Origin of the Stokes–Einstein deviation in liquid Al–Si
Ni Zhan and John R. Kitchin, “Origin of the Stokes–Einstein deviation in liquid Al–Si”, Molecular Simulation 48(4), 303–313 (2022; published online December 2021). https://doi.org/10.1080/08927022.2021.2012572
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Question
The Stokes–Einstein–Sutherland (SES) equation links a single-particle property, the self-diffusion coefficient D, to a continuum property, the viscosity η, through an effective diameter d proportional to T/(Dη). For many liquids d is constant over a range of temperatures, which makes the relation useful for estimating one hard-to-measure property from the other. Near and below the melting point, however, d often stops being constant, and there is no consensus on why. Proposed explanations include a local microviscosity that differs from the bulk value, growth of five-fold symmetric (icosahedral) clusters, and dynamic heterogeneity or decoupling of species diffusion. This paper asks whether the SES breaks down in Al-rich liquid Al–Si near the eutectic composition and, if so, what atomic-scale structure is responsible (Zhan 2021).
Approach
The authors ran classical molecular dynamics in LAMMPS with an angular-dependent interatomic potential for Si–Al, on 1099-atom cells of Al0.90Si0.10 and Al0.94Si0.06. NPT cooling runs at 1, 0.1 and 0.01 K/ps mapped out crystallization, solid–liquid interface runs gave the melting points, and long NVT runs (10 to 20 ns, seven independent repeats) supplied transport data. Diffusion came from the slope of the mean squared displacement with a Yeh–Hummer finite-size correction. Viscosity came from Green–Kubo integration of the stress autocorrelation function; because the autocorrelation decays slowly at low temperature, they fitted it with an analytic decay form plus a small neural network on the residuals, which kept the viscosity variance low, the dominant term in the uncertainty of d. They also fitted the fractional Stokes–Einstein form D/T ∝ η(-κ), where κ = 1 recovers the SES (Zhan 2021).
For local structure, atoms with a perfect icosahedral Voronoi index <0,0,12,0> and their 12 neighbours were grouped into icosahedral clusters using single-linkage agglomerative clustering from scikit-learn on a precomputed neighbour matrix, so that icosahedra sharing atoms (vertex, edge, face or intercross sharing) merge into one cluster. Clusters were tracked across frames when at least 13 atoms persisted. Finally, they introduced per-atom estimates of diffusion and of viscosity (an Einstein-type formula built from per-atom stresses and Voronoi volumes), so that atoms inside clusters could be excluded from the transport averages (Zhan 2021).
Key findings
The computed melting points were 630 K for Al0.90Si0.10 and 700 K for Al0.94Si0.06. With the slowest cooling rate, crystallization occurred at 435 K and 504 K respectively; faster cooling gave lower crystallization temperatures or an amorphous solid. Si diffused faster than Al, and diffusion followed a single Arrhenius law over the liquid and supercooled range, with activation energies of 19.0 and 19.6 kJ/mol. Diffusion coefficients agreed reasonably with earlier ab initio values except for Si in Al0.90Si0.10. Viscosities were close to experimental data for eutectic Al0.88Si0.12 but needed two Arrhenius regions, breaking at 645 K and 700 K (near the melting points), with activation energy rising from about 11.4 to 16.2 kJ/mol and 11.8 to 15.2 kJ/mol on cooling (Zhan 2021).
The effective diameter was constant from about 1400 K down to the melting point and changed below it (630 K for Al0.90Si0.10, 690 K for Al0.94Si0.06), with fitted κ = 0.97 ± 0.01 for both compositions, so viscosity grows faster than the SES allows. The number of atoms in icosahedral clusters and the size of the largest cluster grew on cooling, reaching expected values of about 140 atoms (12.7%) and about 40 atoms just before crystallization. Most clusters lived less than 1 ps, but the long-lived fraction rose with undercooling (7.5% at 510 K and 4.3% at 570 K for the two compositions), and the longest lasted up to 8 ps. Per-atom diffusion was the same with or without cluster atoms, whereas per-atom viscosity was always lower when cluster atoms were removed, with the gap widening on cooling. Recomputing d without cluster atoms removed the deviation: κ = 1.00 ± 0.02 and 0.99 ± 0.02 without clusters, against 0.96 ± 0.02 and 0.95 ± 0.02 with all atoms (Zhan 2021).
Significance
The study gives a quantitative mechanism for SES breakdown in a technologically important casting alloy: transient icosahedral clusters thicken the liquid without slowing self-diffusion, because atoms move in and out of clusters. It also shows two transferable tools, an agglomerative clustering definition of shared-atom icosahedral clusters and a per-atom decomposition of viscosity that matches the Green–Kubo result when all atoms are included and could, the authors note, be applied to liquids in contact with solids (Zhan 2021). Data and code are archived on Zenodo (doi:10.5281/zenodo.5554968).