In the last 40 years, crystal nucleation in silicate glasses has been addressed by counting crystal number densities in the volume of glass specimens subjected to double-stage heat treatments (Fokin et al., 2006). The techniques bears the advantage of direct determination of the crystal nucleation rate from calculating the first time derivative of the number density curve but has the drawback that it is limited to a narrow temperature range near the glass transformation temperature (T_g ≤ T ≤ 1.2 T_g) where driving forces are high and crystal nucleation is occurring frequently in the volume. Another drawback is the dissolution of nuclei that are critical at the first stage but subcritical at the second stage. In the case of surface crystallization, the accessible temperature range becomes broader but the time period of crystal nucleation is mostly limited due to fast saturation of active sites. A short nucleation period merely allows for an indirect determination of the crystal nucleation rate. Thus, birth- (from crystal size distribution) (Müller et al., 2000) and impingement-times (Krüger and Deubener, 2015a) of crystals at the surface have been analyzed to deduce their rates of nucleation. In each of these methods (and also for analyzing high-temperature X-ray diffraction data (Dressler et al., 2011, 2014), an ensemble of crystals is analyzed in order to derive nucleation parameters. The criterion applies for large undercoolings but becomes ineffective at higher temperatures where crystal grow rates are generally high and the entire liquid volume is shortly consumed by the first nucleated crystal. In consequence, nucleation data for temperatures T > 1.2 T_g are lacking for both binary/ternary model glasses and for technical multi-component glass systems. The occurrence of a substantial gap in the reported nucleation data is surprising and unexpected if one recalls the large number of glasses produced in industrial practice by forming during quench. Furthermore, the analysis of a crystal ensemble does not allow to prove randomness in the nucleation events, a necessary condition in the framework of CNT. By contrast, the observation of single crystal events helps in discriminating partition of long-range diffusion fluxes (Kelton, 2000).

In the light of the above, this study aims at establishing nucleation kinetics by its time average of repeated single crystal nucleation events rather than using the ensemble average of crystals of double-stage heat treatments. The so-called stochastic approach to crystal nucleation has been introduced for undercooled low viscous (mPa s) liquids, such as metals and hydrous suspensions (Toschev and Markov, 1967; Toschev and Gutzow, 1972; Toschev et al., 1972; Toschev, 1973; Morton et al., 1994; Barlow and Haymet, 1995; Uttormark et al., 1997; Heneghan et al., 2001, 2002; Heneghan and Haymet, 2002; Wilson et al., 2005; Wilde et al., 2006, 2009; Yang et al., 2009, 2011, 2013). Our results show that it is applicable to high viscous (kPa s) silicate melts and it results in heterogeneous (HET) crystal nucleation rates from which TTT representations and critical cooling rates can be derived.

The formation of supercritical nuclei is a stochastic process. Nucleation events occur randomly and independent from each other. Already in the 1960s, Toschev and Markov (1967) showed that the electrolytic deposition of cadmium on platinum single crystal electrodes follows a Poisson distribution. Thereby, both the number of nuclei in a given time interval and the time necessary to form a certain number of nuclei are linked to probability distributions (Toschev and Gutzow, 1972; Toschev et al., 1972; Toschev, 1973).

Later probability distribution functions in the formation of the first supercritical nucleus have been studied for other liquid metals, such as niobium and zirconium (Morton et al., 1994), aluminum (Uttormark et al., 1997), gold and copper (Wilde et al., 2006, 2009), and tin (Yang et al., 2009, 2011, 2013) but also in water (pure and seeded) (Barlow and Haymet, 1995; Heneghan et al., 2001, 2002; Heneghan and Haymet, 2002; Wilson et al., 2005) and in gas hydrates (Maeda et al., 2011, 2012). In all these experiments, a single sample was repeatedly undercooled from above the melting temperature, while either the time (at an isothermal hold) or the temperature (at an isochronal cooling) of crystallization was detected. These experiments provided primary information on the average lag time and the mean undercooling, from which the crystal nucleation rate and the critical cooling rate can be derived. Secondary information include the activity of seeds and the purification of the liquid upon thermal cycling. Especially the ALTA (“automated lag time apparatus”) concept developed for studying the freezing behavior of water (Barlow and Haymet, 1995) is acknowledged since it has been designed to permit time-saving through a feed-back control that heats the sample immediately upon freezing.

In the past years, non-adiabatic fast scanning calorimetry has been developed to in situ measure the response of single metallic drops to temperature changes in a large range of cooling rate spanning four orders of magnitude (Yang et al., 2009, 2011, 2013). In these experiments, a small drop (volume ≈ 10⁻⁶ mm³) is repeatedly heated and cooled on a thin film sensor with rates up to 10⁴ K s⁻¹. Unfortunately, current technology of the so-called flash- or chip-calorimeters is limited to a maximum sensor temperature of about 700 K, which is much lower than the liquidus temperature of lithium disilicate glass [1306 K (Kracek, 1930)].

