The microscopic origin of the extreme glass-forming ability of Albite and B2O3

Understanding the conditions that favour crystallisation and vitrification has been a longstanding scientific endeavour. Here we demonstrate that the extremely high glass-forming ability of unseeded supercooled Na2O·Al2O3·6SiO2 (Albite) and B2O3—known for decades as “crystallisation anomaly”—is caused by insufficient crystal nucleation. The predicted temperatures of the maximum homogeneous nucleation rates are located well below their glass transition temperatures (Tg), in a region of very high viscosity, which leads to extremely long nucleation time-lags and low nucleation rates. This behaviour is due to the remarkably small supercoolings where the glass transition occurs for these liquids, which correspond to a very small driving force for crystallisation at and above the Tg, where crystallisation is normally observed. This meagre nucleation ability is caused by the significant difference in the structures of the supercooled liquids and their isochemical crystals. These findings elucidate the cause behind the crystallisation anomaly, and could be used for the design of other oxide glasses that are extremely stable against crystallisation.

m = dlog(η (T))/d(T g /T)| T = Tg , where η is the equilibrium viscosity and T g is the glass transition temperature. Note that this fragility index is not related to the mechanical properties of the material.
As an example of the types of experimental tests performed with these glasses, in the fifties Schairer and Bowen 6 showed how difficult it is to crystallise Albite from its own melt; one sample was held at 1025 °C (close to the temperature of the predicted maximum growth rate) for five years without crystallising 7 . Uhlmann 8 coined the term crystallisation "anomaly" when trying to decipher this unusual behaviour, which that is also shown by B 2 O 3 glass, which only crystallises after seeding and under very high (GPa) pressures 9 . Albite has been shown to crystallise at atmospheric pressure only when Albite crystal seeds are introduced [10][11][12] . In a very recent article, Siqueira et al. 13 demonstrated that even a (carbon-free) gel-derived Albite glass-and gels easily devitrify-did not crystallise when heated at the temperature of maximum growth rate. One of us (EDZ) has tried to crystallise an unseeded B 2 O 3 glass with different thermal treatments without success.
These experiments indicate that the main challenge preventing crystallisation is in the formation (nucleation) of a crystalline phase (Albite and B 2 O 3 ), and perhaps even in its growth (B 2 O 3 ). This crystallisation anomaly has been a matter of debate for many decades!
The typical stoichiometric oxide glass forming compositions (such as Albite and B 2 O 3 ) normally crystallise in laboratory time scales when heated at any temperature somewhat below their melting points and above their T g , and also during non-isothermal DSC (differential scanning calorimetry) runs. However, crystallisation does not occur in Albite and B 2 O 3 within the time scales of DSC experiments.
Therefore, understanding the reasons why these two glass-formers display this abnormal reluctance to crystallization in heating experiments and outstanding glass-forming ability (in the cooling path from the molten state) is very relevant from the physics, chemistry, and materials science point of view. From a broader scientific perspective, this study could also provide further insight into the nature of the vitreous state and be used for the design of very stable glasses.
In this article we dwell on and calculate three kinetic properties of supercooled liquids that control crystal nucleation and crystal growth: the steady-state homogeneous crystal nucleation rates, I st (T); the nucleation time-lags, τ (T); and the crystal growth rates, U(T), for Albite and B 2 O 3 . Because the homogeneous nucleation kinetics are too slow to be measured in these liquids, we rely on estimated values of I st (T) and τ (T). Finally, we use literature data to compare the similarities and differences between the atomic structures of the parent liquids and their isochemical crystal phases. We sought for any thermodynamic, kinetic, or structural signs that could shed light on the lack of observable crystallisation in the supercooled Albite and B 2 O 3 liquids.

Thermodynamic and Kinetic Parameters that Control Crystal Nucleation and Growth
Reluctant crystallisation could be due to either exceptionally low steady-state crystal nucleation rates, very long induction periods for nucleation, and/or extremely low crystal growth rates. We will now discuss the governing equations for these three kinetic properties, and at their main controlling parameters, that can be calculated or measured, which are: the crystallisation driving force, the nucleus-melt interfacial energy, and the effective diffusion coefficient. We will then start with a brief definition of these parameters, before discussing the crystal nucleation and crystal growth models.
