Decoupling between calorimetric and dynamical glass transitions in high-entropy metallic glasses

Glass transition is one of the unresolved critical issues in solid-state physics and materials science, during which a viscous liquid is frozen into a solid or structurally arrested state. On account of the uniform arrested mechanism, the calorimetric glass transition temperature (Tg) always follows the same trend as the dynamical glass transition (or α-relaxation) temperature (Tα) determined by dynamic mechanical analysis (DMA). Here, we explored the correlations between the calorimetric and dynamical glass transitions of three prototypical high-entropy metallic glasses (HEMGs) systems. We found that the HEMGs present a depressed dynamical glass transition phenomenon, i.e., HEMGs with moderate calorimetric Tg represent the highest Tα and the maximum activation energy of α-relaxation. These decoupled glass transitions from thermal and mechanical measurements reveal the effect of high configurational entropy on the structure and dynamics of supercooled liquids and metallic glasses, which are associated with sluggish diffusion and decreased dynamic and spatial heterogeneities from high mixing entropy. The results have important implications in understanding the entropy effect on the structure and properties of metallic glasses for designing new materials with plenteous physical and mechanical performances.

M etallic glass (MG), or, vitrified metal, is formed by rapidly quenching a liquid melt to bypass the crystallization process, and finally yields a configuration frozen state on a laboratory timescale [1][2][3][4][5] . Owing to the disordered atomic structure and out-of-equilibrium state, MGs exhibit unique and divergent thermodynamic and dynamic characteristics, especially when approaching the glass-transition temperature (T g ) 3,6,7 . Conventionally, calorimetric T g is defined as the temperature at which the specific heat has an abrupt jump and is commonly determined by calorimetric or thermal mechanical approaches [8][9][10] . From a dynamical relaxation perspective, diverse relaxation responses emerge with increasing temperature, from local reversible β-relaxation to global irreversible α-relaxation or a dynamic glass transition that is accompanied by the activation and subsequent percolation of shear transformation zones (STZs) [11][12][13][14] . Generally, since the consistency in the fundamental mechanism of glass transitions, the calorimetric T g follows the same trend as that of the dynamic α-relaxation temperature (T α ), i.e., MGs with high T g always exhibit high T α [15][16][17][18] .
Conventionally, the traditional MGs design strategy is based on one principal element by adding secondary and more elements around deep eutectic points in the phase diagrams, which may restrict the discovery of numerous combinations with unique and divergent physical and mechanical properties [19][20][21] . Recently, an intriguing alloying approach was developed to design new type of near-equiatomic solid solution metallic alloys, defined as multiple-principal-element alloys or high-entropy alloys (HEAs), which broke the conventional metallurgy development strategy, and exhibited great potentials for developing advanced structural and functional materials [22][23][24][25][26][27][28][29][30][31][32] . Owing to the equimolar concentration of each component, HEAs have distinctive physical and mechanical properties [26][27][28][29][30][31][32] arising from the effects of high configurational entropy, large lattice distortion, and sluggish diffusion as well as the cocktail effect 27,33 . Inheriting the distinct properties of MGs and HEAs, the new combinative glass-formed systems termed as "high entropy metallic glasses (HEMGs)" present high thermostability with depressed crystallization kinetics and superior magnetic properties, which reflect the theme of "more is different" [34][35][36] . However, the core critical behaviors of HEAs, especially the effect of sluggish diffusion on the thermodynamic and dynamic properties of HEMGs have not been systematically studied.
