0 . C a O . 3 S i O 2, NazO.SiO2, 3BaO.5SiO 2 and C a O . S i O 2. In the first three cases classical homogeneous nucleation theory provides a satisfactory description of both the temperature dependence and magnitude of the nucleation rate if the crystal-liquid interracial energy o is temperature dependent. All seven compositions showed striking similarities. The values of TMAX/Tm, where TMAx is the temperature of m a x i m u m nucleation and Tm the melting point (or effective melting point) were all in the range 0.54 to 0.59. TMAx was always at, or somewhat above, Tg. Values of T,t/T m where TO is the "just detectable" nucleation temperature ( I = 106 m - 3 s - 1) were 0.62, 0.64 and 0.66 in the first three systems. These represent somewhat higher undercoolings for the onset of homogeneous nucleation than observed in "droplet" nucleation studies of non-silicate systems. However. values of the Turnbull a parameter calculated from the o results were in general accord with droplet studies for non-metals. Values of W * / k T (W* is the thrmodynamic barrier to nucleation) in the various systems also showed close similarities being - 30 at TMAx and - 50 at Td. The remarkably consistent pattern of the results suggests that the nucleation observed is predominantly homogeneous. Reasons for the failure to observe homogeneous nucleation in other "simple" compositions are discussed.

I. Introduction

The kinetics of crystal nucleation and growth are of crucial importance in determining the glass forming abilities of melts. They also determine the glass forming systems which are suitable to be converted into glass ceramics by controlled heat treatment. While numerous studies of crystal growth rates in undercooled melts have been carried out, relatively few studies of nucleation kinetics have been made, particularly in oxide system. The greater experimental difficulties in obtaining accurate nucleation data may be responsible for this. However, in recent years a number of quantitative studies have appeared and it is an appropriate time to consider the current "state of the art". In a recent extensive review of crystal nucleation in glass forming systems by the present author [1] various topics were discussed including steady state and transient nucleation, the roles of nucleating agents and the effects of amorphous phase separation on crystal nucleation. In the present paper the discussion is restricted to "simple" compositions. A "simple" composition is defined here as one in which the crystallising phase has the same composition as the parent glass, i.e. it is 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

518

P.F. James / Kinetics of crystal nucleation in silicate glasses

effectively a single component system. Only "self nucleation" in the volume is considered, i.e. nucleation that occurs internally and not on external surfaces, and without the addition of nucleating agents. We shall see that for the systems studied this " v o l u m e " nucleation is probably homogeneous. Also it will be shown that the available data fit into a clear, consistent and interesting pattern.

2. Nucleation in simple silicate compositions

2.1. Summary of classical nucleation theory The steady state rate of homogeneous crystal nucleation I in a one component supercooled liquid is related to absolute temperature T by the standard expression (see for example refs. [2-4]), I = A e x p [ - (w* + AGD)/kT],

(1)

W* and AG D are respectively the thermodynamic and kinetic free energy barriers to nucleation and k the Boltzmann constant. The pre-exponential factor A is often expressed as A = 2 n v V l / 3 ( kT/h )(o/kT)l/2,

(2)

where n v is the number of atoms or, more strictly, "formula units" of the crystallizing component phase per unit volume of the liquid, V the volume per formula unit, o the crystal-liquid interfacial free energy per unit area and h Planck's constant. A may be considered as approximately constant over the temperature range used for nucleation measurements and to a good approximation

A = n v ( k T / h ).

(3)

For a spherical nucleus

W* = 16¢ro3vz/3AG2,

(4)

where AG is the bulk free energy change per mole in crystallization and Vm the molar volume of the crystal phase (the free energy change per unit volume AGv is AG/Vm). In the analysis of experimental nucleation rates in terms of the theory, accurate thermodynamic data are required. AG for a single component system at temperature T is given by AG= __AHf(T m _ T)/Trn

-fT aCpdT+rfT (ACp/T)dT, T~

Tm

(5)

where A Hf is the heat of fusion per mole, Tm the melting temperature and ACp the difference in specific heats between the crystal and liquid at temperature T. If ACp can be assumed constant we have

AG= - a H f ( r m - T ) / T m -

ACp[(Trn-

T)-- T ln(Tm/T)].

(6)

P.F. James / Kinetics of crystal nucleation in silicate glasses

If

ACp is

519

taken as zero

A G = - A H f ( T m - Y ) / Y m.

(7)

When ACp is an unknown constant the following equation obtained by Hoffmann [5] has often been used: AG=

-AHr(T,.- T)T/T~.

(8)

Both expressions (7) and (8) are usually only applicable for small undercoolings (Tm - T). Calorimetric measurements for alkali disilicate [6] indicate that eqs. (7) and (8) overestimate and underestimate respectively the magnitude of AG, the discrepancies being larger at high undercoolings. For lithium disilicate eq. (7) is much closer to the experimental data than eq. (8). When ACp is small ,~G is linear with temperature to a good approximation over limited temperature intervals (say 100°C) and for a specified temperature range we can write [7]

AG = -AH~(T,- T)/T~,,

(9)

where A H x and T are numerically derived values. Thus AG is related to an effective supercooling ATe(= 7" - T) without neglecting ACp. The kinetic barrier AG D can be expressed in terms of an effective diffusion coefficient in the liquid given by D = Do e x p ( - A G D / k T ) where DO= kTX2/h and X is a quantity of the order of atomic dimensions (the "jump distance"). Various authors have related D to the viscosity of the liquid ~ by the Stokes Einstein relationD = kT/3crX~l, obtaining

I = (Ah/3qrX3~)

exp(-

W*/kT).

(10)

From eq. (3) we have

I= ( AcT/vl) e x p ( - W*/kT), where A c = Nvk/3~rX3. Hence if experimental available a plot of ln(I~/T) against l/TAG 2

(11)

I'= I(1+ 2 ~ (-1)" exp(-n2t/r))

(12)

data for 1, ~ and AGv are should, according to theory, produce a straight line, the slope yielding o and the intercept yielding A,,. Thus, in principle, A can be determined to within a few orders of magnitude. Most of the uncertainty lies in the use of the Stokes-Einstein equation. However, in justification of the procedure, it may be pointed out that experimental growth rates in one component systems can be predicted to within an order of magnitude or so using this equation [8]. According to theory the steady state nucleation rate I in a supercooled liquid is not attained immediately at a given temperature. Time is required to approach a steady state size distribution of crystal embryos in the liquid. The transient rate I' at time t is given by Kashchiev [9] as

520

P.F. James / Kinetics of crystal nucleation in silicate glasses

where r is an induction time and n an integer. The number of nuclei N ( t ) at time t is oo

N(t)/Ir=t/r-rr2/6-2

E [(-1)"/n2] exp(-n2t/r) •

(13)

n=l

For t > 5r this reduces to the simple equation N ( t ) = I ( t - 7rZr/6).

