Lag time to crystal nucleation of supercooled lithium disilicate melts: A test of the classical nucleation theory

Journal of Non-Crystalline Solids 426 (2015) 1–6

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Lag time to crystal nucleation of supercooled lithium disilicate melts: A test of the classical nucleation theory S. Krüger, J. Deubener ⁎ Institute of Non-Metallic Materials, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany

a r t i c l e

i n f o

Article history: Received 26 March 2015 Received in revised form 15 June 2015 Accepted 18 June 2015 Available online xxxx Keywords: Heterogeneous nucleation; Lag time; Lithium disilicate glass; Classical nucleation theory

a b s t r a c t A noticeable discrepancy between the observed lag times to crystal nucleation of continuous cooling experiments and the predictions of the classical nucleation theory (CNT) is evident. In particular, in these experiments nucleation at free surfaces and in contact with noble metal is delayed by many orders of magnitude with respect to CNT even if one neglects the heterogeneous character of the phase transition in the kinetic analysis. In this paper it is proposed that delayed nucleation is a consequence of a smaller Gibbs free energy of the evolving critical nucleus as compared to the growing macrocrystal. Considering lag times in scales of the reduced melting temperature T/Tm from 0.5 to 0.92 a difference in the free energy of crystallization of 6.7 kJ mol−1 and a melting point depression of 146 K is approximated. The results are in line with the description of heterogeneous systems as introduced by the generalized Gibbs approach but can be also a hint to metastable polymorphs at the nanoscale. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Liquid-to-crystal nucleation of supercooled lithium disilicate glass melt has been a subject of extensive experiments [1–5] (see also review in [6]), theory [7,8] and simulations [9–11]. In most of the experimental reports nucleation was studied upon annealing at temperatures above the glass transition of a prior glass melt that was quenched. In particular, double-stage heat-treatments were carried out to allow nucleation during the first hold at the nucleation temperature Tn and to grow these nuclei to observable crystal sizes during the second hold at the development temperature Td, with Td N Tn. We note that the size of the crystals to be developed at Td depends on the observation technique used. Frequently, optical microscopy is used, which requires crystals at the micrometer scale. In order to determine the number densities of nuclei from counting crystals of much larger size, negligible small growth rates and nucleation rates are assumed at Tn and Td, respectively (no overlap of the Tammann rate curves I0(T) and U0(T) at Tn and Td, respectively with I0(T) = temperature-dependent stationary nucleation rate and U0(T) = temperature-dependent crystal growth rate). Typical crystal number density curves obtained from the double-stage heattreatment technique show an initial time period, where no crystals are observed. Then a non-stationary period, where the number density increases non-linearly with time, followed finally by a stationary nucleation behavior (linear increase of the crystal number density with time) [12]. The non-linear period has been related to an experimental induction time tind as determined by the intercept of the stationary nucleation part with the time axis. Using classical nucleation theory ⁎ Corresponding author. E-mail address: [email protected] (J. Deubener).

http://dx.doi.org/10.1016/j.jnoncrysol.2015.06.023 0022-3093/© 2015 Elsevier B.V. All rights reserved.

Collins and Kashchiev [13,14] proposed that the ratio of inherent lag time τ and tind is constant. The inherent lag time τ equals the time needed for the cluster to diffuse through the near-critical size space of (Gn⁎ − Gn) ≤ kT, where Gn⁎ and Gn are the cluster energies of the critical and sub/super-critical cluster, respectively, and k is the Boltzmann constant [15]. This process results in the transient character of the crystal number density curve. Later Shneidman [16–18] derived tind as the sum of the inherent lag time τ to nucleation and the time to grow nuclei to observable sizes. In this approach the lag time is [17] τ¼

