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Title:  Unique cluster expansion for reliable firstprinciples prediction of alloy thermodynamics and phase diagrams 
Author(s):  Tan, Teck L. 
Director of Research:  Johnson, Duane D. 
Doctoral Committee Chair(s):  Johnson, Duane D. 
Doctoral Committee Member(s):  Bellon, Pascal; Trinkle, Dallas R.; Zuo, JianMin 
Department / Program:  Materials Science & Engineerng 
Discipline:  Materials Science & Engr 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Cluster Expansion
FirstPrinciples Monte Carlo Simulation Thermodynamics Phase Diagrams Alloy Cluster Meanfield Theory Transition Temperatures Ordering Phasesegregating Coarsegraining Subspace Projection Fractional Factorial Design 
Abstract:  A theoretical multiscale 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 firstprinciples 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 welldefined 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 nonunique truncated CE sets that are extracted from an unnecessarily large number of firstprinciples data. In fact, the CE is directly related to the wellstudied 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 firstprinciples 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 nonunique 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 phasesegregating and ordering alloys via SI, using DFTdetermined structural energies from a selected set of ordered structures as input. The optimal CE set is used in Monte Carlo (MC) simulations and/or meanfield theories (MFTs) to construct temperaturecomposition phase diagrams for FCC CaSr, PdRh and AgAu (ordering), explaining various features and comparing them to available experiment data. The ECI of AgAu 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 AgAu to illustrate how critical ECI become systematically linearly independent with augmentation of configuration subspaces. Importantly, with only onethird 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 clusterbased MFT, generalized to arbitrary cluster sizes, is introduced to serve as a quick and reliable way to calculate phase diagram. Via a clusterlattice Fourier transform, the cluster MFT obeys selfconsistent relations between cluster and coarsegrained lattice correlations. Already with a singlesite 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 finitesize 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. 
Issue Date:  20110525 
URI:  http://hdl.handle.net/2142/24227 
Rights Information:  Copyright 2011 by Teck Leong Tan. All rights reserved. 
Date Available in IDEALS:  20110525 
Date Deposited:  201105 
This item appears in the following Collection(s)

Dissertations and Theses  Materials Science and Engineering

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois