Computational Methods for Microstructure-Property Relationships
Somnath Ghosh
Dennis Dimiduk
Editors
Computational Methods for Microstructure-Property Relationships
ABC
Editors Somnath Ghosh Ohio State University Dept. Mechanical Engineering W. 19. Ave. 201 43210 Columbus Ohio USA
[email protected]
Dennis Dimiduk Wright-Patterson Air Force Base Materials & Manufacturing Directorate Air Force Research Lab. 45433-7702 Dayton Ohio USA
[email protected]
ISBN 978-1-4419-0642-7 e-ISBN 978-1-4419-0643-4 DOI 10.1007/978-1-4419-0643-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010935949 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
This book is dedicated to all those individuals whose discontent with the present state of knowledge and vision for the future make the research wheels turn
Preface
Design of and with materials plays an intrinsic role in today’s challenging world of high performance structural components and applications. They constitute an integral part of comprehensive structural design, given the opportunity offered by optimal materials design for structural performance and life enhancement. These opportunities impose high demands on effective modeling and simulation methodologies to establish quantitative relations between the material microstructure and physical properties at different length scales. The rapid advances in computer and computational sciences enable sophisticated simulations that unravel the underpinnings of complex material microstructure on behavior. In concert with outstanding advances in experimental methods, these computational tools are increasingly able to enhance the fundamental understanding of microstructure–property relations, thus enabling materials and process design for improved performance and life. The field of computational materials modeling transcends traditional disciplinary boundaries between mechanics, materials science, physics and chemistry, mathematics and computer science. In addition, it is creating a true synergy between experiments and modeling in terms of incorporation of physics, calibration, and validation. The results of these unified efforts at various scales are yielding unprecedented levels of rigor and accuracy in predictions of complicated phenomena that have previously eluded scientists and engineers. The role of microstructure on physical properties and performance is emerging as a quantitative discipline with broad and direct implications on material design. This book “Computational Methods for Microstructure–Property Relationships” is an attempt to capture this rapid advancement at a period of time, with a glimpse of what is yet to come in this very dynamic emergent field of science and technology. It introduces state-of-the-art advances in computational modeling as well as experimental approaches for materials structure–property relations. Representing a body of collected works by well-known professionals in the field, it covers topics ranging from materials modeling principles with a multiscale perspective to materials design. It presents the current state of knowledge for a wide collection of research areas related to materials assessment in structures–materials interactions. The collection aims at establishing the necessity of a robust integrated computational mechanics and computational materials science framework, vii
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together with an experimental validation protocol, that treats heterogeneous materials at microstructural and continuum scales. Selectively encompassing both computational mechanics and computational materials science disciplines, it offers an analysis of current techniques and selected topics important to industry researchers, such as deformation, creep,. and fatigue of primarily metallic materials. It emphasizes modeling at continuum and heterogeneous microstructural scales, e.g. crystalline or grain scales, validated with experimental observations. This book in intended for researchers in academia, industry, and government laboratories to understand the issues and challenges involved in predicting performance and failure in materials, with a focus on the engineering structure–materials interaction. Researchers, engineers, and professionals involved with predicting the performance and failure of materials are expected to find this book a valuable reference. This book is topically divided into four essential parts. The first part deals with 3D image-based materials structure data collection and representation and microstructure builders for mechanical response simulations. Chapter “Serial Sectioning Methods for Generating 3D Characterization Data of Grain-and Precipitate-Scale Microstructures” introduces serial-sectioning methods for 3D characterization of grain- and precipitate-scale microstructures. It focuses on the use of serial-sectioning methods and associated instrumentation as a means for collecting microstructural, crystallographic, and chemical data. Chapter “Digital Representation of Materials Grain Structure” discusses the state-of-the-art methods in the field of microstructure representation with focus on the following: data collection, feature identification, mesh generation, quantitative descriptors, and synthetic structure generation. Chapter “Multi-Scale Characterization and Domain Partitioning for Multi-Scale Analysis of Heterogeneous Materials” discusses the development of a multiscale characterization methodology leading to a microstructural morphology-based domain partitioning method for materials having nonuniform heterogeneous microstructure. The set of methods is intended to provide a concurrent multiscale analysis model with the initial computational domain that delineates regions of statistical homogeneity and heterogeneity. The method is intended as a preprocessor to multiscale analysis of mechanical behavior and damage of heterogeneous materials. Chapter “Coupling Microstructure Characterization with Microstructure Evolution” discusses the synergistic coupling of quantitative microstructure characterization via experimental imaging techniques, with computer simulations of microstructural evolution using the phase-field method. Having experimental images as inputs, the chapter describes uses of the phase-field method at different length scales to explore mechanisms of microstructural evolution, extract material parameters, conduct physics-based repairs of experimentally reconstructed microstructures, and evolve the microstructure for different time, temperature, stress, etc. regimes. Part of this volume is devoted to materials constitutive laws and kinematical approaches, together with their coupling of material structure to responses via simulation codes. These are presented in next six chapters. Chapter “Representation of Materials Constitutive Responses in Finite Element Based Design Codes” surveys
