Development of Computational Algorithms for Materials

Simulation and Design

Qiang Du Penn State University

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Outline •  An unauthorized story about “Materials Genome”

•  A brief review of our recent algorithmic development works supported by: –  NSF-DMR ITR: Computational Tools for Multicomponent

Materials Design 2002-2007

–  NSF-IIP Center for Computational Materials Design (CCMD) Phase I, 2005-2010

–  NSF-IIP I/UCRC CGI: Center for Computational Materials Design (CCMD), Phase II, 2010-2015

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A Glimpse of Materials Genome from Happy Valley

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Materials Genome as seen from outside

Ceder group, MIT

Zikui Liu, PSU

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General Manager: Dr.Zikui Liu, Prof of MSE Penn State

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Media

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Fun activities at Penn State

Working with State College High Math Club: outreach activities of NSF-CCMD 2010

NSF-ITR project group photo 2005

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Story as told on Wiki

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Performance

Property

Structure

Processing

Chemistry

Performance

Defects

Crystallo-graphy

Kinetics

Thermo-Dynamics

Materials Engineering and Science

Top-Down, Inverse Design Bottom-Up, Forward Simulation Courtesy: Zikui Liu, Penn State/Materials Genome

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MatCASE Project •  An Information Technology Research (ITR) Project

supported by NSF-DMR (2002 to 2007) Computational Tools for Multicomponent Materials Design •  PI Zikui Liu (MSE) •  Co-PIs: Chen (MSE), Du (MATH), Raghavan (CSE) •  http://www.matcase.psu.edu (Materials Computation And Simulation Environment)

•  It involved researchers from the Pennsylvania State University, Ford Motor Company, and National Institute of Standard and Technology.

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Four Major Steps 1.  First principles calculations to determine thermodynamic

properties, lattice parameters, and kinetic data of unary, binary and ternary compounds

2.  CALPHAD data optimization to extract thermodynamic properties, lattice parameters, and kinetic data of multicomponent systems combining results from the first-principle calculations and experimental data

3.  Multicomponent phase-field modeling to produce a microstructure in one to three space dimensions

4.  Finite element analysis to generate the mechanical response from the simulated microstructure

Liu, Z.K., Chen, L.Q., Raghavan, P., Du, Q., Sofo, J., Langer, S. & Wolverton C., An integrated framework for multi-scale materials simulation and design, J. Comput-Aided Mater. Des., 11, 2004, 183–199.

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NSF ITR: MatCASE Project

Experimental data

Mechanical responses of simulated

microstructures

Interfacial energies, lattice parameters and elastic constants

Kinetic data

Bulk thermodynamic data

Database for lattice parameters, elastic constants and interfacial energies

Kinetic database

Bulk thermodynamic database

Plasticity of phases Microstructure in 2D and 3D Elasticity of phases

First-principles calculations

CALPHAD

OOF: Object-oriented finite element

analysis

Phase-field simulation

Our focus

Our focus

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Going beyond ITR: IUCRC CCMD

Director: Zikui Liu (Penn State) Co-Director: David McDowell (GaTech)

Phase I: 2005-2009 Phase II: 2011-2015 NSF IIP PI: Zikui Liu, Co-PI: Chen, Du, Raghavan

NSF IUCRC: CCMD

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Algorithmic development 1.  Efficient phase field simulation codes + Mathematical

and numerical analysis –  Iterative solver for inhomogeneous elasticity, variable mobility –  Exponential time integrator for higher order time marching –  Adaptive resolution via moving spectral methods –  Saddle point search for nucleation (a fundamental problem)

Du-Chen-Yu-Zhu-Feng-Hu-Dai-J.Zhang-L.Zhang-JY.Zhang 1.  Robust codes for phase diagram computation

–  Automated phase diagram calculation –  Statistical analysis of input date, uncertainty quantification (critical for making computation predictive tools)

Du-Liu-Emelianenko-Saal-JY.Zhang

Joint work with Maria Emelianenko and Zikui Liu 16

Automated computation of phase diagrams

Some typical binary and ternary phase diagrams

Phase diagrams are maps of the equilibrium phases associated with various combinations of temperature and composition

