Tunable chemical complexity to control atomic diffusion in alloys

Medium- and high-entropy alloys, a class of single-phase concentrated solid solution alloys (SP-CSAs), offer tunable chemical complexity and attractive properties including mechanical strength, corrosion resistance, thermal stability, and radiation tolerance. Despite extensive experimental data, predictive models explaining why closely composed concentrated alloys exhibit markedly different diffusion, mechanical behavior, and radiation tolerance remain incomplete. Prior studies have explored roles of mixing enthalpy, defect energetics, local lattice distortions, stresses, electronic and magnetic disorder, with limited predictive success. The existence and origin of vacancy-based sluggish diffusion in CSAs were recently clarified by atomistic simulations, showing dependence on site percolation and composition-dependent vacancy migration energies rather than configurational entropy alone. In binary Ni-Fe alloys, maximum sluggish vacancy diffusion occurs near Fe atomic fraction ~0.22, close to the fcc site percolation threshold, and is associated with chemically-biased diffusion. Because sluggish and chemically biased diffusion govern segregation, creep, defect clustering, and microstructure evolution, a more general framework is needed. Here, the authors propose that a quantifiable measure of chemical complexity—local formation energies of migrating defects—controls interstitial-atom-based diffusion, leading to percolation-driven, chemically biased, and sluggish diffusion in Ni-Fe alloys. The study aims to model and verify this mechanism using μs-scale MD, a mean-field diffusion model, and lattice kinetic Monte Carlo (kMC), thereby linking defect energy distributions to macroscopic transport and suggesting a route to design radiation-tolerant alloys by tuning chemical complexity.

The literature debates the presence and origin of sluggish diffusion in concentrated alloys. Some works report sluggish diffusion in high-entropy alloys (HEAs) and CSAs (e.g., Tsai et al., Rohrberg et al., Dabrowa et al.), while others question its universality (Miracle & Senkov; Paul; Kucza et al.). Recent simulation studies (Osetsky et al., 2018) attribute vacancy-based sluggish diffusion to site percolation combined with composition-dependent migration energies, not to configurational entropy. Additional reported CSA features include nontrivial mixing enthalpies, unusual defect energetics, local lattice distortions, and electronic/magnetic disorder, which correlate variably with properties but lack predictive models. For interstitial diffusion, prior DFT and MD studies in equiatomic alloys (Barnard & Morgan; Osetsky et al., 2016; Zhao et al., 2017) offered hints of slow and chemically biased diffusion but did not provide a full composition-dependent picture, percolation thresholds, or a chemical-complexity-based metric. Classical percolation theory estimates thresholds for different lattices and mechanisms (e.g., Bocquet, 1994 for interstitial dumb-bell diffusion on fcc with p_d-b ≈ 0.72; Xu et al., 2014 for general percolation), and prior vacancy diffusion in Ni-Fe matched fcc site percolation (p_site ≈ 0.18). However, for interstitial mechanisms in real alloys, deviations indicate additional chemistry-driven effects beyond classical percolation.