(14)

The following relation between r and temperature has been obtained [9,10] r = (16h)~Zo/rr2V2AG~) e x p ( a G ' D / k T ) ,

(15)

where AG~ is the activation free energy for self diffusion in the liquid. AG~ may be identical to the kinetic barrier AG o in eq. (1), although this remains to be established. Finally, it is appropriate to briefly consider the heterogeneous nucleation rate on a substrate in the supercooled liquid, which is given [4] by Ihet = n ~ ( k T / h ) e x p [ - (W~'et + A G D ) / k T ] ,

(16)

where n~ is the number of atoms or "formula units" of the liquid in contact with the substrate per unit area and W~'~t=f(O)W* where f(O} is a function of the contact angle between the crystal nucleus and substrate [4], f(O) < 1 for 0~<0~<~r. 2.2. Experimental evidence 2.2.1. Lie0 -2SiO e (LS2) 2.2.1.1. Earlier work. Lithium disilicate glass has been intensively studied by several authors (see for example [7,10-14]). One reason for the importance of this system is that detailed AG data are available for the nucleation range. Although crystalline LS2 melts incongruently, the LS2 composition is completely liquid I°C above the incongruent melting point. Hence the theoretical congruent melting point must be almost identical with the incongruent one and the system can be treated as effectively one component with a melting point at 1034°C. The various studies [10-14] are in substantial agreement. Volume (internal) nucleation is observed in the range 425-530°C. The maximum nucleation rate (4.25 × 109 m -3 s - l ) occurs at about 454°C. The experimental method used by James and other workers is discussed fully elsewhere [1]. At higher temperatures, usually well above the nucleation maximum, a single stage heat treatent can be used. The number of crystals per unit volume Nv can be readily obtained by reflection optical microscopy of polished and lightly etched sections through the glass samples. At lower nucleation temperatures a double stage heat treatment is used. After nucleation treatment at T~ the glass is heat treated for a short time at a higher "growth" temperature TG to grow the crystals to observable dimensions for counting by

P.F. James / Kinetics of crystal nucleation in silicate glasses

521

optical microscopy. Accurate and reproducible nucleation rates are obtained by this procedure provided it is used correctly [1]. The basis of the method has been investigated in detail and fully justified by Kalinina and co-workers [15-17]. James [10] observed pronounced non-steady state nucleation particularly at lower temperatures below the nucleation maximum. However, for isothermal heat treatment the nucleation rate approached a constant, steady state value at longer times. An important result was that the experimental N,, versus time curves were accurately described by Kashchiev's theory [9] (eq. (13)). The temperature dependence of the induction time was also described satisfactorily by the theory (eq. (15)). A plot of logm~" versus 1 / T was a straight line (fig. 1). The induction times ~" were derived from N v versus time plots by measuring the intercepts on the time axis and using eq. (14). The steady state nucleation rates I for LS 2 were analysed by Rowlands and James [7], who discussed various ways of determining the parameters A, AGD and o in the theory using the experimental data. In one method of analysis A G D in eq. (1) was expressed as A H D - T A S D where A H D and AS D are the activation enthalpy and entropy respectively. Taking A H D, A S o and o as constants over the temperature range studied eq. (1) was fitted to the nucleation data of James [10] using published thermodynamic data for AG [6,18]. Assuming a spherical nucleus a value of 222 mJ m 2 for the interfacial energy o was obtained. This method had the advantage that no viscosity data were

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1/T K-1 xl03 Fig. 1, Relation between induction times (~') for crystal nucleation and absolute temperature T for lithium disilicate glass [10]. Plot of lOgl0~-(min ) versus 1/T.

522

P.F. James / Kinetics of crystal nucleation in silicate glasses

required although it was realised that assuming AH D and AS o to be constants was an oversimplification. Also the pre-exponential constant A could not be determined experimentally by this method since the value of AS D was unknown. However, A may be determined from eq. (11) using viscosity data. A plot of lOglo(IT1/T ) versus 1~TAT2 was made using the viscosity data of Matusita and Tashiro [19] and the AG data from ref. [18]. ATx is related to AG through eq. (9). A good straight line fit was obtained over most of the temperature range (445 to 527°C) as predicted by eq. (11), the slope giving o = 190 mJ m -2. The classical theory thus appeared to give a good description of the temperature dependence of I over most of the range. However, the description was poor at low temperatures. Furthermore, the experimental value of A from the intercept of the plot was about 20 orders of magnitude higher than the theoretical value from eq. (2). Neilson and Weinberg [20] independently performed a similar analysis and reached almost identical conclusions. A plot of ln(ITI/T) versus 1/AG2T using the same viscosity data [19] but the AG data from ref. [6] is shown in fig. 2(a) (lAG[ = 53 3 7 0 - 3 9 . 3 7 T J m o l . - 1 over the range involved). The result is very similar to the original plot of Rowlands and James [7], a good straight line fit being obtained over a wide range. The o derived from the slope is 198 mJ m -2 in good agreement with the previous value. The value of A from the intercept is 5.9 × 1060 m -3 s-1, again much higher than the theoretical value (about 1019 times). Rowlands and James [7] discussed possible reasons for the apparent disagreement between the values of A from theory and experiment. Heterogeneous nucleation was very unlikely since the pre-exponential factor for heterogeneous nucleation is expected to be many orders of magnitude smaller than that for homogeneous nucleation (not greater, as observed). Independent experimental results indicating that the volume nucleation in this glass is predominantly homogeneous were obtained by James et al. [21]. The nucleation kinetics were determined in a number of glasses all close to the LS 2 composition but prepared under different conditions. Glasses melted in platinum crucibles were compared with glasses melted under completely platinum-free conditions in silica crucibles. Also glasses prepared from ordinary purity batch materials were compared with glasses from very high purity materials. No significant differences in nucleation were observed indicating that heterogeneous nucleation on platinum particles, possibly introduced during melting, was negligible for the conditions used. Also the results suggested that minor impurities in the levels present did not significantly affect the nucleation behaviour. Another possible reason for the discrepancy in A values could be the neglect of certain statistical mechanical contributions in the classical derivation of IV*, which might lead to an increase in the theoretical value of A [22]. The magnitude of this effect is very difficult to assess in the present case. However, a simpler way of explaining the discrepancy is to postulate a temperature-dependent interfacial free energy. Rowlands and James [7] found the A value was in agreement with classical theory for LS 2 at temperatures above the nuclea-

