Quantifying the origin of metallic glass formation

The waiting time to form a crystal in a unit volume of homogeneous undercooled liquid exhibits a pronounced minimum τX* at a ‘nose temperature' T* located between the glass transition temperature Tg, and the crystal melting temperature, TL. Turnbull argued that τX* should increase rapidly with the dimensionless ratio trg=Tg/TL. Angell introduced a dimensionless ‘fragility parameter', m, to characterize the fall of atomic mobility with temperature above Tg. Both trg and m are widely thought to play a significant role in determining τX*. Here we survey and assess reported data for TL, Tg, trg, m and τX* for a broad range of metallic glasses with widely varying τX*. By analysing this database, we derive a simple empirical expression for τX*(trg, m) that depends exponentially on trg and m, and two fitting parameters. A statistical analysis shows that knowledge of trg and m alone is therefore sufficient to predict τX* within estimated experimental errors. Surprisingly, the liquid/crystal interfacial free energy does not appear in this expression for τX*.


Supplementary Fig. 2
Plot of log(dmax 2 ) as a function of Senkov's F1 parameter as given in eqn. (7) of the Supplementary Methods below. Data used to determine the F1 parameter are given in Supplementary  Table  I Variation of alloy liquidus temperature TL with the atomic fraction, x , of Ni in the ternary (Pd1--xNix)80P20 alloys. See Supplementary  Table  I,  entries  20--23 and the Supplementary Methods section below for a detailed description of the data used to construct the plot. The solid curve is a parabolic fit to the data similar to that used by Chen [22,23] to describe the composition variation of the VFT parameters in the ternary system.
Variation of the Turnbull parameter trg with x, the atomic fraction of Ni in the ternary (Pd1--xNix)P20 glass forming alloys. See Supplementary Table I, entries 20--23, and the Supplemental Methods section below for description of the data. Solid curve is a parabolic fit to the data. See Supplementary Methods section below for a discussion of the data.

Supplementary Table I: Metallic Glass Database
A compiled database for 42 metallic glass forming alloys. A description of the entries in the database and the methods used assess the data compiled are provided in the accompanying Supplementary Methods text below. References used as sources for data (last column in the table) are included in the list of references for the Supplementary Information. 25 [27,28] 25 15 [22,[27][28][29][30][31][32][33]37,38] see discussion in SI and Fig.4  Footnotes for Supplementary Table I # Values extrapolated using the composition dependence of m (measured data for 1.5--6 at.% boron are extrapolated to 0 % boron) using fitting function in ref. [1] * These values of m are estimated from high temperature data only and based on the early studies of equilibrium liquid viscosity as reported by Davies [4]. These data were not used in Figs. 2 and 3 since such values of m are known to be overestimated compared with those based on low temperature viscosity data. See discussion in SI text. In general, when low temperature viscosity data are reported in terms of VFT fitting parameters (D and T0), the Angell m parameter has been analytically determined from the identity relation:

Supplementary Methods Methods for Compiling the Metallic Glass Database
The data used to compile Supplementary Table I  Reference numbers are also indicated in other columns to assist the reader in identifying the source of specific parameter values. Column 10 also contains comments regarding the entries. Below, we describe the procedures and methods used to assess the literature data.

A. Assessment of Viscosity data (Tg and m)
Low temperature viscosity data in the literature near and above Tg are obtained using techniques such as Parallel Plate Rheometry [77], Beam Bending [64], and Creep Rate studies on wires and ribbons [21]. Data typically cover the range of viscosity from 10 6 --10 14 Pa--s. The lowest measureable viscosity is generally limited by intervening crystallization of the metallic glass. Data for alloys with poor stability against crystallization (small ΔT = TX-Tg, where TX is the crystallization temperature) are often restricted to viscosities above 10 10 Pa--s. Reported data are most often fit using the Vogel--Fulcher--Tammann (VFT) equation: where D, T0, and η0 (the liquid viscosity in the high temperature limit) are fitting parameters [78]. It is frequently assumed that η0 ≈ 10 --3 Pa--s. For most studies, best values of D and T0 and η0 are tabulated. As seen in eqn. (1), the Angell Fragility parameter, m, is the slope of a plot of log(η) vs. Tg/T evaluated at Tg [78]. It is easily shown that m can be expressed in terms of the VFT parameters as: Tabulated values of m in Table I were derived from this relationship for cases where the VFT--parameters were reported. Experimental uncertainty in fitting parameters D, T0, and the rheological Tg are propagated and give a corresponding uncertainty in m. The typical error in determining m is estimated below. Low temperature viscosity data can also be fit using other model functions such as the Cooperative Shear Model [79]. In this case, the m values are determined from: where n is an index describing the exponential decay of the flow barrier W(T) vs.
T/Tg [79]. Clearly, m can also be obtained directly from the slope of a log(η) vs. Tg/T plot at T = Tg. This direct method was used to obtain m in cases where digitized viscosity--temperature data are available. For some entries in the Table, Table I). Based on those data, we shall establish an empirical scaling relation between τX * and dmax 2 as described in the next section. This relation can be used to roughly convert each measure of GFA to the other.