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FEM-based tools for simulating materials behavior and reviews the material models available in commercial codes. Chapter “Accounting for Microstructure in Large Deformation Models of Polycrystalline Metallic Materials” analyzes the influence of microstructure on large-strain mechanical behavior for metallic polycrystalline materials. Results of a macroscale continuum internal state variable-based model for tantalum are compared with those from a multiscale polycrystalline plasticity approach having explicit representation of the polycrystalline aggregate. In Chapter “Dislocation Mediated Continuum Plasticity: Case Studies on Modeling Scale Dependence, Scale-Invariance, and Directionality of Sharp Yield-Point”, the authors discuss a field dislocation dynamics theory to account for the emergence of inhomogeneous dislocation distributions at mesoscopic length scales, as well as their coupling to initial and boundary conditions and consequences on mechanical behavior. Size effects and scale-invariant intermittency are interpreted through field dislocation dynamics. Anisotropy of strain hardening induced by the emergence of internal stress fields is also reviewed in this chapter. Chapter “Dislocation-Mediated Time-Dependent Deformation in Crystalline Solids” shows a methodology for incorporating the effects of slip gradients associated with intra-grain deformation heterogeneity in crystal-plasticity-based finite element simulation. The treatise quantifies the orientation dependence of the misorientation field in the polycrystalline microstructure and introduces a modification of the kinematic decomposition that accommodates distortions arising from the presence of a static dislocation distribution. Chapter “Modeling Heterogeneous Intra-Grain Deformations Using Finite Element Formulations” is a review of two crystal plasticity-based methodologies for the prediction of microstructure–property relations in polycrystalline aggregates. These include a mean-field, second-order viscoplastic self-consistent method and a Fast Fourier Transform-based full-field method. Numerical examples demonstrate that models like the FFT-based formulation can explicitly account for interaction between individual grains. Finally, chapter 11 discusses multiscale modeling of plastic deformation and strength in crystalline materials with emphasis on models and experiments below the grain level. Specifically, the chapter deals with experimental advances and theoretical models for characterizing dislocations at the subgrain level. The third part introduces computational mechanics for time dependency of materials with links to fracture mechanics and multi-time scaling methods for fatigue in next three chapters. Chapter “Stochastic Upscaling for Inelastic Material Behavior from Limited Experimental Data” develops time-dependent plastic deformation and creep models for crystalline solids using dislocation-level mechanics. The theory uses microstructural information to develop broad quantitative mechanistic relationships that match the observed phenomenology. The discussion includes mobility-controlled systems, where dislocations move through the crystal under stress and interaction of dislocations with discrete obstacles for a range of alloys. Chapter “DDSim: Framework for Multiscale Structural Prognosis” introduces a prototype hierarchical computational simulation system called damage and durability simulator (DDSim) for prognosis of fatigue life of airframe components. While this prototype focuses on fatigue cracking, the framework can be extended to other
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modes of damage. Chapter “Modeling Fatigue Crack Nucleation Using Crystal Plasticity Finite Element Simulations and Multi-Time Scaling” addresses two important aspects of predicting fatigue crack nucleation in polycrystalline alloys under dwell cyclic loading. The first is a microstructure-sensitive criterion for dwell fatigue crack initiation in polycrystalline titanium alloys, while the second part of this chapter discusses a wavelet transformation-based multi-time scaling (WATMUS) algorithm for accelerated crystal plasticity finite element simulations. The WATMUS algorithm significantly enhances the computational efficiency for fatigue life prediction. Finally, the fourth part of this book deals with some additional emerging topics in the next three chapters. Chapter “Challenges Below the Grain Scale and Multiscale Models” examines selected experimental methods at different length scales that are important tools in building models for location specific design. While special experimental techniques are needed to probe the material at finer scales to assess the local behaviors, testing methods at all scales are discussed to demonstrate the breadth of experimental capability available at each scale of the material. A stochastic upscaling approach for strain-hardening plastic materials from limited experimental data based on random matrix theory is introduced in chapter “Emerging Methods for Matching Simulation and Experimental Scales”. The uncertainty characterized by constitutive tangential matrices can be construed as a reflection, on the coarse scales, of fluctuations of the fine scale features from which constitutive matrices are constructed. Finally, chapter “Simulation-Assisted Design and Accelerated Insertion of Materials” introduces some emerging concepts for robust design of materials and challenges for the synthesis of modeling and simulation and materials design. The distinction between materials design and multiscale modeling is elucidated in this chapter with emphasis on top-down requirements on material structure and performance to meet product requirements. The editors note that this work would not have been possible without continued financial and technical support from their employers, namely The Ohio State University and the Air Force Research Laboratory, Materials and Manufacturing Directorate. They also gratefully acknowledge the research support from various sponsoring agencies, viz. the Defence Advanced Research Projects Agency (Program Director: Dr. Leo Christodoulou), The Air Force Office of Scientific Research (Directors: Drs. Lyle Schwartz and Tom Russell; Program Directors: Drs. Craig Hartley and David Stargel), The Army Research Office (Program Director: Dr. Bruce Lamattina), and the Office of Naval Research (Program Director: Dr. Julie Christodoulou). In closing, the editors would like to extend their sincere thanks and appreciation to all the contributing authors of this volume for embracing our template vision and providing excellent state-of-the-art articles on different topics in the general field. They are also thankful to the Springer editorial staff, particularly Alex Greene and Andrew Leigh, for their tremendous support with the production of this book. Somnath Ghosh expresses his love and deep appreciation to his wife Chandreyee, son Anirban, and mother Lalita for their constant encouragement and support throughout this project. Dennis Dimiduk offers his deepest appreciation to his wife Lisa
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whose love and support made this work possible. He also extends thanks to the current and past members of the advanced metals team at AFRL who helped to form aspects of the vision represented in this book. Columbus, Ohio Dayton, Ohio
Somnath Ghosh Dennis M. Dimiduk April 2010
Contents
Microstructure–Property–Design Relationships in the Simulation Era: An Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Dennis M. Dimiduk
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Serial Sectioning Methods for Generating 3D Characterization Data of Grain- and Precipitate-Scale Microstructures.. . . . . . .. . . . . . . . . . . . . . . . . 31 Michael D. Uchic Digital Representation of Materials Grain Structure . . . . . . . . .. . . . . . . . . . . . . . . . . 53 Michael A. Groeber Multiscale Characterization and Domain Partitioning for Multiscale Analysis of Heterogeneous Materials . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 99 Somnath Ghosh Coupling Microstructure Characterization with Microstructure Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .151 Chen Shen, Ning Ma, Yuwen Cui, Ning Zhou, and Yunzhi Wang Representation of Materials Constitutive Responses in Finite Element-Based Design Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .199 Yoon Suk Choi and Robert A. Brockman Accounting for Microstructure in Large Deformation Models of Polycrystalline Metallic Materials.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .239 C.A. Bronkhorst, P.J. Maudlin, G.T. Gray III, E.K. Cerreta, E.N. Harstad, and F.L. Addessio Dislocation Mediated Continuum Plasticity: Case Studies on Modeling Scale Dependence, Scale-Invariance, and Directionality of Sharp Yield-Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .277 Claude Fressengeas, A. Acharya, and A.J. Beaudoin