Nt ind ; I 0 E1 ½expð−γ Þ

ð1Þ

where Ntind is the crystal number density at tind, E1 is the first exponential integral and γ is the Euler constant. Additionally, Slezov and Schmelzer [19] introduced τ from a different solution to the transient nucleation problem. In literature it was reported that the thermal history of glasses affects the induction time of nucleation [20–23]. Since the growth rate U0 at T ≥ Tn in lithium disilicate is positive [24] (= small overlap of Tammann curves) the subsequent heating at a finite rate between Tn and Td will change the non-stationary crystal number density curve (undercounting). In particular, a survival size of nuclei, which deviates from those at Td, will alter the deduced temperature dependences of the critical size and of the interfacial energy [25]. On the other hand, in cases of considerable overlap of the Tammann curves, the number density of crystals can be measured directly after single-stage treatments at Tn. Then tind and the obtained crystal number density curve will be shifted to larger times by the time period, which is needed to grow nuclei at Tn to observable size, since U0(Tn) b U0(Td) [6,26,27].

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In general for any supercooling temperature ΔT (ΔT = Tm − Tn with Tm = melting temperature (1306 K [28]) and Tn b Tm) the classical nucleation theory CNT predicts an inherent lag time τ to nucleation as it is the time to “collect” enough energy via fluctuations to pass the nucleation region (near-critical size space). According to [13,14], the lag time for critical cluster formation (spherical shape) at Tn can be written as τ¼z

kT n σ cl ΔGv 2 a2 Dτ

;

ð2Þ

where z is a numerical factor (=16 / π [13,14], =80 / 3 [19]), σcl is the specific free energy of the critical nucleus–liquid interface, a is the mean size of a “structural unit”, ΔGv is the thermodynamic driving force for crystallization, i.e. the difference between the free energies of the liquid and crystal per unit volume of the crystal and Dτ is the diffusivity of the moving unit through the critical nucleus–liquid interface. Usually the latter process is described by thermally activated diffusion via: Dτ ¼


 a2 kT n ΔGD0 ; exp − h kT n

ð3Þ

where h is the Planck constant, ΔGD′ is the activation energy for the transfer of a structural unit from the liquid to a nucleus (kinetic barrier to nucleation). The Stokes–Einstein (SE) and Eyring (EY) relations are frequently used to correlate diffusion and shear viscosity η in silicate melts [29]. Both the SE relation and the EY relation require the size of a (atom, molecule, cluster or activated complex), which, in principle, involves the problem in selecting the correct size of the moving unit. In the hydrodynamic SE approach, valid for dilute molecular suspensions, friction (friction factor f = 6πηr for non slip between the diffusing units) exerts a force on a particle of radius r undergoing Brownian movement through the surrounding medium of viscosity η controlling its mobility. The friction factor itself is related to the self diffusion coefficient D by f = kT / D. For a spherical particle much larger than the solvent molecule rearranging gives [30]: D¼

kT ; λSE η

ð4Þ

where λSE = 6πr is the characteristic length in the SE approach. Based on the independent movement of particles in dilute suspensions SE proved to be an effective link between viscosity and diffusivity in monoatomic liquids, but was inadequate for describing ion diffusivity in polymerized liquid silicates [31,32]. The SE relation therefore is a strong test for polymerization in liquids. Since the SE relation does not hold for small, fast moving ions [33] and the connection of the diffusion of network formers with the viscosity of the melt is suitable [34–36] we used the EY model in our analysis. In the EY model the diffusion of a particle (atom, ion) is based on a hop or a jump, which results when the nearest neighbors of the diffusive particle are pushed aside. EY treats the short-range order in liquids as a quasi-lattice structure in small regions and allows the particle to jump over the potential barrier into its adjacent hole. According to EY reaction rate theory the size of the activated complex λEY, containing CN + 1 particle, is related to the quotient kT / Dη by [37]:
λEY ¼ CN