Driving Force for Crystallisation (Δ μ ). Supercooled liquids always have a higher Gibbs free energy (G l ) than their stable isochemical crystal phases (G c ). The driving force for crystallisation is defined as Δ μ ≡ − Δ G = G l − G c and is positive for temperatures below the melting point 14,15 . In addition, the absolute value of Δ μ increases with decreasing temperature from the melting point (that is, increasing undercooling). Equation (1) provides a way to calculate Δ μ for a closed system under isobaric condition 14,15 .
In the above equation, Δ H m is the molar enthalpy of melting, C p,l is the molar heat capacity of the supercooled liquid, C p,c is the molar heat capacity of the crystal, and T is the absolute temperature. When the driving force for crystallisation is expressed in units of volume rather than in moles of substance, the conversion requires the knowledge of the molar volume, V m : Figure 1 shows a normalized plot of Δ μ (T g )/Δ H m versus T g /T m (calculated and proposed here) for a series of glass-forming oxides, where three regions can be observed: the red area depicts the region and oxide glasses for which internal homogeneous crystal nucleation is readily observed; the region of the two glasses for which no crystallisation is observed is blue. All the other points refer to oxide glasses that only crystallise heterogeneously starting on their external surfaces. This plot is in agreement with and reinforces Turnbull's 16 observation that liquids with T g /T m > 2/3 can be easily undercooled to the glassy state if they are free of crystalline seeds.
Nucleus-Liquid Interfacial Energy (σ cm ). Unfortunately the nucleus-melt interfacial energy, σ cm , cannot be directly measured. However, it can be estimated using the Skapski-Turnbull 17-19 expression: where α is a non-dimensional constant and N A is Avogadro's number. From both theoretical arguments and the fitting of the Classical Nucleation Theory equation to experimental crystal nucleation rates in oxide glass-formers that show measurable homogeneous nucleation, α is expected to be in the range of 0.4-0.6 20 . For glasses that do not display homogeneous nucleation α may be even higher.
Viscosity and Diffusivity. The effective diffusion coefficient controlling viscous flow and supposedly crystal nucleation and growth in supercooled liquids can be calculated by the Eyring 21 equation: where D η is the diffusion coefficient, k B is Boltzmann's constant, d is the jump distance, which is equivalent to the diameter of the diffusing molecules, and η is the viscosity. As d varies significantly less with temperature than η, for practical purposes it can be considered a constant equal to d o ∼ (V m /N A ) 1/3 . Several viscosity equations are available in the literature [22][23][24][25] . In this work we considered the VFT 26-28 which corresponds to Eq. (5). This equation together with Eq. (6) (also known as CWAM 29,30 ) fit the viscosity data of oxide liquids between the T g and T m very well, but yield upper and lower bounds, respectively, for the extrapolated values of the equilibrium viscosity below T g . They are both used in this article.
VFT VFT 10 10 , 0 In these equations, η ∞ is the asymptotic value of the equilibrium viscosity when T → ∞, and A, B, C, and T 0 are adjustable parameters. Figure 2 shows viscosity data and fitted curves for Albite and B 2 O 3 in an Oldekop-Laughlin-Uhlmann-Angell [31][32][33] (OLUA) plot. The values for the adjustable parameters in this figure are shown in Table 1.
