Disorder viscosity correction approach to calculate spinodal temperature and wavelength

TL;DR

This work tackles parameter-free prediction of spinodal decomposition in multi-component materials by introducing the Disorder Viscosity Correction (DVC), which leverages small, finite POCC tiles to compute a cumulant-expanded free energy $F(oldsymbol{x},T)$ and a self-interaction energy $\mathcal{E}_ ext{si}$ that mitigate long-range fluctuations. The authors define a self-consistent correction $\chi_ ext{mix}\mathcal{E}_ ext{si}$ to obtain a physically meaningful halting of unbounded phase separation and to preserve local concavity necessary for interface stabilization, enabling calculation of the spinodal temperature $T_ ext{sp}(oldsymbol{x})$ and maximum wavelength $oldsymbol{ ext{λ}}_ ext{sp}(oldsymbol{x},T)$. They validate the approach against binary and ternary experimental data, showing good agreement for spinodal temperatures and reasonable estimates for wavelengths, and demonstrate compatibility with high-throughput and machine-learning workflows for exploring high-entropy materials. Overall, DVC offers a scalable, parameter-free pathway to screen and understand spinodal-driven microstructure formation in complex disordered systems, complementing existing interatomic potentials and CALPHAD-type approaches.

Abstract

Spinodal decomposition, a key mechanism to microstructure formation in materials, has long posed challenges for predictive modeling, due to the need for parameter-free approaches that accurately capture local energy landscapes. In this work, we propose an approach to predict spinodal behavior by introducing a disorder viscosity correction to bulk free energies computed from finite, small, representative cells. We approximate the energy penalty required to transition into a disordered state to enable the stabilization of locally concave bulk free energy regions - essential for interface formation - while suppressing long-range concentration fluctuations. This approximation circumvents the complexity of full ab initio parameterization of interfacial properties and is well-suited for high-throughput and machine-learning frameworks. Our approach captures the necessary physics underpinning spinodal kinetics, offering a scalable route to predict spinodal regions in compositionally complex and high-entropy materials.

Figures (6)

• Figure 1: Disorder viscosity correction workflow for calculating spinodal temperature and wavelength. The workflow describes the approach to calculate the spinodal temperature and wavelength using the disorder viscosity correction ( DVC). (a) We begin by generating POCC tiles of the minimal size congruent with a concentration grid $\{\mathbf{x}_1,\cdots,\mathbf{x}_{N_{\substack{\hbox{grid}}}}\}$ of $N_{\substack{\hbox{grid}}}$ points and calculating their ab initio energies. (b) Then, using the AFLOW standard (Methods), we calculate the DFT energies of all the POCC tiles. (c) Next, at each concentration grid point, the moments $M_\nu$, cumulants $K_\nu$ and self-interacting energy of the disordered system $\mathcal{E}_\mathrm{si}$ are computed. The self-interacting energy is extremely difficult to calculate, and therefore, in the zeroth approximation and infinite-temperature limit, we approximate it by the ensemble average energy of the disordered system --- e.g., $\mathcal{E}_\mathrm{si}\approx\sum_iP_iE_i$, where $P_i$ and $E_i$ are the probability and energy of the $i$-tile, respectively. (d) Afterwards, we fit multivariate polynomials in $\mathbf{x}$ for $K_\nu$ and $\mathcal{E}_\mathrm{si}$. (e) This allows us to obtain analytical expressions for $F$ and $\mathcal{E}_\mathrm{si}$. As discussed in the text, the choice in the size of POCC tiles leads to an over-stabilization of the spinodal temperature $T_\mathrm{sp}$. To remedy this issue, we correct $F$ by a fraction of the self-interaction energy $\chi\mathcal{E}_\mathrm{si}$, where $\chi\in [0,1]$. We call this the disorder viscosity correction. (f) The DVC is done self-consistently through the calculation of $\chi_\mathrm{mix}$ (orange dashed box). (g) We determine $\chi_\mathrm{mix}$ as the average (1/2) between no correction ($\chi\equiv0$) and the value at which the system is completely miscible (the minimum $\chi$ value at which $T_\mathrm{sp}$ is zero for the whole concentration space). (h) With $\chi_\mathrm{mix}$, we calculate the concentration-Hessian of the corrected free energy $F-\chi_\mathrm{mix}\mathcal{E}_\mathrm{si}$. (i) Finally, we use this to compute the corrected $T_\mathrm{sp}$ and spinodal wavelength $\lambda_\mathrm{sp}$.
• Figure 2: Spinodal decomposition in binary systems. (a–b) Spinodal temperature, as a function of composition, for AuPt and CuNi. (c–d) Maximum spinodal wavelength, as a function of the undercooling temperature, for 40/60 at.% Au/Pt and 30/70 at.% Cu/Ni alloys. The labels DFT and DFT+DVC correspond to the results obtained without and with the DVC to the free energy, respectively.
• Figure 3: Spinodal decomposition in ternary systems. (a–d) Spinodal temperature, as a function of composition, for (Ti,Zr)C, (Ga,In)N (Ca,Sr)O, and, (K,Na)Cl. The labels DFT and DFT+DVC correspond to the results obtained without and with the DVC to the free energy, respectively.
• Figure 4: Cumulant magnitude decay. At equi-concentration, the magnitude of the cumulants decays as a function of the index for the disordered systems considered in this study. The large decrease in the magnitude is very general and observed at all concentrations.
• Figure 5: Cumulant fitting. Fitting of the cumulants using multivariate polynomials. The points are results obtained from the ab initio calculations, while the lines are the polynomial fits to the data.