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Dislocation-Mediated Time-Dependent Deformation in Crystalline Solids .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .311 Michael Mills and Glenn Daehn Modeling Heterogeneous Intragrain Deformations Using Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .363 Paul Dawson, Jobie Gerken, and Tito Marin Full-Field vs. Homogenization Methods to Predict Microstructure–Property Relations for Polycrystalline Materials . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .393 R.A. Lebensohn, P. Ponte Casta˜neda, R. Brenner, and O. Castelnau Stochastic Upscaling for Inelastic Material Behavior from Limited Experimental Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .443 Sonjoy Das and Roger Ghanem DDSim: Framework for Multiscale Structural Prognosis . . . .. . . . . . . . . . . . . . . . .469 John M. Emery and Anthony R. Ingraffea Modeling Fatigue Crack Nucleation Using Crystal Plasticity Finite Element Simulations and Multi-time Scaling . . . . . . . . . .. . . . . . . . . . . . . . . . .497 Somnath Ghosh, Masoud Anahid, and Pritam Chakraborty Challenges Below the Grain Scale and Multiscale Models . . .. . . . . . . . . . . . . . . . .555 Hussein M. Zbib and David F. Bahr Emerging Methods for Matching Simulation and Experimental Scales . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .591 Andrew H. Rosenberger Simulation-Assisted Design and Accelerated Insertion of Materials . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .617 D.L. McDowell and D. Backman Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .649
Contributors
A. Acharya Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA,
[email protected] F.L. Addessio Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Masoud Anahid Department of Mechanical Engineering, The Ohio State University, W496 Scott Laboratory, 201 West 19th Avenue, Columbus, OH 43210, USA D. Backman Worcester Polytechnic Institute, Mechanical Engineering, Worcester, MA, USA David F. Bahr School of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA A.J. Beaudoin Department of Mechanical Sciences and Engineering, University of Illinois at Urbana Champaign, Urbana, IL 61801, USA R. Brenner Laboratoire des Propriet´es M´ecaniques et Thermodynamiques des Mat´eriaux, Universit´e Paris XIII, Av J.-B.Clement, 93430 Villetaneuse, France Robert A Brockman Universal Energy Systems, Dayton, USA C.A. Bronkhorst, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA,
[email protected] ˜ P. Ponte Castaneda Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA O. Castelnau Laboratoire des Propriet´es M´ecaniques et Thermodynamiques des Mat´eriaux, Universit´e Paris XIII, Av J.-B.Clement, 93430 Villetaneuse, France E.K. Cerreta Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Pritam Chakraborty Department of Mechanical Engineering, The Ohio State University, W496 Scott Laboratory, 201 West 19th Avenue, Columbus, OH 43210, USA xv
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Contributors
Yoon Suk Choi Universal Energy Systems, Dayton, USA,
[email protected] Yuwen Cui Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA Glenn Daehn Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA Sonjoy Das Massachusetts Institute of Technology, Cambridge, MA 02139,
[email protected] and Department of Civil Engineering, University of Southern California, Los Angeles, CA, USA Paul Dawson Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA,
[email protected] Dennis M Dimiduk Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA,
[email protected] John Emery Solid Mechanics Division, Sandia National Laboratories, Albuquerque, NM 87113, USA,
[email protected] Claude Fressengeas LPMM, Universite Paul Verlaine - Metz/CNRS Ile du Saulcy, 57045 Metz Cedex 01, France,
[email protected] Jobie Gerken ANSYS, Inc., Canonsburg, PA 15317, USA Roger Ghanem Department of Civil Engineering, University of Southern California, Los Angeles, CA 90089, USA,
[email protected] Somnath Ghosh Department of Mechanical Engineering, The Ohio State University, W496 Scott Laboratory, 201 West 19th Avenue, Columbus, OH 43210, USA,
[email protected] G. T. Gray III Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Michael A. Groeber Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA,
[email protected] E. N. Harstad Engineering Sciences Division, Sandia National Laboratories, Albuquerque, NM 87185, USA Anthony R. Ingraffea Cornell Fracture Group, 643 Rhodes Hall, Cornell University, Ithaca, NY 14853, USA,
[email protected] Deepu S. Joseph Department of Mechanical Engineering, The Ohio State University, W496 Scott Laboratory, 201 West 19th Avenue, Columbus, OH 43210, USA
Contributors
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R.A. Lebensohn Materials Science and Technology Division, Los Alamos National Laboratory, MS G755, Los Alamos, NM 87545, USA,
[email protected] Ning Ma Corporate Strategic Research, ExxonMobil Research & Engineering Company, 1545 Route 22 East, Rm LB248, Annandale, NJ 08801, USA P. J. Maudlin Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Tito Marin Department of Industrial Engineering, University of Parma, Parma, Italy D.L. McDowell Georgia Institute of Technology, GWW School of Mechanical Engineering, Atlanta, GA, USA,
[email protected] Michael Mills Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA,
[email protected] Andrew H. Rosenberger Materials & Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA,
[email protected] Chen Shen GE Global Research, 1 Research Circle, Niskayuna, NY 12309, USA Michael D. Uchic Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA,
[email protected] Yunzhi Wang Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA,
[email protected] Hussein M. Zbib School of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA,
[email protected] Ning Zhou Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210, USA
Microstructure–Property–Design Relationships in the Simulation Era: An Introduction Dennis M. Dimiduk
Abstract Computational methods are affecting a paradigm change for using microstructure–property relationships within materials and structures engineering. This chapter examines the emergent use of quantitative computational tools for microstructure–property–design relationships, primarily for structural alloys. Three major phases are described as a historical “serial paradigm,” current “integrated computational materials engineering” and, future “virtual materials systems” emerging from advances in multiscale materials modeling. The latter two phases bring unique demands for integrating microstructure representations, constitutive descriptions, numerical codes, and experimental methods. Importantly, these approaches are forcing a fundamental restructuring of materials data for structural engineering wherein data centers on a hierarchy of model parameterizations and validations, rather than the current application-specific design limits. Examining aspects of current research on microstructure-sensitive design tools for single-crystal turbine blades provides an accessible glimpse into future computational tools and their data requirements. Finally, brief descriptions set context and interrelationships for the remaining chapters of the book.