VM nN A

1=3 ¼

kT ; Dη

ð5Þ

where VM and CN are the molar volume and the coordination number of the moving particles in the cluster, respectively. NA is Avogadro's number (mol−1) and n is the number of atoms per formula unit. Experimental crystal number density curves of the deeply supercooled lithium disilicate melts of a narrow temperature range (from approx. 700 to 760 K) close to the glass transition (Tg = 724 K

[38]) have been analyzed to gain τ and to test CNT [5,6,26,39,40]. These studies showed that CNT fails to predict the correct (experimentally observed) lag time. In particular, the classical nucleation theory overestimates the work of critical cluster formation and underestimates the value of the steady-state nucleation rate I0 and, thus, overestimates τ. Accordingly, the kinetic barrier, the size of the moving structural unit and the thermodynamic driving force were adopted, which will be explained in detail below: Nascimento et al. [41,42] investigated the effects of a decoupling at the crossover temperature Tc (Tc ≈ 1.1 Tg) of the dynamics of crystal nucleation and growth from those of viscous flow, i.e. the validity of the SE and EY relations at T b Tc. They concluded from analyzing lag times in a narrow temperature interval (0.53 ≤ T/Tm ≤ 0.59) through Eq. (2) that the effective diffusion coefficient for the transport of structural units through the liquid–crystal boundary Dτ is larger than the SE and EY diffusivity D by a factor of 1.7 for T N Tg but increases exponentially for T b Tg in the temperature range from 693 to 763 K [42]. In their analysis the SE and EY jump length was used as an adjustable parameter with λ = 270 pm. The temperature-dependent ratio Dτ/D can be approximated by the equation:
 Dτ T ; ¼ A1 þ A2 exp − D A3

ð6Þ

where A1 = 1.7, A2 = 1.3 × 1032 and A3 = 9.7 K. Fokin et al. [40] followed a generalized Gibbs approach (GGA) [8] and explained the discrepancy between I0, τ and CNT by a thermodynamic driving force for the formation of an evolving nucleus, which is smaller than for the growth of a macroscopic crystal by a constant value of 7.83 kJ mol−1 (fit through I0 + U0 data) and 10.89 kJ mol−1 (fit through I0 + τ data). The reduction of ΔGv corresponds to a = 588 pm and results, dependent on the used data set, in the specific interfacial energies of σcl = 0.122 Jm−2 and σcl = 0.105 Jm−2. Their analysis was restricted to a narrow temperature interval (0.53 ≤ T/Tm ≤ 0.59). By contrast to homogeneous volume nucleation, when annealing a lithium disilicate glass above Tg, continuous supercooling from Tm provokes heterogeneous nucleation of the melt in contact with platinum metal (container wall [43–45], thermojunction [46–48] and particles [49]) or at free surfaces of levitated droplets [49]. The inherent lag time of nucleation in those experiments is then τ ¼ ΔT  q−1 ;

ð7Þ

where q = cooling rate. These studies showed that the first nucleus grew fast (approximately 7–8 orders of magnitude faster than at the glass transition range [24]), which impedes the formation of a nuclei ensemble (the homogeneous nucleation rate is extremely low at temperatures close to Tm) and allows one to detect the liquid-to-crystal transformation in-situ from the onset of the exotherm using differential scanning calorimetry [43,44,49] or from visual inspection using a highspeed video camera [49]. In any case large parts of the small liquid volume (DSC–pan, levitated droplet) will be consumed by the first crystal during a time interval 1–2 orders of magnitude smaller than τ. Recently, it has been shown that due to the stochastic nature of the nucleation (nucleation is not a deterministic process) τ has to be determined from an ensemble of cooling runs of the same liquid volume [44]. Further, it was shown that repeatedly supercooling of the same volume of liquid results in the same distribution of lag times as isothermal holds at a fixed supercooling temperature. This confirms that the ergodicity of the system (volume average is equal to the time average) at relative low degrees of supercooling is valid [50–53]. Experimental lag times of continuous cooling regimes have not been compared with theoretical predictions. The use of these data will represent a more rigorous approach since the range of supercooling the lithium disilicate melt is extended considerably from a small interval around Tg (0.53 ≤ T/Tm ≤ 0.59) to 0.53 ≤ T/Tm ≤ 0.92. Therefore the present study aims to compile τ data of homogeneous and

