Size-dependent vitrification in metallic glasses

The vitrification (glass transition) of liquids supercooled below their melting temperature remains a central open problem in condensed matter physics. The manner in which vitrification occurs, and the resulting glass age, strongly influence glass properties and their evolution over time. Metallic glasses (MGs) are technologically important due to their superior mechanical properties and corrosion resistance, and these properties are sensitive to vitrification conditions. Fracture toughness correlates with the enthalpic state of the glass, commonly parameterized by the fictive temperature Tf, which depends on cooling rate. Conventional understanding ties vitrification to the main α relaxation with super-Arrhenius temperature dependence; thus cooling-rate-dependent Tf should mirror the temperature dependence of the α-relaxation time τ. However, prior work has shown that, at least for some Au-based MG, vitrification at low cooling rates proceeds with a weaker temperature dependence than α relaxation, implying additional mechanisms beyond α relaxation participate in vitrification. A further long-standing question is whether reducing sample size alters vitrification and relaxation. While nanoscale perturbations of dynamics have been seen in molecular and polymer glasses, and free-surface effects are confined to nanometric scales in MGs, significant size effects on vitrification kinetics have been reported in polymers at micrometer scales. Whether similar size-dependent vitrification exists in metallic glasses has remained largely unexplored, despite potential implications for mechanical behavior (e.g., ductile-to-brittle transitions, size-dependent fracture morphology). This work addresses how reducing sample size (micron scale) affects vitrification kinetics and atomic mobility in archetypal Au- and Pt-based MGs.

Early confinement studies in small-molecule glass formers (e.g., o-terphenyl, benzyl alcohol) found Tg depressions in nanopores below ~70 nm. Thin polymer films (e.g., polystyrene) exhibited Tg reductions below ~50 nm thickness, with extensive subsequent work revealing that any changes in α-relaxation dynamics typically occur only below ~10 nm. Electron correlation microscopy in MGs shows surface-perturbed dynamics at sub-nanometer to few-nanometer scales, indicating the α-relaxation length scale remains very small. In contrast, vitrification kinetics (as characterized by Tg or Tf) can show significant reductions at much larger length scales, even exceeding micrometers in polymeric systems without strong interfacial adsorption. To explain such observations despite bulk-like α-relaxation, models invoking equilibration via diffusion and annihilation of free-volume holes at free interfaces have been advanced and visualized directly in colloidal glasses. In metallic glasses, low-temperature excess endotherms linked to fast, non-α relaxation processes have been reported across various glassy systems (metals, plastic crystals, polymers, glucose, phase-change materials), especially after deep aging far below Tg. Prior Au-based MG work indicated vitrification kinetics may decouple from α relaxation at low cooling rates, implicating non-α processes.

• Materials: Two bulk metallic glass systems were investigated: Au49Cu26.9Si16.3Ag5.5Pd2.3 at.% ribbons (7 ± 1 μm thickness), produced by tilt casting then melt spinning; and Pt57.5Cu14.7Ni5.3P22.5 at.% nanofibers prepared via thermoplastic press-and-pull through a steel mesh at 543 K. Amorphous structure was confirmed by XRD. Short sections (3–2000 ng) were harvested for calorimetry.
• Sample sizing and geometry: Five specimens per composition with different sizes were prepared by cutting ribbons or nanoneedles. SEM on FSC chips (Hitachi TM3000 for Au-based; Zeiss Sigma VP for Pt-based) characterized morphology. A characteristic equivalent length leq = V/A (volume-to-free-surface-area ratio) was used, independent of geometry. For spherical-like samples, V and A were derived from diameters; for film-like samples, A from SEM and V from mass and density. Mass was estimated from the heat-flow step ΔHF near Tg using m = ΔHF/(Cp q), with Cp from conventional calorimetry and q the heating rate.
• Calorimetry and dynamic measurements: Fast scanning calorimetry (FSC) covered heating/cooling rates from 0.5 to 5000 K s−1. Step-response protocols (small temperature jumps) were used to obtain thermal susceptibility and, after Fourier transform, the complex specific heat c = c′ + i c″. The frequency-dependent reversing specific heat Cp,rev (≈ c′) was measured; the α relaxation manifests as a step in Cp,rev. The mid-step defines a characteristic α-relaxation time τ at each temperature.
• Vitrification scans: Specific heat scans were performed at selected heating rates qh after cooling at various rates q to probe vitrification behavior and endothermic overshoots. Tg values were determined using the Moynihan method, with liquid and glass Cp obtained from linear fits to Cp,rev in respective temperature regions.
• Data analysis: Temperature-dependent τ(T) was fitted with the Vogel–Fulcher–Tammann (VFT) equation τ = τ0 exp(D′ T0/(T − T0)); reported example parameters: D = 9.8, T0 = 311 K (Au-based MG); D = 7.4, T0 = 426 K (Pt-based MG). Cooling-rate-dependent Tg(q) at high rates was also fitted with VFT using the same D and T0 as τ(T). At low rates and small sizes, vitrification kinetics were analyzed with Arrhenius behavior of q−1 to extract activation energies of fast non-α processes. Size dependence was tested against the free-volume-hole diffusion (FVHD) model, which predicts log(qc−1) ~ 2 log(leq) − log[2D(Tg)] (i.e., qc−1 ∝ leq2 at fixed Tg). Insets of relevant figures confirmed scaling of qc−1 with leq2 at selected Tg values. Identical leq but different geometries were compared to test geometry independence in the FVHD framework.
• Crystallization control: Data exhibiting reduced Cp steps at the glass transition (indicative of partial crystallization) at the lowest rates were excluded from Tg analysis.