1 Microstructure–Property–Design Relationships and Structural Materials Engineering Present-day advancements in microstructure–property relationships are coming about via computational methods. The efforts largely recognize that microstructure– property relationships evolve over a wide range of scales and that both technical and computational advances must occur for adequate representations of these
D.M. Dimiduk () Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA e-mail:
[email protected]
S. Ghosh and D. Dimiduk (eds.), Computational Methods for Microstructure-Property Relationships, DOI 10.1007/978-1-4419-0643-4 1, c Springer Science+Business Media, LLC 2011
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relationships within predictive tools. However, many of these efforts fall short of full recognition that engineered materials are systems. What is needed is a computational methodology and framework for systems engineering of materials and the sciences that support such an approach. The systems engineering of materials within a simulation environment will provide advances to both the materials utilization and representations for usefully advancing quantitative microstructure–property relationships. To better understand what is needed from the computational framework, it is useful to briefly examine materials in present-day engineering. About 100 years ago, a defining aspect of materials science and engineering (MSE) had its origin in the first microscopy studies of materials structure; yet, nearly a half-century would pass before their impact evolved into the MSE discipline (Smith 1988; Cahn 2001; Olson 1997). With the study of microstructure (including defect structure or statistical extrema), the materials engineer gained an important tool by which processes and properties are controlled. Microstructure– properties science was born and has expanded ever since. MSE now recognizes four major disciplines of practice that identify the unique character of the MSE field: processing–structure–properties–design, irrespective of the material type, class, or design application (see Fig. 1). However, unlike other mainstream engineering
Fig. 1 The four defining disciplines of practice for materials science and engineering (MSE) represented as a tetrahedron. In this context, “Processing” refers to composition, synthesis, and processing in general. “Structure” refers to all aspects of microstructure, including both intrinsic and extrinsic defects from the atomic to macroscopic scales. “Properties” and “Design” refer to materials performance and behavior and, to engineering design rather than materials design, respectively. Selected examples of the types of studies and activities that tend to link the major disciplines are shown about the periphery of the figure
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disciplines (e.g., civil, chemical, electrical, and mechanical) that largely came into existence as the quantitative frameworks for them emerged, the same cannot be said for MSE. More broadly speaking, structural alloys tend to be defined from two different perspectives. Materials producers (and patent law) associate them with compositions of matter and the prescribed synthesis and process paths by which they are formed into engineered products. Alternatively, from a structures-design engineering perspective, materials are viewed as contextual databases containing representative measured values of property bounds, including statistical minima, as functions of selected variables, such as temperature or state of stress. These are often represented within models. The contextual aspect of those databases usually relates to specific application products and manufacturing processes. Additionally, the structures engineer also attributes the businesses and practices that make materials available as products to the materials engineering discipline. Historically, the metallurgist, ceramist or chemical engineer and, more recently the materials-scientist or engineer carried out the onerous task of melding these perspectives into unified activities and practices for safe and affordable structures. Within this engineering reality, the notion of microstructure–property relationships is only implicit at best. While both materials and process engineers and part designers recognize that structural materials have significant microstructural variations, there are few quantitative tools and standards that permit integration of that knowledge into the broad engineering process, especially in a predictive manner. Consequently, outside of the MSE community, processing–properties–design relationships (not including structure) are generally recognized via the “allowables” (i.e., distributions of property values including statistically defined minima) for using a material for a given design. For most of today’s products, one typically defines an application and then seeks to define and document a processing specification through specific suppliers by which a selected composition of matter will reproducibly lead to properties (performance) for that application. Material microstructure descriptors, such as grain size (ASTM 2002), are only used as specifications of the material for process assurance. For higher value engineered structures (e.g., a gas turbine engine disk), a database of processing–properties relationships typically develops in which the data are reduced to phenomenological constitutive laws that are linked to the application design process via finite element method simulations of the part configuration. With few exceptions, these constitutive relationships are assumed to hold over volumes of material that are essentially on the scale of the part (meter scale), even though every metallurgist or materials scientist understands that heterogeneities or defects that affect properties exist over many length scales from the full part to the atomic level. Clearly, there is a disconnection between MSE, and the broader engineering community as the notion of processing–structure–properties–design essentially does not exist beyond MSE. This disconnection is a rational result of the fact that even after a century of development the quantitative links between processes and structure and, structure and properties, are insufficiently advanced to permit direct systems-oriented
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optimization of materials and products (in “Simulation-Assisted Design and Accelerated Insertion of Materials” by McDowell and Backman). There is simply too much complexity associated with the kinetics of processes to quantitatively define the resulting microstructures within the equally complex hierarchy of length and timescales of the applications. For electronic materials, the length and timescales may be extremely small fractions of seconds and nanometer dimensions, while for structural composites they may be at the scale of the components and system dimensions (meters) over timeframes of years. Fortunately, current advances in computing capabilities and MSE tools bring opportunities for not only expanding the quantitative basis for processing–structure– property–design relationships within simulation environments but also for redefining aspects of MSE within those simulation environments. In so doing, MSE becomes a quantitative engineering discipline for structural materials and several aspects of its relationship to other engineering disciplines will be redefined. Full recognition of this opportunity stems from considering aspects of the use of computer modeling and simulations along the evolutionary path of MSE.
