recent studies of internal and surface nucleation in...

Recent Studies of Internal and Surface Nucleation in Silicate GlassesAuthor(s): Edgar D. Zanotto and Vladimir M. FokinSource: Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Vol.361, No. 1804, Nucleation Control (Mar. 15, 2003), pp. 591-613Published by: Royal SocietyStable URL: http://www.jstor.org/stable/3559191Accessed: 21-11-2017 11:02 UTC

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rM THE ROYAL 10.1098/rsta.2002.1150 MMI SOCIETY

Recent studies of internal and surface

nucleation in silicate glasses

BY EDGAR D. ZANOTTO AND VLADIMIR VM. FOKIN Vitreous Materials Laboratory (LaMa V), Department of Materials Engineering,

Federal University of Sdo Carlos, 13565-905 Sdo Carlos-SP, Brazil

Published online 29 January 2003

This article reviews recent findings on internal and surface nucleation in silicate glasses. The internal homogeneous nucleation rates sharply decrease and the induction times increase with the Tg/TL ratio (Tg is the glass-transition temperature and TL is the liquidus temperature). Only systems that have Tg/TL < 0.58 display measurable internal nucleation rates on a laboratory time-scale. Numerous tests of the classical nucleation theory have demonstrated that the theory fails to describe nucleation rates in glasses quantitatively. Possible explanations for this failure are tested and discussed. Surface nucleation depends strongly on the surface quality, e.g. tips, cracks and scratches, elastic stresses, foreign particles and surrounding atmosphere. The mechanisms of surface nucleation are still not fully understood, but some of the key factors are now known and the surface-nucleation density can thus be controlled for the development of sintered glasses or glass ceramics.

Keywords: nucleation; crystallization; silicate glass; glass surface

1. Introduction and objectives

When any glass is heated for a long enough time above the glass-transition range, Tg, devitrification readily starts from surface nuclei. Surface crystallization is, therefore, often related to the undesired devitrification of glass articles during fabrication or use, or to the manufacture of some types of glass ceramics via sintering followed by crystallization of glass powders. Therefore, to prevent spontaneous devitrification, or to produce sintered glasses or glass ceramics, processing variables such as particle size distribution, heating rate and temperature must be strictly controlled to avoid premature crystallization, which would arrest viscous flow sintering (e.g. Zanotto & Prado 2001; Prado et al. 2002). Even in the absence of catalysers, which are usually employed in the manufacture of commercial (cast) glass ceramics, a few glasses display spontaneous internal nucleation (in addition to surface nucleation). Some of these special systems are discussed in the first part of this review. In the second part, we summarize relevant tests of the classic nucleation theory (CNT) in glasses and discuss the possible reasons for its failure to describe nucleation rates. Finally, in the third part, we show evidence of the key factors that control the kinetics of surface nucleation in silicate glasses. Different theories, and modifications of the CNT, do indeed exist (e.g. Granasy & James 1998, 1999a, b, 2000); however, due to

One contribution of 15 to a Discussion Meeting 'Nucleation control'.

Phil. Trans. R. Soc. Lond. A (2003) 361, 591-613 ? 2003 The Royal Society 591


592 E. D. Zanotto and V. M. Fokin

lack of space, in this review we focus mostly on our own research, which concentrated on the use and testing of the CNT.

2. Theoretical background

The steady-state homogeneous nucleation rate of spherical crystals can be written as (e.g. Gutzow & Schmelzer 1995)

( W* + AGD) kT a2o1/2 Ist Ioexp - W + GD , Io = 2N 2 1/2 , (2.1)

167 a.3 W* = (2.2)

3 AG2 v ' where W* is the thermodynamic barrier and AGD is the kinetic barrier for nucleation, o is the crystal-melt surface energy per unit area of crystal, N1 is the number density of 'structural' units of size a, k and h are Boltzmann's and Planck's con- stants, respectively, and AGv is the thermodynamic driving force per unit volume of crystal. Within the classical theory of nucleation the temperature of maximum nucleation rate, Tmax, is determined mainly by thermodynamic and kinetic barriers. Fokin et al. (2002a) expressed these quantities as

W* 1 167r a3AHm T = C1Z , CI Tr Tr (2.3) kT Tr(1 - Tr)2' 3 RTm Tm'

AGD(T) - AG(T) C2 2 2.31B kT kT Tr - Tr Tm (2.4)

Tm

where Tm and AHm are the temperature and molar heat of melting, and a is an empirical coefficient obtained from fitting nucleation experiments to the CNT. With the approximation of size- and temperature-independent surface energy, a varies from 0.40 to 0.55 for silicate glasses (e.g. Manrich & Zanotto 1995). To and B are empirical coefficients in the Vogel-Fulcher-Tammann equation that correspond to a temperature-dependent activation free energy for viscous flow AG,(T), and C2 ' 30.

To obtain equations (2.3) and (2.4), we assumed that AGD " AG,. In addition, we used Turnbull's equation for the thermodynamic driving force, which neglects the difference in specific heat between the crystalline and liquid phases, and a semi- empirical equation linking the crystal-melt surface energy with the molar heat of melting.

3. Trends on crystal nucleation in silicate glasses

Based on experimental nucleation data for silicate glasses, James (1989), Zanotto (1987) and Zanotto & Weinberg (1989) observed that stoichiometric glasses having a reduced glass-transition temperature Tgr = Tg/TL (Tg is the glass-transition temperature and TL is the liquidus temperature) higher than 0.58-0.60 display only surface (mostly heterogeneous) crystallization, while glasses showing internal homogeneous

Phil. Trans. R. Soc. Lond. A (2003)


Internal and surface nucleation in silicate glasses 593

15 (a)

2Na2O. CaO.3SiO2 Na20.2CaO.3SiO2

X BaO.2SiO2 10 -

Li20.2SiO2

CaO.SiO2

o 6 (b)

5

+ 0

2

0.50 0.55 0.60

0 ITgr 0.50 0.52 0.54 0.56 0.58

Tg Tm/L

Figure 1. (a) Maximum nucleation rates against Tgr for 51 glasses of stoichiometric compositions and non-stoichiometric compositions. (b) Time-lag of nucleation at Tmax. (After Fokin et al. (2002a).)

nucleation have Tgr < 0.58-0.60. Since, at temperatures T < TL the nucleation rate is always positive, the absence of volume nucleation data for glasses having Tgr > 0.58 indicates undetectable nucleation on a laboratory time-scale. When the reduced glass-transition temperature is relatively high (Tgr > 0.58), the work of critical cluster formation at T ? Tg is still too great to produce measurable nucleation. However, close to or on interfaces, the work of critical cluster formation and the viscosity are typically lower than bulk values causing surface crystallization. The transition from glasses demonstrating only surface crystallization to glasses showing volume nucleation is explained by an increase in the volume nucleation rate with decreasing Tgr, verified by Deubener (2000) and extended by Fokin et al. (2002a).

(a) Experimental nucleation-rate data

Figure la (Fokin et al. 2002a) summarizes the data for the maximum nucleation rates, Imax- I(Tmax), versus Tgr for 51 glasses of stoichiometric and non- stoichiometric compositions belonging to eight silicate systems clearly showing the trend that Tgr increases with decreasing Imax. In the relatively narrow range of Tgr (0.50 to 0.58) exhibited by these 51 glasses, the nucleation rate drops by




(a)

2 0 - .......... .. .

0--

I -20

15 2 CD (b)

0

-60

_ 2a-. -80

0.50 0.54 0.58

Tg /ITmIL

-100

0.4 0.6 0.8 1.0 T,,

Figure 2. Maximum nucleation rate as a function of reduced glass-transition temperature. (a) The lines are calculated from the CNT with To0r = 0.4 (1, 2) and C2 = 4.5 (la, 2a) for

different thermodynamic barriers: C1 - 5 (1, la), C1 - 7 (2, 2a). (b) Points refer to experimental data. The lines are calculated from the CNT with C1 = 4.5 (1), C1 - 6.5 (2). Solid lines, C2 = 4.5 and Tor = Tgr - C2/30; dashed lines, Tor - 0.4. (After Fokin et al. (2002a).)

ca. 12 orders of magnitude! Hence, nucleation becomes practically undetectable at Tgr > 0.58. This finding corroborates the proposals of James (1989), Zanotto (1987) and Zanotto & Weinberg (1989). The drastic drop in Imax is accompanied by an increase in the time lag of nucleation at Tmax (see figure ib) (Fokin et al. 2002a).

(b) Analysis with CNT

Because equation (2.4) has two independent parameters, C2 and Tor, the viscosity and, correspondingly, Tgr may be varied in two different ways, by keeping either C2 or Tor fixed. However, in the most interesting range of temperatures (0.5 < Tr < 0.6), these different ways of varying Tgr lead to similar results. Thus, Tgr can be treated as the decisive quantity for an understanding of general trends regarding the nucleation rate.

The lines in figure 2 show the values of I(Tmax) against Tgr, taken from I(Tr) curves calculated by (2.1), (2.3) and (2.4) for different Tgr and, correspondingly, for different kinetic barriers, using the theoretical pre-exponential term Io = 1042 m-3 S-1

According to the CNT, a decrease in temperature produces two effects: a decrease in the thermodynamic barrier (leading to a higher nucleation rate), and an increase in




the kinetic inhibition of nucleation due to an increase in viscosity (resulting in a lower nucleation rate), producing a maximum in the steady-state nucleation rate well below Tm. Within CNT, Tr = 1/3 is a lower limit of Trmax obtained when the value of the kinetic barrier tends to zero. However, the kinetic inhibition of nucleation in glass- forming silicate melts is high. If Tgr increases, the kinetic inhibition of nucleation occurs at higher temperatures and, thus, at higher values of the thermodynamic

barrier to nucleation, shifting Trmax = Tmax/TL above 1/3 and decreasing I(Tmax) (e.g. Filipovich et al. 1983). This is shown in figure 2a. This trend is independent of how the parameter Tgr is changed. Quantitatively, the results obtained by two different methods of varying Tgr (figure 2a, curves 1, 2 and la, 2a) show significant differences, which are, however, most pronounced for Tgr > 0.7-0.8. In the range of interest (0.5 < Tgr < 0.6), however, the differences are very slight. Figure 2b shows that the experimental data on homogeneous nucleation are bounded by log Imax versus Tgr lines, calculated using reasonable values of C1 and C2 and different ways

of varying Tgr. The experimental AG are bounded by the expressions of Turnbull (upper bound,

used here) and Hoffman (lower bound). Moreover, nucleation of metastable phases is possible, such as BaO-2SiO2, shown by Lewis et al. (1979). The data point relating to this glass is quite distant from the others. Additionally, thermodynamic barriers (see equation (2.3)) vary substantially for different glasses. The trends observed must therefore be analysed within reasonable limits. Nevertheless, both the experimental data and the calculated curves show a clear decrease in Imax with Tgr. Thus, although the CNT does not provide a perfect explanation of the Imax-versus-Tgr dependence, it does give a correct trend.

Only ca. 10% of the glass-forming substances vitrified at normal cooling rates have Tgr < 0.6 (Gutzow & Schmelzer 1995); hence, only these 10% show volume nucleation on a laboratory time-scale. In the case of glass-forming metallic alloys, however, most compositions are located at Tg/TL < 0.6, thus nucleating copiously. The high nucleation and growth rates explain why metallic alloys are reluctant glass-formers compared with the typical glass-formers.

4. Testing the classical nucleation theory for internal homogeneous nucleation

Over the past 20 years, the CNT has been exhaustively tested by several authors, with some stoichiometric silicate glasses that show internal nucleation (e.g. James 1982, 1989; Manrich & Zanotto 1995). In these tests, it was generally assumed that viscous flow controls molecular transport at the nucleus-matrix interface; for this case, equation (2.1) can be written as

!st -- K, exp , (4.1 a) 1 N2/3(kTo)1/2 h

K E 2 13 = Io 413, (4.1 b) where I has a value of the order of the Si-O bound length and K, is only weakly dependent on temperature.