P.F. James / Kinetics of crystal nucleation in silicate glasses

40

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Fig. 2. (a) ln(l*l/T) versus 1/AG2T for L i 2 0 . 2 S i O 2 glass. I in m - 3 s i (data from ref. [10]). */ in N s m -2 (data from ref. [19]). AG in J mol. ] (from ref. [6]). T in K. (b) Similar plot to (a) using the same 1 and AG data but new viscosity ('O) data from ref. [26].

tion m a x i m u m Tin.~ provided o was expressed in the form o = o r = 64.0 + 0.109 T(mJ m 2),

(17)

where T is in K. This gives o r = 148 mJ m -2 at 500°C, less than the value of 190 mJ m -2 from the previous plot. This procedure, which involves a negative interracial entropy term, has been used to explain discrepancies of a similar

524

P.F. James / Kinetics of crystal nucleation in silicate glasses

type for metallic systems [23,24]. A temperature dependent o for the lithium disilicate system has also been suggested by Fokin et al. [13]. 2.2.1.2. Recent work. Recently James and coworkers [25-27] have re-examined

the data on lithium disilicate. The previous conclusions were largely confirmed except in one respect. In the work described above [7] the viscosity data of Matusita and Tashiro [19] were used for a glass of nominally the same LS 2 composition as studied by James [10]. Gonzalez-Oliver [25] measured the viscosity of the identical glass prepared by James [10] over the range of temperatures used for nucleation. The results were much higher (a factor of 10 or more depending on temperature) than those of Matusita and Tashiro [19]. The disagreement may be due to differences in experimental technique, in base compositions of the glasses or possibly water content of the glasses. Zanotto [26] independently measured the viscosity of a nearly stoichiometric lithium disilicate glass and obtained good agreement with the results of Gonzalez-Oliver [25]. A Fulcher fit to Zanotto's data in the nucleation range above 450°C gave log~0*/= 1.81 + [1346.6/( T - 594.8)],

(18)

where ~ is in Pa s, T in K. In fig. 2(b) l n ( I ~ / T ) is replotted versus 1/zaG2T using the original nucleation data of James [10], AG data from ref. [6] and the new viscosity data represented by eq. (18). As before a good straight line fit is obtained over a wide temperature range, but a better fit is obtained at lower temperatures than in the previous plot. In fact, all the points lie close to the line except for the data point at 425°C. However, the I value at 425°C was approximate (see table 3 in ref. [10]) and was possibly an underestimate due to non-steady state effects. It should be noted that the viscosities at lower temperatures were obtained by extrapolation from eq. (18) and may be subject to some errors. Nevertheless, it is reasonable to conclude that the temperature dependence of the nucleation rates is now described satisfactorily by the theory (eq. (10)) even at lower temperatures. The slope of the plot in fig. 2(b) gives o = 207 mJ m-2. This is similar to the previous value, although somewhat higher. The experimental value of A from the intercept is still much higher than the theoretical value (by a factor of over 10 z5 times). The nucleation rates in L i 2 0 . 2 S i O z glass were redetermined recently [26,27]. A glass was prepared from extremely high purity starting materials and was much purer than that of James [10]. However, the results were in good agreement with those of James and other workers [13-15]. 2.2.1.3. A possible reconciliation with classical theory. The discrepancy between

the experimental and theoretical values of A can be removed by postulating a temperature dependece o r as explained above. It is assumed that A is given by theory (eq. (2)) and W* by eq. (4). This was done by substituting the I, AG and new viscosity data (fig. 2(b)) into eq. (10). The value of o r was then obtained by successive approximation since it also enters the more accurate expression

P.F. James / Kinetics of crystal nucleation in silicate glasses

525

for A (eq. (2)). Vm was taken as 61.34 × 10 -6 m 3 mo1-1 [28]. Calculation of n V and V depends on the choice of the fundamental "building unit" for nucleation. Here V was taken as the volume per L i 2 0 . 2 S i O 2 formula unit and n,. was equated to V - ] . Also it was assumed that ~3 = V since the exact value of ~, which occurs in the Stokes-Einstein equation is uncertain. Fortunately, the calculations of 07- and W * / k T are insensitive to the exact choice of values for n v and 2~. The values of o r and W*/kT calculated in this way to fit exactly the nucleation data are shown in figs. 3(a) and 3(b). In the nucleation range a T increases with temperature from about 144 to 158 mJ m 2. Over most of the range the following linear relation applies o T = 29.11 + 0.1617 T (mJ m - 2 ) ,

(19)

where T is in K. The values are somewhat higher than those found in earlier work (eq. (18)) partly because of the new viscosity data. The theoretical value of A varied only slightly, from 5.3 × 10 41 m 3 s i at 454°C to 5.8 × 10 41 m 3 s i at 527°C. For comparison the approximate value of A from eq. (3) was 1.5 × 1041 m 3 s 1 at 454°C. As expected the value of W * / k T (assuming the theoretical A) increased with temperature (fig. 3b) varying from 29 to 47 in the range of nucleation measurements. At first sight the temperature dependent o of eq. (19) might appear to be inconsistent with the good straight line of ln(I~l/T) versus 1/AG2T in fig. 2(b) which implies a constant temperature independent o. That no inconsistency exists can be readily demonstrated as follows. To a good approximation AG can be expressed by eq. (9) and o r = a + bT where a, b are constants (eq. (19)). The slope of the plot in fig. (2b) at temperature T is thus Slope

d ln( I~l/T) d(1/AG2T )

(16~rV2m)