In this study, we utilized the strategy of the equivalent substitution elements to design La(Ce)-based, Pd(Pt)-based, and Ti (Zr)-based MGs and HEMGs, as the model systems to investigate the high-entropy effect on the structure and dynamics of glassforming alloys.  38 . Figure 1b presents the entropy of mixing of the three MGs systems, calculated by 4S mix ¼ ÀR∑ n i¼1 x i lnx i , where R is the gas constant and x i is the mole fraction of the ith element 27 . The designed HEMGs yield the values of 4S mix with 1.38 R, 1.67 R, and 1.57 R for LaCe-HEMG, PdPt-HEMG, and TiZr-HEMG, respectively, which are higher than the value generally defined as high entropy (1.36 R) 33 . Differential scanning calorimeter (DSC) traces of La 55 Ni 20 Al 25 , Ce 55 Ni 20 Al 25 , and LaCe-HEMG obtained at a heating rate of 0.33 K s −1 are shown in Fig. 1c. The obvious glass transition and crystallization signals confirm the glassy state of the three samples. Calorimetric T g is determined from the intersection of the tangent lines of the onset of transformation and is indicated by arrows (Fig. 1c). LaCe-HEMG with the highest 4S mix yields an intermediate T g , PdPt-HEMG and TiZr-HEMGs show similar tendencies ( Supplementary  Figs. 2a and 3a), which are consistent with the rule of mixture 39,40 . In general, MGs exhibit multicomplex relaxation dynamics and αrelaxation as the main relaxation mode is directly related to viscous flow and the glass transition 3,41 . Figure 1d presents the temperature dependence of the loss modulus (G " ) of La 55 Ni 20 Al 25 , Ce 55 Ni 20 Al 25 , and LaCe-HEMG at 1 Hz with a constant heating rate of 0.05 K s −1 (normalized by the maximum peak value). Two relaxation modes are clearly exhibited, which are termed as the β-relaxation and α-relaxation, from the low to high temperatures, respectively. With half of La replaced by Ce, LaCe-HEMG exhibits the highest α-relaxation temperature (T α ), which is not synchronous with the calorimetric T g (Fig. 1c). According to conventional wisdom, a glass with a higher calorimetric T g always poses higher difficulty in activating α-relaxation and thus has a larger value of T α 15,42 . Besides, T α presents a monotonous tendency in the case of regular element substitution 15 and follows the rule of mixture, which is the same as the calorimetric T g . Nevertheless, LaCe-HEMG, with a modest thermodynamic devitrification behavior, exhibits the uppermost dynamic glass-transition process (inset of Fig. 1d). Identical results were observed in the other two MGs systems, viz. Pd(Pt)CuNiP and Ti(Zr)CuNiAl (Supplementary Fig. 2b and Supplementary Fig. 3b). In addition, the consecutive decreased storage modulus excludes the possible crystallization near the α-relaxation process as shown in Supplementary Fig. 4, which indicates that the α-relaxation peak is not terminated by the early crystallization. Therefore, our findings indicate a decoupling between the calorimetric and dynamical glass transitions of HEMGs.

Decoupling of calorimetric
Fragility and viscosity of HEMGs. Fragility and viscosity are the most fundamental properties, reflecting the temperaturedependent relaxation and dynamic heterogeneity of glasses in supercooled liquid regions. Therefore, the divergent calorimetric and dynamical glass-transition behaviors of the HEMGs are expected to be disclosed by fully understanding the kinetic behaviors near the glass transition. Fragility can be determined by the variation in T g as a function of heating rate 42,43 . Figure 2a shows the DSC traces with the heating rates ranging from 25 to 600 K s −1 for La(Ce)-HEMG system. T g and the crystallization temperature (T x ) shift to higher temperatures with the increase of heating rate, which arises from the involvement of thermal activation in these kinetic processes. Figure 2b presents the heating rate dependence of T g for La 55 Ni 20 Al 25 , Ce 55 Ni 20 Al 25 , and LaCe-HEMG. The particular average structural relaxation time, τ, can be obtained by τ ¼ is the width of the glass transition with a constant value of 30 K, and q H is the cooling rate. Furthermore, τ can be well described by the Vogel-Fulcher-Tammann (VFT) equation 43 : where τ ∞ is the relaxation time at infinite temperature with a constant value of 1 × 10 −14 s, T 0 is the VFT temperature, D is the fragility parameter, and fragility m = 16 + 590/D 44 Fig. 2b with Eq. (1). In comparison with other MGs, LaCe-HEMG exhibits a strong glass behavior in terms of the lowest fragility. Figure 2c presents the nonequilibrium viscosity of the La(Ce) NiAl MGs as a function of temperature, which was measured by thermomechanical analysis (TMA). As the temperature increases above T g , the viscosity decreases rapidly. While, the viscosity of LaCe-HEMG is obviously higher than the others, suggesting the higher dynamic stability and more sluggish dynamic behavior of HEMG upon heating. Therefore, a plausible origin of the retarded α-relaxation for LaCe-HEMG could be sluggish diffusion from the high-entropy effect in HEAs 24,25 .