Glasses in the binary Li₂O–SiO₂ system, such as lithium disilicate (Li₂Si₂O₅) have been the object of extensive research (see, for example, Tomozawa, 1972, 1973; Matusita and Tashiro, 1973; James, 1974, 1985; Fokin et al., 1981; Weinberg and Neilson, 1985; Barker et al., 1988; Deubener et al., 1993; Ota et al., 1997; Burgner and Weinberg, 2001; Nascimento et al., 2011) since their crystal nucleation and growth rates are relatively low at high undercoolings (T < 1.2 T_g) and thus conveniently measurable from crystal number densities and crystal sizes of optical microscopy images (James, 1974, 1989; Rowlands and James, 1979). Furthermore, detailed thermodynamic data are available for this system (Takahashi and Yoshio, 1973) which allow to test the validity of basic nucleation models from comparisons of predicted rates with measured data (Neilson and Weinberg, 1979; Rowlands and James, 1979; Zanotto and James, 1985; Weinberg and Zanotto, 1989; Fokin et al., 2010a).

Li₂Si₂O₅ crystals nucleate homogeneously in the volume (James et al., 1978; James, 1985; Zanotto and James, 1985; Zanotto, 1987; Barker et al., 1988) disregarding the controversy of metastable phase formation prior to the crystallization of stable Li₂Si₂O₅ crystals (Hench et al., 1971; Tomozawa, 1972; Barker et al., 1988; Deubener et al., 1993; Zanotto, 1997; Burgner et al., 1999; Iqbal et al., 1999) but also heterogeneously at suspended particles, such as platinum (Cronin and Pye, 1986; Narayan et al., 1996; Ray and Day, 1996; Ray et al., 1996; Mishima et al., 2006; Ranasinghe et al., 2007) and at the surfaces with the surrounding gas atmosphere (James, 1989; Ray and Day, 1996; Ray et al., 1996; Ranasinghe et al., 2007; Fokin et al., 2010b). Ranasinghe et al. (2007) performed heat treatments on pure and platinum doped Li₂Si₂O₅ glasses and found that the critical cooling rate increases and the activation energy of crystallization decreases when Li₂Si₂O₅ glass is heated in a container as compared to containerless processing (levitation) and that in both cases platinum promotes the crystallization of lithium disilicate glasses. However, the temperature dependence of preferred HET crystal nucleation leading to observable “coast-island” microstructures [see Fig. 1C in Krüger and Deubener (2015a) and Fig. 14 in Fokin et al. (2010b)] is not well understood with respect to isothermal heat treatments and for ramping at constant rates from above melting temperature and below glass transformation temperature, respectively.

The present paper reports on novel experiments of HET crystal nucleation, which have been performed by the stochastic approach under isothermal conditions at small undercooling (1.6 T_g). In particular, the time elapsed before the formation of the first supercritical nucleus is detected from repeated measurements in a PtRh-crucible and the HET steady-state nucleation rate is derived from the first-order kinetics of the probability distribution function. Using previous data (Krüger and Deubener, 2014, 2015a), a TTT diagram is produced from which the HET and homogeneous (HOM) crystallization can be clearly distinguished. The TTT curves of lithium disilicate help in understanding the development of coast-island microstructures observed in practice during cooling from above the liquidus temperature. We assume that the here established method is able to clarify also the crystallization mechanisms in other glass-forming systems.

A glass of nominal composition Li₂Si₂O₅ was prepared by melting a batch of the raw materials SiO₂ and Li₂CO₃ (both of analytical grade). The batch was mixed and melted three times in a platinum crucible at 1673 K for 1 h under ambient conditions. The melt was subsequently quenched between two steel plates, which resulted in cooling rates of ~150–250 K min⁻¹. The glass was analyzed by X-ray fluorescence (S4 Pioneer; Bruker AXS, Karlsruhe, Germany) while the Li₂O content was determined by atomic emission spectroscopy (ICP-OES Vista MPX; Varian). The analyzed chemical composition was 77.4 SiO₂, 21.0 Li₂O, and 1.6 impurities (all in percent by weight), which slightly deviates from the nominal stoichiometry of Li₂Si₂O₅.