The actual diffusion coefficients controlling crystal nucleation and growth, D I and D U , respectively, are unknown for oxide glass formers. Hence they have been largely calculated by the Eyring or Stokes-Einstein equations, D η 34-39 , which yield good results for various oxide glass formers 38 at relatively high temperatures not far below the liquidus. Accurate crystal growth studies revealed that this equation breaks down at deep supercoolings, at T d ∼ 1.1-1.2T g 38,39 . Equilibrium viscosity and crystal growth kinetics are thus "coupled" from the melting point down to T d , but they diverge significantly below T d . For this reason, T d is often denominated decoupling Figure 1. Reduced driving force for crystallisation evaluated at T g versus reduced T g for a number of oxide glass-formers. The red points refer to compositions that reveal internal homogeneous crystal nucleation, whereas the blue points refer to the two known stoichiometric oxide liquids that do not show any signs of nucleation (B 2 O 3 and Albite). The remaining points refer to supercooled liquids that only reveal surface (heterogeneous) crystal nucleation. When available, the typical uncertainties are shown for some compositions. Thermophysical data were obtained from various sources 46,47,[63][64][65][66][67][68][69][70][71][72][73][74][75][76] . Values of T g were obtained by fitting viscosity data collected from the SciGlass database 77 .
Scientific RepoRts | 7:43022 | DOI: 10.1038/srep43022 (or breakdown) temperature. However, to the best of our knowledge, signs of a similar breakdown of the crystal nucleation kinetics have not been clearly shown to-date. Therefore, in this article, we use the Eyring equation to approximate the diffusion coefficient controlling nucleation, and thus neglect the effects of a (possible) breakdown. In addition, as the crystal growth calculations are carried out above T d , as we will show later, there is no need to consider a (possible) breakdown. Finally, at least Albite is a very "strong" liquid for which the breakdown is much less likely.

Theories and Models of Crystallisation and their Applicability
Crystallisation may be divided into the following two processes: the formation of stable crystalline nuclei, and their volume change with time and temperature. The first is termed crystal nucleation and the second is termed crystal growth. In the following subsections we will briefly introduce the main models for these two processes.
Classical Nucleation Theory. The Classical Nucleation Theory (CNT) provides the physical description of the crystal nucleation process. Here we introduce the basic equations for the steady-state homogeneous crystal nucleation rate, I st , st c For spherical nuclei, the work of the formation of a nucleus with a critical size, often denoted as the thermodynamic barrier for crystal nucleation, W c is given by Eq. (10).
A nucleus is called "critical" when its growth is spontaneous resulting in a decrease of the system's Gibbs free energy. These relations will be used in the present study to estimate the crystal nucleation rates in the framework of CNT. In the above equations, σ cm is the crystal-liquid interfacial energy of the critical nucleus (in units of J/m 2 , assumed to be the same used in the macroscopic growth equations); k B is the Boltzmann constant; T is the absolute temperature; Δ μ v is the driving force for crystallisation in units of J/m 3 ; and D I is the effective diffusion coefficient determining the processes of aggregation of "structural units" to the crystalline clusters, estimated here by the Eyring expression (Eq. 4). Hence, the expression for the nucleation rate becomes The actual magnitudes of I st cannot be calculated with a reasonable degree of accuracy due to the uncertainty in σ cm 40,41 and its cubic exponent inside an exponential. However, the temperature dependence of I st and the location of its maximum, T max , can be adequately calculated using properly estimated values of σ cm , as we will do in this article.
Certain defect sites and solid impurities can reduce the thermodynamic barrier and the time-lag for nucleation. The formulation presented above can be maintained with correction factors 42 given by Eqs (12) to (15).
In the above equations, W c,het is the thermodynamic barrier for heterogeneous crystal nucleation, φ and ξ are functions that depend on the contact angle between the nucleus and the substrate, θ , and τ het is the time-lag for heterogeneous nucleation.
A key parameter that can be measured for glass-forming systems is the nucleation time-lag, τ , as a function of temperature, as shown in ref. 43. For all oxide glass-forming liquids tested to-date, these nucleation time-lags only become detectable (reach laboratory time scales) in the neighbourhood of their respective T g and keep increasing upon further supercooling. Figure 3 shows the experimental values of τ at the temperatures of maximum crystal nucleation rate, T max . These are data for 12 substances that show homogeneous nucleation in laboratory time scales. It is quite clear that τ increases dramatically (please note the logarithm scale) with increasing values of the reduced glass transition temperature T gr = T g /T m . An extrapolation of this plot for the cases of Albite (T gr = 0.78) and B 2 O 3 (T gr = 0.76) gives an estimated time-lag of more than 10 10 seconds (about 300 years). Figure 4(a) and (b) shows the cupola shape of the crystal nucleation rate curves for glassy Albite and B 2 O 3 . The homogeneous nucleation (θ = π rad, blue line) rate maxima are located well below the glass transition temperatures, which suggest that the crystal nucleation rates at and above T g (where the kinetic phenomena start to be measurable for most glass-forming systems) could be extremely small.