Atomic structure. The relaxation behaviors of MGs are related to their spatial heterogeneity in the nanoscale domains 12,45 . Fig. 3a, d, g presents high-resolution transmission electron microscope (HRTEM) images of LaCe-HEMG, La 55 Ni 20 Al 25 , and Ce 55 Ni 20 Al 25 , respectively. The homogeneous mazelike features and the diffraction halo in the selected area electron diffraction (SAED) patterns (inset in Fig. 3a, d, g) confirm the amorphous nature of the samples. Apparently, no clear atomic structural differences and phase separation could be identified in the three MGs down to the sub-nanoscale from the phase-contrast images. However, when scanning transmission electron microscopy (STEM) analysis with a high-angle annular dark-field (HAADF) detector was performed to image the local atomic structures (Fig. 3b, e, h), the HAADF-STEM images present obvious contrast variations with domain sizes of several nanometers. It appears that the contrast variation of LaCe-HEMG is slightly ambiguous, together with a concomitant decrease in the domain size. The averaged domain sizes, measured by a rotationally averaged fast Fourier transform method 46 , correspond to feature lengths of 2.491 ± 0.297 nm, 2.380 ± 0.248 nm, and 1.851 ± 0.206 nm for La 55 Ni 20 Al 25 , Ce 55 Ni 20 Al 25 , and LaCe-HEMG, respectively (Supplementary  Table 1). Accordingly, LaCe-HEMG with the highest value of 4S mix possesses the least contrast variation and the smallest domain size. Since the HAADF contrast indicates local chemical or density variations, STEM energy-dispersive spectroscopy (EDS) with a spatial resolution of~3 Å was applied to further reveal the elemental distributions. As shown in Fig. 3f Fig. 3c. Therefore, the contrast in the HAADF images is attributable to the nanoscale chemical fluctuation. The higher T α of HEMGs plausibly originates from these relatively inconspicuous chemical variations with less pronounced spatial Fig. 1 The mixing of entropy, calorimetric, and dynamical behaviors. a ΔH mix (kJ mol −1 ) for the constituting atomic pairs in the La(Ce)NiAl system. b Calculated entropies of mixing for the three MG systems. c DSC traces of the La(Ce)NiAl system with a heating rate of 0.33 K s −1 . The arrows indicate the calorimetric glass-transition temperature (T g ). d Temperature dependence of the loss modulus of the La(Ce)NiAl system at 1 Hz with a constant heating rate of 0.05 K s −1 , normalized by the maximum peak value. The arrows indicate the α-relaxation temperature (T α ), and the inset shows the evolution of T g and T α with the components. contrast heterogeneity and sluggish diffusion, which may depress the dynamical relaxation behaviors near the glass-transition temperatures.
Crystallization behavior. HEMGs with higher T α suggest sluggish cooperative atomic movement in the moderately supercooled liquid states, which, in principle, should also influence the crystallization behavior. Figure 4a shows the XRD spectra of crys-  (Fig. 4d), which appears to be formed by heterogeneous crystallization. In addition, the SAED patterns of the three heat-treated specimens still exhibit amorphous diffraction halos, suggesting the insufficient crystallization processes (the inset of Fig. 4b-d). Figure 4e shows the HRTEM image of the crystallization region for LaCe-HEMG (dashed red frame in Fig. 4d), the large precipitate is comprised of nanocrystals with an average grain size of~8 ± 1.7 nm, which are embedded in the amorphous matrix and smaller than the crystallites in the crystallized La 55 Ni 20 Al 25 and Ce 55 Ni 20 Al 25 MGs ( Supplementary Fig. 5 and Supplementary Table 1). It appears that HEMGs demonstrate a suppressed crystal nucleation and growth and, consequently, the enhanced thermal stability of the supercooled liquids.

