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

Abstract

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 (T_g), 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 T_g, 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.

Introduction

The most stable thermodynamic state of matter below the melting point or liquidus is the crystalline state, where the atoms are well organized in a periodic lattice at short-, intermediate-, and long-range distances. The vast majority of natural materials on Earth’s crust are crystalline. To make glasses one has to deceive the natural solidification path: crystallisation¹. There has been a long-standing quest to understand the conditions that favour crystallisation and vitrification of supercoooled liquids (and the nature of the vitreous state) ever since glasses became a major topic within condensed matter physics and materials science and engineering.

Most ionic, covalent, van der Waals, mixed bonded, metallic, and inorganic liquids crystallise within laboratory time scales when they are cooled down somewhat below their respective equilibrium melting points or liquidus temperatures (T_m). Turning to the realm of inorganic glasses, researchers and engineers frequently struggle against unwanted spontaneous crystallisation (also known as devitrification) when they try to make glasses in their laboratories or industries; whereas others try to design and control crystallisation in glass article interiors to produce a type of nanoporcelain known as glass-ceramics^2,3,4. In fact, “crystallisation” is so important that it is the most frequent keyword used by researchers in the last 200 years of glass science history⁵.

However, to the best of our knowledge and to our great surprise, crystallisation has never been reported to occur in pure unseeded supercooled Na₂O·Al₂O₃·6SiO₂ (Albite) and B₂O₃ liquids, at atmospheric pressure. These materials display outstanding glass-forming ability (when cooled from the liquid state), and are likely the most stable stoichiometric oxides against crystallisation (on heating from the glassy state) discovered to-date! In the glass science field, a “stoichiometric glass” is one that has a thermodynamically stable isochemical crystalline phase. This definition should not be confused with the traditional “stoichiometry”, that is, the calculation of relative quantities of reactants and products in chemical reactions.

In this article we will focus on both glasses: in Albite glass because it is also outstandingly chemically durable and may, for instance, provide an excellent matrix for the immobilization of radioactive fission products; and on B₂O₃ because this is one of the main oxide glass-formers used in a plethora of commercial glass products. In addition, Albite is a strong liquid, whereas B₂O₃ is fragile, and enough information on their structures and properties is available to perform sufficient calculations to develop a strong argument to explain their outstanding glass-forming ability. A liquid may be strong or fragile depending on the value of its fragility index m, defined by 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 Bowen⁶ 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 crystallising⁷. Uhlmann⁸ coined the term crystallisation “anomaly” when trying to decipher this unusual behaviour, which that is also shown by B₂O₃ glass, which only crystallises after seeding and under very high (GPa) pressures⁹. 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 B₂O₃ 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 B₂O₃), and perhaps even in its growth (B₂O₃). This crystallisation anomaly has been a matter of debate for many decades!

The typical stoichiometric oxide glass forming compositions (such as Albite and B₂O₃) 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 B₂O₃ 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 B₂O₃. 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 B₂O₃ 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 (Δμ)

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 Δμ ≡ −ΔG = G_l − G_c and is positive for temperatures below the melting point^14,15. In addition, the absolute value of Δμ increases with decreasing temperature from the melting point (that is, increasing undercooling). Equation (1) provides a way to calculate Δμ for a closed system under isobaric condition^14,15.

In the above equation, ΔH_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 Δμ(T_g)/ΔH_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’s¹⁶ 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.

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 (B₂O₃ 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 database⁷⁷.

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^17,18,19 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 0.4–0.6²⁰. 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 Eyring²¹ 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^26,27,28 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.

In these equations, η_∞ is the asymptotic value of the equilibrium viscosity when T → ∞, and A, B, C, and T₀ are adjustable parameters. Figure 2 shows viscosity data and fitted curves for Albite and B₂O₃ 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.

Figure 2: Viscosity data for Albite^{78,79,80,81,82,83,84,85,86} and B₂O₃^87,88,89,90 and fitted curves using the VFT equation.

These are examples of “strong”, almost linear (Albite) and “fragile”, highly curved (B₂O₃) liquids.

Table 1 Fitted values of the viscosity parameters for both liquids.

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^{34,35,36,37,38,39}, which yield good results for various oxide glass formers³⁸ 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 (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,

and the time-lag for crystal nucleation, τ (time necessary for the establishment of a stationary embryo size distribution up to the critical size),

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/m², assumed to be the same used in the macroscopic growth equations); k_B is the Boltzmann constant; T is the absolute temperature; Δμ_v is the driving force for crystallisation in units of J/m³; 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 factors⁴² 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 B₂O₃ (T_gr = 0.76) gives an estimated time-lag of more than 10¹⁰ seconds (about 300 years).

Figure 3: Experimental values of the nucleation time-lags at T_max for 12 oxide compounds that show homogeneous nucleation in laboratory time scales.

Data from ref. 43. The dashed vertical lines refer to the expected values for B₂O₃ and Albite from a linear extrapolation.