3 b T ( T ' - T) a3 1 + (a+b~3T---T~)

) '

(20)

We can also put Slope = -

(16~rV2m/3k) Oa~,

(21)

where oa is an " a p p a r e n t o " calculated from the slope at a given temperature. Substituting a, b and ~E~ for LS 2 (from the AG data [6] in the temperature range involved T~ is 1355 K) we find at 454°C, oa = 207.4 mJ m 2 and o r= 146.6 mJ m -2. At 527°C, o, = 209.2 mJ m -2 and o r= 158.4 mJ m 2. Thus o r has increased by 8% whereas a, has increased by less than 1% in this range. It is clear that % is greater than o T. Also the slope has changed only slightly ( < 3%). This explains the constancy of slope observed in fig. 2(b). Hence the good straight line fit is consistent within experimental error with a varying o r . A n analysis similar to the above was used first by Turnbull [23] to explain a discrepancy between theory and experimental A values for the crystallisation

P.F. James / Kinetics of crystal nucleation in silicate glasses

526

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thermodynamic free energy barrier to nucleation.

of supercooled mercury (the experimental value of A w a s 107 times the theoretical value). Recently, the possible effect of a temperature dependent o on metallic glass formation has been discussed [29]. Spaepen [30] has proposed a model for the structure of the liquid-crystal interface in metallic systems in which o is largely entropic in origin and decreases approximately linearly with temperature. The above result for LS z likewise indicates a large contribution to o from the negative entropy term (eq. (19)). The brigin of the effect is unclear at present. It has been suggested [13] that the apparent increase in o with

P.F. James / Kinetics of crystal nucleation in sificate glasses

527

temperature may be caused by a sharpening of the liquid-crystal interface and that the effect is associated with the increase in critical nucleus size with rise in temperature. 2.2.2. Na20 -2CaO -3SiO 2 (NC2S3)

Volume nucleation was observed in this composition without addition of nucleating agents in the range 550-700°C [25,31-33]. Non-steady state nucleation effects were found at the lower end of the range, as in L S 2. Steady state nucleation rates I were analysed using measured viscosities and thermodynamic data [31]. The maximum rate (5.60 × 1011 m-3s 1) occurred at about 595°C, somewhat higher than the glass transformation temperature at 565°C (corresponding to a viscosity of 10 t2 Pa s). These results were confirmed by a DTA study of Marotta et al. [34]. NC2S 3 is a congruently melting compound (m.pt. 1289°C). The crystallising phase has the same composition as the parent liquid or glass. A l n ( I ~ / T ) versus 1 / A G 2 T plot of the results from ref. [31] is shown in fig. 4. AG was calculated from eq. (6) using the measured values of AHf and AC e. A straight line is obtained, the slope indicating a o (or more strictly oa from eq. (21)) of 182 mJ m -2. In ref. [31] 180 mJ m -2 is quoted but the difference is not significant. The "experimental" pre-exponential factor A determined by the "intercept" method was 109°m 3s 1 compared with the theoretical value of about 1041m 3s 1. Although the "intercept" value of A is sensitive to the value of AG [1] it seems unlikely that this large discrepancy with theory is due to an error in AG. A more probable explanation is that o is temperature dependent as suggested for LS 2. Fig. 5(a) shows a plot of interfacial free energy o r versus temperature, o r was calculated at each temperature assuming A was given by theory (eq. (2))

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528

P.F. James / Kinetics of crystal nucleation in silicate

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and substituting I, 77 andAG into eq. (10), as described previously for LS 2. Vm was taken as 126.6 × 10 -6 m 3 mol. -~ [31]. V was taken as the v o l u m e per N a 2 0 - 2 C a O - 3SiO 2 formula unit. Again n v was equated to V -1 and ~3 to V. In the range studied o r increases approximately linearly with temperature from about 108 to 118 mJ m -2 and can be represented by o r

= 9.39 + 0.1141 T(mJ m-2),

(22)

P.F. James / Kinetics of crystal nucleation in silicate glasses

529

where T is in K. The value of A from eq. (2) varied only slightly from 3.1 × 1041 m 3s-1 at 595°C to 3.4× 1041 m-3s -1 at 680°C. These results are similar to those obtained for LS2. Again o v is smaller than oa from the slope of the "nucleation plot" (fig. 4). Substituting appropriate values in eqs. (20) and (21) (T~ is 1884 K for the range involved) we find at 6 4 1 ° C o a = 1 8 2 . 5 m J m 2 a n d o r = 1 1 3 . 9 m J m -2 and at 680°C o ~ = 1 8 1 . 3 m J m -2 and o r = 117.9 mJ m 2. Thus the slope is almost constant and gives a o~ value much greater than o T. The "intercept" value of A is also much higher than the theoretical value. Values of W * / k T calculated as in fig. 3(b) for LS 2 are plotted in fig. 5(b). 2.2.3. B a O - 2 S i O 2 (BS2)

Barium disilicate is a congruently melting compound (m.pt. 1420°C). A glass of this composition exhibits volume nucleation without deliberate additions of nucleating agents. Crystal nucleation rates for the barium disilicate crystal phase from a BaO. 2SiO 2 glass have been measured in the range 662 to 780°C [35,36]. The nucleation rates were higher than in lithium disilicate glass and a maximum rate of 1.87 × 1012 m-3s -1 was observed at 700°C. DTA gave a glass transition temperature Tg of 688°C. James and Rowlands [36] analysed the I versus T curve using the method described earlier for LS 2 in which AG D is expressed as A H D - T A S D and AH D, AS D are taken as constants (method 2 in ref. [7]). Assuming a spherical nucleus o was found to be 132 mJ m -2 Unlike LS 2 no detailed specific heat data were available and AG was calculated from the approximate eq. (7) using the known heat of fusion [37]. Recently the viscosity of BS2 glass was measured in the nucleation range [26,27]. The following Fulcher fit was obtained lOgl0~ = 1.83 + [1701.9/( T - 795.6)],

(23)

where ~/is in Pa s, T in K. The glass was of higher purity than used in ref. [35]. However, the nucleation rates in this glass were in good agreement with the pevious study, In fig. 6 l n ( I ~ / T ) versus 1 / A G Z T is plotted using the nucleation data in refs. [35,36], the viscosity from eq. (23) and AG from eq. (7) with AHf = 37.5 kJ mol-1 [37]. Only the nucleation rates I at 700°C and above are used since the rates below 7000C were probably underestimated due to non-steady state effects. A straight line is obtained as found in the previous studies yielding a o (or o~) of 136 mJ m -2, in quite good agreement with the value obtained in ref. [36]. The value of A from the intercept is 2 × 1058 m-3s -1, again very high compared with the theoretical value of about 10 41 m - 3 s -1. This discrepancy must be viewed as tentative since the calculated z~G is approximate. However, Zanotto [26] has shown that when a reasonable allowance is made for the specific heat correction using eq. (6) the discrepancy is even larger, although the value of o~ is only slightly higher. Since the discrepancy is similar to that found in the previous two systems, it is reasonable to suggest once more that o is temperature dependent.