Compare these figures with Fig. 4(c), which shows experimental data and calculations for lithium disilicate (LS2), a supercooled liquid that undergoes internal homogeneous nucleation. The position of the homogeneous crystal nucleation peak for LS2 in the respective T/T m scale is not much different from the same peak for Albite, but the value of T g /T m for LS2 is equal to its homogeneous crystal nucleation maximum. We will discuss this issue in more detail below.
Experimental data for heterogeneous crystal nucleation of LS2 44,45 in a supercooled liquid contained in a Pt pan is well described using a value of θ = π /7 rad. When we used this value of θ in the calculations for Albite and B 2 O 3 an interesting picture emerged: the heterogeneous nucleation peak is in fact well above the glass transition temperature for these two compositions, a temperature region where the expected nucleation time-lags are much shorter. This means that heterogeneous crystal nucleation may be possible in these materials with a powerful nucleation agent (such as seeding with their own isochemical crystals).
To calculate the nucleation curves of Fig. 4(a) and (b), we used Eq. (11) with d 0 calculated from the molar volume, as shown in Subsection 2.3. In this context we calculated σ cm via Eq. (3) with α = 0.5 and Δ μ v by Eqs (1) and (2). Table 2 shows all the physical and thermodynamic parameters used in the calculations. The heat capacity for the crystalline phase is shown in Eq. (16) for Albite 46  The values of equilibrium viscosity (rather than the non-equilibrium viscosity), given by Eq. (5) were used even below the T g of each glass because the structural relaxation times, τ rel , are significantly shorter than the crystal nucleation time-lags, which is similar to the average time of formation of the first critical nucleus, < τ > . In fact, the temperature where τ rel = < τ > is the so-called supercooled liquid metastability limit (SCLML) or kinetic spinodal, T  . Our estimates of T KS for three oxide liquids (to be published) demonstrate that T KS /T m < 0.50, whereas T max /T m for Albite is 0.55 and is 0.65 for B 2 O 3 . Therefore, these two glasses should relax to the supercooled liquid state before the first critical nucleus is formed. Hence, the use of equilibrium viscosity to calculate nucleation kinetics above T KS is a very reasonable assumption. However, in any case, if instead the non-equilibrium viscosity were used in the calculations, the resulting T max would be located at even deeper supercoolings. In that case, the conclusion so far reached (long nucleation time-lags) would be reinforced. We have also used Eq. (6) for the equilibrium viscosity in the above discussed calculations and the results did not significantly change.
A key element for the rationale of the present article is that the maximum experimental steady-state homogenous crystal nucleation rates, I st (T max ) drastically decrease with increasing T gr of the parent glass. The overall picture is shown in Fig. 5 and has also been corroborated by theoretical calculations as discussed thoroughly by Fokin et al. 43 . This figure indicates that the homogeneous nucleation rates in Albite (T gr ∼ 0.78) and B 2 O 3 (T gr ∼ 0.76) are likely extremely small.
Apart from three outliers, the theoretical prediction is sufficient to describe all the other experimental data within the uncertainty (that is, the range considering C 1 from 5.0 to 6.5.) The data scatter makes it difficult to affirm that the temperature dependence of the theoretical prediction is different from the experimental data. Perhaps the experimental data show a steeper decrease than the theoretical prediction, but if this is the case, our conclusions would not be affected.