Figure 4(a) and (b) shows the cupola shape of the crystal nucleation rate curves for glassy Albite and B₂O₃. 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.

Figure 4

Normalized crystal nucleation rate curves versus reduced temperature for the cases of homogeneous nucleation (θ = π rad) and heterogeneous nucleation (θ = π/7 rad) for (a) Albite and (b) B₂O₃. A plot for lithium disilicate is shown in (c) to compare our calculations for Albite and B₂O₃ with experimental homogeneous^{91,92,93,94,95,96,97,98,99,100,101,102,103,104} and heterogeneous^44,45 crystal nucleation data. The value of I_max for the heterogeneous crystal nucleation data was taken from Fig. 6 of ref. 45. The vertical dashed line is the estimated T_g (where the viscosity is 10¹² Pa.s.).

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 B₂O₃ 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 d₀ 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 Δμ_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 Albite⁴⁶ and Eq. (17) for B₂O₃⁴⁷:

Table 2 Some properties of glassy and crystalline Albite and B₂O₃.

with C_p,c in J/K.mol and T in Kelvin.

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_KS^48,49,50. 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 B₂O₃. 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 B₂O₃ (T_gr ∼ 0.76) are likely extremely small.

Figure 5: Maximum experimental steady-state homogeneous crystal nucleation rates at T_max versus reduced T_g (T_gr) for 51 oxide glass-formers that undergo internal homogeneous nucleation.

Data from ref. 43. The full line is the predicted I_st(T_max) using Eqs (3), (9), (16) and (17) from Gupta et al.¹⁰⁵ In these calculations we considered reasonable values of C₂ ≡ C_VFT/T_m = 4, I₀ = 10⁴² m⁻³ s⁻¹, and C₁ ≡ (Δμ/ΔH_m)²W_c/k_BT_m equal to 5.0 and 6.5. The vertical dashed blue line refers to the T_gr of B₂O₃ and the red line to Albite.

Apart from three outliers, the theoretical prediction is sufficient to describe all the other experimental data within the uncertainty (that is, the range considering C₁ 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 growth²⁰. 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 entropy⁵¹, ΔH_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 crystal growth. The parameter f can be estimated using Eq. (19) if one considers screw dislocations that form Archimedean spirals⁵² at the advancing liquid–crystal interface.

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 book²⁰.

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 d₀, 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.

Figure 6

Simulations with the screw dislocation crystal growth model for (a) Albite and (b) B₂O₃. The experimental data points in (a), referring to the growth of crystalline layers in seeded Albite glass observed by Uhlmann et al.¹⁰, match the curve calculated with d_o = 1 Å. The horizontal dashed line refers to the value of growth rate for which the crystals would grow 1 μm in 1 day.

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 B₂O₃. 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 B₂O₃), 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 B₂O₃ 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 B₂O₃ 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 empirically⁵³ that the densities of the parent glass and corresponding crystal phase are quite similar (Δρ/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 Δρ/ρ_glass. In the case of Albite, Δρ/ρ_glass = 15% and for B₂O₃ the density mismatch is approximately 34%.

The Δρ/ρ_glass ratio gives clear signs of structural similarity or dissimilarity between the parent glass and its isochemical crystal, hence Albite and B₂O₃ 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, Δμ_V, will be weak. However, as the crystal nucleation rates exponentially diminish with an increase in σ_cm³/Δμ_V² (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 B₂O₃, 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 surfaces⁵⁴. 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 Brown⁵⁵ 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, ΔM₂/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 B₂O₃ Glass and Crystal

High-resolution nuclear magnetic resonance (NMR) measurements of Oxygen-17 in B₂O₃ glass⁵⁶ 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 B₂O₃ 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 (B₂O₃-I). The ease of vitrification of B₂O₃ 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 B₂O₃ glass sample sprinkled with B₂O₃-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 B₂O₃ glass with B₂O₃-II crystals, but then the density mismatch would be even higher (3.11 g/cm³, giving almost 70% difference) or other polymorphs.

One important remark is that the structure of B₂O₃-I is exclusively made by BO₃ triangles⁶¹, which is significantly different from the ∼ 70% boroxol rings that the glass is made. In a recent publication, Wright and Vedisheva⁶² suggested that the excellent glass-forming ability of B₂O₃ is due to the need to either break up the boroxol groups to form crystalline B₂O₃-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 boroxols⁶².

This finding for the inorganic oxide glass-formers Albite and B₂O₃ 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.

Table 3 Physical parameters controlling crystal nucleation in oxide glass formers that undergo homogeneous nucleation in comparison to Albite, B₂O₃, and LS2.

Finally, it is quite interesting to note that Albite liquid is strong; its fragility index m is 26. However, other strong liquids, such as SiO₂ and GeO₂ easily crystallise when heated above T_g. On the other hand, B₂O₃ 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 (B₂O₃) 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 Δρ/ρ_glass and large T_g/T_m.