P.F. James / Kinetics of crystal nucleation in silicate glasses

530 50

J

i

f

i

i

i

i

J

/.,5

c

4O

4'0

4-2

4.4 4"6 ( 1/&G2T )xlO12

4.8

Fig. 6. I n ( I T / T ) versus 1/AG2T for BaO.2SiO 2 glass. I in m - 3 s -1 (data from refs. [35,36]. ~/ in N s m -2 (data from ref. [26]). AG in J mol. -1 (data from ref. [37]). T in K.

Figs. 7a and 7b show ~he calculated values of o r and W * / k T assuming A is given by theory (eq. (2)), and using eq. (10) as described previously. Vm was taken as 73.34 × 1 0 - 6 m 3 mol. -1 [38]. V was taken as the volume per BaO. 2SiO 2 formula unit. As before n v was equated to V-1 and ~3 to V. The value of o r increases with temperature from 100.9 mJ m -2 at 700°C to 107.4 mJ m -2 at 780°C. It appears that o r is not linear with T in this case. Again a large contribution to o from a negative interfacial entropy term is indicated. The value of A varied only slightly from 4.5 × 1 0 41 m - 3 s - t at 700°C to 4.9 × 1 0 41 m - 3 s - 1 at 780°C. Hence, as in the case of LS 2 and NC2S3, o T is smaller than the apparent interfacial energy (aa) from the slope of the "nucleation plot" (fig. 6). Once more it is readily shown by substituting appropriate values into eqs. (20) and (21) that the slope of the nucleation plot is almost constant in the temperature range involved giving a oa of 136 m -2. 2.2.4. Comparison o f L S 2, N C 2 S 3 and B S 2

A comparison of information from the preceding sections is shown in tables l(a) and l(b). TMAx is the temperature of maximum nucleation rate. For the present purpose Tg is taken as the temperature for a viscosity of 1012 Pa s. To is the temperature corresponding to a volume nucleation rate I of 1 c m - 3 s - 1 (or 106 m - 3 s - 1 ) . The latter rate is chosen arbitrarily as the "just detectable" limit. The To values were obtained by extrapolating the plots in figs. 2(b), 4 and 6 to higher temperatures. Knowing l n ( l ~ / T ) as a function of 1 / A G Z T and AG and 71 as functions of T, Td could be determined. A striking feature of table l(a) is the closeness of the values of T M A x / T m for the three compositions. The values of T J T m and T J T m are also close. TMAx is equal to T~ for LS z and is somewhat above T~ for N C z S 3 and BS 2. Td is well above T~ in each case.

P.F. James / Kinetics of crystal nucleation in silicate glasses

106

531

i

106 o-t

mJm-2 104

10")

I0C

I

I

I

980

I

1000

I

I

I

I

1020

1040

I

1060

T(K)

~0

i

35 kT

30

25

I

~

I

I

I

1000

I

I

1020

I

I

1040

I

1060

T(K)

Fig. 7. (a) Crystal-liquid interfacial free energy o T (mJ m 2) as a function of temperature T(K) for BaO.2SiO 2 (see text). (b) W * / k T as a function of temperature T(K) for BaO-2SiO 2.

Table 1 Summary of data for Li 2 0 . 2 S i O 2 (LS 2), Na 2 0 . 2 C a O . 3SIO2(NC2S 3 ) and BaO. 2SiO 2 (BS 2 ) Table l(a) System

LS 2 NC2S 3 BS 2

Reduced temperatures ( T in K)

Temperatures (°C) Tm

TMAx

Td

Tg

1034 1289 1420

454 595 700

535 730 846

454 565 689

TMAx Tm 0.556 0.556 0.575

Td ~m

T~ ~m

0.618 0.642 0.661

0.556 0.536 0.568

P.F. James / Kinetics of crystal nucleation in silicate glasses

532 Table l(b) System

LS 2 NC2S 3 BS 2

or

o~

a

aI

W*/kT

W*/kT

(nO m - 2 ) (at T = TMAx )

(mJ m - 2 )

(from o r at T = TMAx )

(from %)

(at T=/'MAX)

(at T = Td)

146 108 101

207 182 136

0.31 0.25 0.40

0.45 0.42 0.54

32 30 27

49 53 52

The values of W * / k T and o r were calculated as described previously using eq. (10) and assuming A was given by theory (eq. (2)). The values of W * / k T show a striking similarity being - 30 at T = TMAx and - 50 at T = Td for all three systems. It is of interest to calculate the W * / k T required to give "just detectable" nucleation at Tg in a "typical system". Substituting I = 106 m - 3s- 1, 71 = 1012 Pa s and typical values for A (from theory) and V we find W * / k T <_40 to give detectable nucleation. The parameter a in table 1 is given by Ct = O( N A V 2 ) I/3 / A H f ,

(24)

where N A is Avogadro's number. The quantity o(NAV2m) 1/3 is defined as the " m o l a r interfacial energy" [39]. a is dimensionless. Nucleation experiments using droplets indicate a values between ~ and ½ for a range of materials [2,40], with a - 0.4 to 0.5 for metals and a - 0.3 typical for most non-metals. The a values in table 1 (0.31, 0.25 and 0.40) were obtained by using the o r results at T = TMAx in eq. (24). The values denoted a ' were obtained from the oa results. The former values using o T appear to be in best agreement with the droplet nucleation studies for non-metals. 2.2.5. Other systems The kinetics of volume nucleation have been measured in only a few systems apart from those discussed above. It appears that "self nucleation" in the volume, i.e. without addition of nucleating agents, is relatively rare. In contrast, nucleation from the glass surface is readily observed in most compositions. Some available data on systems where volume "self nucleation" has been observed are included in tables 2a and 2b. These systems are all stoichiometric and crystallise without change of composition as in the previous examples (most of the data for N a z O . SiO 2 apply to a non-stoichiometric glass, see below). There is evidence that there is a greater tendency towards volume nucleation at stoichiometric crystal phase compositions because of the higher AG available at these compositions [1]. It should be noted that certain systems such as A1203-SIO 2 which are not represented in table 2(a) also exhibit volume crystallisation but no quantitative nucleation data or even values of TMAx are available at present.