Table 1. Ratio of experimental and theoretical pre-exponential and surface energy calculated by the CNT for different glasses

(Values for ACp = 0 calculated using Turnbull's approximation.)

ACp = 0 C, = f(T)

glass log a (J m-2) log he (Jm-2) Li20.2SiO2 15 0.19 19 0.20

Na20.2CaO.3SiO2 18 0.17 72 0.19

2Na20.CaO.3SiO2 27 0.15 139 0.17

75 O O O

65-

2 x 10-20 4 x 10-20 6 x 10-20

/ITAGV, (m6 j-2 K-l)

Figure 3. Plots of ln(Isttind AG2) against 1/TAG2 for three stoichiometric glasses. Linear fit for T > Tg. AGv corresponds to ACp = f(T).

The results of a few such tests were summarized by Manrich & Zanotto (1995). Using ?r(T) and AG(T) data, and assuming a constant surface energy, which is obtained by fitting the temperature of maximum nucleation rate to the theoretical Tmax, the CNT predicts the temperature dependence of the nucleation rates well, but shows a colossal discrepancy in the actual nucleation rates. The theoretical and experimental nucleation rates differ by more than 20 orders of magnitude!

Weinberg & Zanotto (1989) assumed that the temperature dependence of the induction times, tind, reflects that of molecular transport in the nucleation process, making it possible to test CNT while avoiding assumptions about the viscosity. Therefore, equation (2.1) reads

Ist = K G- V exp (- (4.2 a) Phil. tindTrans. R. Soc. Lod. A (2003)




where

K, = N,2(kT)2U)1/2 3/2 a84I, (4.2 b) and K, is weakly dependent on temperature. Figure 3 shows ln(IsttindAG2) versus 1/TAG2 plots for three stoichiometric glasses. Table 1 presents the Kexp/Kthheo ratio and the surface energy o, calculated from figure 3, for temperatures above the glass- transition temperature (there is an inflection, not shown, at Tg). The original form of CNT, which assumes a temperature- and size-independent surface energy, was used in this study. Different approximations were employed for the thermodynamic driving force

AHm Tm TM AC AG = (Tm-T) - ACpdT' + T T dT', (4.3) Tm IT T7 T

where AGv = AG/V, AHm is the latent heat of melting per mole, Tm is the melting point, AC, CC - C, (< 0) is the difference in specific heat between the crystalline and liquid phases and V is the molecular volume. The Kexp/K!heo ratio characterizes the discrepancy between theory and experiment, and it is strongly affected by the choice of AG (ACp = 0 or ACp = f(T)). The value of K~exp/Kheo dramatically increases as one passes from Turnbull's to Hoff- man's approximation for AG(T). Turnbull's approximation, which corresponds to the upper limit for AG(T), gives the lowest discrepancies. In any case, with any reasonable approximation for AG(T), the discrepancy between theory and experiment is quite considerable. If one assumes a constant nucleus-liquid surface energy and uses either the viscos-

ity or the induction time to account for the molecular rearrangements at the interface, drastic discrepancies result between theoretical and experimental nucleation-rate values, regardless of the expression used to estimate the driving force. These findings also hold true for the experimental AG, which is bounded by Turnbull's and Hoff- man's equations. In the following sections we test three other possible explanations for the CNT failure: metastable phase precipitation in the early stages of nucleation; compositional shifts of the crystal nuclei; and a possible dependence of the surface energy on temperature and nucleus size.

(a) Metastable phase formation

A possible explanation for the apparent failure of CNT is the early precipitation of metastable crystalline phases having a lower surface energy, which may induce heterogeneous nucleation of the stable phase. Alternatively, however, metastable phases may nucleate independently and transform to the stable phase at later times. Some authors have suggested the appearance of metastable phases in Li20.2SiO2 (LS2) glass, for example, in contrast to others, who were unable to detect any metastable phase prior to stable LS2 crystal. Although many techniques have been used, e.g. small-angle X-ray scattering, dielectric relaxation, Raman spectroscopy and X-ray photoelectron spectroscopy (Hench et al. 1971; Freiman & Hench 1968; Joseph & Pye 1986, respectively), they provide only indirect evidence, which contributes to the continuing uncertainty. Reviews on this issue were published by Zanotto (1997) and Burgner et al. (2000).




Even studies focusing on a direct technique, transmission electron microscopy (TEM), are controversial. James & Keown (1974), for instance, studied the nucleation of an (almost) stoichiometric lithium disilicate glass heat treated in the temperature range of 450-490 'C for periods of up to 150 h. Electron diffraction patterns showed only the stable phase LS2. On the other hand, Deubener et al. (1993) concluded that a transient phase appears as a precursor to the stable LS2 phase in a (slightly hyper-stoichiometric) 33.5 mol.% glass treated at 454 'C for 7 h and 40 h. However, J. Deubener (1994, personal communication) later mentioned that their results were subject to uncertainty due to fast degradation of the crystals under the electron beam. A phase different from LS2 was also observed by Soares (1997) and Zanotto (1997) in a hypostoichiometric 32.5 mol.% Li20 nucleated at 454 'C for 5-20 h. However, the diffraction patterns were obtained along only one zone axis, which did not allow for the proper identification of the crystals. In this case also, the crystals became amorphous in a few minutes. Therefore, until recently, there was some evidence that different metastable phases formed during the early stages of crystallization in LS2 glasses, but the nature and role that metastable phases played with respect to the crystallization mechanism remained unclear. Regarding this latter issue, in a detailed study of the overall crystallization kinetics of stoichiometric LS2 glass, Zanotto & Leite (1996) concluded that any metastable phase, if present at all, had no significant impact on the crystallization path of the stable LS2 crystal.

Recently, Soares et al. (2002) investigated the early (less than 1% crystallized fraction) and intermediate (ca. 5-10%) stages of crystallization of hypostoichiometric, stoichiometric and hyper-stoichiometric lithium disilicate glasses by TEM/selected area diffraction (SAD). In all their samples, treated from 2.5 h to 120 h at 454 'C (Tg is close to the temperature of maximum nucleation), two different crystalline phases, LS2 and LS, were clearly indexed by SAD. These are stable phases in the hyper- stoichiometric glass, but hypostoichiometric and stoichiometric glasses may follow two possible paths during nucleation: (i) LS2 crystals may nucleate heterogeneously on top of the LS crystals. This path may partly explain the failure of the CNT. (ii) There is simultaneous homogeneous nucleation of LS2 and LS, but LS disappears during the heat treatment. Thus, LS is a metastable phase that does not interfere with the nucleation path of the LS2.

Despite the poor statistics (only half-a-dozen crystals were detected in each TEM sample), there was no evidence that LS crystals induce the heterogeneous nucleation of LS2. In addition, there was plenty of glass remaining even after 120 h; thus, thermodynamic equilibrium had not yet been reached. However, LS crystals should be metastable, since the calculations of B. A. Shakhmatkin and N. M. Vedishcheva (2001, personal communication) demonstrate that the thermodynamic driving force for LS crystallization in LS2 glass is ca. 10% less than the one for LS2 crystals; oth- erwise, the phase diagram would be incorrect!

TEM then provided clear evidence that a second phase, LS, nucleates concurrently with the stable phase, LS2, in stoichiometric and hypostoichiometric lithium disilicate glasses treated at Tg ~ 454 ?C for long times. No phases other than di- and metasilicates appeared. There was no evidence of heterogeneous nucleation of lithium disilicate on top of the lithium metasilicate. So this mechanism does not explain the discrepancy of the CNT.




60

OO ?Q O - ---------- C --- 0 -- 0 0 --------- 0 -------0-------

oO 4 tv Na

50 + Si 5 Ca

-----?f~--~----(D -------- ------ -------~] Ca

10

15 -

10 - 2Vv

0 0.2 0.4 0.6 0.8 1.0 a

Figure 4. Glass 1Na20.2CaO.3SiO2: compositional variation against volume fraction crystallized at 650 'C. Glass (points) measured by EDS; crystals (dotted lines 1 and 2 denote Na and Ca, respectively) calculated from the parent glass composition and EDS measurements. Solid lines fit the experimental data. Dashed lines 3 (Si) and 4 (0) correspond to the parent glass composition. (After Fokin et al. (2002b).)

(b) Compositional shifts of crystal nuclei

While metastable phases have crystallographic structures unlike those of the stable phase, within certain limits, solid solutions ('ss's) can continuously change the composition of a given crystallographic system during phase transformation. Hence, generally speaking, the composition and, consequently, the properties of the critical nuclei may deviate considerably from those of the corresponding macro phase.

Fokin et al. (1999, 2002b) recently observed that such is, indeed, the case in a stoichiometric 1Na20.2CaO.3SiO2 glass (N1C2S3) that displays homogeneous internal nucleation. Phase transformation in this glass proceeds through the formation of ss crystals, which are enriched in sodium relative to the parent glass. During crystallization, the composition of the ss crystals continuously approaches the stoichiometric one. The deviation of the nuclei composition from that of the parent glass forms sodium-poor diffusion fields in their vicinity, and nucleation terminates in an early stage of phase transformation. This is consistent with the finding of Potapov et al. (2000), who demonstrated that the nucleation rate in glasses close to N1C2S3 is very sensitive to the parent glass composition and decreases with decreasing sodium oxide content.

The evolution of crystal and glass compositions and the corresponding change of the lattice parameter are shown in figures 4 and 5. Extrapolating the change of crystal composition to zero time (or zero volume fraction crystallized, a 0= ) suggests that the critical clusters are also enriched in sodium. The deviation of the nuclei

composition from the stoichiometric one diminishes the thermodynamic driving force for crystallization, AGv, and has to magnify the kinetic barrier for nucleation. Since the nucleation of sodium-rich ss crystals occurs instead of the expected stoichiometric




10.54

10.52 --

10.50

10.48 -"O

10.46

10.44 0 0.2 0.4 0.6 0.8 1.0

Figure 5. Parameter a of the hexagonal crystal cell against volume fraction crystallized at 650 'C. (After Fokin et al. (2002b).)

crystals, the decrease in the driving force for crystallization must be compensated for by a decrease in surface energy in equation (2.2) for the thermodynamic barrier. The present interpretation of the nucleation kinetics in NIC2S3 glass is consistent

with Ostwald's rule of stages, generalized by Schmelzer et al. (2000) as

those classes of critical clusters determine the transformation process corresponding to a minimum work of critical cluster formation-as compared with all other possible alternative structures and compositions which may be formed at the given thermodynamic constraints.

Since ss formation is a common phenomenon in silicate systems, it is important to keep in mind when analysing nucleation kinetics that the composition of the critical nuclei and, correspondingly, their thermodynamic driving force and surface energy, may differ considerably from those of the final macrophase. In some cases, as in the N1C2S3 glass, this fact may partly explain the failure of the CNT to predict nucleation rates.

(c) Temperature- and size-dependent nucleus-liquid surface energy

We have already mentioned that CNT fails when the nucleus-liquid surface energy is treated as a size-independent (capillarity approximation) and temperature- independent property, i.e. a(r, T) = uo. This discrepancy can be avoided by calcu- lating a surface energy from experimental nucleation-rate data at each temperature, using the theoretical pre-exponential factor. The surface energy obtained by this method slightly increases with temperature (da/dt - (0.06-0.16) x 10-3 J m-2 K), as observed for metals (Turnbull 1952; Miyazawa & Pound 1974) and silicate glasses (e.g. Fokin & Zanotto 2000).

Since the surface energy calculated from nucleation data refers to that for critical nuclei whose size depends on temperature, the fitted a(T) dependence arises from two factors, the temperature dependence of surface energy for a planar interface, dauo/dT, and the size dependence of surface energy. However, according to Rusanov




1.5 x 10-4

A 2 o 3

050-.

