Unique cluster expansion for reliable first-principles prediction of alloy thermodynamics and phase diagrams

"A theoretical multi-scale approach based on the cluster expansion (CE) has been developed to aid materials design and discovery for scientific and engineering applications. Using structural energies from first-principles electronic theory, an effective (CE) Hamiltonian that is suitable for large scale systems is constructed, enabling the calculation of thermodynamic quantities and prediction of phase diagrams. The CE Hamiltonian is expanded in terms of correlation functions of all geometric cluster entities on a fixed lattice, forming a basis set that spans the entire alloy configuration space on the lattice. The coefficients in the CE are the system's effective cluster interactions (ECI), which are physically well-defined and have unique values for each alloy system. Yet the CE is useful only when the ECI are truncated, permitting the ECI to be obtained from known energies in a configuration subspace. Such ECI are biased by those ECI not in the truncated set, because when projected onto an arbitrarily selected subspace (by truncation), linear dependencies exist between cluster functions. The dependencies are overlooked in current methods, resulting in non-unique truncated CE sets that are extracted from an unnecessarily large number of first-principles data. In fact, the CE is directly related to the well-studied Walsh transformation and the Hadamard matrices, whose properties are utilized in fractional factorial design principles (also known as Design of Experiment) to evaluate reliably the variables from only a selected subset of known data. Via these concepts, we develop a systematic choice of configuration subspaces that identify clearly the linear dependencies, allowing a unique truncation of CE that keeps the critical ECI. These concepts lead to the subspace projection (SSP) method, which gives a physically and mathematically sound approach to select a unique alloy CE via structural inversion (SI), starting from a subset of known energies evaluated from first-principles methods such as Density Functional Theory (DFT). In contrast, current methods for optimally truncated CE set are based on some ""chosen"" statistical measure of predictive capability of structural energies. This leads to non-unique truncated CE sets, although the issue, as we had shown, is partially addressed when the CE basis is enforced to be compact and locally complete. The theoretical methods we proposed were implemented in the Thermodynamics Toolkit (TTK). We use TTK to construct a CE for various phase-segregating and ordering alloys via SI, using DFT-determined structural energies from a selected set of ordered structures as input. The optimal CE set is used in Monte Carlo (MC) simulations and/or mean-field theories (MFTs) to construct temperature-composition phase diagrams for FCC Ca-Sr, Pd-Rh and Ag-Au (ordering), explaining various features and comparing them to available experiment data. The ECI of Ag-Au system also compare well to those extracted directly from the DFT electronic charge density at the dilute limit, elucidating their electronic structure origins. We also use our SSP method to construct a CE for Ag-Au to illustrate how critical ECI become systematically linearly independent with augmentation of configuration subspaces. Importantly, with only one-third the number of DFT structural energies to extract ECI, we achieve a unique and physical set of ECI via SSP, significantly reducing the amount of computational effort required. Lastly, a cluster-based MFT, generalized to arbitrary cluster sizes, is introduced to serve as a quick and reliable way to calculate phase diagram. Via a cluster-lattice Fourier transform, the cluster MFT obeys self-consistent relations between cluster and coarse-grained lattice correlations. Already with a single-site cluster, the proposed MFT results in topologically correct phase diagrams for Ising models on frustrated systems, which traditional MFTs fail to achieve. Phase transition temperatures from MC and series expansion are recovered upon finite-size scaling. Together these techniques permit a rapid, unique and reliable approach to materials characterization, design and discovery. In the future we will apply these novel methods and tool to more complex alloy systems."

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Materials Science & Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Tan, Teck L.

Contributors dc:contributor

• Johnson, Duane D.
• Bellon, Pascal
• Trinkle, Dallas R.
• Zuo, Jian-Min

• Cluster Expansion
• First-Principles
• Monte Carlo Simulation
• Thermodynamics
• Phase Diagrams
• Alloy
• Cluster Mean-field Theory
• Transition Temperatures
• Ordering
• Phase-segregating
• Coarse-graining
• Subspace Projection
• Fractional Factorial Design

Statement dc:rights

• Copyright 2011 by Teck Leong Tan. All rights reserved.

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/24227

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/24227