Crystal Growth Models. Experimental crystal growth rates in oxide glass-formers are reasonably well described by one of the following three classical models: normal or continuous growth, screw dislocation growth, and 2D secondary nucleation growth 20 . The first two growth modes are quite common for oxide glass-formers and share the same expression: The difference between these two growth models is on the fraction of sites on the growing crystal interface available for atomic attachment, f. For materials with a small melting entropy 51 , Δ H m /T m < 2R (R is the gas constant), f is independent of the temperature and approximately 1. In such case, normal growth prevails. Otherwise, if 0 < f < 1 and is temperature dependent, then the screw dislocation mechanism governs    In the above equations, U is the crystal growth rate, D U is the effective diffusion coefficient controlling crystal growth, and σ is the crystal-liquid surface energy (normally assumed to be size and temperature independent and equal to σ cm ). When f = 0 (perfect crystals) then crystal growth must occur via 2D secondary nucleation, each crystal layer being formed by a secondary nucleus on the top of the primary nucleus and so on. The expression for this mechanism is well documented in Gutzow & Schmelzer's book 20 .
To calculate the crystal growth rates curves shown in Fig. 6, we used Eqs (18) and (19) and the same value for σ used in the Subsection 3.1 and two values for d 0 , of the order of magnitude of a chemical bond length. The experimental data points refer to the growth of crystalline layers observed by Uhlmann et al. 10 after dusting an Albite glass surface with crystalline Albite mineral crystals.
Therefore to grow a crystal of 1 micron (that is resolvable under an optical microscope) one would need many years at the temperatures of maximum homogeneous nucleation, T max ∼ 0.55T m (Albite) and ∼ 0.65T m for B 2 O 3 . However, for temperatures well above T g , measurable growth occurs in a timeframe of hours, as shown in Fig. 6(a), where the four data points of Uhlmann et al. 10 (obtained with seeded Albite glass) fit quite well the calculated curve with d o = 1 Å. In summary, the maximum crystal growth rates are slow (U max ∼ 1 Å/s for Albite and ∼ 50 Å/s for B 2 O 3 ), but growth can and has been observed above T g and is not the problem hindering the crystallisation in supercooled Albite liquid, as shown by the experiments with seeded samples [10][11][12] . For B 2 O 3 the theoretically estimated growth rates (by the screw dislocation mechanism) are even higher than those of Albite. However, we cannot draw firm conclusions about its growth kinetics because, to our knowledge, no published growth data are available for growth of B 2 O 3 crystals at ambient pressure. One could also argue the growth rates in this glass could be described by another growth mechanism.
In any case, the exceptional resistance to crystallisation of both liquids clearly resides in their poor crystal nucleation kinetics, which was theoretically analyzed in the preceding sessions. In the following session, we dwell on the structures of the glass and their isochemical crystal phases in a quest for a structural reason for this poor nucleation ability.

Structural Aspects of the Glasses and their Isochemical Crystals
It is known empirically 53 that the densities of the parent glass and corresponding crystal phase are quite similar (Δ ρ /ρ glass < 8%) for all stoichiometric oxide glasses that display internal homogeneous crystal nucleation in laboratory time/size scales. However, several stoichiometric oxide glasses that only display surface (heterogeneous) nucleation, or no nucleation at all, show higher values of Δ ρ /ρ glass . In the case of Albite, Δ ρ /ρ glass = 15% and for B 2 O 3 the density mismatch is approximately 34%.
The Δ ρ /ρ glass ratio gives clear signs of structural similarity or dissimilarity between the parent glass and its isochemical crystal, hence Albite and B 2 O 3 seem to have a significantly different structure from their crystalline phases. Let us consider two extreme cases: if the structure of the liquid is very close to that of its crystal form, then their densities must be similar and the necessary structural reorganization for crystallisation is facilitated by the reduction in the kinetic barrier for nucleation and reduction of the surface energy, but the thermodynamic driving force, Δ μ V , will be weak. However, as the crystal nucleation rates exponentially diminish with an increase in σ cm 3 /Δ μ V 2 (Eq. 10), the role of σ cm is stronger and prevails. Structural similarity (lower σ cm ) is thus favourable for higher nucleation rates. The cases of Albite and B 2 O 3 , for which the densities (and inferentially the structures) of the liquids strongly differ from those of the isochemical crystals, correspond to a high surface energy, which lead to smaller nucleation rates. As the values of α (Eq. 3) are unknown for glasses displaying heterogeneous nucleation, we cannot estimate the values of the surface energy for these two glasses.