5.01x 0,-5 E-4

-5.0 x 10-5

0 5 10 15 20

26 (A)

Figure 6. Average values of dur,/dT against Tolman's parameter for Li20.2SiO2 (1, 2) and Na20.2CaO.3SiO2 (3) crystals in parent glasses having the same compositions. The kinetic barrier for nucleation was estimated from time lag of nucleation (2, 3) and viscosity (1). (Reproduced with permission from Fokin & Zanotto (2000).)

(1978) and Gutzow et al. (1985), when the molar volume of the liquid phase is higher

than that of the crystal phase, as is most typical, dru,/dT should be negative. To decouple size and temperature effects, Fokin & Zanotto (2000) took into account

the size dependence of a using, as a first approximation, an equation derived by Tolman (1949) for a liquid drop with radius R,

o(R) =- " (4.4) (1 + 26)/R'

where 6 is Tolman's parameter, which characterizes the (unknown) width of the interfacial region between the coexisting phases.

The work of forming a spherical nucleus may thus be written as

W = 4 rR3 AGv - 4 (4.5) 3 R + 26*

The critical radius can be found from the condition (OW/OR)R=R = 0, where

(ao, - 2AGv6) + (a 2 + 2AGv 63u)1/2 R -v.(4.6) AGy

Figure 6 shows the average values of docu/dT versus Tolman's parameter. The fits of cra to experimental nucleation data for L20.2SiO2 and Na20.CaO.3SiO2 glasses were performed for different values of 6. As 6 increased, doa,/dT progressively decreased, becoming negative at 6 > 2.6 x 10-10 m (LS2 glass) and 8 x 10-10 m (NC2S3 glass). Thus, reasonable values of the Tolman parameter may be chosen so that oa, increases with decreasing temperature, in line with the predictions of Rusanov (1978) and Gutzow et al. (1985).

Essentially all methods to determine the nucleus-liquid surface energy are based on nucleation experiments, involving certain additional assumptions. Recently, Fokin




et al. (2000) estimated surface energy without using the nucleation-rate data directly, but instead using the dissolution of subcritical nuclei with an increase in temperature. However, the values of the surface energy determined by this method, using the thermodynamic driving force of crystallization of the macro phase, were much higher than those obtained from a direct fit of nucleation-rate data to CNT. This discrepancy may be eliminated if a reduction of thermodynamic driving force for crystallization of near-critical size clusters is allowed for. This conclusion is of sin- gular importance for the analysis of nucleation data. One last issue concerns the possible effect of elastic stresses on nucleation (Schmelzer & Gutzow 2001). This, however, is a matter for further developments.

5. Surface nucleation in silicate glasses

As we have shown, ca. 90% of silicate glasses have Tgr > 0.60 and consequently display only surface crystallization. Close to or on interfaces, both thermodynamic and kinetic barriers for nucleation are typically lower than bulk values, causing a pre- dominance of surface nucleation. However, its mechanisms and kinetics have drawn little attention and are not well understood so far. A detailed review of research

aimed at understanding surface nucleation was recently published by Miiller et al. (2000). Our attention here focuses mainly on the properties of nucleation sites, i.e. the mechanisms of surface nucleation.

(a) Surface-nucleation kinetics

Since surface nucleation occurs mostly on some active sites (e.g. tips, cracks, scratches and foreign particles), the kinetics of surface nucleation are governed by the exhaustion of these sites due to crystal nucleation and growth. Hence, the kinetic curve N(t) generally reveals a saturation, N(t) -+ Ns (figure 7a). As follows from the drastic drop of Ns (figure 7b, curve 1) with increasing temperature, potential nucleation sites can disappear during heat treatment, thus causing no nucleation. Different Ns(T) dependencies (figure 7b) also illustrate that different crystal phases (here, X-phase and pu-cordierite) can be induced by different kinds of nucleation sites and that quite a distinct nucleation kinetics occurs in each case.

The full N(t)-curve, including its saturation, is needed to calculate the nucleation rate. Owing to experimental limits (high Ns value, short saturation time due to high nucleation activity, difficulties of surface quality reproduction, etc.), such data are very rare. Often, the combination of a small number of nucleation sites and high activity leads to saturation before the achievement of visible sized crystals. In this case, Koster's (1988) method may be applied to estimate the nucleation kinetics. This post-mortem method is based on the analysis of crystal size distribution, caused by the difference in the time at which the crystals nucleated, which, in turn, leads to the time difference for growth.

The most comprehensive surface-nucleation kinetic data reported on so far were obtained for cordierite glass (Pannhorst 2000). A very important characteristic fea- ture of surface-nucleation kinetics is the position of its maximum, Tmax, which considerably exceeds the glass-transition temperature and is often close to the temperature of maximum crystal growth rate. One should recall that, in the case of easily measured homogeneous volume nucleation, Tmax is close to T,. A compilation of available




(a)

0 0 O ----------------------Ns

150 - , 0,' b

I1

,0'0 I o

S4-

a 3 ,o

2-

, 800 850 900 oQ.-O T(oC) 0 500 1000 1500 2000

t (h)

Figure 7. Nucleation kinetics on a cordierite glass surface polished by cerium oxide. (a) Num- ber density of X-phase crystals against heat-treatment time at 800 oC. (b) Saturation level of

the number density of (1) X-phase and (2) /t-cordierite crystals as a function of temperature. (Reproduced with permission from Filipovich et al. (1996).)

values of AT - Tmax - Tg for different glasses and types of surfaces show variations of 70-300 'C. According to the CNT, the high temperature of the surface-nucleation maximum is caused mainly by a reduced thermodynamic barrier.

(b) Surface nucleation sites

Because surface-nucleation kinetics are governed by the presence of nucleation sites, the knowledge and understanding of their properties is a key factor in control- ling surface crystallization. Various difficulties complicate the study of the nucleation mechanism. Experimental limitations preclude micro-analytical evidence of the former nucleation sites (e.g. healing of cracks, dissolution of dust particles). The experimental limits of the determination of N(t) often hinder the measurement of the saturation number Ns. This quantity must be known for a comparison of nucleation data from different systems and for the understanding of external parameters, such as surface conditions or annealing atmosphere. Several kinds of nucleation sites, differing in number density and activity, may occur simultaneously (Zanotto 1991b; Deubener et al. 1992; Filipovich et al. 1996). Additionally, the glass surface cannot be described using interior properties (e.g. due to unsaturated chemical bonds, evaporation, cor- rosion). Finally, the statistical nature of surface-nucleation sites (e.g. scratches, dust) causes data scatter, which can obscure the effect of many parameters, and requires




the control of reproducible surface preparation and ambient annealing conditions. We consider here surface roughness, dust particles and annealing atmosphere.

(c) Surface roughness

Mechanical damage is known to promote surface nucleation of glass. Table 2, taken from Miiller et al. (2000), summarizes surface-nucleation density (N) for damaged glass surfaces, revealing a strong dependence of N upon the degree of surface roughness. Smaller N, down to ca. 5 x 10-8 pm-2, frequently occur at freshly fractured surfaces. Low N are also typical for fire-polished glass surfaces. Medium N, between 10-3 gm-2 and 10-6 gm-2, are evident on mechanically polished or fractured surfaces in a standard laboratory atmosphere. The data scatter is probably due to the presence of dust particles. Larger N of up to 10-1 ?m-2 are reported for ground or glass powder surfaces.

(d) Corners, edges and tips

The nucleation mechanism related to mechanical damage has not been well understood to date. Tabata (1927) observed, in several silicate glasses, that surface nucleation is promoted by 'sharp edges or cicatrices'. Ernsberger (1962) showed, for plate- glass, that crystals are preferably located at the cross-points (and, therefore, at the edges) of macro-cracks. Recently, it was confirmed, for cordierite, diopside, lithia- alumina-silica and float glasses, that sharp edges or tips are favoured nucleation sites (Miiller et al. 1996; Schmelzer et al. 1995; Reinsch et al. 1994). Surface crystals preferentially occur along surface edges caused by fracturing, along

the edges of cracks caused by rapid cooling of glass ribbons or along diamond-made surface scratches. Sometimes, double chains of crystals form along the scratch edges. Glass particles dusted on a fractured glass surface, or those caused by sample fracturing, triggered crystallization at the contacted glass surface (Miiller et al. 1992; Schmelzer et al. 1995; Fokin & Zanotto 1999). Under clean conditions, particles having the composition of the parent glass were

detected at the centre of most of the very few crystals nucleated at the fractured surface of a cordierite glass. For a given glass powder sample (all particles were exposed to identical milling conditions), the surface-nucleation density, N, increased with decreasing particle size because small glass particles provide a larger number of sharp edges per surface area (Miiller et al. 1995). For p-cordierite crystals nucleating on mechanically damaged cordierite glass surfaces, other experiments show that N is correlated to the number density of surface edges or tips, whereas the condition of damaging exerts a minor influence. A controlled reduction of N can be obtained by surface smoothing. Thus, HF etching of ground surfaces prior to the thermal treatment sharply decreases both the number of these edges and N. The value of 21 and the mean linear distance between surface tips (detected by mechanical surface- profiling) are reduced quite similarly (Miiller et al. 1996). Accordingly, polishing of SiC-ground glass surfaces reduces N with polishing time, approaching invariance at t > 5 min (Reinsch et al. 1999). Optical and electron micrographs show that, for t > 5 min, all the tips and edges of the ground surface are eliminated.



-i CIO

0n

Table 2. Surface-nucleation densities, N, of differently damaged glass surfaces

surface glass crystal N (m-2) 21 (grm)a reference powder M2A2S5 p-cordierite 10-3-10-1 30-3 Miiller et al. (1995)

ground M2A2S5 p-cordierite 0.2-7 x10-2 22-4 Miiller & Thamm (1990) CZA2S willemite 3 x 10-2 6 McMillan (1982)

mechanically M2A2S5s p-cordierite 1-10 x 10-4 100-30 Miiller & Thamm (1990) polished 2 x 10-4 70 Yuritsyn et al. (1994a)

CZA2S willemite 1-3 x 10-3 30-18 McMillan (1982) diopside diopside 6-10 x 10-2 1-3 Zanotto (1991a) anorthite anorthite 1-4 x 10-3 30-7 Wittman & Zanotto (1999) NC3S6 devitrite 1 x10-1 3 Zanotto (1991b)

non-stoich. 3-wollastonite 5-10 x 10-5 140-100 Zanotto (1991b) devitrite

tridymite 5-10 x10-5 140-100 Zanotto (1991b) devitrite 1.5-7 x 10-4 80-38 Zanotto (1991b)

mic. slide devitrite 1-6 x 10-4 100-40 Zanotto (1991b)

as-received float devitrite 3-30 x 10-4 58-18 Zanotto (1991b) diopside 5 x 10-5 140 Zanotto (1991b) tridymite 1 x10-5 316 Zanotto (1991b)

mic. slide devitrite 8-30 x 10-4 35-18 Zanotto (1991b)

fractured M2A2S5 p-cordierite 10-3-10-8 30-5000 Reinsch et al. (1994) 10-5-10-6 300-1000 Yuritsin et al. (1994a)

fire-polished mic. slide devitrite '0' 'oo' Zanotto (1991b) M2A2S5 p-cordierite 'lowest' Yamane & Park (1990)

aThe average distances between crystals, 21 - N-1/2, taken from Miiller (1997), are presented for the sake of illustration.