Overview: The study combines (i) microsecond-scale classical molecular dynamics (MD) simulations to measure tracer diffusion coefficients and analyze interstitial dumb-bell configurations across Ni-Fe compositions and temperatures, (ii) a mean-field diffusion model linking migrating defect energies to composition-dependent diffusion and percolation behavior, and (iii) a lattice kinetic Monte Carlo (kMC) model with on-the-fly transition state estimation incorporating local dumb-bell formation energy differences as a tunable measure of chemical complexity. Interatomic potentials: Empirical interatomic potentials for fcc Ni-Fe (ref. 33), consistent with DFT defect properties, were used for MD and to extract migration barriers in pure elements (E_m^Ni ≈ 0.35 eV, E_m^Fe ≈ 0.27 eV). MD simulations: Thermally activated migration of interstitial atoms (self-interstitial atoms, SIAs) in the <100> dumb-bell configuration was modeled for pure Ni, pure fcc Fe, and 17 binary alloys with Fe atomic fractions C_Fe = 0.025, 0.05, 0.1, 0.2, 0.25, 0.35, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.9, 0.95, 0.975 over T = 500–1100 K. Microsecond-scale trajectories captured up to ~10^6 IA jumps and ~5 μs physical time to ensure statistical convergence amid strong jump correlations driven by Ni-Ni dumb-bell stability. Trajectory analysis employed two decomposition techniques to estimate diffusion coefficients and correlation scales: TJD (Trajectory Jumps Decomposition) with equal-jump segments N_i and TTD (Trajectory Time Decomposition) with equal time segments t_s. Saturation tests (equiatomic NiFe at 500 K) showed consistent diffusivities for segments N > 300 and t_s > 1.5 ns, yielding D_d = (6.2 ± 0.3) × 10^−11 m^2/s. Tracer diffusion coefficients D*, D*_Ni, D*_Fe were obtained from linear atomic square displacement vs time; Arrhenius parameters at equiatomic composition: D0 = 3.5 × 10^−6 m^2/s, activation energy E = 0.43 eV. Mean-field theoretical model: A binary alloy is modeled with a migrating defect having two ground states (A and B, analogous to Ni-Ni and Fe-Fe dumb-bells) and transitions between them. Key parameters: migration energies E_a^A, E_a^B and ground-state energy difference ΔE = E_a^A − E_a^B. The diffusion coefficient is D = D0 [p_A⟨β⟩_A + p_B⟨β⟩_B], with state occupation probabilities p_i = (C_i B_i)/(C_A B_A + C_B B_B), B_i = exp(−β E_i^f), and mean-field averaging over local environments, ⟨B⟩ = C_A B_A + C_B B_B. The main result (after combining equations) gives D normalized by D_BB: D ∝ (C_A + (1 − C_A) B_SD)/(1 + (1 − C_A) B_SD), where B_SD = exp(−βΔE_SD) and ΔE_SD = E_A^f − E_B^f. The model predicts a minimum in D at C_min ≈ √(B_SD) under conditions relevant to Ni-Fe IA diffusion, and clarifies roles of ΔE^m = E_A^m − E_B^m and ΔE^f = E_A^f − E_B^f: to locate the diffusion minimum between p_c and 1 − p_c, ΔE^m must exceed ΔE^f and ΔE_SD must not be too large; otherwise the minimum aligns with percolation thresholds. Calculations used T = 500 and 1100 K, ΔE^m ≈ 0.08 eV (from pure-element barriers) and a range of ΔE^f consistent with kMC-derived formation energy spectra. Lattice kMC with on-the-fly transition states: The <100> dumb-bell has three possible compositions (Ni-Ni, Ni-Fe, Fe-Fe) and eight possible transitions when atoms A1 or A2 jump to a neighboring site, changing the dumb-bell composition (A1-A2 → A2-A2, A1-A1, or A1-A2). Transition state energies are estimated during kMC as E‡ = E_m(atom) + ΔE, where E_m(atom) is the migration barrier of the jumping species in the corresponding pure element (Ni or Fe), and ΔE is the configuration energy change computed via Bell-Evans-Polanyi considerations from statistically averaged ground-state formation energies of dumb-bells ⟨E^f_NiNi⟩, ⟨E^f_NiFe⟩, ⟨E^f_FeFe⟩. These formation energies were obtained by static modeling over >700 random atomic distributions per composition (C_Fe = 0.1–0.9). In equiatomic NiFe, mean formation energies (relative to global minimum) were ≈3.62 eV (Ni-Ni), 4.06 eV (Ni-Fe), 4.28 eV (Fe-Fe), implying an affinity for Ni-Ni dumb-bells and negative ΔE for Fe-Fe → Ni-Fe conversions (e.g., ⟨ΔE1⟩ = ⟨E^f_NiFe⟩ − ⟨E^f_FeFe⟩ ≈ −0.22 eV). Composition dependence of ⟨ΔE1⟩ and ⟨ΔE2⟩ = ⟨E^f_NiFe⟩ − ⟨E^f_NiNi⟩ was quantified and found weakly composition dependent (~0.1–0.2 eV differences). Chemical complexity parameter α was introduced to scale the contribution of local energy differences to barriers: E‡ = E_m + αΔE, with α ∈ [0,1]. kMC steps: initialize 10a × 10a × 10a fcc cell with random Ni/Fe at target composition and a central <100> dumb-bell; identify nearest neighbors of dumb-bell atoms; enumerate possible transitions and calculate E‡; select and execute move with rate weighting at temperature; repeat. A constant attempt frequency 2.5 THz was used, fitted to match MD diffusivity of <100> dumb-bell in pure Ni at 500 K. Assumptions and analysis: Initial states are random solid solutions (no explicit local chemical order, LCO). Theoretical model ignores explicit percolation to isolate energy-driven effects; percolation’s qualitative shifts of D minima relative to p_c are discussed. MD trajectory segmentation ensures convergence and captures jump correlations induced by chemical preferences.