2 Computational Materials Science for Microstructure Computers and simulation were available essentially since the origins of MSE as a recognized discipline. Several phases of their use in MSE are linked to growth in computational capacity and databases. In the 1950s and 1960s, the computer was commonly used to model specific phenomena, usually within a mean-field, especially where numerical solutions to differential equations were necessary. From a materials engineering perspective, perhaps the best example of this is the computer calculation of phase diagrams or the “CalPhad” method that was well developed by the end of the 1960s (Kaufman and Bernstein 1970). During that period foundations were built for materials-oriented computer simulations that last to this day (see additional diverse examples such as computing diffraction contrast of transmission electron microscopy images (Head et al. 1973) and plasticity analysis for metal deformation (Mandel 1973; Kocks 1987) to name but two others. Importantly, even though the foundational sciences were known more than 40 years ago, neither the computational capacity nor the necessary databases were sufficiently developed for the CalPhad method to have significant engineering impact at that time for alloy or process development. Only about 10 years ago did the method begin to add value to engineered products and the practices of MSE. Today, after more than a decade of sustained development investments for engineering, CalPhad techniques are becoming a part of standard industrial methods (NMAB 2008; Backman et al. 2006). A second phase in the maturation of materials computational methods occurred during the 1970s and 1980s through research in process modeling. Simulation codes evolved that are still in use today (ProCAST http, DEFORM http). These codes, based on continuum fields and state variables without treatments of microstructure, are essential to design engineering of aerospace and other industrial parts
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and components. Also during this period, methods for solving a range of materials challenges from the electronic structure of materials to techniques for plasticity and stress analysis continued to advance (Hafner 2000; McDowell 2000). Methods for simulating plasticity under crystallographic constraint within the finite-element method gave new insights into behavior at the mesoscopic scale, including strain localization during crystal slip (Asaro 1983). One could say that during the late 1970s through mid 1980s, computational materials science (CMS) came into its own as a discipline of study. Here, the term CMS refers to the activities of a widespread community of investigators that are developing simulation tools to represent unit mechanisms exhibited by materials. These include such techniques as electronic structure methods for selected material properties and thermodynamic quantities (Hafner 2000; van de Walle et al. 2002; Liu et al. 2006); empirical atomistic methods that offer insight into understanding dislocation core structures, surfaces and grain boundaries (Daw and Baskes 1984; Vitek 1985; Tschopp et al. 2008); dislocation dynamics methods (Devincre et al. 2001; Ghoniem et al. 2000); phase field methods and, many others. A good compendium of such methods may be found in the work edited by Yip (2005). However, the majority of the CMS-based advances in understanding mechanisms of materials behavior had little or no impact on materials engineering. While the quantitative nature of simulated results improved, too frequently they lacked comprehensive context or sufficient accuracy for use in engineering design. The few applications of simulation-based methods to industrial-world microstructure– properties engineering tended to use simulation results to provide qualitative insight into existing engineering processes (see for example Dimiduk 1998); however, there were notable exceptions (Shercliff and Ashby 1990). There are numerous reasons for this, but obvious among them was insufficient computational capacity together with the integrated data available during that time. Throughout the 1980s and 1990s, many materials simulation efforts were performed in relative isolation within the MSE, physics, mechanics, and chemistry communities having few linkages to engineering techniques or design tools. Unlike other engineering disciplines, one might suppose that the role of simulations within MSE was viewed as only interesting or important for understanding qualitative behavior trends since so little community-wide work was carried out to establish standard techniques and methods as foundations for industrial practice. Consequently, simulation-centric materials engineering methods continued to evolve in a piece-wise fashion within proprietary corporate communities. Throughout this period, there were few efforts outside of the process-modeling discipline that attempted to integrate mechanistic or heuristic knowledge within simulations to understand the microstructure–property relationships in engineered products, as a standard methodology of practice. Although some researchers recognized that the quantitative aspects of microstructure–property relationships were underdeveloped (Cedar 2000), CMS was often characterized as simply “applied quantum mechanics” (Bernholc 1999). During this period, CMS was essentially a “cottage industry” of models and modelers of and to itself (Dimiduk et al. 2004a).
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During the mid and late 1990s, a few industry, government and academic leaders began to see the limiting aspects of this state of CMS (see for example Olson 1997; Christodoulou DARPA-AIM http; Fraser CAMM http). These leaders recognized that CMS approaches to materials modeling typically originated from the “bottom up” of the length and time scales and that such approaches rarely made an impact on the practices or efficiencies of design engineering, especially for structural materials. Further, there was recognition in the MSE community that significant computing capability was becoming sufficiently widespread that new approaches to simulationbased materials engineering should be attempted from the “top down.” As a result, two notable new initiatives in computational-based materials engineering were initiated in the first year of the new millennium (NMAB 2008).
3 Integrated Computational Materials Engineering 3.1 Materials Readiness and the Evolving Microstructure–Properties–Design Paradigm To best understand the uniqueness of the integrated computational materials engineering (ICME) approach and its impact on the practices associated with microstructure–properties–design relationships, it is useful to first understand the concepts of materials engineering readiness. Materials development and process engineering involves significant open-ended risk and cost. To manage and mitigate that risk, the MSE community adopted various frameworks for assessing readiness along the pathway toward product application. These frameworks are similar to ones used for other engineering but are tailored (especially within major manufacturing companies) to materials and processes disciplines. Figure 2 illustrates the highest-level structured “stage-gate” process that exists within most materials and processes practice. Typically, ten levels of readiness are defined and the progression of application-specific technologies through these levels occurs within well-defined engineering templates. These templates demand specific test data, cost assessments, manufacturing source qualification, etc. that gain fidelity and scope at each stage of development. This serially staged paradigm of materials and processes technology maturity to some degree reflects the learning curve that innately exists for anything new. Unfortunately, the expanding scope required at each step is a key limiter to this paradigm that adds significant risk, quite often cost, and certainly time. Reviews of case studies of materials development that follow a serial paradigm have shown that it leads to serious challenges for materials development and limits the opportunities for coupling materials and process advancements within mainstream engineering design practice (NMAB 2008; NMAB 2004; Lipsitt et al. 2001; Dimiduk 2001; Dimiduk et al. 2003). The serial paradigm leads to what has been called the “valley-of-death” for new materials and processes. That valley exists for several reasons including funding gaps, long time requirements for experimental
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Fig. 2 General technology readiness levels (TRL) for materials. The ten stages of materials readiness are adopted from the broader engineering readiness metrics used for products and systems. Historically, achieving the transition from TRL 3 to TRL 5 is the most difficult step. The reason for this is that technical risks typically remain high at TRL 3; however, the financial outlays required to mitigate them also grow much more substantially at this stage by comparison to the lower levels. Better materials and processes simulation tools are needed throughout, but especially for risk mitigation through the TRL 3–5 maturity levels. For aerospace materials, evolutionary advances (such as modified alloy compositions within established applications) are known to require 7–12 years to reach first use. For more challenging completely new materials, such introducing ceramic composites or TiAl alloys in turbine engines, the time span for achieving fist-use readiness exceeded 26 and 36 years, respectively