A chip (= 21.4 mg) of the glass was inserted in a lidded PtRh-crucible and subjected to high-temperature differential scanning calorimetry (DSC 404 F3 Pegasus; Netzsch, Selb, Germany) under nitrogen atmosphere. A scheme of the operating mode of the DSC with the characteristic repeatedly performed heating-dwelling–cooling-dwelling cycles (red dashed-dotted line) is shown in Figure 1. In particular, the glass chip was heated at 0.167 K s⁻¹ to a temperature of 1353 K, which is 47 K above T_m [T_m = 1306 K (Kracek, 1930)], dwelled for 300 s to ensure fully melting of the sample, cooled by ~1.25 K s⁻¹ to the dwell temperature of 1173 K (undercooling ΔT = 133 K) and dwelled for 3360 s [the dwell time was set to ca. 1 h, a reasonable time in accordance with the results of Krüger and Deubener (2014)]. After dwelling the sample was immediately heated again to start the next run. In total 284 (= N₀) runs were performed. In each run, the time elapsed before the first supercritical nucleus is formed (δ_N and δ_N+1) was calculated by subtracting the beginning of the isothermal hold (t_b(N) and t_b(N+1)) from the crystallization onset time (t_N and t_N+1) of the isothermal exothermal DSC signal (blue solid line in Figure 1). We assume that the recorded crystallization onset times follow the same distribution as δ_N and δ_N+1 since the time for crystal growth is negligible at this high temperature.

The basic result of our experiment is a histogram of the time of the formation of the first supercritical nucleus for each run δ_N in dependence of the run number N (Figure 2). This histogram provides the following information: first, 26 runs were crystallized before the dwell temperature at ΔT = 133 K was reached (δ_N < 0 s, blue dashed-dotted line of Figure 2) and 21 runs remained uncrystallized due to the programmed maximum dwell time of 3360 s (δ_N > 3360 s, red dashed-dotted line of Figure 2). Second, the histogram shows a random distribution of δ_N which indicates that the preferred HET nucleation site is continuously active during the entire experiment. A fast Fourier transformation (not shown) proved absence of any diurnal or other periodic behavior. One may assume that the crystal nucleates at the three-phase boundary between the silicate melt, the crucible wall, and the nitrogen atmosphere, but this is not known. Furthermore, the silhouette of Figure 2 (black outline of the time elapsed before the first supercritical nucleus is formed) seems to be horizontal, which indicates that compositional changes of the melt due to possible nitrogen dissolution and lithium evaporation at the surface are negligible. Furthermore, no shift in the integrated signal values was observed (not shown) which confirms the absence of any traceable mass loss during the entire period of the experiment.

In order to determine the HET steady-state nucleation rate I_{0 (HET)} from this isothermal experiment, the relative frequencies of the number of runs remained unfrozen after a certain time are calculated. The procedure is similar to the calculation of the survival curve in Barlow and Haymet (1995) and Krüger and Deubener (2014). The δ_N data of Figure 2 are first arranged according to their length. At the shortest δ_N (= 0 s), there are 258 runs uncrystallized (26 runs crystallized during cooling to the dwell temperature). In all other runs, crystallization occurred at larger δ_N. For δ_N = 0, the uncrystallized fraction is N(δ_N = 0)/N₀ = 258/284 = 0.908. For the second shortest δ_N, the fraction uncrystallized is 257/284 = 0.905 and so on. The last N(δ_N)/N₀ fraction calculated in that way is 21/284 = 0.074 at δ_N = 3342 s since 21 runs were still uncrystallized at the end of the programmed isothermal hold. This fraction is set to 0. The calculated fractions N(δ_N)/N₀ are plotted as a function of δ_N (Figure 3). The distribution curve shows a simple exponential decay that illustrates an increase in the nucleation probability with longer δ_N. The nucleation probability can be described by a simple first-order rate equation (1)N(δN)N0=Aexp(−I0(HET)δN) and its logarithmic form (2)lnN(δN)N0=ln(A)−I0(HET)δN where the constant A accounts for the initially crystallized fraction at δ_N = 0. Equation 2 is shown by the insert of Figure 3. The apparently linear dependence of the data on δ_N confirmed first-order reaction kinetics, and fitting the data (green solid line in Figure 3) results in a HET crystal nucleation rate of I_{0 (HET)} = (9.19 ± 0.04) × 10⁻⁴ s⁻¹. We note that exclusion of the 26 runs crystallized before the isothermal hold is reached results in the same value of I_{0 (HET)} but the parameter A of Eq. 1 will become unity. However, we think that the consideration of all data points in the probability distribution as shown in Figure 3 is the most appropriate linking to the stochastic experiment.

The most striking feature of the results is that the found exponential decay of Figure 3 is similar to those reported for seeded water (Barlow and Haymet, 1995; Heneghan et al., 2001, 2002). This underlines the fundamental character of crystal nucleation as a stochastic process obeying first-order reaction kinetics. However, one should note that due to the experimental constraints, such as the limited cooling rate to reach the dwell temperature and the restricted dwell time of 3360 s, 47 of the 248 runs are not captured by the probability distribution. Thus, a second crystal nucleation mechanism might be active with characteristic time scales t_N < t_{b (N)} and t_N > δ_N (3360 s). This assumption is supported by the fact that the long time fractions of Figure 3 show a negative deviation from the proposed linearity. On the other hand, the negative deviation can also arise from an insufficient number of cooling runs preventing good statistics and, therefore, underestimating N(δ_N)/N₀ at long times.

