Neural network kinetics for exploring diffusion multiplicity and chemical ordering in compositionally complex materials

Abstract

Diffusion involving atom transport from one location to another governs many important processes and behaviors such as precipitation and phase nucleation. The inherent chemical complexity in compositionally complex materials poses challenges for modeling atomic diffusion and the resulting formation of chemically ordered structures. Here, we introduce a neural network kinetics (NNK) scheme that predicts and simulates diffusion-induced chemical and structural evolution in complex concentrated chemical environments. The framework is grounded on efficient on-lattice structure and chemistry representation combined with artificial neural networks, enabling precise prediction of all path-dependent migration barriers and individual atom jumps. To demonstrate the method, we study the temperature-dependent local chemical ordering in a refractory NbMoTa alloy and reveal a critical temperature at which the B2 order reaches a maximum. The atomic jump randomness map exhibits the highest diffusion heterogeneity (multiplicity) in the vicinity of this characteristic temperature, which is closely related to chemical ordering and B2 structure formation. The scalable NNK framework provides a promising new avenue to exploring diffusion-related properties in the vast compositional space within which extraordinary properties are hidden.

Fig. 1. Schematic illustration of neural network kinetics (NNK) framework.

a The on-lattice structure and chemistry representation of the entire system. A vacancy and its local atomic environment are encoded into a digital matrix (neuron map). b NNK framework consists of a neural network that outputs vacancy migration barriers, and a neuron kinetics module that implements neuron jump (diffusion jump) based on kinetic Monte Carlo (kMC). See “Methods” for details on neuron kinetics. Vacancy jumps and chemical evolution are efficiently modeled by swapping of neurons and neuron map evolution.

Fig. 2. Predicting diffusion barrier spectra in the entire composition space of Nb–Mo–Ta.

a Creation of unique neuron maps and feature vectors for each individual diffusion path $\mathrm{P}$, which enables the prediction of eight path-dependent barriers from a vacancy. The symbol $\mathrm{V}$ represents the vacancy. b Performance of neural network in predicting diffusion barrier spectrum in concentrated, Nb₃₃Mo₃₃Ta₃₃, and dilute, Nb₉₀Mo₅Ta₅, solutions. c Diffusion barrier diagram generated by the neural network. The nonequimolar Nb₁₅Mo₆₅Ta₂₀ alloy exhibits the highest barrier in the Nb–Mo–Ta system.

Fig. 3. Diffusion kinetics-mediated local chemical order in the equimolar NbMoTa alloy.

a Variation of chemical order ${\delta }_{\mathrm{ij}}$ obtained at different annealing temperatures displays a critical temperature that divides the map into two characteristic regimes, denoted as diffusion-favored (I) and diffusion-limited (II). b Development of Mo–Ta order, ${\delta }_{\mathrm{Mo}-\mathrm{Ta}}$, as a function of diffusion jumps from $2\times {10}^4$ to $2\times {10}^7$. The inset shows that the jump number dependence of peak temperature converges to the critical value ~800 K below which the chemical ordering is suppressed.

Fig. 4. Jump randomness and diffusion multiplicity of an equimolar NbMoTa alloy.

a Schematics of two limiting lattice jump modes. One of the eight paths is predominated in directional jump (jump randomness $R=0$), while all eight paths have the same hopping probability $p$ in random jump ($R=1$). b Spatial and statistical distributions of lattice jump randomness, $R$, at three representative temperatures. At 3000 K the distribution of $R$ ($R_{\mathrm{peak}}$ = 0.7) indicates highly random diffusion, while at 400 K the lattice jumps transform to directional (selective) diffusion mode ($R_{\mathrm{peak}}$ = 0.0). Lattice jumps at 800 K exhibit highly heterogeneous diffusion modes, shown by the broad distribution of $R$. c Diffusion multiplicity $Var\left(R\right)$ as a function of temperature reveals a critical temperature (~850 K) at which diffusion is more heterogeneous (widest distribution of $R$). Moving to the two ends, diffusion approaches simple random and directional modes at ultimate high- and low temperatures, respectively.

Fig. 5. B2 structure nucleation and growth kinetics during annealing in NbMoTa.

a B2 cluster size evolution with the number of diffusion jumps. b–d Spatial distributions of growing B2 cluster at ${1\times 10}^6$, ${5\times 10}^6$, and ${1\times 10}^7$ diffusion jumps. Clusters are color-coded by their size.