However, other oxide glasses also have similar high values of density mismatch and still crystallise, although they always do with the crystal growth starting on heterogeneous nucleation sites that exist on the sample surfaces 54 . Hence, this large density mismatch is only an indication of the structural dissimilarity. One should then dive deeper into the structural details of the parent glass and crystal phases for further insight, as we will indeed do below.

Structures of Albite Glass and Crystal.
Ordering at short-length and, in some cases, at intermediate-length scales is a universal feature of the glassy state. Taylor and Brown 55 indicated that the structure of Albite glass consists predominantly of 6-membered Si-O rings, whereas the structure of the isochemical crystalline compound is exclusively constructed of 4-membered rings.
According to Zanotto et al. 54 the dipolar second moments, Δ M 2 /M 2C , describing the spatial correlation between the network modifiers in oxide glasses and their isochemical crystals vary from approximately 4 to 16% for oxide systems that undergo internal homogenous crystal nucleation, whereas for those which only display surface nucleation, this parameter may reach 60%. However, for Albite glass and crystals, the Na-Na second moment differs by more than 130%! This is by far the largest difference ever reported for the modifier-modifier second moment among all oxide glass formers and clearly points to significant structural differences between the glass and crystal phases.
These results further support our working hypothesis that the intermediate and medium range structure of Albite glass and its corresponding crystal phase widely differ. However, this significant structural difference seems to hinder crystal nucleation only in said composition, not crystal growth, because Uhlmann et al. 10 , Selvaraj et al. 11 , and Liu et al. 12 were able to crystallise Albite glass at high temperatures using crystalline seeds. This insensitiveness of crystal growth to the structural mismatch results from growth being normally measured at much higher temperatures (T ∼ T m ) where bond breaking and rearrangement is much easier than at deep supercoolings, T ∼ T g , where the nucleation rates are normally measured.

Structures of B 2 O 3 Glass and Crystal.
High-resolution nuclear magnetic resonance (NMR) measurements of Oxygen-17 in B 2 O 3 glass 56 provided evidence for structural units responsible for ordering on short-and intermediate-length scales. At the molecular level, planar BO 3/2 units accounted for the local ordering. Oxygen-17 NMR spectra resolved detailed features of the inclusion of these units in boroxol rings, oxygen bridging two rings, and oxygen shared between two nonring BO 3/2 units. On the basis of these and corroborative boron-11 NMR and scattering results, the authors concluded that boron oxide glass consists of domains that are rich or poor in boroxol rings. Approximately 70% of the boron atoms are located in boroxol-rich regions. The radius of the ring-rich domains was estimated to be about 20 Å or larger, which is quite similar to the sizes proposed from independent light scattering experiments and the Boson peak observed in Raman spectra. These domains were proposed to be the structural basis of intermediate-range order in glassy boron oxide.
A not yet well understood question is the relationship between the glassy and the various possible crystalline forms a system may adopt. By means of ab initio calculations, Ferlat et al. 57 discovered the existence of B 2 O 3 crystalline polymorphs with structural properties similar to the glass and formation energies comparable to the known ambient crystal. According to those authors, the energy degeneracy of the crystals, which is high at ambient pressure and suppressed under pressure, provides a framework to understand the system's ability to vitrify and the origin of the crystallisation anomaly. Their main contribution was to evidence these novel crystals, some of them sharing greater structural similarity with the glass than do the experimentally synthesized crystals (B 2 O 3 -I).
The ease of vitrification of B 2 O 3 was related to the competition for crystallisation between numerous low-energy crystals (some with boroxols, some without). The authors considered that their work reaffirms the role played by polymorphism in a system's ability to vitrify 58,59 .