Discussion
HEMGs exhibit retarded α-relaxation (dynamical glass transition) and distinct crystallization resistance, which may originate from the sluggish diffusion hypothesis on account of the effect of high mixing entropy. β-relaxation (or Johari-Goldstein relaxation) is considered the dynamical precursor of α-relaxation and plays a promising role in understanding the dynamical glass transition 47,48 . The frequency dependence of β-relaxation evolution was explored by dynamic mechanical analysis (DMA) with an amplitude of 0.1% and testing frequencies (f) ranging from 0.5 to 16 Hz (inset of Fig. 5a and Supplementary Fig. 6). Figure 5a shows the frequency dependence of the β-relaxation peak temperatures  16,38 . LaCe-HEMG exhibits moderate activation energy for β-relaxation which follows the same tendency as calorimetric T g . Since β-relaxation only involves local atomic motion, the high-entropy effect appears not to influence short-range atomic diffusion. Accordingly, the temperature-dependence of α-relaxation is shown in inset of Fig. 5b and Supplementary Fig. 7. The activation energy values of α-relaxation (E α ) ( Fig. 5b  Remarkably, LaCe-HEMG shows the highest α-relaxation activation energy, suggesting that HEMGs have the highest energy barrier for the dynamical glass transition, and thus the highest T α . Figure 5c, d shows the Kissinger plots of thermal T g and T x at different heating rates from the flash DSC traces (Fig. 2a). The measured activation energies for the calorimetric glass transition and crystallization are listed in Table 1. Again, LaCe-HEMG has intermediate activation energy for the glass transition and the highest energy for crystallization, which is in line with the calorimetric T g and crystallization behavior.  These structural characterization and dynamic measurements build up a comprehensive picture of the decoupling between the calorimetric T g and DMA T α of HEMGs as illustrated in Fig. 6. HEMGs with nearly equimolar components and high configuration entropy, could suppress the dynamic heterogeneity in the supercooled liquids on account of the reduced Gibbs free energy, and then inherit to the glassy state with relatively small and homogeneous nanoscale domains (Figs. 3b and 6a) 49 . These nanoscale spatial domains with an average size of~2 nm are analogous to the size of STZs, and therefore might be associated with the structural origin of the β-relaxation process 14 (Fig. 6b). Consequently, HEMGs with small nanoscale domains possess vibrant potential STZs on account of the favorable cooperative transition probability by "Adam-Gibbs" theory 50 , with the minor activation energy of β-relaxation ( Fig. 5a and Table 1).
Furthermore, β-relaxation could be associated with the activation of isolated STZs, which are confined within the elastic matrix and do not involve long-range diffusion 14 . Therefore, the β-relaxation behavior of HEMGs is nearly unaffected by the sluggish diffusion effect. When the density and size of STZs reach a critical point with increasing temperature, their percolation of STZs contributes to distinct α-relaxation 51 . Figure 6b, c illustrates the relationship between β-relaxation and α-relaxation based on the structural and energy perspectives. In the α-relaxation process, the expansion and percolation correspond to an increased mismatch penalty between STZs, which relates to long-range cooperation rearrangement and atomic diffusion 52,53 (Fig. 6b). Small STZs in HEMGs might expect to consume excess energy to reach a critical transition size 52,53 , and subsequent percolation with long-range propagation could plausibly be suppressed by sluggish diffusion of HEMGs. Therefore, HEMGs with relatively homogeneous structures (Figs. 3b and 6a) inhibit large-scale atomic migration in the process of α-relaxation, resulting in the delayed α-relaxation transition and elevated activation energy ( Fig. 1d and Table 1). Analogously, the STEM-EDS mappings of crystallized MGs (Fig. 4f and Supplementary Fig. 5) verify a longrange diffusion and migration, suggesting that sluggish diffusion affects the crystallization behaviors of HEMGs and results in higher activation energy (Figs. 5d, 6c and Table 1).
