Real-time observation of the isothermal crystallization kinetics in a deeply supercooled liquid

Below the melting temperature Tm, crystals are the stable phase of typical elemental or molecular systems. However, cooling down a liquid below Tm, crystallization is anything but inevitable. The liquid can be supercooled, eventually forming a glass below the glass transition temperature Tg. Despite their long lifetimes and the presence of strong barriers that produces an apparent stability, supercooled liquids and glasses remain intrinsically a metastable state and thermodynamically unstable towards the crystal. Here we investigated the isothermal crystallization kinetics of the prototypical strong glassformer GeO2 in the deep supercooled liquid at 1100 K, about half-way between Tm and Tg. The crystallization process has been observed through time-resolved neutron diffraction for about three days. Data show a continuous reorganization of the amorphous structure towards the alpha-quartz phase with the final material composed by crystalline domains plunged into a low-density, residual amorphous matrix. A quantitative analysis of the diffraction patterns allows determining the time evolution of the relative fractions of crystal and amorphous, that was interpreted through an empirical model for the crystallization kinetics. This approach provides a very good description of the experimental data and identifies a predator-prey-like mechanism between crystal and amorphous, where the density variation acts as a blocking barrier.


Instrument resolution
The instrument resolution function ∆ was estimated by considering the width of the Bragg peaks in KBr powder. This material has a f cc structure with a lattice parameter a = 6.598Å that provides peaks even at relatively low scattering angles. In particular, for λ = 0.724Å, the (111) reflection is at 2θ = 10.9 • , a value that ensures a reliable determination of the instrument resolution also in the region of the first peaks of the α-quartz GeO 2 .
The sample was loaded into an Al cylindrical cell and measured at T = 373 K to avoid any water contamination of the KBr powder. Data were analyzed according to Refs. [S2,S3]. The empty cell contribution was not subtracted and Al Bragg peaks were thus included in the analysis. The so-obtained diffraction pattern is shown in Fig. S1.
The measured intensity was fitted as the sum of three components S(2θ) = S T (2θ) + S B (2θ) + bkg. (S1) The first term S T (2θ) accounts for the thermal diffuse scattering, S B (2θ) represents the Bragg peaks pattern of KBr and Al, whereas bkg is a flat background due to the incoherent contributions. The TDS contribution S T (2θ) was estimated using a very simple approximation from Ref.
where exp (−2W ) = exp −2B( sin θ λ ) 2 is the Debye Waller factor. The value B = (2.33 ± 0.09)Å 2 is taken from Ref. [S4]. The Bragg peaks contribution is written as a sum of Gaussian functions and it turns out to be: (S3) * Electronic address: marco.zanatta@univr.it The position of the ith peak 2θ i is given by the structure, whereas its area A i is fitted independently for each peak. Conversely, we assume that the full width at half maximum of each peak is completely given by the instrument resolution ∆. Following Ref. [S5] we can write that where W 2 , W 1 and W 0 depend on the instrument collimation and they can be obtained by fitting Eq. S1 to the data.
Fig . S1 shows the results of the fit on KBr data. The fit was refined by including both KBr and Al peaks, red and black lines in Fig. S1(b) respectively. The instrument resolution determined according Eq. S4 is reported in the inset of Fig. S1.
where X(t) is the fraction of transformed volume and k an effective rate constant depending on the nucleation and growth rates. The exponent n is termed Avrami exponent and is expected to assume integer or half an integer values. The Avrami exponent depends on the characteristics of the process. As an example, for continuous nucleation and 3D spherical growth, the Avrami exponent is n = 4. The JMAK equation implies a complete crystallization of the amorphous medium, which appears in contrast with the observed time evolution of the crystallized fraction A c . Figure S2 shows a fit of A c with Eq. S5 using k and n as free parameters. The results is represented with a blue dashed line and it does reproduce neither the shape nor the long-time behavior.
In order to improve the agreement at long t, we can introduce a constant scale parameter X c accounting for the incomplete crystallization. Consequently, Eq. S5 can be rewritten as The result of the fit with this modified JMAK equation is shown in Fig. S2 (solid blue line). The scale parameter results X c = 0.76 ± 0.01 while the Avrami exponent is n = 1.92 ± 0.02. Equation S6 provides a considerably better agreement than the JMAK equation, in particular in the first stages of the crystallization processes. However, as the crystallized fraction increases, the model is still not able to accurately reproduce the shape of A c as the approach proposed in the paper.  As a matter of fact, at the beginning of the crystallization process, the hypotheses of random nucleation and similar growth of spherical regions hold. Increasing the transformed fraction in a non-diffusive environment (T T m ), the density difference between crystallized and amorphous region influences both the nucleation and the growth mechanism, thus producing the observed non-complete crystallization.