or empirical iterations and, what may loosely be called a “point contact” interface between present-day design engineering and materials engineering. To further illustrate this point of contact, Fig. 3 schematically depicts the broad engineering procedural steps that may be used to select the geometric configuration of a manufactured aerospace metal component. The figure also shows selected materials and processes procedural steps that are taken to assure appropriate microstructure–property relationships are maintained in the final product. Inspection of the figure reveals that the primary interface between the design process and the materials development process lies in the steps needed to assure that validated constitutive descriptions (or minima curves and allowables) are available for the design optimization procedures. Thus, within this schematic depiction, the interface between the communities is a point contact. This point of contact includes not only the constitutive laws that reside within component design codes, but also their empirical validation against databases that must sufficiently encompass the variations of microstructure–property relationships judged to be important to the specific
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Fig. 3 Schematic representation of activities within today’s experiment-intensive processing– properties–design serial paradigm for materials engineering. The methodology has no explicit consideration of microstructure. Microstructural effects are only implicitly considered when extracting specimens and as selected specifications for parts. Microstructural effects/variation is represented through expensive, time-consuming testing and multiple full-scale process and test iterations are usually required
design. Given that the allowables databases are produced from application-specific, full-scale development hardware, this serial approach inevitably leads to a conservative estimation of material performance and does so through a costly process. Since part-specific and feature-specific microstructures and properties cannot be accounted for within the design system, the observed “worst-case” uncertainties are assigned to all parts at all locations (Christodoulou and Larsen 2004). Consequently, microstructure–property relationships are specified and controlled in the context of their application databases alone, usually via testing of full-scale prototype parts. However, further advancement in the design process demands a less conservative and more realistic, probabilistic approach (McClung et al. 2008; Millwater and Osborn 2006). That new demand is driving the MSE community toward developing predictive tools for location-specific properties that can be used within probabilistic design tools. Herein lies one major hurdle for microstructure–properties sciences and materials development in general. As long as materials behavior can only be indirectly defined within the very specific contexts of their applications, via extensive testing of samples excised from full-scale prototypes that may not even directly capture the design or materials-limiting features of interest, materials development will always entail long development times and high costs. That fundamental limitation in the procedure for obtaining and representing materials performance data presently
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places the whole of MSE into a unique domain that is outside of those of the other engineering disciplines. The time scales, cost structures and design tools are simply mismatched, while the risk is high. Today one develops empirical knowledge of materials response to chemistry and process iterations within the stage-gated templates described previously, such that learned practitioners of the engineering disciplines can support design judgments. Those judgments inevitably entail reasonable assurances to business managers that the financial investments in scale-up and advances in technology readiness are affordable within business plans and product timing. For the future, materials development needs to be achieved via a new materials-todesign paradigm. Essential to that paradigm is that the materials readiness structure (readiness templates) be re-cast to maximize the scope of readiness information at the earliest stages; then, to expand only their fidelity with added development investments and time. Fortunately, efforts toward building these are well underway.
3.2 Accelerated Insertion of Materials, Virtual Aluminum Castings, and the ICME Paradigm Today, the computational tools that facilitate quantitative support for the development and investment judgments required for materials scale-up are just emerging. Examples of these exist within the ICME demonstration efforts that occurred during this decade (NMAB 2008). Essentially, the underlying concept behind the efforts is that having simulation tools for all aspects of new product and materials development will reduce development time while lowering costs and risks. Two notable examples of the ICME paradigm will now be discussed. Within the aerospace sector, the Accelerated Insertion of Materials (AIM) program was sponsored by the US Defense Advanced Research Projects Agency (DARPA) and the United States Air Force, to examine and restructure the paradigms for metal and organic-composite materials development (NMAB 2008; Backman et al. 2006; Dimiduk et al. 2003, 2004a, b). Similarly, within the automotive sector, the Virtual Aluminum Castings (VAC) program was sponsored by Ford Motor Company (NMAB 2008; Allison et al. 2006). In the specific sense of microstructure– property relationships, the efforts showed that representing the microstructural aspects of materials (especially including kinetics and mechanical behavior), via models that are integrated within design-engineering optimization protocols and software, yields dividends to the product development cycle. Importantly, the case studies showed that even elementary theory and empirical models have a substantial positive impact on the design engineering process when fully integrated within a computational environment (NMAB 2008; Dimiduk et al. 2003). Somehow, that important payoff to engineering was missed by most of the CMS and MSE research communities and to this day is not developed as an integrated materials engineering standard practice. Also, when viewed from the perspectives of these demonstrations, there is now a clear justification for expanding the fidelity
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Fig. 4 Schematic representation of activities within ICME paradigm for materials engineering (see text for explanation). Methodology explicitly includes microstructural-based design via microstructure evolution within process models and, mechanical property models being applied to various regions of designed part. Including microstructure–property relationships via simulations means that the domains of design and materials engineering overlap much more significantly within ICME than within the historical paradigm for materials engineering. The ICME paradigm includes the early cases of explicitly using processing–structure–properties–design within closed-loop engineering frameworks
of microstructure–property representations and predictive capabilities and, also a somewhat general template for both focusing those developments and then integrating them into the product value stream as they occur. The materials and processes development paradigm has changed with the evolution of ICME. Figure 4 shows a similar schematic as the one previously described in Fig. 3, but with modifications that reflect broad procedural changes brought about via the ICME approach as it was applied in the AIM program. Two aspects of the new procedure are noteworthy. First, as shown by the expanded activities associated with step “C2” (in the upper right-hand side of the figure), specific simulation tools focused on microstructure–property relationships enter into the development paradigm. Second, utilizing such tools fundamentally changes the experimental activity that currently takes place to empirically assure the manufactured products perform in the desired fashion. Rather than many full-scale synthesis and processing trials followed by sectioning and testing, many of the results of such efforts are now anticipated via simulations. Having models, even in empirical form, integrated with the design process permits iteration and optimization via design tools and minimizes the time-consuming and expensive procedures associated with fullscale prototype product development. Thus, the overlap between engineering design and MSE fields of practice has expanded. That expansion is the direct result of
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using simulation tools to provide a more quantitative and structured description of the microstructure–property relationships of materials. A widespread acceptance by a peer group of engineers, systematic reductions in the types, cost and quantity of data needed and, the predictive nature or capabilities of the microstructure– properties relationship tools used within such a paradigm, are all direct measures of the quantitative advance of the field. Future advances in computational methods for microstructure–property relationships should be evaluated by those metrics.
