Metallic glasses exhibit secondary (β-type) mechanical relaxations whose structural origin is only partly understood. A 2020 Acta Materialia study of La-based metallic glasses tied the fast secondary relaxation to low-electron-density, liquid-like regions left behind by the formation of string-like solute-centered clusters, and the slow secondary relaxation to medium-range atomic order, within a generalized Maxwell picture of independent localized flow units. That study's framework leaves two linked questions unanswered, and they define this problem: (1) how to separate the geometric (atomic-packing) contribution from the chemical (electronic-bonding, charge-density) contribution in determining the size and population of these liquid-like regions upon alloying, and (2) how to extend the independent-flow-unit picture to a theory with inter-unit coupling that reproduces the cooperative relaxation modes — shoulders and excess wings on the β feature — arising in large connected string-like solute-cluster networks. Substantial progress on the structure–dynamics correlation has been reported — liquid-like atoms as carriers of the fast relaxation, electronic-heterogeneity and entropy effects on the β feature — but no published delineation of geometric versus chemical contributions to flow-unit formation and no structure-specific coupled-mode theory, so the problem remains open.

Background

Metallic glasses (MGs) are metals frozen into a disordered, liquid-like atomic configuration by extremely fast cooling; lacking a crystal lattice, they deform through localized atomic rearrangements rather than dislocation motion. When an MG is subjected to a small periodic stress in dynamic mechanical analysis (DMA), the phase lag between stress and strain defines a loss modulus whose spectrum shows a slow, cooperative α relaxation (the viscous flow associated with the glass transition) and, at lower temperature or higher frequency, a secondary β relaxation — also called Johari–Goldstein relaxation — produced by localized rearrangements of small groups of atoms. In many glass formers the β feature is itself composite: it can appear as a resolved shoulder on the α loss peak, or degenerate into an 'excess wing' — a broad, featureless tail of loss on the high-frequency flank of a relaxation peak rather than a resolved secondary peak. In La-based MGs an additional, even faster 'fast secondary relaxation' appears below the ordinary β. A standard way to read these spectra is the generalized Maxwell model: an array of viscoelastic units with a broad distribution of relaxation times. In this picture the sub-glass-transition features are attributed to 'flow units' or liquid-like regions — small, loosely packed patches of locally low viscosity and locally low atomic (and electron) density embedded in an elastic glassy matrix. The physical origin of these flow units, and of the β spectral shape, is a long-standing open problem: prior work had linked β relaxation variously to locally favored icosahedral-like packing (Liu et al. 2014), to dynamic-spatial heterogeneity (Zhu et al. 2016), to string-like collective rearrangements (Yu et al. 2017), to the caged vibration of small atoms in La-based MGs (Wang et al. 2018), to atomic-mass contrast between constituents (Cui et al. 2018), and to electronic heterogeneity in high-entropy MGs (Jiang et al. 2021) — mutually non-equivalent indicators, none of which separates the roles of packing geometry and electronic bonding. The β relaxation is also practically important: its strength correlates with the brittle-to-ductile transition and cryogenic toughness of MGs (Wang et al. 2017). The source study (Huang, Yuan et al., Acta Materialia 196 (2020) 88–100) combined synchrotron X-ray diffraction with ab initio molecular dynamics (MD — molecular dynamics driven by density-functional-theory forces, which resolve electronic charge density but only for small cells and very fast simulated quenches) on La-based MGs alloyed with different solute elements. It found solute-atom-centered clusters connected into string-like medium-range order (MRO — atomic correlations beyond nearest neighbors), whose formation leaves behind dispersed low-electron-density regions (LEDRs — patches of depleted valence charge density, the structural signature of the liquid-like flow units). The activation energy of the fast secondary relaxation increased as LEDRs were reduced, the slow β tracked La–La next-nearest-neighbor distances and the size of the string-like clusters, and the fast mode was identified with the activation of a small population of extremely low-viscosity liquid-like regions. The interpretation used a generalized Maxwell model plus free-volume picture (the classical assumption that local flow requires excess free volume around the moving atoms) that treats the flow units as independent. What this framework cannot do — separating how the flow units form from how their dynamics couple — is the substance of the present problem.

Problem Statement

Turn the empirical correlation between electronic/atomic structure and secondary relaxation in metallic glasses — established for La-based glasses as the prototype system by the source study — into a quantitative, predictive account, through two linked deliverables sharing one scientific objective (a structure-based theory of the secondary-relaxation spectrum of metallic glasses): (1) Decomposition of flow-unit origins: a validated separation of the geometric (atomic-packing) contribution from the chemical (electronic-bonding, charge-density) contribution that determines the size and population of low-electron-density, liquid-like flow units when the base glass is alloyed — expressed as a model, simulation protocol, or designed experimental series that predicts, from composition, the LEDR statistics and hence the activation energy of the fast secondary relaxation, with the geometry-only and chemistry-only contributions individually evaluated (for example via counterfactual structural models that suppress charge transfer while preserving packing, or alloy series that vary bonding at nearly constant packing efficiency and vice versa). (2) Coupled-mode theory: an extension of the independent-flow-unit generalized Maxwell description to a model with explicit inter-unit coupling that reproduces the observed β spectral shape — including the shoulders and excess wings produced by cooperative relaxation modes in large, connected string-like solute-cluster networks — predicting the loss-modulus spectrum from the same structural inputs. Deliverable (1) supplies the structural input (where the flow units are and why they form) and (2) supplies the dynamics (how their interactions shape the measured spectrum); the two are complementary and are to be confronted with the experimental LEDR statistics and relaxation spectra of the La-based alloy series. A downstream application question additionally arises — the degree to which the liquid-like regions directly control shear-band nucleation, propagation and cryogenic ductility (shear bands being the narrow, localized deformation bands along which metallic glasses fail) across wider composition and strain-rate ranges — which is a natural consequence test of the same objective, remains open, and is recorded here as a separable extension rather than a completion requirement of the theoretical core.

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