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3

heterogeneous nucleation and compare both sets of experimental data with the predictions of CNT. Primary motivation of the analysis is to answer the question, whether the adoptions of the diffusivity (kinetic barrier) and of the driving force to nucleation (thermodynamic barrier) are also consistent with the findings of τ at higher temperatures. We show that considering a smaller thermodynamic driving force connected with the nanoscale of the evolving nucleus can help to match experiment and theory while other parameters such as the jump distance, the interfacial energy and the contact angle are inefficient in explaining the existing discrepancy between the lag times to heterogeneous nucleation and CNT. 2. Data and analysis 2.1. Experimental lag time data Lag times are reported for continuously cooling a small volume of lithium disilicate liquid (ca. 20 mg) in a Pt/Rh–pan of a DTA/DSC once [49], 5–6 times [43], and 332 times [44]. Fig. 1 shows the first step of our analysis that nucleation is delayed for 5 to 38 min. If one plots the nucleation temperature Tn of the three studies as a function of the cooling rate multiplied by the lag time, which is equal to the supercooling temperature ΔT, all nucleation events collapse to a single cooling curve (insert of Fig. 1). Surprisingly, the supercooling temperatures observed in Refs. [43,49] were covered by those of Ref. [44]. We note that ΔT was examined in our previous study with statistical significance while in Refs. [43,49] a lack of adequate statistics is evident. In the second step we sort the lag times in increasing order using only the data set of Ref. [44] (total number of runs N0 = 332). By that a sigmoidal curve of the fraction nucleated N / N0 was generated, which ranges from zero at zero lag time when all samples were still liquid (unnucleated) to unity at some larger lag time when all samples were nucleated (Fig. 2). To approximate the nucleation probability (most frequent lag time) the first derivative, i.e. d (N/N0) / dτ was calculated (insert of Fig. 2). Inspection of the probability curve reveals a strong maximum at 1769 s (ΔT = 147 K). The spread of these nucleation events is 384 s if the width of the sigmoidal curve within the 10 and 90% fractions nucleated is used. The full width at the half maximum of the probability curve (insert of Fig. 2) is 368 s. Nucleation studies under the above cooling regime have been performed also at free surfaces of levitated drops using pure and doped

Fig. 1. Temperature of nucleation Tn versus lag time τ of the continuously supercooled lithium disilicate melt in contact with Pt/Rh container running the same volume once (green triangle [49]), five to six times (red square [43]), and 332 times (blue circle [44]). The symbols indicate Tn of each run. The thin dashed lines show the actual temperature of each experiment. The insert shows the temperature to nucleation as a function of the cooling rate multiplied by the lag time, which for a constant cooling rate q is the supercooling temperature ΔT in units of K. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Fraction of nucleated runs N/N0 and nucleation probability (insert) as a function of lag time. The symbols indicate N/N0 of each run. The lines are the cumulative distribution function of N/N0 and the nucleation probability (d (N/N0) / dτ) (insert) as calculated utilizing data of Ref. [44] with N0 = 332.

(Pt) lithium disilicate, respectively [49]. For each case only Tn of a single cooling run (using 4 different cooling rates) was reported. In the absence of adequate statistics we analyze these nucleation events by calculating their average lag time, which is (1533 ± 278) s for pure and (341 ± 146) s for Pt-doped lithium disilicate. Lag time data of homogeneous nucleation in the volume of lithium disilicate melts are utilized from one of the author's previous study [26]. Lag times for both types of nucleation (heterogeneous and homogeneous) were sort by their reduced temperature T/Tm and compiled in Tab. 1. 2.2. Input parameter for computation of lag times To compute the lag time as a function of the reduced temperature T/Tm after Eq. (2) a set of parameters were used: The thermodynamic

Table 1 Lag times of homogeneous and heterogeneous nucleation in supercooled lithium disilicate melts. Type of nucleation

T/Tm

τ (s)

Ref.

Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom Hom hom Hom Hom Hom Hom Hom hom Het (air) Het (Pt) Het (Pt)

0.535 0.538 0.538 0.542 0.546 0.546 0.546 0.548 0.550 0.550 0.555 0.557 0.558 0.561 0.564 0.565 0.565 0.567 0.570 0.574 0.581 0.584 0.592 0.851 ± 0.061 0.888 ± 0.012 0.919 ± 0.013

88,500 51,300 51,420 ± 6720 29,940 19,320 19,320 15,360 ± 2160 14,940 10,440 9600 5220 4440 3840 2280 ± 360 1500 1380 1080 ± 120 1020 780 480 228 ± 300 126 51.6 1533 ± 278 1769 ± 192 341 ± 146

[26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [26] [49] [44] [49]

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driving force ΔG was derived by fitting the calorimetric data of Takahashi and Yoshio [54] in the range from 0.2 to 1 T/Tm through the Eq. (8) [7]

 T w T 1− ; ΔG ¼ T m ΔSm 1− 1− Tm 2 Tm

ð8Þ

where TmΔSm = ΔHm = melting enthalpy (61,090 J mol− 1 [54]) and w = Δcp(Tm) / ΔSm = 0.35. The molar volume VM (6.154 × 10− 5 m3 mol− 1), the mean size of a structural unit a (4.675 × 10− 10 m) and the number density of these units N (9.785 × 1027 m− 3) were derived from crystal density ρ (2.438 g cm−3 [5]) and molar mass M (150.06 g mol−1) of the lithium disilicate at room temperature (neglecting thermal expansion) and using the simple calculus: VM ¼

M ; ρ

N¼

NA ; VM

a ¼ N−1=3 :

ð9Þ

The diffusivity of the moving unit was calculated from Eq. (5) by assuming EY is valid (D = Dτ) and using a Vogel–Fulcher–Tammann dependence of viscosity (log η0 = A + B / (T − T0)) with the parameter A = −2.37, B = 3248.62, T0 = 500.24 (viscosity η0 in Pa s and K) [38] as well as from Eq. (6), which takes into account the breakdown of EY formalism in the description of the kinetic barrier at T b Tc. For the former an activated volume was varied from CN = 1 as a lower limit, which results in the effective jump length λEY = 225 pm, to CN = 4 as an upper limit (λEY = 900 pm). Alternatively, σcl was varied within the limits from 0.1 to 0.25 J m− 2 [6,42], while z was set for both computations as 16/π [13,14]. In a second series of calculations the thermodynamic driving force was varied in accordance with the general Gibbs approach (GGA) [8]. Therefore it is assumed that the formation of critical cluster is connected with a lower free energy than the growth of the macroscopic crystal. The difference is introduced by a constant excess energy ΔGex = ΔGmacrocrystal(T) − ΔGnanocrystal(T) [40], which will be explained in detail below. In a third type of modeling heterogeneous nucleation on a substrate is assumed, which lowers the thermodynamic barrier of nucleation considerably. In the heterogeneous case the lag time depends on the degree of wetting, which is expressed by the contact angle φ between critical nucleus and the substrate as [55]: τ het 1 1 ¼ 1− cosφ− cos2 φ; τhom 2 2