3.3 The Evolving Needs for Materials Data Another important aspect of the ICME paradigm for materials not explicitly shown in Fig. 4 was a significant aspect of the both AIM and VAC feasibility demonstrations. That aspect pertained to the development of models and the nature of experimental data. Within the historical processing–properties–design paradigm for materials, critical design data exists almost entirely in the form of measured mechanical properties obtained from production-scale hardware – again, having little explicit connection to microstructure. However, the ICME paradigm changes the structure and types of data that are essential to design. Under the ICME approach, data must be associated with models and supported simulation codes. Also, specific types of data are collected for the primary purpose of validating codes. That data often extends outside of the ranges typically associated with prototype parts and may be associated with certain pedigree-type materials and microstructures. Fortunately, just as simulation tools and models are becoming more advanced, test and evaluation procedures are becoming automated and miniaturized. Critical data can increasingly be measured from small-scale samples prepared to validate kinetics or mechanical behavior domains for models. The last paragraph discusses points that are nontrivial and merit further comment. For example, the nature of data intrinsic to an expanding ICME paradigm is data associated with simulation tools and their validation. Those tools by their inherent architectures and operative material models define the data required for their use. In this respect, the ICME paradigm is in its infancy and aspects of data taxonomy and efficiencies must be developed for the purposes of supporting simulations. However, even from the initial case studies just described, the ICME paradigm suggests a different view of data and materials informatics than the one described by recent reports on the subject (Cebon and Ashby 2006; Arnold 2006). Those reports essentially describe higher-fidelity extensions of the classical MSE-design paradigm – a paradigm constrained by the empirical development of handbook materials allowables. Within that paradigm, the role of the computer is “passive” in that it primarily facilitates the organization of greater quantities of information. When data sets are sufficiently large and too complex for typical human interrogation, this paradigm may not exclude cases of the “blind discovery” of new relational knowledge (data mining) in a more “active” mode.
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However, computational tools and simulation environments are beginning to synthesize data that may be fused with conventional empirical measurements (Liu et al. 2006; van de Walle et al. 2002). The practice is likely to spread far beyond its present use within alloy thermodynamics. Yet, there is little readiness for this within the old processing–properties–design paradigm and the practice is limited even within the current ICME paradigm. The MSE and design communities have a formidable task ahead of them to define appropriate data architectures and a taxonomy that will not only permit full “active” utilization of materials simulations in the design process but also maintain efficient certifiable engineering practices throughout the new simulation era. Within an emergent paradigm called here “virtual materials systems” that taxonomy and the actual data are facets of the substantially expanded and quantitative nature of microstructure–property relationships. Finally, the new ICME paradigm suggests that the materials allowables view of data will change to more effectively utilize the active power of materials simulations for “synthesizing” data and providing quantitative insights into materials response.
3.4 ICME: Lessons Learned There is value to considering lessons learned from the initial case studies of AIM and VAC. The recent report by the US National Materials Advisory Board discusses some of these lessons (NMAB 2008), but a selected three global aspects are highlighted here. First, for the longer term, the contrasting primary attributes of engineering design and MSE must be bridged. For engineering design, those attributes include a simulation-centric community of practitioners, education structures that convey such practices, well developed and supported simulation tools that are integrated with heuristic data and, the expectation that many rapid-time-frame simulations will be carried out as a routine part of the design process. Conversely, the primary attributes of MSE in this regard currently include long lead times for experiment results within a data focused community of practitioners, an educational system that is just now grappling with an appropriate treatment of ICME and its tools, relatively few established and supported simulation codes that are still too separated from heuristic data and, a general expectation that when simulations are done they will commonly be characterized by relatively few large-scale simulations performed in a supercomputing environment. As aspects of the previous discussion and portions of this book support, the gap between these communities exists in no small part because of the still underdeveloped quantitative sciences and standards associated with materials microstructure–properties kinetics and mechanical behavior. A second lesson contained in ICME is that the engineering design paradigm needs to evolve to explicitly include material heterogeneity within engineered parts (read microstructure–property–design relationships). In present day design practice, those aspects of heterogeneity not broadly included in databases or represented in analytical and simulation tools tend to be captured via heuristic rules that constrain the design process. For example, heterogeneities within materials lead to a
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variation in the performance for identically designed parts and populations of those parts perform differently. Consequently, that variability in parts often leads to the costly replacements of part populations based upon time in service, rather than conditional replacement tied to specific part behavior (Christodoulou and Larsen 2004). In the longer term the development of simulation tools must strive to mitigate the need for heuristic rules by integrating sensor measurements of the service history and environment into materials response models. In that way, the design and user communities would also gain tools to assess variations in the part lives that result from variations in their application environments. Conversely, it is important for the ICME and CMS communities of practice to recognize that in most cases of engineering design, there is an incomplete understanding of details of the use or operational environment even with sensor measurements. Thus, heuristic rules will always be a part of the design system to varying degrees and both design and materials must strive for robustness. A third key lesson focuses on the notion that all engineering design proceeds from representations of the desired properties, behavior phenomena, part-geometry, design constraints and, the materials from which parts are constructed. The materials engineer should ask of every item of interest “how should this item be modeled and represented in the optimization framework for design?” The whole of this book focuses on selected aspects of microstructure–property representations. However, engineering design demands representations for many additional aspects of the product value-stream that could interact effectively within simulation-based materials properties tools, including product cost (in “Simulation-Assisted Design and Accelerated Insertion of Materials” by McDowell and Backman). Thus, the nature of the representation used in simulation for a selected attribute or property is one clear measure of present-day understanding, relative importance and tractability of the attribute within the design and simulation environment. The examples of ICME to date suggest that simulation-based approaches to microstructure–properties relationships can be effectively used in the design process to add value to engineered products. Thus, it is reasonable to expect that the fidelity of those tools for the representations of microstructure–property relationships will also grow. Consequently, it is useful to consider where the expansions of microstructure–property science may lead within the modeling and simulation era.
4 Multiscale Materials Modeling, Materials Systems Simulation Science, and Virtual Materials Systems Present-day advancements in microstructure–property relationships are coming about via the techniques of multiscale materials modeling, especially concurrent multiscale modeling. Those efforts largely recognize that microstructure–property relationships evolve over a wide range of scales and that advances must occur for adequate representations of these within predictive tools. However, even most of those efforts fall short of full recognition that engineered materials are systems and,
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engineering demands standards of practice. What is needed for MSE and CMS is a systems approach to materials simulations and the sciences that supports such an approach. The systems engineering of materials within a simulation environment will provide the usefully structured advances to both materials utilization and the tools for quantitatively representing microstructure–property relationships. Given the context of materials engineering discussed previously, it is useful to peer into the future of microstructure–properties science and engineering. This book captures one view of that future through a selected look at a few of the advanced techniques in the field as well as some of the pacing state-of-the-art capabilities and challenges. The set of techniques is drawn from the editors’ viewpoint that microstructure–properties relationships science is headed toward the development of “virtual materials systems” in every sense of the term. That is, just as the biological sciences are slowly evolving toward computer-based representations of systems (such as humans for example) that somewhat virtually function in the same ways as their real-world counterparts, so too MSE should strive to supply computer-based systems representations of materials that mimic the real-world behavior at all scales (Wikipedia Virtual Human http). This view of the future demands a full embrace of simulation tools as an integrating theme and, in some respects, a defining aspect of the quantitative microstructure–property sciences. The view also requires that research embrace the notion that materials rarely perform outside of a systems context (for example, a turbine engine airfoil system, or an automotive engine valve system, etc.).