P.F. James / Kinetics of crystal nucleation in silicate glasses

533

Table 2 Summary of data for 2 N a 2 0 . C a O . 3 S i O 2 (N2CS3), N a 2 0 . S i O 2 (NS), 3BaO.5SiO 2 (B3S5) and CaO. SiO 2 (CS) Table 2(a) System

Reduced temperatures (T in K)

Temperatures (°C) Tm

TMAx

Tg

N2CS 3

= 1175

NS B3S5 CS

1089 = 1433 1544

505 460 700-725 = 790

470 410 -690 = 757

TMAX

Tg

Tm

~m

0.54 0.54 0.57-0.59 - 0.58

0.51 0.50 = 0.56 = 0.57

Table 2(b) System

o r ( m J m - 2) (at T = TMAX)

a (from o r at T = TMAX)

W* / k T (at T = TMAX)

N2CS 3 NS

173

0.36

~ 25 32

Volume nucleation rates in the 2 N a 2 0 . CaO-3SIO2(N2CS3) systems were determined by Kalinina et al. [7]. No detailed analysis on the previous lines can be made due to the absence of thermodynamic data. This composition melts incongruently at 1149°C and has a liquidus temperature of about 1200°C [41]. The metastable congruent melting point must lie between these limits and a value of 1175°C was adopted for Tm in table 2(a). Using this value TMAx/Tm is very similar to the results in table l(a). We have made an approximate estimate of W * / k T at irMAx (table 2(b)) using eq. (10) and assuming A to be given by theory (eq. (3)). /MAX w a s taken as 4.3 x 1013 m - 3 s 1 [17] and 71 at irMAx was estimated as - 10 l° Pa s from data in ref. [17] and Vm was taken as 129.7 x 10 -6 m 3 mol. -1 using the published density [42]. We obtained W * / k T - 25, again similar to the results in table l(b). Filipovich et al. measured the rate of nucleation of crystals of sodium metasilicate in a 46 N a 2 0 . 5 4 S I O 2 (tool.%) glass [43]. A maximum nucleation rate of 6.0 × 1011 m - 3 s 1 occurred at 460°C. In table 2(a) it is assumed that TMAx for the stoichiometric N a 2 0 . SiO z also occurs at 460°C. Sodium metasilicate melts congruently at 1089°C [44]. The Tg was estimated as 410°C [45,46]. We have calculated Or and W * / k T at TMAX using eq. (10) and assuming A to be given by theory (eq. (2)). /MAX was taken as 6.0 × 1011 m - 3 s 1, Vm as 46.2 × 10 6 m 3 tool.-1 [47], and ~ at TMAx was estimated as 10 l° Pa s from ref. [451. AG was obtained approximately from eq. (7) with AHf = 51.8 kJ mol. -1 [18]. o r was found to be 174 mJ m -2. The values of c~ and W * / k T were once more similar to the results in table l(b).

534

P.F. James / Kinetics of crystal nucleation in silicate glasses

MacDowell [48] and Freiman et al. [49] demonstrated that 3BaO. 5SiO 2 glass crystallises internally without a catalyst. The compound 3BaO-5SiO 2 melts incongruently at 1423°C. The liquidus occurs at 1443°C [50]. The metastable congruent melting point must lie between these limits and we have adopted 1433°C for the effective Tm in table l(a). From ref. [49] TMAx is 700 to 725°C. Tg is not available, but it is probably similar to the value for barium disilicate (i.e. - 690°C). Recently very low volume crystal nucleation rates were observed in 50CaO • 50SiO 2 (mol.%) glass [51]. The maximum rate occurred at about 790°C, only just above Tg (757°C from DTA). The compound C a O - S i O 2 (Wollastonite) melts congruently at Tm = 1544°C [52]. Very low volume nucleation rates have also been observed in a glass of composition SrO. SiO 2, again close to Tg [51]. The results in tables 2(a) and 2(b) show a striking similarity to those in tables l(a) and l(b). Of particular note is the remarkably consistent values of TMAx/Tm for the seven systems. They are all in the range 0.53-0.59. The consistent pattern of these results, the high undercoolings required for volume nucleation in these systems, and the work of James et al. [21] provide support for the view that the nucleation in most of these systems is predominantly homogeneous. It is interesting to note that droplet nucleation experiments on metals, alkali halides, organic liquids and other systems indicate that observable homogeneous nucleation begins at reduced undercoolings (AT/T m) of 0.15 to 0.25 [39,40]. Clearly the onset of volume nucleation occurs at much higher undercoolings than this in the present silicate systems (typically 0.33-0.38 from table l(a)). Let us consider data on other silicate systems which do not appear to be consistent with the above picture. Klein et al. [53] studied N a 2 0 - 2SiO 2 glass. The time required to obtain a given volume fraction crystallised was measured at various temperatures using X-ray diffraction. Assuming the JohnsonMehl-Avrami equation and knowing the crystal growth rate a nucleation rate was deduced. A In I~/ versus 1/AG2T plot was linear indicating a o of 55 mJ m -2 and a pre-exponential factor in reasonable agreement with classical homogeneous nucleation theory. The range of reduced undercoolings in this study were 0.15 to 0.33, which are small compared with those observed by other workers (see tables 1 and 2). However, Klein et al. [53] interpreted their results in terms of volume nucleation whereas in other studies of N a 2 0 - 2SiO 2 glass [12,54] only surface nucleation was observed. Experimentally, surface nucleation appears to occur more easily than volume nucleation and usually can be observed at smaller undercoolings [55]. In a similar study to that of Klein et al., Cranmer et al. [56] determined the nucleation kinetics in a glass of the anorthite (CaA12Si2Os) composition. Again the results appeared to indicate good agreement with classical homogeneous nucleation theory, the range of reduced undercoolings used being 0.26 to 0.33. A surface energy of 190 mJ m -2 and an ct of 0.21 were deduced. However, other workers [57,58] were not able to observe volume nucleation in anorthite glass after heating at a wide range of temperatures, including long