From an experimental perspective, 80 years ago Kracek et al. 60 prepared a B 2 O 3 glass sample sprinkled with B 2 O 3 -I crystal seeds (this phase is not explicitly stated in their paper, but this is in agreement with its reported melting point of 450 °C). After several months at various temperatures, the crystals remained suspended and did not grow. As far as we know, no one has ever tried to seed B 2 O 3 glass with B 2 O 3 -II crystals, but then the density mismatch would be even higher (3.11 g/cm 3 , giving almost 70% difference) or other polymorphs.
One important remark is that the structure of B 2 O 3 -I is exclusively made by BO 3 triangles 61 , which is significantly different from the ∼ 70% boroxol rings that the glass is made. In a recent publication, Wright and Vedisheva 62 suggested that the excellent glass-forming ability of B 2 O 3 is due to the need to either break up the boroxol groups to form crystalline B 2 O 3 -I, which has a structure based on ribbons of independent BO3 triangles, or to break up and reform boroxol groups to yield a suggested, but so far undiscovered, ambient pressure crystalline polymorph with a structure based solely on boroxols 62 .
This finding for the inorganic oxide glass-formers Albite and B 2 O 3 is analogous to what is known by the polymer science community. A few high molecular weight organic materials, in particular atactic polymers (e.g., polystyrene and PMMA) are very resistant to crystallisation because their intricate chain entanglements prevent the required molecular reorientation to organize themselves into a crystalline lattice. Summary of Properties and Structural Aspects. Table 3 summarizes the main parameters that control the crystal nucleation rates. They are quite unfavourable for the two oxide glasses vindicating our hypothesis to explain the lack of observable crystallisation in these glasses.
Finally, it is quite interesting to note that Albite liquid is strong; its fragility index m is 26. However, other strong liquids, such as SiO 2 and GeO 2 easily crystallise when heated above T g . On the other hand, B 2 O 3 can be considered a fragile glass-former (m ∼ 50) and is also notoriously reluctant to crystallise, whereas many other fragile liquids easily crystallise. Hence, the abnormal crystal nucleation behaviour is not linked to the liquid fragility.

Conclusions
Our analysis clearly demonstrates that the lack of detectable crystallisation ("crystallisation anomaly") in both supercooled liquids is indeed meagre homogeneous crystal nucleation, solving an enigma that persisted for 80 decades since the pioneering work of Kracek. This scenario is linked to the relatively small supercoolings where the glass transition takes place (T g /T m = 0.76-0.78) in these compositions. The latter leads to a very small crystallisation driving force at T ≥ T g , where crystallisation is normally observable in laboratory timescales. Thus, a deeper supercooling is necessary to achieve a high enough crystallisation driving force, leading to a T max that is well below the respective T g . Associated with this, we demonstrated that the crystal nucleation time-lags are extremely long at T max , effectively prohibiting that such experiments be conducted in laboratory time scales. However, for heterogeneous crystal nucleation catalysed by efficient nucleating agents, T max is shifted to higher temperatures above the respective T g . In this setting, the predicted nucleation time-lags become measurable. These calculations explain the experimental results confirming that these glasses can be crystallised only when seeded with powerful nucleating agents. In the unseeded and ambient pressure conditions, they will only crystallise in geological time scales.
Moreover, this anomalous crystallisation behaviour is not linked with the liquid fragility, as both strong (Albite) and fragile (B 2 O 3 ) liquids demonstrate this phenomenon. Ultimately, the major differences in the atomic structures of the parent liquids and their isochemical crystal phases are the cause for poor nucleation.
From a broader scientific and technological perspective, the present results not only debunk a long-standing mystery regarding the crystallisation anomaly, but also provide further insight into the connection between the structure and crystal nucleation ability of two important oxide glass forming liquids. In addition, these results could be used by glass researchers and engineers for the compositional design of oxide glasses that are stable against crystallisation. To that end, oxide glass-forming compositions should be selected that have a large Δ ρ /ρ glass and large T g /T m .  Table 3. Physical parameters controlling crystal nucleation in oxide glass formers that undergo homogeneous nucleation in comparison to Albite, B 2 O 3 , and LS2.