ð10Þ

Fig. 3. Lag times to nucleation in supercooled lithium disilicate melts as a function of the reduced temperature T/Tm. A) Lines are calculated from CNT by Eq. (2) and using the EY approach Eq. (5) in the description of the kinetic barrier with CN = 1 (λEY = 225 pm), 2 (λEY = 450 pm) and 4 (λEY = 900 pm) as well as taking into account faster dynamics for T b Tc, i.e. the breakdown of EY (Eq. (6)). σcl and z were set as 0.15 J m−2 and 16/π, respectively. B) Lines are calculated from CNT by Eq. (2) and using the EY approach Eq. (5) in the description of the kinetic barrier with CN = 2 (λEY = 450 pm) but for a different crystal liquid interfacial energy σcl = 0.1, 0.15 and 0.25 J m−2. z was set as a constant (16/π). Data: Homogeneous nucleation of Ref. [26] (red triangle) and heterogeneous nucleation of Ref. [49] (blue diamond and square) and Ref. [44] (blue circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where τhet and τhom are the lag time of heterogeneous and homogeneous nucleation, respectively. 3. Results U-shaped curves are obtained of increasingly larger lag times for supercooling towards the glass transition as well as towards the melting point. Fig. 3A shows that the experimental lag times of homogeneous nucleation are in agreement with the predictions of CNT. By contrast the calculated lag times underestimate the lag times of heterogeneous nucleation observed in practice by 6–7 orders of magnitude. In particular, an activated volume, which consists of three atoms (CN + 1), i.e. one moving and two adjacent atoms, fits the data for homogeneous nucleation best, if the interfacial energy and contact angle are set as 0.15 J m−2 and 180°, respectively. The analysis reveals further that a temperature-independent activation energy (Arrhenian type) is even in better agreement with experiments, which confirms the proposed decoupling of the dynamics of nucleation from those of viscosity at low temperatures. Irrespective of the choice of the jump length CNT fails in predicting lag times of heterogeneous nucleation in lithium

disilicate. The same result was obtained if the specific interfacial energy was varied (Fig. 3B) since σcl and τ are only linearly interdependent in Eq. (2). Fig. 4A shows that decreasing the thermodynamic quantities of the critical nucleus from the values of the stable macroscopic crystal can explain the observed lag times of both heterogeneous and homogeneous nucleation in supercooled lithium disilicate melts. Considering the most reliable data set of Ref. [44], an excess energy of ≈ 6.7 kJ mol−1 is obtained. Fig. 4B shows the effect of heterogeneous nucleation on lag time. With decreasing contact angle lag times are shortened. The contribution of heterogeneity on the high temperature flank of the curve is negligible since the τ function tends rapidly to infinity as T/Tm → 1. 4. Discussion The most striking effect of the analysis is that only the reduction of the thermodynamic driving force seems to match lag times of

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Fig. 5. Free energy of the stable macrocrystal, metastable nanocrystal and metastable melt . Thermodynamic data of lithium disilicate from Ref. [54]. for T b Tmacrocrystal m

radius r of the hypothetically melting nanocrystal can be approximated as: 2σ cl V M !"

r nanocrystal ¼ − T m ΔSm 1−

T nanocrystal m

T macrocrystal m

1−

w T nanocrystal m 1− macrocrystal 2 Tm

!# :

ð11Þ

Fig. 4. Lag times to nucleation in supercooled lithium disilicate melts as a function of the reduced temperature T/Tm. A) Lines are calculated from Eq. (2) and using the EY approach Eq. (5) in the description of the kinetic barrier with CN = 2 (λEY = 450 pm) but with an excess energy ΔGex = 0 (CNT), 4.9, 6.7 and 8.9 kJ mol−1 connected with the formation of nuclei. B) Lines are calculated from Eq. (2) and using the EY approach Eq. (5) in the description of the kinetic barrier with CN = 2 (λEY = 450 pm), the excess energy ΔGex = 6.7 kJ mol−1 connected with the formation of nuclei of critical size and assuming a heterogeneous character of the nucleation (Eq. (10)) with a contact angle φ = 50° and 30°. Experimental τ data as in Fig. 3.