?-

C,

C,,

CC

C, C3

C,'



(e) The effect of elastic stresses

A simple nucleating mechanism of convex edges or tips was suggested by Schmelzer et al. (1993), based on the viscous elastic nature of glass-forming melts. These authors have shown that elastic stresses caused by the growing crystalline clusters, whose density deviates from that of liquid, can considerably hinder nucleation. Calculations are based on equation (5.1), which describes the elastic energy required for the addition of one monomer of volume, vc, in the interior of the glass, 0o:

9(1 - ) (5.1)

The term E/9(1 - y) represents the effect of the elastic properties (E is Young's modulus, y is Poisson's number) and 6 (PC - PG)/PG is the relative density difference between the melt and the crystal. Neglecting stress relaxation, all elastic energy is accumulated to the cluster size a according to 0(a) = aO0. Hence, the molar free energy change of crystallization is Ap/() = Ap - NA~o and the corrected volume nucleation rate, I(E), is given by (Schmelzer et al. 1993)

IF")--Ilexp AA) - IW . (5.2) As 0o decreases with the ratio between the surface distance and the cluster radius, nucleation is easier close to the surface than in the interior, due to the less-intensive stress field in its vicinity (Schmelzer et al. 1995). This effect of the elastic stresses is greater for sharp convex surface curvatures such as surface edges or tips. This hypothesis offers a good explanation of all of the experimental results men-

tioned. Other observations support this assumption more directly.

(i) Large cracks caused by quenching of cordierite glass ribbons were completely healed during a thermal treatment (Schmelzer et al. 1995; Miiller et al. 1992). The former tips did not cause nucleation. This effect confirms the model pro- posed by Schmelzer et al. (1993) because the elastic response of the glass matrix at the crack tips is nearly the same as for any other point in the interior.

(ii) In contrast to the sharp edges of fractured glass surfaces, smooth or wavy surface curvature causes less-intensive surface nucleation (Miiller et al. 1996). Schmelzer et al. (1995) have shown that, correspondingly, the energy of elastic deformation depends on the angle of convex surface tips or edges.

(iii) The nucleation activity observed along former edges of cracks (Schmelzer et al. 1995), scratches or Vickers indentations is greater for the largest values of 6. The nucleation on two glass surfaces after Vickers indentation were studied by Reinsch et al. (1999). Less numerous cristobalite crystals grow at the edges of the indentation on the float glass surface (6 m 10%) than diopside crystals growing at the indentation edges on the diopside glass surface (6 m 15%). In the case of u-cordierite growing on the cordierite glass surface (6 2%), indenting did not cause additional surface nucleation.




10-2 fire clay

OO 0 0

10-3 Q dense corundum

U0

10-4 - O U

10-5 quartz glass 0 V

10-6 O OO O 1 * O

- U

10-8 sample no.

Figure 8. Nucleation density of /t-cordierite at cordierite glass surfaces, fractured and annealed under different conditions. O denotes fire-clay furnace, air; E, corundum tube furnace, air; K, corundum tube furnace, dust protected; 0, corundum tube furnace, dry air; *, quartz-glass tube furnace, air; 0, quartz-glass tube furnace, vacuum. (Reproduced with permission from Muller (1997).)

(f) Solid particles

In the case of smooth glass surfaces (as-received, polished or fractured), N is strongly affected by the presence of solid particles. As early as 1739, R6amur triggered surface nucleation of glassware by contamination with foreign solid particles. Particles of pre-crystallized glass were used to increase crystallization of powdered (Rabinovich 1979) or solid glasses (Ding et al. 1994). Figure 8 shows that there is a strong scatter in N due to the random occurrence of dust for different furnace materials, dust protection, or vacuum. A scatter of seven orders of magnitude in N occurred in fire-clay furnaces, while this scatter was smaller using a gas-dense quality corundum furnace and much smaller when silica glass was used as the furnace material.

The impact of solid foreign particles is not merely dependent on their number but is also determined by their nucleating activity. For example, microscopic observations show that numerous solid particles resting on glass surfaces during the crystallization treatment did not cause nucleation. The nucleating efficiency of solid metal substrates, 0, has been studied for heterogeneous volume nucleation (Gutzow 1980a, b). A comprehensive model is given in Dobreva & Gutzow (1993), connecting 0 with the cohesion forces in the substrate and lattice misfit. The activity of foreign surface particles should follow that general concept. However, additional effects of the ambient atmosphere must be expected.




Oxide dust particles, which are in most cases thermally stable, were found to provide active nucleating substrates for p-cordierite. Thus, ZrO2, known for its thermal stability, was found to be the most efficient surface-nucleation catalyser. Nucleating activity was affected by its crystal morphology, indicating epitaxial phenomena. Sim- ilarly, the nucleation activity of dusted p-cordierite, a-cordierite, quartz and fire-clay powders (the latter is a mixture of corundum, a-cordierite, mullite and quartz) may be caused by the lattice misfit between these powdered crystals and cordierite crystal. Correspondingly, larger N are evident for agate (quartz) ball-milled cordierite glass powders (Miiller et al. 1995) because of the slight lattice misfit with cordierite. We note, however, that the stability (and nucleation activity) of oxide particles can be limited by dissolution into liquid cordierite. Thus, dusting with TiO2 and WO3 reduces N, as is the case of dusting with unstable compounds such as W and WC. Non-oxide dust particles, in most cases less thermally stable, did not increase N

except in a vacuum (p < 10-4 mbar), where better oxidation stability is guaranteed. The inertness of SiC and Si3N4 is probably due to the formation of an amorphous SiO2 surface layer. The increased nucleation activity of these particles under vacuum supports this explanation. In the case of the larger concentrations of unstable particles (W, WC, B4C or less-stable oxides such as W03 and B203), a rippled glass surface and the occurrence of foreign crystalline phases were observed, indicating chemical interaction with the liquid (dissolution). In the worst case, the latter effect can essentially change the composition of the glass surface layer. Thus, intensive dusting with B4C triggered crystallization of needle-like foreign crystals and caused a rippled surface morphology. These effects can be explained assuming an enrichment of B203 in the near-surface area due to oxidation of B4C and to the well-known decrease in surface tension by the addition of B203 (Scholze 1977). This observation is confirmed by the chemical inertness of B4C under vacuum. W and WC show a similar inertness. Their oxidation is most probably followed by dissolution of the oxide into the melt, which modifies the surface morphology. A rippled surface and the total absence of p-cordierite were also caused by dusting with K2SO4. X-ray analyses of a heat-treated mixture of K2SO4 and cordierite glass powder showed leucite as the crystal phase (Miiller et al. 2000), indicating dissolution. In summary, the nucleating activity of solid sites is predominantly limited by

their chemical stability. Chemical reactions with the glass and with the ambient atmosphere can decrease their nucleating efficiency. Accordingly, all thermally stable furnace-tube materials (e.g. fire clay or alumina) turned out, unfortunately, to provide the most active nucleation seeds. The same holds true for oxide-milling materials (Miiller et al. 1995).

(g) Ambient atmosphere

A strong effect of ambient gases (e.g. water vapour) on surface crystallization is frequently stressed in the older literature. The effect, however, is more important on the crystal's growth than on the number of nucleation sites (Miiller et al. 2000). Experiments under glove-box conditions showed no effect of ambient humidity on the surface-nucleation density of 1-cordierite, while the crystal growth rate substantially increased with increasing humidity (Miiller et al. 1996). It should also be noted that very low N, down to N 10-s im-2, occasionally occur for freshly fractured surfaces and that such small N are typical for fire-polished surfaces, even in air atmosphere.




6. Overall conclusions

The internal homogeneous nucleation rates of silicate glasses sharply decrease and the induction times increase with the Tg/TL ratio. Only systems that have Tg/TL < 0.58 display measurable internal nucleation rates on a laboratory time-scale and can be used to analyse nucleation theories.

Using the capillarity approximation, numerous tests of the CNT have demonstrated that the theory fails to quantitatively describe the nucleation rates in glasses. Possible explanations for this failure, such as the use of induction time instead of viscosity to account for molecular transport, the precipitation of metastable phases, compositional shifts of the crystal nuclei, and the possible variation of the surface energy with nucleus size and temperature, were tested and discussed. Of these, the last two appear to be the most relevant. The possible effect of elastic strain is a matter for future studies.

Surface nucleation depends strongly on surface quality: tips, cracks and scratches, elastic stresses, foreign particles and surrounding atmosphere. Despite the fact that the mechanisms of surface nucleation are still not fully understood, some of the key factors are now known, thus allowing for control of the surface-nucleation density and sintering with concurrent crystallization. This improved knowledge has been used in the manufacture of sintered glass ceramics.

The authors thank several collaborators from over the past five years: M. C. Weinberg, J. W. Schmelzer, P. C. Soares Jr, M. O. Prado, E. B. Ferreira, N. S. Yuritsyn and 0. V. Potapov. Stimulating discussions on surface crystallization with R. Miller, G. Volksch, W. Pannhorst, W. Holand (members of Technical Committee 7 of the ICG) are deeply appreciated. We also thank P. F. James, L. Granasy and K. Kelton for reviewing the manuscript. E.D.Z. acknowledges the funding provided by CNPq, Cyted, PRONEX and FAPESP (Brazil) in the period 1998-2002, when most of the studies were performed.

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Discussion

A. BOCCACCINI (Department of Materials, Imperial College of Science, Technology and Medicine, London, UK). You presented a very interesting overview of recent studies on internal and surface nucleation in glass. The experimental results you show are obtained from glass samples having large dimensions in comparison with the scale of crystals nucleated. It would be interesting to extend your findings to consider the behaviour of nanoparticles of glass having a very high surface-area/volume ratio. Is it possible to separate the phenomenon of internal and surface nucleation in nanoparticles experimentally?

E. D. ZANOTTO. In order to separate internal and surface nucleation in glass nanoparticles one might try a few experiments. We have shown in this paper that internal (homogeneous) nucleation is experimentally observed in bulk glasses if Tg/TL < 0.58 (temperatures in kelvin). At such high undercoolings the expected critical nucleus radius is only ca. 1 nm, thus we believe that internal nucleation could occur in particles 50-100 nm in diameter. At such small scale, however, the problem is how to detect some crystals after they have grown to a few tens of nanome- tres, internally or on the particles surface. One could try to use TEM methods for direct observation, or non-isothermal techniques, such as differential thermal analysis or differential scanning calorimetry, for indirect examination. In the latter case,




comparison of the height and temperature of the crystallization peaks obtained for identical experimental conditions, for nanoparticles and millimetric particles of the same glass, could give important clues. For instance, if internal crystallization pre- dominates, the traces of the two widely differing particle sizes would be quite similar and vice versa.



, published 15 March 2003, doi: 10.1098/rsta.2002.1150361 2003 Phil. Trans. R. Soc. Lond. A Edgar D. Zanotto and Vladimir M. Fokin glassesRecent studies of internal and surface nucleation in silicate

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10.1098/rsta.2002.1150

Recent studies of internal and surfacenucleation in silicate glasses

By Edgar D. Zanotto and Vladimir M. Fokin

Vitreous Materials Laboratory (LaMaV), Department of Materials Engineering,Federal University of Sao Carlos, 13565-905 Sao Carlos-SP, Brazil

Published online 29 January 2003

This article reviews recent findings on internal and surface nucleation in silicateglasses. The internal homogeneous nucleation rates sharply decrease and the induc-tion times increase with the Tg/TL ratio (Tg is the glass-transition temperature andTL is the liquidus temperature). Only systems that have Tg/TL < 0.58 display mea-surable internal nucleation rates on a laboratory time-scale. Numerous tests of theclassical nucleation theory have demonstrated that the theory fails to describe nucle-ation rates in glasses quantitatively. Possible explanations for this failure are testedand discussed. Surface nucleation depends strongly on the surface quality, e.g. tips,cracks and scratches, elastic stresses, foreign particles and surrounding atmosphere.The mechanisms of surface nucleation are still not fully understood, but some of thekey factors are now known and the surface-nucleation density can thus be controlledfor the development of sintered glasses or glass ceramics.