In order to exclude memory effects between adjacent runs of the histogram (Figure 2), a Pearson correlation test (Barlow and Haymet, 1995; Heneghan et al., 2001) was performed (not shown). A Pearson correlation coefficient r = 0.01 was obtained, which confirms that the probability of having correlation between adjacent runs is close to 0. The exclusion of memory effects is illustrated by different onset times of adjacent runs (Figure 4). Herein, the runs 6 and 7 exhibit significantly different crystallization onset times. Moreover, the individual evolution of the heat release of each run of Figure 4 reveals growth of one or only few crystals (Krüger and Deubener, 2014). A larger ensemble of growing crystals would lead to an averaged signal shape (Gaussian) as it is typical for heating lithium disilicate glasses in DTA/DSC experiments from temperatures below glass transformation (Krüger et al., 2013).

Additionally to the visual inspection of the signal’s shape the broadness is analyzed. As an example, the signal of run 249 exhibits a broadness of 600 s (Figure 4). Assuming only one growing crystal and a crystal size that corresponds to the circumference of the crucible (the final geometry of the crystal after 284 runs is ring-shaped due to wetting of the crucible walls by the Li₂Si₂O₅ melt) leads to a crystal growth rate of 3.4 × 10⁻⁵ m s⁻¹. The approximated growth rate from the signal’s length is very close to the reported value (5.1 × 10⁻⁵ m s⁻¹) of Burgner and Weinberg (2001) for the same temperature that supports the assumption that the entire liquid volume is consumed by only one or few crystals. Few crystals would result in a somewhat smaller broadness (see, e.g., run 126 of Figure 4). A low number of crystals growing simultaneously at the surface are further supported by the observations of Ranasinghe et al. (2007). They found that only up to two crystals were simultaneously growing at the surface of levitated Li₂Si₂O₅ drops. Thus, in accordance with our DSC protocol (Figure 1), assigning the onset time of the crystallization peak to the time of the formation of the first supercritical nucleus seems to be reasonable.

In order to link I_{0 (HET)} to our previous stochastic experiment (Krüger and Deubener, 2014), which was run under constant cooling conditions (cooling rate = 0.083 K s⁻¹), nucleation rate data have to be derived from the undercooling-dependent N(ΔT)/N₀ vs. ΔT distribution curve (Figure 5). In this case, the probability curve with the fractions of runs remaining unfrozen N(ΔT)/N₀ at a certain undercooling is deconvoluted considering the temperature-dependent I_{0 (HET)}: (3)I0(HET)=C1ηexp−C2TΔT2 where C₁ and C₂ are temperature-independent constants and η is the viscosity of the Li₂Si₂O₅ melt. In order to determine C₁ and C₂ from the isochronal experiment, Eq. 3 is inserted in Eq. 1 and the parameter A is set to unity. By doing so, we assume that the crystal nucleation process is independent on the cooling rate. Rearranging the equation and taking the logarithm results in: (4)lnηδNlnN0N(ΔT)=ln(C1)−C2TΔT2. Figure 5

Probability distribution of the fraction of runs uncrystallized (N(ΔT) N₀⁻¹) in dependence of the undercooling taken from Krüger and Deubener (2014) and deconvoluted to determine the temperature-dependent nucleation rate curve (insert). The green solid line of the insert shows the best fit of Eq. 4 through the data points using the adjustable parameters C₁ = 391 Pa and C₂ = 2 × 10⁸ K³. Note the renaming τ_N ≡ δ_N.

The most striking outcome of the present study is the “double-nose” TTT diagram of lithium disilicate due to a decoupling of the nucleation mechanisms of HOM volume nucleation of crystals at large undercoolings (0.53–0.61 T_m) and HET surface nucleation of crystals at smaller undercoolings (0.62–0.92 T_m). This behavior is the origin for the evolution of coast-island microstructures of lithium disilicate in practice. Due to the stochastic experiments at small undercoolings HET nucleation is measurable and the corresponding rate with a maximum at 1071 K is quantified for the first time. Based on the calculated TTT diagram the critical cooling rate is about 73 K s⁻¹.

The results emphasize that exploring the time average of nucleation events by thermal cycling is a novel and highly attractive route to study HET crystal nucleation in silicate glasses. In particular, they point out that the rare data on HET nucleation rates and critical cooling rates that are based on one-time cooling experiments have to be reconsidered to include statistical certainty.

SK performed experiments. SK and JD prepared manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.