• Atomic mobility (α relaxation) is size independent: Temperature- and frequency-dependent Cp,rev steps overlap across sizes (from bulk-scale to several micrometers), yielding identical τ(T) for different leq. VFT fits of τ(T) are composition-dependent but size-invariant (e.g., D = 9.8, T0 = 311 K for Au-based; D = 7.4, T0 = 426 K for Pt-based).
• Pronounced size-dependent vitrification kinetics: At high cooling rates, Tg(q) follows the α-relaxation VFT dependence, independent of size. At lower cooling rates, Tg deviates significantly from α-controlled behavior, with stronger deviations for smaller samples (micron scale). The smallest samples exhibit Tg (Tf) depressions exceeding 40 K relative to bulk at the lowest rates avoiding crystallization (~1 K s−1), for both Au- and Pt-based MGs.
• Non-α fast equilibration processes dominate at low rates: Decoupling between Tg(q) and α relaxation increases with decreasing size, indicating additional faster processes assist vitrification. In Au-based MG, a pronounced low-temperature excess endotherm grows with decreasing cooling rate and size; in Pt-based MG, a weaker but detectable excess is found in Cp excess relative to a high-rate reference.
• FVHD model validation: For fixed Tg deep in the non-α regime (e.g., 370 K for Au-based, 500 K for Pt-based), qc−1 scales as leq2, as predicted by free-volume-hole diffusion to a free interface. Samples with identical leq but different geometry show identical Tg(q), supporting the V/A control predicted by FVHD.
• Activation energies of fast processes: Arrhenius analysis of qc−1 in the smallest samples yields Ea ≈ 75 ± 5 kJ/mol (Au49Cu26.9Si16.3Ag5.5Pd2.3; Tg ≈ 372 K) and Ea ≈ 180 ± 10 kJ/mol (Pt57.5Cu14.7Ni5.3P22.5; Tg ≈ 505 K), consistent with the slow Arrhenius process (SAP) proposed to govern equilibration in amorphous materials.
• Universality across compositions: When Tg depression relative to bulky samples is plotted versus cooling rate and leq, both MG systems display similar size and rate dependence.
• Practical implication: Micron-scale samples can reach thermodynamic states (Tf) far below bulk values within seconds, accessing low-energy glassy states otherwise requiring prolonged aging in bulk.

The study disentangles size effects on spontaneous equilibrium dynamics (α relaxation) from those on vitrification kinetics. The α relaxation, characterized by Cp,rev steps and τ(T), remains bulk-like across micrometer sizes, consistent with its nanometric cooperative length scale. In contrast, vitrification kinetics are strongly size dependent at low cooling rates, revealing that fast non-α mechanisms govern equilibration when α relaxation becomes too slow. The observed Tg depressions (>40 K) and qc−1 ∝ leq2 scaling substantiate a mechanism wherein free-volume holes diffuse to free interfaces and are removed (FVHD), accelerating equilibration in smaller samples due to larger surface-to-volume ratio. The low-temperature excess endotherm (strong in Au-based, weaker in Pt-based) further supports active fast processes deep below Tg. Extracted activation energies align with the slow Arrhenius process (SAP), suggesting liquid-like zones or shear transformation zones facilitate relaxation independently of the α process. These findings rationalize reported size-enhanced ductility in MGs: smaller samples more effectively equilibrate structural disorder under non-linear mechanical loading, reducing shear localization. Thus, while small samples achieve low Tf (thermodynamically stable), they can be kinetically more responsive to external stimuli, implying that ductility cannot be inferred from Tf alone without considering size-dependent kinetic stability.

This work demonstrates that metallic glasses exhibit size-independent α-relaxation dynamics but pronounced, micrometer-scale size dependence in vitrification kinetics. Using FSC across 0.5–5000 K s−1, the authors show that smaller samples, especially at low cooling rates, vitrify at much lower Tg/Tf (by >40 K) than bulk, due to fast non-α equilibration mechanisms. The data validate a free-volume-hole diffusion model (qc−1 ∝ leq2) and indicate activation energies consistent with a slow Arrhenius process governing low-temperature equilibration. These insights provide a route to rapidly access ultra-low-energy glassy states in MGs by modestly reducing sample size, with implications for mechanical behavior and for exploring fundamental questions such as the approach to the ideal glass. Future research should extend aging protocols on micron-sized MGs to create and study ultra-stable states, generalize to other MG chemistries, and directly correlate size-dependent vitrification pathways with mechanical responses under controlled non-linear stimuli.

• Crystallization at very low cooling rates limited the accessible Tg range; data showing reduced Cp steps (partial crystallization) were excluded.
• Arrhenius fits for activation energies were reliably performed only for the smallest samples; for larger samples the α to non-α crossover restricted the fitting window.
• The FVHD analysis used a one-dimensional diffusion approximation justified by small free-volume size relative to curvature; deviations from ideal geometries may introduce uncertainties.
• Mass and volume estimations relied on Cp values from conventional calorimetry and SEM-derived geometries, which may contribute systematic errors in leq.
• The study focused on two MG compositions; broader generality across MG families, different surface chemistries, and environmental conditions remains to be verified.