Keywords: nucleation; crystallization; silicate glass; glass surface

1. Introduction and objectives

When any glass is heated for a long enough time above the glass-transition range, Tg,devitrification readily starts from surface nuclei. Surface crystallization is, therefore,often related to the undesired devitrification of glass articles during fabrication oruse, or to the manufacture of some types of glass ceramics via sintering followed bycrystallization of glass powders. Therefore, to prevent spontaneous devitrification,or to produce sintered glasses or glass ceramics, processing variables such as particlesize distribution, heating rate and temperature must be strictly controlled to avoidpremature crystallization, which would arrest viscous flow sintering (e.g. Zanotto& Prado 2001; Prado et al . 2002). Even in the absence of catalysers, which areusually employed in the manufacture of commercial (cast) glass ceramics, a fewglasses display spontaneous internal nucleation (in addition to surface nucleation).Some of these special systems are discussed in the first part of this review. In thesecond part, we summarize relevant tests of the classic nucleation theory (CNT) inglasses and discuss the possible reasons for its failure to describe nucleation rates.Finally, in the third part, we show evidence of the key factors that control the kineticsof surface nucleation in silicate glasses. Different theories, and modifications of theCNT, do indeed exist (e.g. Granasy & James 1998, 1999a, b, 2000); however, due to

One contribution of 15 to a Discussion Meeting ‘Nucleation control’.

Phil. Trans. R. Soc. Lond. A (2003) 361, 591–613591

c© 2003 The Royal Society




lack of space, in this review we focus mostly on our own research, which concentratedon the use and testing of the CNT.

2. Theoretical background

The steady-state homogeneous nucleation rate of spherical crystals can be writtenas (e.g. Gutzow & Schmelzer 1995)

Ist = I0 exp(

−W ∗ + ∆GD

kT

), I0 = 2N1

kT

h

(a2σ

kT

)1/2

, (2.1)

W ∗ =16π

3σ3

∆G2v, (2.2)

where W ∗ is the thermodynamic barrier and ∆GD is the kinetic barrier for nucle-ation, σ is the crystal–melt surface energy per unit area of crystal, N1 is the numberdensity of ‘structural’ units of size a, k and h are Boltzmann’s and Planck’s con-stants, respectively, and ∆Gv is the thermodynamic driving force per unit volumeof crystal. Within the classical theory of nucleation the temperature of maximumnucleation rate, Tmax, is determined mainly by thermodynamic and kinetic barriers.Fokin et al . (2002a) expressed these quantities as

W ∗

kT= C1

1Tr(1 − Tr)2

, C1 ≡ 16π

3α3∆Hm

RTm, Tr ≡ T

Tm, (2.3)

∆GD(T )kT

≡ ∆Gη(T )kT

=C2

Tr − T0r, C2 ≡ 2.31B

Tm,

T0r ≡ T0

Tm, C2 = C ′

2(Tgr − T0r),

(2.4)

where Tm and ∆Hm are the temperature and molar heat of melting, and α is anempirical coefficient obtained from fitting nucleation experiments to the CNT. Withthe approximation of size- and temperature-independent surface energy, α variesfrom 0.40 to 0.55 for silicate glasses (e.g. Manrich & Zanotto 1995). T0 and B areempirical coefficients in the Vogel–Fulcher–Tammann equation that correspond to atemperature-dependent activation free energy for viscous flow ∆Gη(T ), and C ′

2 ≈ 30.To obtain equations (2.3) and (2.4), we assumed that ∆GD ∼= ∆Gη. In addition,

we used Turnbull’s equation for the thermodynamic driving force, which neglectsthe difference in specific heat between the crystalline and liquid phases, and a semi-empirical equation linking the crystal–melt surface energy with the molar heat ofmelting.

3. Trends on crystal nucleation in silicate glasses

Based on experimental nucleation data for silicate glasses, James (1989), Zanotto(1987) and Zanotto & Weinberg (1989) observed that stoichiometric glasses having areduced glass-transition temperature Tgr = Tg/TL (Tg is the glass-transition temper-ature and TL is the liquidus temperature) higher than 0.58–0.60 display only surface(mostly heterogeneous) crystallization, while glasses showing internal homogeneous





Tgr

0.50 0.55 0.60

6

4

2

log

( (

Trm

ax)/

s)τ

0.50 0.52 0.54 0.56 0.580

5

15

Tg / Tm/L

log

(Im

ax/(

m−3

s−1

))

2Na2O.CaO.3SiO2Na2O.2CaO.3SiO2

CaO.SiO2

Li2O.2SiO2

BaO.2SiO2

(a)

(b)

10

Figure 1. (a) Maximum nucleation rates against Tgr for 51 glasses of stoichiometric compo-sitions and non-stoichiometric compositions. (b) Time-lag of nucleation at Tmax. (After Fokinet al . (2002a).)

nucleation have Tgr < 0.58–0.60. Since, at temperatures T < TL the nucleationrate is always positive, the absence of volume nucleation data for glasses havingTgr > 0.58 indicates undetectable nucleation on a laboratory time-scale. When thereduced glass-transition temperature is relatively high (Tgr > 0.58), the work of crit-ical cluster formation at T ∼ Tg is still too great to produce measurable nucleation.However, close to or on interfaces, the work of critical cluster formation and theviscosity are typically lower than bulk values causing surface crystallization. Thetransition from glasses demonstrating only surface crystallization to glasses showingvolume nucleation is explained by an increase in the volume nucleation rate withdecreasing Tgr, verified by Deubener (2000) and extended by Fokin et al . (2002a).

(a) Experimental nucleation-rate data

Figure 1a (Fokin et al . 2002a) summarizes the data for the maximum nucle-ation rates, Imax ≡ I(Tmax), versus Tgr for 51 glasses of stoichiometric and non-stoichiometric compositions belonging to eight silicate systems clearly showingthe trend that Tgr increases with decreasing Imax. In the relatively narrow rangeof Tgr (0.50 to 0.58) exhibited by these 51 glasses, the nucleation rate drops by





2a 1a

2

1

2

1

0.50 0.54 0.580

5

15lo

g (I

max

/(m

−3 s

−1))

(b)

(a)

log

(Im

ax/(

m−3

s−1

))

20

0

−100

Tgr0.4 0.6 0.8 1.0

Tg / Tm/L

−20

−40

−60

−80

Figure 2. Maximum nucleation rate as a function of reduced glass-transition temperature.(a) The lines are calculated from the CNT with T0r = 0.4 (1, 2) and C2 = 4.5 (1a, 2a) fordifferent thermodynamic barriers: C1 = 5 (1, 1a), C1 = 7 (2, 2a). (b) Points refer to experimen-tal data. The lines are calculated from the CNT with C1 = 4.5 (1), C1 = 6.5 (2). Solid lines,C2 = 4.5 and T0r = Tgr − C2/30; dashed lines, T0r = 0.4. (After Fokin et al . (2002a).)

ca. 12 orders of magnitude! Hence, nucleation becomes practically undetectable atTgr > 0.58. This finding corroborates the proposals of James (1989), Zanotto (1987)and Zanotto & Weinberg (1989). The drastic drop in Imax is accompanied by anincrease in the time lag of nucleation at Tmax (see figure 1b) (Fokin et al . 2002a).

(b) Analysis with CNT

Because equation (2.4) has two independent parameters, C2 and T0r, the viscosityand, correspondingly, Tgr may be varied in two different ways, by keeping either C2or T0r fixed. However, in the most interesting range of temperatures (0.5 < Tr < 0.6),these different ways of varying Tgr lead to similar results. Thus, Tgr can be treated asthe decisive quantity for an understanding of general trends regarding the nucleationrate.

The lines in figure 2 show the values of I(Tmax) against Tgr, taken from I(Tr) curvescalculated by (2.1), (2.3) and (2.4) for different Tgr and, correspondingly, for differentkinetic barriers, using the theoretical pre-exponential term I0 = 1042 m−3 s−1.

According to the CNT, a decrease in temperature produces two effects: a decreasein the thermodynamic barrier (leading to a higher nucleation rate), and an increase in





the kinetic inhibition of nucleation due to an increase in viscosity (resulting in a lowernucleation rate), producing a maximum in the steady-state nucleation rate well belowTm. Within CNT, Tr = 1/3 is a lower limit of Tmax

r obtained when the value of thekinetic barrier tends to zero. However, the kinetic inhibition of nucleation in glass-forming silicate melts is high. If Tgr increases, the kinetic inhibition of nucleationoccurs at higher temperatures and, thus, at higher values of the thermodynamicbarrier to nucleation, shifting Tmax

r = Tmax/TL above 1/3 and decreasing I(Tmax)(e.g. Filipovich et al . 1983). This is shown in figure 2a. This trend is independentof how the parameter Tgr is changed. Quantitatively, the results obtained by twodifferent methods of varying Tgr (figure 2a, curves 1, 2 and 1a, 2a) show significantdifferences, which are, however, most pronounced for Tgr > 0.7–0.8. In the range ofinterest (0.5 < Tgr < 0.6), however, the differences are very slight. Figure 2b showsthat the experimental data on homogeneous nucleation are bounded by log Imaxversus Tgr lines, calculated using reasonable values of C1 and C2 and different waysof varying Tgr.

The experimental ∆G are bounded by the expressions of Turnbull (upper bound,used here) and Hoffman (lower bound). Moreover, nucleation of metastable phases ispossible, such as BaO–2SiO2, shown by Lewis et al . (1979). The data point relatingto this glass is quite distant from the others. Additionally, thermodynamic barriers(see equation (2.3)) vary substantially for different glasses. The trends observed musttherefore be analysed within reasonable limits. Nevertheless, both the experimentaldata and the calculated curves show a clear decrease in Imax with Tgr. Thus, althoughthe CNT does not provide a perfect explanation of the Imax-versus-Tgr dependence,it does give a correct trend.

Only ca. 10% of the glass-forming substances vitrified at normal cooling rates haveTgr < 0.6 (Gutzow & Schmelzer 1995); hence, only these 10% show volume nucleationon a laboratory time-scale. In the case of glass-forming metallic alloys, however,most compositions are located at Tg/TL < 0.6, thus nucleating copiously. The highnucleation and growth rates explain why metallic alloys are reluctant glass-formerscompared with the typical glass-formers.

4. Testing the classical nucleation theory forinternal homogeneous nucleation

Over the past 20 years, the CNT has been exhaustively tested by several authors,with some stoichiometric silicate glasses that show internal nucleation (e.g. James1982, 1989; Manrich & Zanotto 1995). In these tests, it was generally assumed thatviscous flow controls molecular transport at the nucleus–matrix interface; for thiscase, equation (2.1) can be written as

Ist = Kη1η

exp(

−W ∗

kT

), (4.1 a)

Kη ≡ 12

N2/31 (kTσ)1/2

l3= I0

h

4l3, (4.1 b)

where l has a value of the order of the Si–O bound length and Kη is only weaklydependent on temperature.





Table 1. Ratio of experimental and theoretical pre-exponential and surface energycalculated by the CNT for different glasses

(Values for ∆Cp = 0 calculated using Turnbull’s approximation.)

∆Cp = 0 Cp = f(T )︷︸︸︷︷︸︸︷glass log

(Kexp

η

Ktheoη

)σ (J m−2) log

(Kexp

η

Ktheoη

)σ (J m−2)

Li2O.2SiO2 15 0.19 19 0.20Na2O.2CaO.3SiO2 18 0.17 72 0.192Na2O.CaO.3SiO2 27 0.15 139 0.17

65

75

2Na 2O

.CaO

.3Si

O2

Na 2O

.2C

aO.3

SiO

2

Li 2O

.2Si

O2

2 × 10−20 4 × 10−20 6 × 10−20

1/T∆GV2 , (m6 J−2 K−1)

ln (

I st t in

d ∆

GV2

(J2

m−3

)

70

Figure 3. Plots of ln(Isttind ∆G2v) against 1/T∆G2

v for three stoichiometric glasses.Linear fit for T > Tg. ∆Gv corresponds to ∆Cp = f(T ).

The results of a few such tests were summarized by Manrich & Zanotto (1995).Using η(T ) and ∆G(T ) data, and assuming a constant surface energy, which isobtained by fitting the temperature of maximum nucleation rate to the theoret-ical Tmax, the CNT predicts the temperature dependence of the nucleation rateswell, but shows a colossal discrepancy in the actual nucleation rates. The theo-retical and experimental nucleation rates differ by more than 20 orders of magni-tude!

