Scientists Put a Famous Atomic Diffusion Scaling Law to the Test in Molten Iron Alloys

Deep inside every steel mill, every casting line, and every molten droplet of iron-based alloy, atoms are constantly jostling, colliding, and migrating through a dense, disordered liquid. How quickly those atoms move—captured by the diffusion coefficient—governs how alloys solidify, how glasses form, and how high-temperature melts behave during industrial processing. Yet measuring diffusion directly in liquids heated well beyond 1500 degrees Celsius is notoriously difficult, which is why physicists have long relied on elegant theoretical shortcuts. One of the most celebrated of these is the universal scaling law proposed by Mikhail Dzugutov in 1996, which claims that a reduced diffusion coefficient depends on a single thermodynamic quantity, the excess entropy, through a simple exponential relationship. A new study published in Results in Physics by R.C. Gosh, Sazia Akter Maria, and Md Tareq Mahmud of the University of Dhaka now puts that law through one of its most demanding tests yet: liquid iron-based transition metal alloys.

The appeal of Dzugutov’s scheme lies in its microscopic foundation. Rather than reducing transport coefficients by macroscopic parameters such as number density and temperature, as Yosenfeld’s earlier excess-entropy scaling did in 1977, Dzugutov used microscopic reduction parameters: an effective hard-sphere diameter for length and the Enskog collision frequency for inverse time. The hard-sphere picture rests on two propositions. First, at high densities, the transfer of momentum and energy between atoms is dominated by short-range repulsive interactions, which behave essentially like binary collisions of billiard-ball-like hard spheres. Second, the frequency of local structural relaxation—the rate at which atoms escape the transient cages formed by their neighbors—can be calculated from Enskog kinetic theory for hard spheres. The result is a remarkably compact formula in which the normalized diffusivity equals 0.049 times the exponential of the excess entropy, expressed in units of the Boltzmann constant.

The excess entropy itself is a subtle quantity. Defined as the total thermodynamic entropy minus the ideal gas entropy, it can be expanded as a series of contributions from pairs of particles, triplets, and higher-order correlations. The leading two-body term, which depends directly on the pair distribution function, is known to contribute more than 85 percent of the excess entropy across most thermodynamic states, and as much as 95 percent near the triple point of simple liquids. Crucially, the pair distribution function can be obtained from x-ray or neutron diffraction experiments or from computer simulations, making Dzugutov’s scaling unusually practical: it connects atomic dynamics to structure and thermodynamics using quantities that are actually measurable. Later work by Jakse and Pasturel showed that an excess entropy derived from the Carnahan-Starling equation of state—the best analytical description of hard-sphere thermodynamics—can also justify the scaling, provided the temperature dependence of the effective hard-sphere diameter is properly accounted for.

What has been missing, the authors argue, is a systematic test for alloys rather than pure metals. Although Yokoyama and colleagues applied the scaling law to binary alloys in 2007, their analysis was restricted to equiatomic compositions. The new study therefore examines cobalt-iron (CoxFe1−x) and nickel-iron (NixFe1−x) alloys across the full concentration range, from iron-rich to cobalt-rich or nickel-rich, at a temperature of 1833 Kelvin. Iron-based transition metal alloys are not an arbitrary choice: they underpin structural steels, magnetic materials, and even models of planetary cores, and their liquid-state transport properties remain poorly constrained by experiment because of the extreme conditions involved.

On the theoretical side, the team combined the Bretonnet-Silbert pseudopotential—a model specifically designed for liquid transition metals that accounts for both s-p and d band contributions, including the sp-d hybridization that simpler pseudopotentials miss—with linearized Weeks-Chandler-Andersen perturbation theory. The latter provides the effective hard-sphere diameters by solving a transcendental equation that equates the Helmholtz free energy of the real interacting system with that of a hard-sphere reference system. Two different local field correction functions, due to Ichimaru-Utsumi and Vashishta-Singwi, were used to screen the electron-ion interaction, allowing the authors to assess which treatment of electron correlation better describes these dense metallic liquids. This combination of Bretonnet-Silbert potential and LWCA theory had never before been applied to calculate diffusion and viscosity in transition metal alloys.