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Gibbs energy minimization

is the number sites in phase i, is volume fraction of k particles

in phase i, is the Gibbs energy of phase i

if!ik

iG

min(fi,!i

k ){G = fi Gi(!i

k )}i=1

n

!

fi =i=1

n

! f0

fi!ik = f0 !0

k

i=1

n

! , k =1,…,K

!  (user-dependence) rely on prior knowledge

!  (stability, reliability) failure in identifying miscibility gap, lack quality assessment

Mathematically, equilibrium analysis of a K-component system with n phases leads to a minimization problem:

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Algorithms for automation and statistical analysis of phase diagram computation

Goals: " Calculate equilibria in multicomponent multiphase systems " Minimize the number of trials, get comparable accuracy of

solution with lower complexity

" Provide quantitative assessment of data quality and reliability of resulting phase diagrams, essential for predictive simulations

Ideas: " Rely on the geometrical properties of the Gibbs energies to find

better starting points, automatically identify miscibility gaps

" Use adaptive approach with effective sampling techniques

" Perform statistical data analysis and error quantification Emelianenko M., Liu Z.K., Du Q., Computational Materials Science, 2005

(with illustrations for ternary Ca-Li-Na system) J. Saal, J.Y. Zhang, V. Manga, M. Carolan, Q. Du and Z.-K. Liu, Review and statistical assessment of La1-xSrxCoO3 ! oxygen nonstoichiometry, 2012

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Critical nuclei morphology in solid state phase transformations

•  Predicting nucleation rate and its dependence on composition, temperature under various conditions is a basic problem in materials design and is of broad interests

•  Recently we studied the effect of elastic energy contribution in nucleation; critical nucleus may exhibit non-convex shapes and/or may have lower symmetry, leading to various crystallographic orientations (structure domains)

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Nucleation and growth with diffuse-interface nuclei

0 50000 100000 150000 2000000.0

0.2

0.4

0.6

0.8

1.0

1.2

Are

a fra

ctio

n (f(i))

Time step (i)

Diffuse-interface nucleation and growth KJMA's interpretation (fixed growth rate) KJMA's interpretation (varying growth rate)

(b)

Comparing to kinetics using the classical nucleation and normal growth theory and the JMAK equation

),3

exp(1)( 32

tuItf !!""=#

),exp( *0 GII !"=

We developed a diffuse-interface model to identify critical nuclei offline and combined with a KMC scheme to study nucleation and growth. L. Zhang-L.Q. Chen-Q. Du, Phy. Rev. Lett, 2007, Acta Mater., 2008, Comm. Comp. Phys., 2010, J. Comp. Phys, 2010; Heo,-Zhang-Du-Chen, Scripta Mater. 2011

Algorithmic development •  Robust algorithms for computing saddle points and

minimum energy paths –  Minimax algorithms –  Constrained string methods –  Shrinking dimer dynamics (SDD, a recently proposed

dynamic system variant of the popular dimer method). E.g.,

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Algorithmic development •  SDD: search index-1 saddle in an extended space

–  Efficiency: uses only force, just like a gradient system, –  Robustness: enjoys local stability, with guaranteed

convergence for some special potential energy

•  SDD can handle high dimensional complex energy landscape –  Rigorous mathematical/numerical analysis –  Demonstration for free energy surfaces associated with

the diffuse interface model –  Demonstration for potential energy surfaces associated

with geometrically constrained particles

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L.Zhang-L.Q. Chen-Q. Du, J. Sci. Compt., 2008; Q. Du-L.Zhang, Comm. Math. Sci., 2009; J.Zhang-Q,Du, J. Comp. Phys. 2012, SIAM Num Anal 2012

Main Collaborators •  Maria Emelianenko, GMU •  Lei Zhang, UCI •  Jingyan Zhang, Penn State •  Zikui Liu, James Saal, Penn State •  Long-Qing Chen, Tae-woo Heo, Penn State

Publications available at: http://www.math.psu.edu/qdu Thank you!

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