Weinberg & Zanotto (1989) assumed that the temperature dependence of theinduction times, tind, reflects that of molecular transport in the nucleation process,making it possible to test CNT while avoiding assumptions about the viscosity.Therefore, equation (2.1) reads

Ist = Kτ1

∆G2v tind

exp(

−W ∗

kT

), (4.2 a)





where

Kτ ≡ 163 N2

1 (kT )1/2σ3/2 =83

hσ

a4 I0, (4.2 b)

and Kτ is weakly dependent on temperature. Figure 3 shows ln(Isttind∆G2v) versus

1/T∆G2v plots for three stoichiometric glasses. Table 1 presents the Kexp

η /Ktheoη ratio

and the surface energy σ, calculated from figure 3, for temperatures above the glass-transition temperature (there is an inflection, not shown, at Tg). The original formof CNT, which assumes a temperature- and size-independent surface energy, wasused in this study. Different approximations were employed for the thermodynamicdriving force

∆G = −∆Hm

Tm(Tm − T ) −

∫ Tm

T

∆Cp dT ′ + T

∫ Tm

T

∆Cp

T ′ dT ′, (4.3)

where ∆Gv = ∆G/V , ∆Hm is the latent heat of melting per mole, Tm is the meltingpoint, ∆Cp = CC

p − CLp (< 0) is the difference in specific heat between the crystalline

and liquid phases and V is the molecular volume.The Kexp

τ /Ktheoτ ratio characterizes the discrepancy between theory and experi-

ment, and it is strongly affected by the choice of ∆G (∆Cp = 0 or ∆Cp = f(T )). Thevalue of Kexp

τ /Ktheoτ dramatically increases as one passes from Turnbull’s to Hoff-

man’s approximation for ∆G(T ). Turnbull’s approximation, which corresponds tothe upper limit for ∆G(T ), gives the lowest discrepancies. In any case, with any rea-sonable approximation for ∆G(T ), the discrepancy between theory and experimentis quite considerable.

If one assumes a constant nucleus–liquid surface energy and uses either the viscos-ity or the induction time to account for the molecular rearrangements at the interface,drastic discrepancies result between theoretical and experimental nucleation-rate val-ues, regardless of the expression used to estimate the driving force. These findingsalso hold true for the experimental ∆G, which is bounded by Turnbull’s and Hoff-man’s equations.

In the following sections we test three other possible explanations for the CNTfailure: metastable phase precipitation in the early stages of nucleation; composi-tional shifts of the crystal nuclei; and a possible dependence of the surface energy ontemperature and nucleus size.

(a) Metastable phase formation

A possible explanation for the apparent failure of CNT is the early precipitationof metastable crystalline phases having a lower surface energy, which may induceheterogeneous nucleation of the stable phase. Alternatively, however, metastablephases may nucleate independently and transform to the stable phase at later times.Some authors have suggested the appearance of metastable phases in Li2O.2SiO2(LS2) glass, for example, in contrast to others, who were unable to detect anymetastable phase prior to stable LS2 crystal. Although many techniques have beenused, e.g. small-angle X-ray scattering, dielectric relaxation, Raman spectroscopy andX-ray photoelectron spectroscopy (Hench et al . 1971; Freiman & Hench 1968; Joseph& Pye 1986, respectively), they provide only indirect evidence, which contributes tothe continuing uncertainty. Reviews on this issue were published by Zanotto (1997)and Burgner et al . (2000).





Even studies focusing on a direct technique, transmission electron microscopy(TEM), are controversial. James & Keown (1974), for instance, studied the nucle-ation of an (almost) stoichiometric lithium disilicate glass heat treated in the tem-perature range of 450–490 ◦C for periods of up to 150 h. Electron diffraction patternsshowed only the stable phase LS2. On the other hand, Deubener et al . (1993) con-cluded that a transient phase appears as a precursor to the stable LS2 phase in a(slightly hyper-stoichiometric) 33.5 mol.% glass treated at 454 ◦C for 7 h and 40 h.However, J. Deubener (1994, personal communication) later mentioned that theirresults were subject to uncertainty due to fast degradation of the crystals underthe electron beam. A phase different from LS2 was also observed by Soares (1997)and Zanotto (1997) in a hypostoichiometric 32.5 mol.% Li2O nucleated at 454 ◦C for5–20 h. However, the diffraction patterns were obtained along only one zone axis,which did not allow for the proper identification of the crystals. In this case also,the crystals became amorphous in a few minutes. Therefore, until recently, therewas some evidence that different metastable phases formed during the early stagesof crystallization in LS2 glasses, but the nature and role that metastable phasesplayed with respect to the crystallization mechanism remained unclear. Regardingthis latter issue, in a detailed study of the overall crystallization kinetics of stoichio-metric LS2 glass, Zanotto & Leite (1996) concluded that any metastable phase, ifpresent at all, had no significant impact on the crystallization path of the stable LS2crystal.

Recently, Soares et al . (2002) investigated the early (less than 1% crystallizedfraction) and intermediate (ca. 5–10%) stages of crystallization of hypostoichiometric,stoichiometric and hyper-stoichiometric lithium disilicate glasses by TEM/selectedarea diffraction (SAD). In all their samples, treated from 2.5 h to 120 h at 454 ◦C (Tgis close to the temperature of maximum nucleation), two different crystalline phases,LS2 and LS, were clearly indexed by SAD. These are stable phases in the hyper-stoichiometric glass, but hypostoichiometric and stoichiometric glasses may followtwo possible paths during nucleation: (i) LS2 crystals may nucleate heterogeneouslyon top of the LS crystals. This path may partly explain the failure of the CNT.(ii) There is simultaneous homogeneous nucleation of LS2 and LS, but LS disappearsduring the heat treatment. Thus, LS is a metastable phase that does not interferewith the nucleation path of the LS2.

Despite the poor statistics (only half-a-dozen crystals were detected in each TEMsample), there was no evidence that LS crystals induce the heterogeneous nucle-ation of LS2. In addition, there was plenty of glass remaining even after 120 h; thus,thermodynamic equilibrium had not yet been reached. However, LS crystals shouldbe metastable, since the calculations of B. A. Shakhmatkin and N. M. Vedishcheva(2001, personal communication) demonstrate that the thermodynamic driving forcefor LS crystallization in LS2 glass is ca. 10% less than the one for LS2 crystals; oth-erwise, the phase diagram would be incorrect!

TEM then provided clear evidence that a second phase, LS, nucleates concurrentlywith the stable phase, LS2, in stoichiometric and hypostoichiometric lithium disil-icate glasses treated at Tg ∼ 454 ◦C for long times. No phases other than di- andmetasilicates appeared. There was no evidence of heterogeneous nucleation of lithiumdisilicate on top of the lithium metasilicate. So this mechanism does not explain thediscrepancy of the CNT.





0 0.2 0.4 0.6 0.8 1.0

3

2

1

ONaSiCa

at (

%)

5

10

15

50

60

α

4

Figure 4. Glass 1Na2O.2CaO.3SiO2: compositional variation against volume fraction crystallizedat 650 ◦C. Glass (points) measured by EDS; crystals (dotted lines 1 and 2 denote Na and Ca,respectively) calculated from the parent glass composition and EDS measurements. Solid lines fitthe experimental data. Dashed lines 3 (Si) and 4 (O) correspond to the parent glass composition.(After Fokin et al . (2002b).)

(b) Compositional shifts of crystal nuclei

While metastable phases have crystallographic structures unlike those of the sta-ble phase, within certain limits, solid solutions (‘ss’s) can continuously change thecomposition of a given crystallographic system during phase transformation. Hence,generally speaking, the composition and, consequently, the properties of the criticalnuclei may deviate considerably from those of the corresponding macro phase.

Fokin et al . (1999, 2002b) recently observed that such is, indeed, the case in astoichiometric 1Na2O.2CaO.3SiO2 glass (N1C2S3) that displays homogeneous inter-nal nucleation. Phase transformation in this glass proceeds through the formationof ss crystals, which are enriched in sodium relative to the parent glass. Duringcrystallization, the composition of the ss crystals continuously approaches the stoi-chiometric one. The deviation of the nuclei composition from that of the parent glassforms sodium-poor diffusion fields in their vicinity, and nucleation terminates in anearly stage of phase transformation. This is consistent with the finding of Potapov etal . (2000), who demonstrated that the nucleation rate in glasses close to N1C2S3 isvery sensitive to the parent glass composition and decreases with decreasing sodiumoxide content.

The evolution of crystal and glass compositions and the corresponding changeof the lattice parameter are shown in figures 4 and 5. Extrapolating the change ofcrystal composition to zero time (or zero volume fraction crystallized, α = 0) suggeststhat the critical clusters are also enriched in sodium. The deviation of the nucleicomposition from the stoichiometric one diminishes the thermodynamic driving forcefor crystallization, ∆Gv, and has to magnify the kinetic barrier for nucleation. Sincethe nucleation of sodium-rich ss crystals occurs instead of the expected stoichiometric





0 0.2 0.4 0.6 0.8 1.010.44

10.46

10.48

10.50

10.52

10.54

a (Å

)

Figure 5. Parameter a of the hexagonal crystal cell against volume fractioncrystallized at 650 ◦C. (After Fokin et al . (2002b).)

crystals, the decrease in the driving force for crystallization must be compensatedfor by a decrease in surface energy in equation (2.2) for the thermodynamic barrier.

The present interpretation of the nucleation kinetics in N1C2S3 glass is consistentwith Ostwald’s rule of stages, generalized by Schmelzer et al . (2000) as

those classes of critical clusters determine the transformation process cor-responding to a minimum work of critical cluster formation—as comparedwith all other possible alternative structures and compositions which maybe formed at the given thermodynamic constraints.

Since ss formation is a common phenomenon in silicate systems, it is important tokeep in mind when analysing nucleation kinetics that the composition of the criticalnuclei and, correspondingly, their thermodynamic driving force and surface energy,may differ considerably from those of the final macrophase. In some cases, as inthe N1C2S3 glass, this fact may partly explain the failure of the CNT to predictnucleation rates.

(c) Temperature- and size-dependent nucleus–liquid surface energy

We have already mentioned that CNT fails when the nucleus–liquid surfaceenergy is treated as a size-independent (capillarity approximation) and temperature-independent property, i.e. σ(r, T ) = σ∞. This discrepancy can be avoided by calcu-lating a surface energy from experimental nucleation-rate data at each temperature,using the theoretical pre-exponential factor. The surface energy obtained by thismethod slightly increases with temperature (dσ/dt ∼ (0.06–0.16) × 10−3 J m−2 K),as observed for metals (Turnbull 1952; Miyazawa & Pound 1974) and silicate glasses(e.g. Fokin & Zanotto 2000).

Since the surface energy calculated from nucleation data refers to that for criticalnuclei whose size depends on temperature, the fitted σ(T ) dependence arises fromtwo factors, the temperature dependence of surface energy for a planar interface,dσ∞/dT , and the size dependence of surface energy. However, according to Rusanov





1 2 3

2 (Å)δ0 5 10 15 20

d ∞

/dT

(J

m−2

K−1

)σ

1.5 × 10−4

−5.0 × 10−5

5.0 × 10−5

Figure 6. Average values of dσ∞/dT against Tolman’s parameter for Li2O.2SiO2 (1, 2) andNa2O.2CaO.3SiO2 (3) crystals in parent glasses having the same compositions. The kinetic bar-rier for nucleation was estimated from time lag of nucleation (2, 3) and viscosity (1). (Reproducedwith permission from Fokin & Zanotto (2000).)

(1978) and Gutzow et al . (1985), when the molar volume of the liquid phase is higherthan that of the crystal phase, as is most typical, dσ∞/dT should be negative.