C. Experimental measurements of τX * and empirical correlation with dmax 2
To measure τX(T) directly, containerless High Vacuum Electrostatic Levitation (HVESL) experiments have been employed [56,63]. Here, the sample is a liquid droplet typically 2.5 mm in diameter. The sample is heated to a temperature above TL, and then allowed to cool by free radiative heat loss (i.e. the Stephan--Boltzmann "T 4 " law). Observed free radiative cooling rates are typically ~5--30 K/s scaling roughly as TL 4 . The internal Fourier thermal relaxation time of a 2.5--mm diameter droplet is less than ~0.1 s [83]. The sample cools sufficiently slowly that internal temperature gradients are small within the sample. The sample is near isothermal. A typical experiment to measure τX(T) involves heating the sample well above TL using power absorbed from a laser beam(s) to achieve an initial steady-state temperature. The laser is suddenly switched off to allow free radiative cooling. Practically, this requires τX * > 1 s for a 2.5 mm diameter liquid drop. Thus, HVESL data for τX * are only for high GFA alloys (roughly 10 entries in Table I).
Supplementary Fig.1 is a plot of dmax 2 versus τX * for 8 alloys where both are measured. The data are plotted on a log--log plot to assess whether power--law scaling τX * d n is appropriate. A best least squares fit gives: τX * = 0.00419 dmax 2.54 (5) where τX * is in seconds and dmax in millimeters This result is strictly empirical. No effort has been made to interpret the exponent n = 2.54 except to note that the above arguments suggest n 2 when the effective volume exposed to the maximum nucleation rate is not considered while one expects n >2 if the effective volume scales with some power of the characteristic sample dimension. In Supplementary   Table I ( [50,56]. In the case of many metal-metalloid glasses, fluxing the melt (e.g. with boron oxide) is observed to significantly reduce heterogeneous nucleants and increase GFA. In this work, the values of dmax in Table I

GFA analysis using Senkov's parameter
Senkov [85] argued that τX * should be proportional to the viscosity of an undercooled liquid at the nose temperature T * . He combined this argument with the VFT equation to obtain a condition for glass formation: Where Rc is the critical cooling rate. The parameter F1 can be expressed in terms of trg and the VFT parameters: Senkov's parameter can be plotted for the ~40 entries in Table  I to test the validity of his hypothesis. Supplementary Fig.2 shows this plot for our database. It can be compared with Figs. 1, Fig.2, and Fig. 3 of the main article. A linear regression accounts for 88% of the variance in the GFA, significantly better than either trg or m alone (compared with Fig. 1 and Fig.2 of the main article where the respective correlation accounts for 60% and 50% of the variance in GFA)

GFA analysis for the ternary Pd--Ni--P system
The data used to construct Fig. 4 of the article are taken from several references [22,27,28,36,37,86--88]. For the ternary alloys (Pd1--xNix)80P20, Chen carried out extensive creep measurements on ribbon samples and obtained viscosity data [22,36]. The Turnbull parameter is obtained from the rheological Tg and the liquidus temperature TL of each alloy. Liquidus data were taken from the ASM ternary phase diagram database (vertical sections of the ternary diagram) [86] and the binary diagrams for the end points as displayed in Supplementary Fig.4. Accurate data from ref. [87] is included for the equiatomic alloy at x = 0.5. A parabolic fit is used for the