To anchor the theory, the researchers performed classical molecular dynamics simulations with the LAMMPS code, using modified embedded atom method (MEAM) interatomic potentials that capture both many-body effects and the angular orientation of bonds—features essential for transition metals. Each simulation placed 30,000 atoms in a cubic box 70 angstroms on a side with periodic boundary conditions. The alloys were first equilibrated at 2500 Kelvin, above their melting points, in a constant-pressure ensemble, then cooled to 1833 Kelvin, and finally equilibrated for a full nanosecond at the target temperature. One thousand configurations were extracted from each run, and the pair correlation functions were computed with the OVITO visualization tool and averaged with a Python code. The team also verified that their results were independent of system size by repeating selected simulations with 20,000 and 40,000 atoms; the reduced diffusion coefficients and viscosities changed only in the second decimal place, consistent with the finite-size correction formula proposed by Khrapak.

The calculated structural quantities proved encouraging. Effective hard-sphere diameters obtained from theory and simulation were comparable and consistent with literature values for pure liquid iron, cobalt, and nickel, and the partial pair correlation functions reproduced the expected concentration trends: the peak height of like-atom pairs rises as that component becomes richer, while cross-pair correlations peak at mid-concentrations. When the reduced diffusion coefficients were plotted against excess entropy, however, a nuanced picture emerged. Simulated data followed Dzugutov’s universal scaling line fairly well across all concentrations of both alloy families. The theoretically calculated data, by contrast, obeyed the scaling only for mid-concentrated alloys, deviating systematically when a single component dominated the mixture. The culprit, the authors show, is the overestimated principal peak height of the calculated partial pair correlation functions in single-component-rich alloys, which inflates the two-body excess entropy.

Notably, the Carnahan-Starling excess entropy proved far more reliable than the two-body approximation, remaining stable across concentrations and closer to established values for pure liquid metals, even in single-component-rich compositions where the two-body entropy varied wildly. This suggests that the Bretonnet-Silbert potential combined with LWCA theory is a sound framework for mid-concentrated liquid transition metal alloys, even if the perturbative treatment of structure needs refinement at the compositional extremes. The Ichimaru-Utsumi local field correction consistently outperformed the Vashishta-Singwi alternative, yielding more reliable inter-diffusion coefficients and shear viscosities, which remained nearly constant across all concentrations and agreed reasonably with molecular dynamics results.

The transport coefficients themselves tell an industrially relevant story. Partial self-diffusion coefficients of cobalt or nickel decrease while those of iron increase as the minority component’s concentration grows, and the inter-diffusion coefficients—computed as concentration-weighted sums of the self-diffusivities—fall in the range of roughly 7 to 9 times 10⁻⁹ square meters per second, comparable to simulation data and to scattered theoretical and experimental values for the pure liquid metals. Shear viscosities derived from the Stokes-Einstein relation with slip boundary conditions, and from Yokoyama’s modified upper-bound formula, came out around 1.6 to 2.8 millipascal-seconds—smaller than experimental measurements, which run nearly twice as high, but closer to experiment than earlier theoretical estimates and consistent in trend with prior work on Fe-Co and Fe-Ni liquid alloys.

Perhaps the most striking result appears in the paper’s appendix, where the authors propose a new scaling law of their own. Because both alloy families deviated from Dzugutov’s line at single-component-rich compositions, they refitted the data with independent prefactor and exponential factors and found that both Co-Fe and Ni-Fe melts obey the same alternative relationship, with a reduced diffusion coefficient proportional to the exponential of minus 0.3 times the excess entropy and a much smaller prefactor of 0.005. Until more sophisticated theories or new experimental data for iron-based liquid alloys become available, the authors suggest, this proposed scheme offers young researchers a practical route to predicting atomic diffusion in these industrially vital melts—while the half-century-old interplay between entropy and atomic mobility continues to reveal fresh surprises in the liquid state.

Subject of Research: Testing Dzugutov's excess-entropy scaling law for atomic diffusion in liquid iron-based cobalt-iron and nickel-iron transition metal alloys using pseudopotential theory and molecular dynamics simulation.

Article Title: Test of scaling law for atomic diffusion of Fe based liquid transition metal alloys

Article References: Test of scaling law for atomic diffusion of Fe based liquid transition metal alloys. (n.d.). https://doi.org/10.1016/j.rinp.2026.108743

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108743

Keywords: atomic diffusion, liquid metals, transition metal alloys, excess entropy scaling, Dzugutov scaling law, molecular dynamics simulation, pseudopotential theory, iron-cobalt alloys, iron-nickel alloys, shear viscosity, hard-sphere model, LAMMPS