To decouple size and temperature effects, Fokin & Zanotto (2000) took into accountthe size dependence of σ using, as a first approximation, an equation derived byTolman (1949) for a liquid drop with radius R,

σ(R) =σ∞

(1 + 2δ)/R, (4.4)

where δ is Tolman’s parameter, which characterizes the (unknown) width of theinterfacial region between the coexisting phases.

The work of forming a spherical nucleus may thus be written as

W = 43πR3 ∆Gv − 4π

R3σ∞R + 2δ

. (4.5)

The critical radius can be found from the condition (∂W/∂R)R=R∗ = 0, where

R∗ =(σ∞ − 2∆Gvδ) + (σ2

∞ + 2∆Gv δσ∞)1/2

∆Gv. (4.6)

Figure 6 shows the average values of dσ∞/dT versus Tolman’s parameter. Thefits of σ∞ to experimental nucleation data for L2O.2SiO2 and Na2O.CaO.3SiO2glasses were performed for different values of δ. As δ increased, dσ∞/dT progressivelydecreased, becoming negative at δ > 2.6 × 10−10 m (LS2 glass) and 8 × 10−10 m(NC2S3 glass). Thus, reasonable values of the Tolman parameter may be chosenso that σ∞ increases with decreasing temperature, in line with the predictions ofRusanov (1978) and Gutzow et al . (1985).

Essentially all methods to determine the nucleus–liquid surface energy are basedon nucleation experiments, involving certain additional assumptions. Recently, Fokin





et al . (2000) estimated surface energy without using the nucleation-rate data directly,but instead using the dissolution of subcritical nuclei with an increase in tempera-ture. However, the values of the surface energy determined by this method, usingthe thermodynamic driving force of crystallization of the macro phase, were muchhigher than those obtained from a direct fit of nucleation-rate data to CNT. Thisdiscrepancy may be eliminated if a reduction of thermodynamic driving force forcrystallization of near-critical size clusters is allowed for. This conclusion is of sin-gular importance for the analysis of nucleation data. One last issue concerns thepossible effect of elastic stresses on nucleation (Schmelzer & Gutzow 2001). This,however, is a matter for further developments.

5. Surface nucleation in silicate glasses

As we have shown, ca. 90% of silicate glasses have Tgr > 0.60 and consequently dis-play only surface crystallization. Close to or on interfaces, both thermodynamic andkinetic barriers for nucleation are typically lower than bulk values, causing a pre-dominance of surface nucleation. However, its mechanisms and kinetics have drawnlittle attention and are not well understood so far. A detailed review of researchaimed at understanding surface nucleation was recently published by Muller et al .(2000). Our attention here focuses mainly on the properties of nucleation sites, i.e. themechanisms of surface nucleation.

(a) Surface-nucleation kinetics

Since surface nucleation occurs mostly on some active sites (e.g. tips, cracks,scratches and foreign particles), the kinetics of surface nucleation are governed bythe exhaustion of these sites due to crystal nucleation and growth. Hence, the kineticcurve N(t) generally reveals a saturation, N(t) → Ns (figure 7a). As follows fromthe drastic drop of Ns (figure 7b, curve 1) with increasing temperature, potentialnucleation sites can disappear during heat treatment, thus causing no nucleation.Different Ns(T ) dependencies (figure 7b) also illustrate that different crystal phases(here, X-phase and µ-cordierite) can be induced by different kinds of nucleation sitesand that quite a distinct nucleation kinetics occurs in each case.

The full N(t)-curve, including its saturation, is needed to calculate the nucleationrate. Owing to experimental limits (high Ns value, short saturation time due to highnucleation activity, difficulties of surface quality reproduction, etc.), such data arevery rare. Often, the combination of a small number of nucleation sites and highactivity leads to saturation before the achievement of visible sized crystals. In thiscase, Koster’s (1988) method may be applied to estimate the nucleation kinetics. Thispost-mortem method is based on the analysis of crystal size distribution, caused bythe difference in the time at which the crystals nucleated, which, in turn, leads tothe time difference for growth.

The most comprehensive surface-nucleation kinetic data reported on so far wereobtained for cordierite glass (Pannhorst 2000). A very important characteristic fea-ture of surface-nucleation kinetics is the position of its maximum, Tmax, which consid-erably exceeds the glass-transition temperature and is often close to the temperatureof maximum crystal growth rate. One should recall that, in the case of easily mea-sured homogeneous volume nucleation, Tmax is close to Tg. A compilation of available





2

3

4

521

800 850 900T (ºC)

log

(Ns/

mm

−2)

Ns

N ×

10−3

(m

m−2

)

150

100

50

0

0 500 1000 1500 2000

(a)

(b)

t (h)

Figure 7. Nucleation kinetics on a cordierite glass surface polished by cerium oxide. (a) Num-ber density of X-phase crystals against heat-treatment time at 800 ◦C. (b) Saturation level ofthe number density of (1) X-phase and (2) µ-cordierite crystals as a function of temperature.(Reproduced with permission from Filipovich et al . (1996).)

values of ∆T ≡ Tmax − Tg for different glasses and types of surfaces show variationsof 70–300 ◦C. According to the CNT, the high temperature of the surface-nucleationmaximum is caused mainly by a reduced thermodynamic barrier.

(b) Surface nucleation sites

Because surface-nucleation kinetics are governed by the presence of nucleationsites, the knowledge and understanding of their properties is a key factor in control-ling surface crystallization. Various difficulties complicate the study of the nucleationmechanism. Experimental limitations preclude micro-analytical evidence of the for-mer nucleation sites (e.g. healing of cracks, dissolution of dust particles). The exper-imental limits of the determination of N(t) often hinder the measurement of thesaturation number Ns. This quantity must be known for a comparison of nucleationdata from different systems and for the understanding of external parameters, such assurface conditions or annealing atmosphere. Several kinds of nucleation sites, differingin number density and activity, may occur simultaneously (Zanotto 1991b; Deubeneret al . 1992; Filipovich et al . 1996). Additionally, the glass surface cannot be describedusing interior properties (e.g. due to unsaturated chemical bonds, evaporation, cor-rosion). Finally, the statistical nature of surface-nucleation sites (e.g. scratches, dust)causes data scatter, which can obscure the effect of many parameters, and requires





the control of reproducible surface preparation and ambient annealing conditions.We consider here surface roughness, dust particles and annealing atmosphere.

(c) Surface roughness

Mechanical damage is known to promote surface nucleation of glass. Table 2, takenfrom Muller et al . (2000), summarizes surface-nucleation density (N) for damagedglass surfaces, revealing a strong dependence of N upon the degree of surface rough-ness. Smaller N , down to ca. 5 × 10−8 µm−2, frequently occur at freshly fracturedsurfaces. Low N are also typical for fire-polished glass surfaces. Medium N, between10−3 µm−2 and 10−6 µm−2, are evident on mechanically polished or fractured sur-faces in a standard laboratory atmosphere. The data scatter is probably due to thepresence of dust particles. Larger N of up to 10−1 µm−2 are reported for ground orglass powder surfaces.

(d) Corners, edges and tips

The nucleation mechanism related to mechanical damage has not been well under-stood to date. Tabata (1927) observed, in several silicate glasses, that surface nucle-ation is promoted by ‘sharp edges or cicatrices’. Ernsberger (1962) showed, for plate-glass, that crystals are preferably located at the cross-points (and, therefore, at theedges) of macro-cracks. Recently, it was confirmed, for cordierite, diopside, lithia–alumina–silica and float glasses, that sharp edges or tips are favoured nucleation sites(Muller et al . 1996; Schmelzer et al . 1995; Reinsch et al . 1994).

Surface crystals preferentially occur along surface edges caused by fracturing, alongthe edges of cracks caused by rapid cooling of glass ribbons or along diamond-madesurface scratches. Sometimes, double chains of crystals form along the scratch edges.Glass particles dusted on a fractured glass surface, or those caused by sample frac-turing, triggered crystallization at the contacted glass surface (Muller et al . 1992;Schmelzer et al . 1995; Fokin & Zanotto 1999).

Under clean conditions, particles having the composition of the parent glass weredetected at the centre of most of the very few crystals nucleated at the fracturedsurface of a cordierite glass. For a given glass powder sample (all particles wereexposed to identical milling conditions), the surface-nucleation density, N , increasedwith decreasing particle size because small glass particles provide a larger number ofsharp edges per surface area (Muller et al . 1995). For µ-cordierite crystals nucleatingon mechanically damaged cordierite glass surfaces, other experiments show that Nis correlated to the number density of surface edges or tips, whereas the conditionof damaging exerts a minor influence. A controlled reduction of N can be obtainedby surface smoothing. Thus, HF etching of ground surfaces prior to the thermaltreatment sharply decreases both the number of these edges and N . The value of 2land the mean linear distance between surface tips (detected by mechanical surface-profiling) are reduced quite similarly (Muller et al . 1996). Accordingly, polishing ofSiC-ground glass surfaces reduces N with polishing time, approaching invariance att > 5 min (Reinsch et al . 1999). Optical and electron micrographs show that, fort > 5 min, all the tips and edges of the ground surface are eliminated.




Internaland

surfacenucleation

insilicate

glasses605

Table 2. Surface-nucleation densities, N , of differently damaged glass surfaces

surface glass crystal N (m−2) 2l (µm)a reference

powder M2A2S5 µ-cordierite 10−3–10−1 30–3 Muller et al . (1995)

ground M2A2S5 µ-cordierite 0.2–7 ×10−2 22–4 Muller & Thamm (1990)CZA2S willemite 3 ×10−2 6 McMillan (1982)

mechanically M2A2S5 µ-cordierite 1–10 ×10−4 100–30 Muller & Thamm (1990)polished 2 ×10−4 70 Yuritsyn et al . (1994a)

CZA2S willemite 1–3 ×10−3 30–18 McMillan (1982)diopside diopside 6–10 ×10−2 1–3 Zanotto (1991a)anorthite anorthite 1–4 ×10−3 30–7 Wittman & Zanotto (1999)NC3S6 devitrite 1 ×10−1 3 Zanotto (1991b)

non-stoich.devitrite

β-wollastonite 5–10 ×10−5 140–100 Zanotto (1991b)

tridymite 5–10 ×10−5 140–100 Zanotto (1991b)devitrite 1.5–7 ×10−4 80–38 Zanotto (1991b)

mic. slide devitrite 1–6 ×10−4 100–40 Zanotto (1991b)

as-received float devitrite 3–30 ×10−4 58–18 Zanotto (1991b)diopside 5 ×10−5 140 Zanotto (1991b)tridymite 1 ×10−5 316 Zanotto (1991b)

mic. slide devitrite 8–30 ×10−4 35–18 Zanotto (1991b)

fractured M2A2S5 µ-cordierite 10−3–10−8 30–5000 Reinsch et al . (1994)10−5–10−6 300–1000 Yuritsin et al . (1994a)

fire-polished mic. slide devitrite ‘0’ ‘∞’ Zanotto (1991b)M2A2S5 µ-cordierite ‘lowest’ — Yamane & Park (1990)

aThe average distances between crystals, 2l ∼ N−1/2, taken from Muller (1997), are presented for the sake of illustration.

Phil.

Trans.

R.Soc.

Lond.