- MD reveals a strong, non-monotonic composition dependence of tracer diffusion in Ni-Fe via interstitial mechanism, with a pronounced minimum in D at C_Fe ≈ 0.50–0.65; at 500 K, the tracer diffusion coefficient decreases by ≈20× relative to pure Ni and ≈500× relative to pure fcc Fe, i.e., min(D_NiFe) ≈ 0.05 D_Ni ≈ 0.002 D_Fe. - Percolation behavior is observed near the composition of minimum diffusivity. At 500 K, the interstitial diffusion percolation threshold is p_c ≈ 0.65–0.70 (C_Fe ≈ 0.7), differing from the ideal fcc <100> dumb-bell percolation threshold p_d-b ≈ 0.72, indicating chemically biased motion. - Partial diffusivities intersect just beyond p_c: D_Ni(C_Fe) and D_Fe(C_Fe) cross when Fe diffusion becomes dominant; concurrently, the chemical composition of migrating dumb-bells switches from Ni-only (Ni-Ni) below p_c to increasing Fe content above p_c, evidencing Fe diffusion percolation and chemically biased transport (Ni preferentially transported even in Fe-rich alloys just below p_c). - Mean-field theory explains the minimum in D as a competition between: (i) preferential Ni transport in Ni-rich alloys due to higher stability (lower formation energy) of Ni-Ni dumb-bells, and (ii) faster Fe migration in Fe-rich alloys due to lower migration barrier (E_m^Fe < E_m^Ni). The minimum lies between p_c and 1 − p_c when ΔE^m > ΔE^f and ΔE_SD is moderate; otherwise the minimum coincides with percolation thresholds. - Static calculations show distributions of dumb-bell formation energies with mean values (equiatomic): ⟨E^f_NiNi⟩ ≈ 3.62 eV, ⟨E^f_NiFe⟩ ≈ 4.06 eV, ⟨E^f_FeFe⟩ ≈ 4.28 eV; energy differences of ~0.1–0.2 eV weakly depend on composition. These quantify chemical complexity and drive bias toward Ni-Ni configurations. - kMC with chemical complexity parameter α reproduces MD trends: α = 1 (high complexity) yields strong percolation, sluggish and chemically biased diffusion, with Fe percolation occurring near C_Fe ≈ 0.70–0.80 (C_Ni ≈ 0.20–0.30). As α decreases (reduced chemical complexity), percolation and sluggish diffusion weaken and the D(C_Fe) curve becomes more linear; at α = 0, D varies linearly with composition as in an ideal solid solution. - Equiatomic NiFe tracer diffusion shows Arrhenius behavior with D0 ≈ 3.5 × 10^−6 m^2/s and activation energy E ≈ 0.43 eV. MD correlation analysis at 500 K indicates saturation of diffusivity estimates for segments >300 jumps or >1.5 ns. - Overall, the distribution of local defect energies emerges as a quantitative metric of chemical complexity linking microscopic energetics to macroscopic transport and percolation phenomena.

The findings demonstrate that atomic transport in concentrated Ni-Fe alloys via interstitial mechanisms is governed by chemical complexity embodied in local defect energies. The observed diffusion minimum and percolation threshold shift relative to ideal lattice predictions cannot be explained by classical percolation alone; instead, the bias toward energetically favorable Ni-Ni dumb-bells in Ni-rich environments and the inherently lower migration barriers for Fe in Fe-rich environments together produce a composition-dependent competition that sets the diffusion minimum between the percolation thresholds of the fast species in each compositional regime. This framework rationalizes chemically biased diffusion—preferential transport of Ni even in Fe-rich alloys below p_c—and explains the strong sluggish diffusion (orders of magnitude reduction in D) as arising from the stability of specific local configurations that trap migrating interstitials and create long-range correlation in jump sequences. By introducing a tunable parameter (α) linking local formation energy differences to transition state barriers in kMC, the study bridges atomistic energetics and mesoscale transport, reproducing MD results and highlighting how reducing chemical complexity recovers ideal, linear compositional dependence. These mechanisms are directly relevant to radiation tolerance: interstitial-vacancy recombination, cluster formation, and interactions with extended defects depend on defect mobilities and biases, implying that tuning chemical complexity can guide microstructure evolution under irradiation and potentially suppress radiation damage.

- The study establishes that variations in local formation energies of migrating defects—a quantifiable measure of chemical complexity—control interstitial diffusion, leading to chemically biased, sluggish diffusion and composition-dependent percolation in Ni-Fe alloys. - A mean-field model, validated by microsecond MD and on-the-fly kMC, explains the non-monotonic D(C_Fe) with a minimum between percolation thresholds when ΔE^m > ΔE^f and moderate ΔE_SD. The model and kMC reproduce the observed percolation and bias transitions in dumb-bell composition. - The work introduces a practical metric for alloy chemical complexity based on distributions of defect formation and migration energies, providing a direct link to macroscopic transport properties. This offers a pathway to design diffusion behavior and radiation tolerance by tuning local energetics through composition and potential local chemical order (LCO). - Future research directions include: incorporating explicit LCO into simulations to quantify its enhancement of effective migration energies and chemical complexity; extending the framework to multicomponent CSAs; exploring defect-defect and defect–extended defect interactions under irradiation using accelerated techniques (e.g., kinetic ART) informed by the proposed energy metrics.

- The mean-field theoretical model neglects explicit percolation effects to isolate energetic contributions; percolation is discussed qualitatively, so quantitative thresholds may shift in real systems. - MD employs classical interatomic potentials; while consistent with DFT benchmarks, accuracy is limited by the potential’s fidelity, especially for defect energetics and magnetic/electronic effects in Fe-rich alloys. - kMC assumes random solid solutions (no explicit local chemical order); the role of LCO is discussed but not directly simulated due to computational constraints. Attempt frequency is fitted (2.5 THz), introducing a calibration parameter. - The mapping from local energy distributions to macroscopic percolation thresholds depends on temperature and the spectrum of accessible configurations; compositional dependence of energy differences is treated statistically and may vary with specific microstructures.