P.F. James / Kinetics of crystal nucleation in silicate glasses

535

heat treatments in the region of Tg. Only surface crystallisation was observed. In view of these contradictory results further work to clarify the situation seems indicated. It is useful to consider the systems N a 2 0 • 2SiO 2, K 2 0 • 2SiO z and anorthite in relation to tables 1 and 2. Table l(a) indicates that just detectable nucleation might be expected at a reduced temperature of - 0 . 6 2 - 0 . 6 6 . A particularly high value of T g / T m, perhaps higher than 0.6, might suggest low or unobservable volume nucleation on the grounds that Tg would be higher than the temperature range expected for just detectable nucleation. The values of Tg/T,n in tables 1 and 2 range from 0.5 to 0.57. Tg for N a 2 0 - 2 S I O : glass is about 453°C I45,46] giving T g / T m - 0.64 (0.66 is quoted in ref. [59]). Tg for anorthite [60] is 843°C from which T g / T m = 0.61. These T g / T m values for sodium disilicate and anorthite are relatively high and may indicate low or undetectable volume nucleation in these systems, in agreement with some workers [12,57,58]. However, the T g / T m should only be used as a rough guide in this way. Thus volume nucleation is absent in K 2 0 - 2 S i O 2 according ref. [12]. From ref. [46] Tg is - 487°C giving T g / T m ~ 0.58, which is only slightly higher than the values given in tables 1 and 2. Clearly, in more general terms, no detectable volume nucleation rate will be observed if W*, which decreases with increasing undercooling, is too high at temperatures between - Tg and Tm. If W* is too high even at Tg (or just below) no nucleation is likely to be observed since the steep rise in viscosity (or kinetic barrier term in eq. (1)) in the transformation range will provide an effective "cut off". The effect is accentuated by non-steady state nucleation which becomes increasingly important for T < Tg [1]. The earlier calculation indicated that W * / k T < _ 40 to give just detectable volume nucleation at T = Tg. Whether or not this can be achieved will depend on the values of o and AG. Matusita and Tashiro [12] suggested that the presence of volume nucleation in Li20 • 2SiO 2 but its apparent absence in N a 2 0 - 2SiO 2 and K 2 0 - 2SiO: may be attributed to the much higher thermodynamic driving force AG in the lithium disilicate case [6]. 2.3. General discussion and conclusions

In this review we have concentrated on volume nucleation in "simple" stoichiometric glasses which crystallize without change in composition. None of the compositions considered undergo amorphous phase separation. Crystal nucleation in more complex systems, the effects of amorphous phase separation on crystal nucleation and other important topics are discussed elsewhere [1,61-63]. Nucleation in Li 2° . 2SiO 2 has been intensively studied since it is one of the few systems where detailed thermodynamic data are available. The results on non-steady state nucleation are in good agreement with the theory [9] both with regard to the time dependence of N v (the number of crystals) at constant temperature and the temperature dependence of the induction times. Rowlands

536

P.F. James / Kinetics of crystal nucleation in silicate glasses

and James [7] analysed the steady state nucleation rates I by plotting ln(171/T) versus 1/AG2T. A good straight line was obtained over most of the temperature range in accordance with classical theory. However, the fit was poor at the lowest temperatures studied. Also the value of the pre-exponential constant A determined experimentally by the intercept method was much higher than predicted by theory. These results were largely confirmed in a recent reexamination of the system. New viscosity data were obtained and used to replot ln(I*l/T) versus 1/AGET. A straight line was obtained over nearly the whole range of temperatures, indicating better agreement with the theory at lower temperatures than previously found. However, the discrepancy between the experimental and theoretical values of A remained. Thus the "nucleation plot" indicated good agreement with theory with respect to the temperature dependence of the nucleation rates but a large discrepancy with theory with regard to the pre-exponential constant A. This analysis assumed a constant interfacial energy o. It was shown (as first suggested in ref. [7] that the discrepancy in A might be explained by postulating a temperature dependent o ( " o r " ) and assuming A was given by theory. Within experimental error, o r was found to increase linearly with rise in temperature, the increase being about 8% in the range 454-527°C. However, this variation produced a negligible effect on the slope of the "nucleation plot". The apparent o from the slope (207 mJ m -2) was larger than o r (147 mJ m -2 at 454°C). The analysis clearly showed that a good straight line plot of In I r l / T versus 1/AG2T does not imply necessarily a constant, temperature independent, o. It could hide an appreciable linear variation of o with T, and indicate a higher interfacial energy than the true value. Moreover the A factor determined from the intercept of the line would be much higher than the true value. Steady state nucleation rates in the systems N a 2 0 . 2 C a O . 3SiO 2 and BaO. 2SiO 2 were also found to yield good straight line plots of ln(I~l/T) versus 1/AG2T. Again the "intercept" values of A were much higher than predicted by theory. Once more a temperature dependent o was shown to provide a possible explanation of the discrepancy. It should be emphasized that the analysis used in this paper to calculate o r depends on two main assumptions. The first is that the classical theory of homogeneous nucleation can be applied. The second is that the Stokes-Einstein equation can be used to relate the kinetic term to viscosity. Although there is some evidence that the latter assumption is a reasonable one [1] it has yet to be proved correct for nucleation. Furthermore, the possibility that the discrepancy in A values between theory and experiment arises from this second assumption and not from a temperature dependent o cannot be eliminated at this stage. The use of the Stokes-Einstein relation is equivalent to calculating AG D in eq. (1) from the viscosity ,/. It is possible that the true kinetic barrier for nucleation may differ from this value. This question is discussed elsewhere [13,15]. However, at present it is considered that a temperature dependent o provides the more plausible explanation of the discrepancy