Besides relaxation, the calorimetric glass transition is accompanied by an increase in heat capacity with the generation of free volumes near T g , which is not sensitive to the heterogeneous domain size 10 . Entropy might induce negligible effects on the thermal glass-transition behaviors, which is consistent with the fact that LaCe-HEMG presents the relatively low activation energy of calorimetric T g (Fig. 5c and Fig. 5 The activation energies for β-relaxation, α-relaxation, glass transition, and crystallization of the La(Ce)NiAl system. a Arrhenius plots of the temperature and frequency dependence of the peak temperature of β-relaxations for the La(Ce)NiAl system. The inset shows the temperature-dependent loss modulus of LaCe-HEMG with testing frequencies ranging from 0.5 to 16 Hz. b Frequency and temperature dependence of the peak temperature of αrelaxations fitted with the Arrhenius relationship. The inset presents the frequency-dependent loss modulus of LaCe-HEMG at different temperatures. c, d Kissinger plots of thermal T g and T x versus heating rate for the La(Ce)NiAl system, respectively. The error bars were obtained by standard deviation from three measurements of T β , T α , and T g , T x on DMA and DSC curves, respectively. T g could be mainly determined by local thermal diffusion and follow the rule of mixture characterized by a monotonous relationship with composition. Base on systematic examinations of the calorimetric and dynamical glass transitions and atomic structures, we can conclude that the high mixing entropy affects the glass transition and crystallization behaviors of HEMGs, as illustrated by the proposed energy landscape (Fig. 6c). It is expected that the HEA 'core effect', i.e., sluggish diffusion on the spatial heterogeneity and the decreased dynamics of supercooled liquids, will provide freedom to solve the fundamental issues, such as the glass transition, relaxations, and aging behaviors, of MGs. Differential scanning calorimetry (DSC) measurements. The Perkin Elmer DSC 8500 was used to perform the thermodynamic transition process of the glassy ribbons with 20 mL min -1 flowing pure argon gas to prevent possible oxidation during heating. The values of glass-transition temperature T g (Fig. 1) were determined using DSC at a constant heating rate of 0.33 K s −1 . Flash DSC (Mettler, Flash DSC 1) with wide heating rates was used to measure the glass transition and crystallization behaviors of La 55 Ni 20  Dynamic mechanical analysis (DMA). The RSA-G2 DMA (TA Instruments) was used to measure the loss and storage moduli of the MGs ribbons. The loss modulus shows a sharp maximum near the glass-transition temperature associated with α-relaxation. The broad maximum at lower temperature represents the β-relaxation. The α-relaxation temperature (T α , Fig. 1) was determined by heating the glassy ribbons from room temperature to over crystallization temperature at 1 Hz with a heating rate of 0.05 K s −1 . The frequency dependence of β-relaxation evolution (Fig. 5) was explored by DMA with an amplitude of 0.1% and testing frequency f ranging from 0.5 to 16 Hz. The temperature dependence of α-relaxation evolution (Fig. 5) explored by DMA with an amplitude of 0.1% and testing temperature ranging near T g .

Methods
Viscosity measurement. The thermomechanical analyzer Q400 TMA (TA Instruments) was used to measure the length change of glassy ribbon samples during heating in the argon fluxion atmosphere (20 mL min −1 ). The measurement of viscosity in Fig. 2 was done with a heating rate of 0.33 K s −1 from room temperature to crystallization temperature with a contrast probe load of 0.1 N. The value of viscosity (η) was calculated from the flow stress by the equation η ¼ σ 3_ ε , where σ is the stress and _ ε is the strain rate.
X-rays diffraction. The phase transformation after annealing the glassy ribbons was characterized by RIGAKU Ultima IV X-Ray diffractometer with monochromatic Cu K α radiation. The ribbons were heated to the onset temperature of crystallization followed by cooling down.
Transmission electron microscopy characterization. Microstructures of the specimens were characterized by a Cs-corrected TEM (JEM-2100F, JEOL, 200 kV) equipped with double spherical aberration correctors for both the probe-forming and image-forming lenses. The high-resolution TEM images were captured by a television-rate camera (Gatan, UltraScan 1000), and the high-angle annular darkfield (HAADF) images were recorded using an annular-type scanning TEM detector. Energy-dispersive spectroscopy (EDS) mappings were acquired at STEM mode with a scan step of 3 Å for elemental analysis.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. Correlation between the evolution of STZs and relaxation behaviors with increasing temperature. The MGs finally yield crystallization state at T x . c Schematic potential energy landscape for MGs, β-relaxation relates a reversible hopping event in an inherent megabasin, while α-relaxation corresponds to an irreversible hopping event across different landscape megabasins.
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