experiment and theory at both low and high supercooling. Of course this includes that the decoupling of diffusivity and viscosity has to be applied at high supercooling to best fit the data below the glass transition (T/Tm b 0.554) as illustrated in Fig. 3A. The other tested parameters such as the size of the activated volume, the interfacial energy and the contact angle appear to be inefficient. While a reduction of the thermodynamic driving force has been proposed for homogeneous nucleation of lithium disilicate in the range from T/Tm = 0.53 to T/Tm = 0.59 [40] it seems to be also valid for heterogeneous nucleation at higher temperatures. In terms of the GGA [8] the results indicate that the evolution of the properties of the nuclei is critical size-independent. In particular, the location of the heterogeneous lag time data on the right hand side of the U-shaped τ curve can be interpreted in terms of a hypothetic melting nanocrystal at a marginally higher T/Tm value, point of the nanocrystal Tm i.e. a melting point depression with respect to the macrocrystal nanocrystal the free energy curves of [56–58]. In order to approximate Tm the macrocrystal, the supercooled liquid and the nanocrystal are computed from the thermodynamic data of Ref. [54] and ΔGex ≈ 6.7 kJ mol−1. Fig. 5 shows that considering ΔGex as a temperature-independent factor nanocrystal macrocrystal /Tm ≈ 0.888 is obtained. Using the classical the ratio Tm thermodynamic analysis (Gibbs–Thomson equation) and Eq. (8) the

Allowing an increase in the molar volume of about 3% with respect to room temperature and setting σcl = 0.15 J m−2 and w = 0.35 in Eq. (11) we obtain rnanocrystal ≈ 2.8 nm. Thus, for nanocrystals of larger size one would expect to have properties, which are connected with those of the macrocrystal while for smaller sized nanocrystals the free energy is higher than for the stable macrocrystal. On the other hand a melting point of crystals of nanosize can be understood also in terms of a phase transition with respect to larger crystals. In this sense the presence of a metastable precursor phase such as a polymorph of the same composition in the early stage of the crystallization of lithium disilicate glass would apply, which has been discussed controversially in the literature [5,59,60]. We note that the actual value of rnanocrystal depends on the choice of the specific interfacial energy of the nanocrystal, which is not known. In general σcl is the missing link in relating experiment to theory. For that reason several analyses were performed in literature in which the crystal–liquid interfacial energy is treated as a free adjustable parameter to match homogeneous nucleation rates and CNT predictions [2,26,61–64]. This is even more true since these tests rely on the introduction of a second adjustable parameter, such as a temperaturedependent coefficient [2,26,62], a size-dependent coefficient [61,64], and a roughness-dependent coefficient [63] of the crystal–liquid interfacial energy. An analysis of σcl with respect to the recently determined heterogeneous nucleation rates at low supercooling, which corresponds here to the studied lag times [44,45] is beyond the scope of this paper and will be addressed in a future work. 5. Conclusions The inclusion of lag time data of heterogeneous nucleation at low supercooling temperatures is a novel approach, which widens considerably the common temperature range used in CNT analyses of lithium disilicate melts. It reveals that apparently the thermodynamic driving force, i.e. the Gibbs free energy connected with the liquid-to-crystal transformation of the evolving critical nucleus, is smaller than the Gibbs free energy of the growing macrocrystal measured by calorimetry. The results are in line with earlier assumptions on size-dependent

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energy effects in the description of nucleation, but can be also a hint to metastable polymorphs at the nanosize. In any case, it is proposed that N 0.888 crystal nucleation in the lithium disilicate melt for T/Tmacrocrystal m becomes vanishingly small (probability tends to zero) regardless of the nature of the nucleation mechanism (homogeneous and heterogeneous) since the thermodynamic driving force for nucleation of the nanocrystal becomes zero, and -vice versa- it appears that the available thermokinetic data are insufficient for the description of lag times to crystal nucleation in supercooled lithium disilicate melts. Acknowledgments

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

J.D. thanks the TC 7 members of the ICG for the fruitful discussions on lag times of crystal nucleation in lithium disilicate based glass–ceramics.

[39]

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