Unraveling strongly entropic effect on $\beta$-relaxation in metallic glass: Insights from enhanced atomistic samplings over experimentally relevant timescales

The Johari-Goldstein secondary $\left(\beta \right)$ relaxation is an intrinsic feature of glasses, which is crucial to many properties of disordered materials. One puzzling feature of $\beta$-relaxation is its wide relaxation peak, which could imply a critical role of entropy. Here we quantify the activation entropy related to the $\beta$-relaxation in metallic glass via well-tempered metadynamics simulations. The activation free energy of the $\beta$-relaxation drastically decreases with increasing temperature, indicating a strongly entropic effect that may contribute a multiplication prefactor up to several orders of magnitude to the frequency. We further argue the entropic effect by linear extrapolation of the temperature-dependent activation free energy to 0 K, which gives rise to activation energy, in agreement with the barrier spectrum explored by the activation-relaxation technique. The entropic effect signifies the multiplicity of activation pathways which agrees with the experimentally found wide frequency domain of the $\beta$-relaxation.

Figure 1

WT-MetaD sampling of $\beta$-relaxation. (a) Time evolution of RMSD. The insets show the first two events, indicated by the sudden jump in CV. Only the atoms with displacement larger than 2 Å are shown. Green and bronze spheres represent Zr and Cu atoms, respectively. The arrows denote the displacement vectors. (b) Free-energy profile versus RMSD. The inset shows the bias potential.

Figure 2

Dependence of frequency and activation free energy on the size of perturbed atom groups. (a) RDF of the glass model. Several characteristic positions are labeled. (b) The predicted frequency as a function of the sampled cluster size. Once the cutoff distance is larger than 4 $\AA$ (including 12 atoms), the frequency starts to increase sharply, which indicates artificial introduction of a softer mechanism other than the surveyed local structural excitation of looplike motion. The insets denote the perturbed clusters along with their average activation free energy.

Figure 3

Convergence of activation free energy with respect to sampling number. (a) Box-and-whisker plot of $\mathrm{\Delta }F$ up to 100 samplings. (b) and (c) Barrier spectra after 50 and 100 samplings, respectively. The curves denote log-normal fits.

Figure 4

Temperature dependence of activation free energy. (a) From top to bottom, the barrier spectra from 200 to 450 K. The curves represent log-normal fits. A dashed curve across the panels indicates decreased $\mathrm{\Delta }F\left(T\right)$ with increasing $T$. (b) Spectrum of activation energy at 0 K via ART. (c)–(e) Temperature-dependent barrier by assuming a parallel mechanism $\mathrm{\Delta }{F}_{para}$, sequential mechanism $\mathrm{\Delta }{F}_{seq}$, and effective mechanism $\mathrm{\Delta }{F}_{eff}$, respectively. The rhombus symbol in (e) denotes ART data.

Figure 5

Arrhenius plot of the occurring frequency of $\beta$-relaxation from well-tempered metadynamics in comparison with experimental data, including ${\mathrm{La}}_{68.5}{\mathrm{Ni}}_{16}{\mathrm{Al}}_{14}{\mathrm{Co}}_{1.5}$ [17, 27, 56] and ${\mathrm{Pd}}_{40}{\mathrm{Ni}}_{10}{\mathrm{Cu}}_{30}{\mathrm{P}}_{20}$ [57] in the literature and ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ in the present work.

Figure 6

Schematic illustration of the topological feature of PEL in terms of the anharmonic effect.