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The Cluster Expansion (CE) Method encounters significant computational challenges in multicomponent systems due to the computational expense of generating training data through density functional theory (DFT) calculations. This work aims to refine the cluster and structure selection processes to mitigate these challenges. We introduce a novel method that significantly reduces the computational load associated with the calculation of fitting parameters. This method employs a Graph Neural Network (GNN) model, leveraging the M3GNet network, which is trained using a select subset of DFT calculations at each ionic step. The trained surrogate model excels in predicting the volume and energy of the final structure for a relaxation run. By employing this model, we sample thousands of structures and fit a CE model to the energies of these GNN-relaxed structures. This approach, utilizing a large training dataset, effectively reduces the risk of overfitting, yielding a CE model with a root-mean-square error (RMSE) below 10 meV/atom. We validate our method’s effectiveness in two test cases: the (Ti, Cr, Zr, Mo, Hf, Ta)B2 diboride system and the Refractory High-Entropy Alloy (HEA) AlTiZrNbHfTa system. Our findings demonstrate the significant advantages of integrating a GNN model, specifically the M3GNet network, with CE methods for the efficient predictive analysis of chemical ordering in High Entropy Materials. The accelerating capabilities of the hybrid ML-CE approach to investigate the evolution of Short Range Ordering (SRO) in a large number of stoichiometric systems. Finally, we show how it is possible to correlate the strength of chemical ordering to easily accessible alloy parameters. 

Abstract We present a systematic investigation of thermodynamic stability, phase-reaction, and chemical activity of Al containing disordered Ti 2 (Al-Ga)C MAX phases using machine-learning driven high-throughput framework to understand the oxidation resistance behavior with increasing temperature and exposure to static oxygen. The A-site (at Al) disordering in  Ti 2 AlC MAX (M=Ti, A=Al, X=C) with Ga shows significant change in the chemical activity of Al with increasing temperature and exposure to static oxygen, which is expected to enable surface segregation of Al, thereby, the formation of Al 2 O 3 and improved oxidation resistance. We performed in-depth convex hull analysis of ternary Ti–Al–C, Ti–Ga–C, and Ti–Al–Ga–C based MAX phase, and provide detailed contribution arising from electronic, chemical and vibrational entropies. The thermodynamic analysis shows change in the Gibbs formation enthalpy (Δ G form ) at higher temperatures, which implies an interplay of temperature-dependent enthalpy and entropic contributions in oxidation resistance Ga doped Ti 2 AlC MAX phases. A detailed electronic structure and chemical bonding analysis using crystal orbital Hamilton population method reveal the origin of change in phases stability and in oxidation resistance in disorder Ti 2 (Al 1−x Ga x )C MAX phases. Our electronic structure analysis correlate well with the change in oxidation resistance of Ga doped MAX phases. We believe our study provides a useful guideline to understand to role of alloying on electronic, thermodynamic, and oxidation related mechanisms of bulk MAX phases, which can work as a precursor to understand oxidation behavior of two-dimensional MAX phases, i.e., MXenes (transition metal carbides, carbonitrides and nitrides).