P.F. James / Kinetics of crystal nucleation in siiicate glasses

537

observed in each of the three systems LS 2, N C 2 S 3 and BS 2. This procedure has also been used to explain similar discrepancies with the classical theory for metallic systems [23,24,30]. The three systems L i 2 0 - 2SiO 2, N a 2 0 . 2 C a O . 3SiO 2 and BaO- 2SiO2 were compared with other systems exhibiting volume nucleation without the presence of nucleation agents, i.e. 2 N a 2 0 . C a O . 3SiO 2, N a 2 0 . SiO 2, 3BaO. 5SiO 2 and C a O . SiO 2. All these systems showed striking similarities. Thus the values of T M A x / T m where TMAx is the temperature of maximum nucleation rate, were all in the range 0.53 to 0.59. For each composition T M A x / T ~ was either slightly greater than, or equal to T g / T m. Values of T J T ~ , where Td is the temperature for "just detectable" nucleation ( I = 1 c m - 3 s - ~ ) , were 0.62, 0.64 and 0.66 for LS 2, NC2S 3 and BS 2 respectively. These represent higher undercoolings for the onset of homogeneous nucleation than have been observed in droplet nucleation studies of metals, alkali halides, organic liquids and other systems (in these systems the reduced undercoolings A T / T m were 0.15 to 0.25). Values of the a parameter (eq. (24)) were calculated using the interfacial energy (aT) results and heats of fusion. The values were 0.31, 0.25, 0.40 and 0.36 for LS 2, NC2S 3, BS 2 and NS respectively, These results are in good agreement with droplet nucleation studies which indicate ~ - 0.3 for nonmetals. For metals the droplet studies indicate higher a values of 0.4 to 0.5 [2,39]. The values of W * / k T for the various systems considered also showed a striking similarity, being - 30 at T = TMAx and - 50 at T = Ta (tables 1 and 2). When these results are considered as a whole they provide a remarkably consistent pattern and support the view that the volume nucleation observed in most, if not all, of these silicate systems is predominantly homogeneous. It is evident that in the examples studied volume nucleation occurs in the vicinity of 7g or just above. This tendency has also been noted elsewhere [43]. However, the striking constancy of the values of T M A x / T m and T o / T m (tables 1 and 2) suggest that certain general rules may apply to all stoichiometric silicate systems. Thus these ratios may be used to "predict" approximate values of TMA x o r Td in systems where no data is currently available. Such values could provide a useful guide to the range of temperatures where volume nucleation may occur. However, more "simple" systems should be studied quantitatively so that further values of TMAX/Tm and T d / T m can be obtained and compared with the present results. Accurate thermodynamic data and viscosities will also be needed in these systems so that further results for o r, W * / k T and a can be obtained. It has been shown that volume crystal nucleation, probably homogeneous in origin, occurs in a significant number of stoichiometric silicate systems that crystallise without change of composition. It is likely that similar behaviour will be found in other "simple" compositions. However, such nucleation is not observed in all systems, and is probably the exception rather than the rule. Thus there is evidence that certain "simple" compositions do not show

538

P.F. James "/Kinetics of crystal nucleation in silicate glasses

detectable volume nucleation [12,54,57,58]. It is tentatively suggested that only surface nucleation and growth may be involved in these cases. There is some indication from the present results that a relatively high value of T g / T m (perhaps > 0.6) may be associated with very low or undetectable volume nucleation, although more data is needed to test this idea. Other factors favouring volume nucleation besides a low Tg/Tm are a high AGv (high A H f / V m and a high Tm - see eq. (7)) and a low o. Likely compositions for homogeneous volume nucleation will probably tend to have a high Tm or liquidus temperature and will often be in the vicinity of the limiting compositions for glass formation ("the glass forming boundaries"). These circumstances should also favour surface nucleation. The present results indicate that fairly high undercoolings are required for homogeneous nucleation and that nucleation rates range from relatively high to undetectable depending on the composition. In contrast surface nucleation is more readily observed in most systems, as mentioned earlier. Surface nucleation is heterogeneous in orgin and is sensitive to the properties and condition of the glass/air interface [64]. However, there have been few quantitative studies of surface nucleation in silicates and little appears to be known about it in relation to glass forming tendency. A comprehensive review on glass formation was recently published [65]. It is interesting to note that the general conclusion that homogeneous crystal nucleation is only observed in certain silicate compositions is consistent with recent studies of geologically important systems [66]. No homogeneous nucleation was observed at temperatures from 400°C to the liquidi in a number of compositions in the diopside-anorthite-forsterite system. Instead, various types of heterogeneous nucleation occurred on, for example, container walls, the glass surface and on cracks in the glass. Let us briefly consider the possible directions of future studies. Many more "simple" systems that crystallise without change of composition should be investigated and those that exhibit volume nucleation studied quantitatively to see if they fit the pattern revealed above. As we have seen, with a few exceptions accurate thermodynamic data, i.e. enthalpy and specific heat measurements from which AG can be determined, are not available. Such data is essential, together with accurate viscosities and nucleation rates, if further comparisons with theory are to be made. A temperature dependent interfacial energy was postulated in the previous discussion to obtain agreement with Classical theory. Unfortunately there appears to be no means of checking this possibility, independently of nucleation studies. The effect may be connected with the small critical nucleus sizes involved (see refs. [1,7]) and with the variation of the critical size with temperature. Theoretical studies may be helpful here. At present the mechanism of atomic, transport across the crystal nucleus-glass interface is not clearly understood. The assumption that the effective diffusion coefficient D can be calculated from viscosity data using the Stokes-Einstein equation requires further investigation. Analysis of the tern-

P.F. James / Kinetics of crystal nucleation in silicate glasses

539

perature dependence of experimentally measured induction times for transient nucleation [1] may provide information on D or AG D. Such an analysis could be aided considerably by a recent detailed theoretical treatment of transient nucleation using numerical simulation by computer [67]. Another approach would be to measure crystal growth rates directly (perhaps by using electron microscopy) in the same temperature range as the nucleation measurements. The latter method would assume that the kinetic barrier for growth was the same as that for nucleation at identical temperatures. Finally, quantitative studies of the kinetics of surface nucleation for comparison with data on volume nucleation would be of considerable interest as discussed previously. In conclusion, the aim of this paper was to provide some insight into the nature of volume nucleation in "simple" silicate glasses. It is hoped that at least partial success was achieved and that a useful framework has been developed for future studies on similar lines.

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