A(2003)

on April 27, 2013

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(e) The effect of elastic stresses

A simple nucleating mechanism of convex edges or tips was suggested by Schmelzeret al . (1993), based on the viscous elastic nature of glass-forming melts. These authorshave shown that elastic stresses caused by the growing crystalline clusters, whose den-sity deviates from that of liquid, can considerably hinder nucleation. Calculations arebased on equation (5.1), which describes the elastic energy required for the additionof one monomer of volume, vC, in the interior of the glass, Φ0:

Φ0 =(

E

9(1 − γ)

)δ2 vC. (5.1)

The term E/9(1 − γ) represents the effect of the elastic properties (E is Young’smodulus, γ is Poisson’s number) and δ = (ρC − ρG)/ρG is the relative density differ-ence between the melt and the crystal. Neglecting stress relaxation, all elastic energyis accumulated to the cluster size α according to Φ(α) = αΦ0. Hence, the molar freeenergy change of crystallization is ∆µ(ε) = ∆µ − NAΦ0 and the corrected volumenucleation rate, I(ε), is given by (Schmelzer et al . 1993)

I(ε) = I exp{

−([(

∆µ

∆µ(ε)

)2

− 1]W

)(1

kT

)}. (5.2)

As Φ0 decreases with the ratio between the surface distance and the cluster radius,nucleation is easier close to the surface than in the interior, due to the less-intensivestress field in its vicinity (Schmelzer et al . 1995). This effect of the elastic stresses isgreater for sharp convex surface curvatures such as surface edges or tips.

This hypothesis offers a good explanation of all of the experimental results men-tioned. Other observations support this assumption more directly.

(i) Large cracks caused by quenching of cordierite glass ribbons were completelyhealed during a thermal treatment (Schmelzer et al . 1995; Muller et al . 1992).The former tips did not cause nucleation. This effect confirms the model pro-posed by Schmelzer et al . (1993) because the elastic response of the glass matrixat the crack tips is nearly the same as for any other point in the interior.

(ii) In contrast to the sharp edges of fractured glass surfaces, smooth or wavysurface curvature causes less-intensive surface nucleation (Muller et al . 1996).Schmelzer et al . (1995) have shown that, correspondingly, the energy of elasticdeformation depends on the angle of convex surface tips or edges.

(iii) The nucleation activity observed along former edges of cracks (Schmelzer etal . 1995), scratches or Vickers indentations is greater for the largest values ofδ. The nucleation on two glass surfaces after Vickers indentation were studiedby Reinsch et al . (1999). Less numerous cristobalite crystals grow at the edgesof the indentation on the float glass surface (δ ≈ 10%) than diopside crystalsgrowing at the indentation edges on the diopside glass surface (δ ≈ 15%). In thecase of µ-cordierite growing on the cordierite glass surface (δ ≈ 2%), indentingdid not cause additional surface nucleation.





10−8

10−7

10−6

10−5

10−4

10−3

10−2

sample no.

N (

µm−2

)

fire clay

dense corundum

quartz glass

Figure 8. Nucleation density of µ-cordierite at cordierite glass surfaces, fractured and annealedunder different conditions. ◦ denotes fire-clay furnace, air; �, corundum tube furnace, air;�, corundum tube furnace, dust protected; �, corundum tube furnace, dry air; �, quartz-glasstube furnace, air; ♦, quartz-glass tube furnace, vacuum. (Reproduced with permission fromMuller (1997).)

(f ) Solid particles

In the case of smooth glass surfaces (as-received, polished or fractured), N isstrongly affected by the presence of solid particles. As early as 1739, Reamur trig-gered surface nucleation of glassware by contamination with foreign solid particles.Particles of pre-crystallized glass were used to increase crystallization of powdered(Rabinovich 1979) or solid glasses (Ding et al . 1994). Figure 8 shows that there isa strong scatter in N due to the random occurrence of dust for different furnacematerials, dust protection, or vacuum. A scatter of seven orders of magnitude in Noccurred in fire-clay furnaces, while this scatter was smaller using a gas-dense qual-ity corundum furnace and much smaller when silica glass was used as the furnacematerial.

The impact of solid foreign particles is not merely dependent on their number butis also determined by their nucleating activity. For example, microscopic observationsshow that numerous solid particles resting on glass surfaces during the crystalliza-tion treatment did not cause nucleation. The nucleating efficiency of solid metal sub-strates, φ, has been studied for heterogeneous volume nucleation (Gutzow 1980a, b).A comprehensive model is given in Dobreva & Gutzow (1993), connecting φ withthe cohesion forces in the substrate and lattice misfit. The activity of foreign sur-face particles should follow that general concept. However, additional effects of theambient atmosphere must be expected.





Oxide dust particles, which are in most cases thermally stable, were found to pro-vide active nucleating substrates for µ-cordierite. Thus, ZrO2, known for its thermalstability, was found to be the most efficient surface-nucleation catalyser. Nucleatingactivity was affected by its crystal morphology, indicating epitaxial phenomena. Sim-ilarly, the nucleation activity of dusted µ-cordierite, α-cordierite, quartz and fire-claypowders (the latter is a mixture of corundum, α-cordierite, mullite and quartz) maybe caused by the lattice misfit between these powdered crystals and cordierite crys-tal. Correspondingly, larger N are evident for agate (quartz) ball-milled cordieriteglass powders (Muller et al . 1995) because of the slight lattice misfit with cordierite.We note, however, that the stability (and nucleation activity) of oxide particles canbe limited by dissolution into liquid cordierite. Thus, dusting with TiO2 and WO3reduces N , as is the case of dusting with unstable compounds such as W and WC.

Non-oxide dust particles, in most cases less thermally stable, did not increase Nexcept in a vacuum (p < 10−4 mbar), where better oxidation stability is guaranteed.The inertness of SiC and Si3N4 is probably due to the formation of an amorphousSiO2 surface layer. The increased nucleation activity of these particles under vac-uum supports this explanation. In the case of the larger concentrations of unstableparticles (W, WC, B4C or less-stable oxides such as WO3 and B2O3), a rippledglass surface and the occurrence of foreign crystalline phases were observed, indicat-ing chemical interaction with the liquid (dissolution). In the worst case, the lattereffect can essentially change the composition of the glass surface layer. Thus, inten-sive dusting with B4C triggered crystallization of needle-like foreign crystals andcaused a rippled surface morphology. These effects can be explained assuming anenrichment of B2O3 in the near-surface area due to oxidation of B4C and to thewell-known decrease in surface tension by the addition of B2O3 (Scholze 1977). Thisobservation is confirmed by the chemical inertness of B4C under vacuum. W and WCshow a similar inertness. Their oxidation is most probably followed by dissolution ofthe oxide into the melt, which modifies the surface morphology. A rippled surfaceand the total absence of µ-cordierite were also caused by dusting with K2SO4. X-rayanalyses of a heat-treated mixture of K2SO4 and cordierite glass powder showedleucite as the crystal phase (Muller et al . 2000), indicating dissolution.

In summary, the nucleating activity of solid sites is predominantly limited bytheir chemical stability. Chemical reactions with the glass and with the ambientatmosphere can decrease their nucleating efficiency. Accordingly, all thermally stablefurnace-tube materials (e.g. fire clay or alumina) turned out, unfortunately, to pro-vide the most active nucleation seeds. The same holds true for oxide-milling materials(Muller et al . 1995).

(g) Ambient atmosphere

A strong effect of ambient gases (e.g. water vapour) on surface crystallization isfrequently stressed in the older literature. The effect, however, is more importanton the crystal’s growth than on the number of nucleation sites (Muller et al . 2000).Experiments under glove-box conditions showed no effect of ambient humidity on thesurface-nucleation density of µ-cordierite, while the crystal growth rate substantiallyincreased with increasing humidity (Muller et al . 1996). It should also be notedthat very low N , down to N ≈ 10−8 µm−2, occasionally occur for freshly fracturedsurfaces and that such small N are typical for fire-polished surfaces, even in airatmosphere.





6. Overall conclusions

The internal homogeneous nucleation rates of silicate glasses sharply decrease and theinduction times increase with the Tg/TL ratio. Only systems that have Tg/TL < 0.58display measurable internal nucleation rates on a laboratory time-scale and can beused to analyse nucleation theories.

Using the capillarity approximation, numerous tests of the CNT have demon-strated that the theory fails to quantitatively describe the nucleation rates in glasses.Possible explanations for this failure, such as the use of induction time instead ofviscosity to account for molecular transport, the precipitation of metastable phases,compositional shifts of the crystal nuclei, and the possible variation of the surfaceenergy with nucleus size and temperature, were tested and discussed. Of these, thelast two appear to be the most relevant. The possible effect of elastic strain is amatter for future studies.

Surface nucleation depends strongly on surface quality: tips, cracks and scratches,elastic stresses, foreign particles and surrounding atmosphere. Despite the fact thatthe mechanisms of surface nucleation are still not fully understood, some of the keyfactors are now known, thus allowing for control of the surface-nucleation density andsintering with concurrent crystallization. This improved knowledge has been used inthe manufacture of sintered glass ceramics.

The authors thank several collaborators from over the past five years: M. C. Weinberg, J. W.Schmelzer, P. C. Soares Jr, M. O. Prado, E. B. Ferreira, N. S. Yuritsyn and O. V. Potapov.Stimulating discussions on surface crystallization with R. Muller, G. Volksch, W. Pannhorst, W.Holand (members of Technical Committee 7 of the ICG) are deeply appreciated. We also thankP. F. James, L. Granasy and K. Kelton for reviewing the manuscript. E.D.Z. acknowledges thefunding provided by CNPq, Cyted, PRONEX and FAPESP (Brazil) in the period 1998–2002,when most of the studies were performed.

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Discussion

A. Boccaccini (Department of Materials, Imperial College of Science, Technologyand Medicine, London, UK ). You presented a very interesting overview of recentstudies on internal and surface nucleation in glass. The experimental results youshow are obtained from glass samples having large dimensions in comparison withthe scale of crystals nucleated. It would be interesting to extend your findings to con-sider the behaviour of nanoparticles of glass having a very high surface-area/volumeratio. Is it possible to separate the phenomenon of internal and surface nucleation innanoparticles experimentally?

E. D. Zanotto. In order to separate internal and surface nucleation in glassnanoparticles one might try a few experiments. We have shown in this paper thatinternal (homogeneous) nucleation is experimentally observed in bulk glasses ifTg/TL < 0.58 (temperatures in kelvin). At such high undercoolings the expectedcritical nucleus radius is only ca. 1 nm, thus we believe that internal nucleation couldoccur in particles 50–100 nm in diameter. At such small scale, however, the problemis how to detect some crystals after they have grown to a few tens of nanome-tres, internally or on the particles surface. One could try to use TEM methods fordirect observation, or non-isothermal techniques, such as differential thermal analy-sis or differential scanning calorimetry, for indirect examination. In the latter case,





comparison of the height and temperature of the crystallization peaks obtained foridentical experimental conditions, for nanoparticles and millimetric particles of thesame glass, could give important clues. For instance, if internal crystallization pre-dominates, the traces of the two widely differing particle sizes would be quite similarand vice versa.




E R R A T U M

Recent studies of internal and surface nucleation in silicate glasses

By Edgar D. Zanotto and Vladimir M. Fokin

Proc. R. Soc. Lond. A361, 591–613 (2003)

The following is the correct form of the first sentence of § 3 a.

Figure 1a (Fokin et al . 2002a) summarizes the data for the maximum nucle-ation rates, Imax ≡ I(Tmax), versus Tgr for 51 glasses of stoichiometric and non-stoichiometric compositions belonging to eight silicate systems clearly showing thetrend that Imax decreases with increasing Tgr.

The following is the correct form of table 1.

Table 1. Ratio of experimental and theoretical pre-exponential andsurface energy calculated by the CNT for different glasses

(Values for ∆Cp = 0 calculated using Turnbull’s approximation.)

∆Cp = 0 Cp = f(T )︷︸︸︷︷︸︸︷glass log

(Kexp

τ

Ktheoτ

)σ (J m−2) log

(Kexp

τ

Ktheoτ

)σ (J m−2)

Li2O–2SiO2 15 0.19 19 0.20Na2O–2CaO–3SiO2 18 0.17 72 0.192Na2O–CaO–3SiO2 27 0.15 139 0.17

The following is the correct form of equation (4.4).

σ(R) =σ∞

1 + 2δ/R. (4.4)

The correct units for N in table 2 are µm−2.

Phil. Trans. R. Soc. Lond. A (2003) 361, 30093009

c© 2003 The Royal Society



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