4.1 Microstructure–Property Representation and Simulation The engineering objective is to represent microstructure–property relationships together with engineered part designs within virtual materials systems. Quantitative predictions of part performance are obtained via numerous statistical instantiations of the material microstructure and the resultant simulated responses of those structures for a current part configuration. To achieve such simulation environments, one must recognize the fact that there are only four primary domains of freedom for simulations that collectively determine the quality and fidelity of the resultant predictions. Figure 5 depicts those domains for representing each aspect of materials microstructure–property relationships in a computational environment. Aspects of concurrent multiscale materials modeling strive to expand these four domains of materials representation by having the structure representation and perhaps even the constitutive description(s) evolve in an adaptive fashion as heterogeneities (such as local deformation or micro-cracking) evolve out of the initial representation. Simulation science involves quantitative management of error metrics, clear descriptions of failure criteria and, an intimate knowledge of the computing environment employed for the simulation set. As depicted in Fig. 5, there must be a multiscale representation of the engineered structure and its microstructure that includes coarse-scale domains of the part,
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Fig. 5 For any property of interest in design, there exists a multiscale hierarchy of microstructural effects that must be represented within simulation codes. However, as this figure depicts, within the computational environment there are only four broad domains of freedom for representing all of those aspects of the material. Recognition of these four domains provides a means for assigning each aspect of the material to the simulation environment and, by doing so, clearly identifies the coarse-graining inherent to the selected technique
extrinsic defect structures, intrinsic microstructure, their statistics at various scales and, even the smallest-scale aspects that affect chemical kinetics. That representation must dovetail with the constitutive descriptions of the system energetics and evolution. For example, the constitutive descriptions may involve pseudopotential formulations for electronic interactions at one lower length-scale, empirical atomic interaction potentials at another scale, mean-field thermally activated process models at a still larger scale, as well as the myriad mechanical behavior descriptions that are captured in present-day property models and design codes. However, to select the most appropriate modeling and simulation development pathway, it is not enough to know the constitutive relationships and structure representation alone; one must also know the context of the system, or design requirements that are to be simulated. Both of these in turn must align with all aspects of computational tractability of the simulation methods, represented in the figure by the domain of numerical schemes. For example, today there is little ability to represent thermally activated processes within parametric dislocation dynamics simulations, making the present form of that numerical environment a poor choice for studying the creep behavior of materials. Similarly, as some of the chapters in this book reflect, there is a growing ability to use explicit grain-level representations of microstructure within continuum constitutive descriptions of flow to examine deformation localization and instabilities for a variety of materials. However, going still further, as these explicit microstructural methods emerge so too must the constitutive descriptions evolve since the present ones tend to coarse-grain at an inappropriate scale. Evolving
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methods for concurrent multiscale simulations will eventually permit localization and time-dependent failure initiation when structure representations and constitutive rules are tailored for those methods of solution. Finally, the aspects of the material that are not represented within the previously described domains of structure representation, constitutive laws or numerical schemes, must be brought to the simulation via measured quantities or empirical calibration parameters. Obviously, there should be recognition that experiments are as much a part of multiscale materials modeling as the simulations themselves. Consequently, there are new quantitative tools emerging for approaching those experimental challenges (Zhao 2006; Uchic et al. 2006; in “Emerging Methods for Matching Simulation and Experimental Scales” by Rosenburger).
4.2 Single-Crystal Turbine Blades: An Emerging Case Study The design and manufacture of turbine engine airfoils is a multibillion dollar-peryear industry. The materials and designs used for the single-crystal high-pressure turbine airfoils represent a limiting aspect of these ubiquitous engines. Since the efficiencies of the engines depend upon the maximum temperature of the gas path, there is a sustained need to find materials and designs that permit continued gains. 4.2.1 A Prototype Challenge In recent years, the operating temperatures of turbine engines have risen beyond the melting point of the Ni-superalloy single-crystal materials used to make the hot section airfoils. Clever designs and manufacturing methods that permit cooling air to flow through the interior of the airfoil, coupled with complex zirconia-based coating systems on the exterior portions exposed to the combustion gases led to such highperformance capabilities (Reed 2006). They have also resulted in complex states of time-dependent stress during service and are an interesting example of applications where dimensional constraint imposed by aero-thermal design interacts with materials at dimensional scales comparable to microstructural dimensions. The continued evolution of these highly engineered hybrid material systems demands advances in design methodology, which today resides principally with anisotropic elasticity and homogeneous descriptions of material response (Meric et al. 1991; Arakere and Swanson 2002; Harrison et al. 2004). What is needed for turbine blade design is a computationally tractable, higher-fidelity design system that permits a better analysis of the spatial–temporal stress state and damage accumulation in a representative environment. Figure 6 shows that the aerodynamic and cooling geometry design features of cooled airfoils (wall thickness, cooling channels, ribs, etc.) are on the scale of the primary material microstructure. Thus any variation of the material may lead to variations in airfoil behavior from region-to-region and from airfoil-to-airfoil, simply as a result of those variations occurring at differing locations relative to the designed geometric features. How does one best use computational methods,
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Fig. 6 (a) Dendritic microstructure representation for a cast single crystal Ni-base superalloy turbine blade produced from serial sectioning and optical metallography. Colored lines map dendrite cores from base to top. Top of figure shows a metallographic section revealing dendrite cores. Note that blade has been “filet” cut to reveal dendrite core locations relative to cooling channels of the airfoil. (b) Backscattered electron image of blade cross-section showing crystal orientation contrast associated with low-angle misoriented grains. (c) Optical micrograph of etched cross section showing dendrites and white eutectic particles. (d) Backscattered electron images of etched cross section showing the finer microconstituents of a typical superalloy blade. Images provided by M. Groeber
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especially for microstructure–property relationships, to permit such assessments within the design process? How might an airfoil designer assess the probability of the weakest-link or life-limiting microstructural feature occurring at the geometric feature that most limits the design? To begin to answer these questions, one must examine not only the nature of the material microstructure–property relationships, but also their representation in design simulation codes, as previously suggested by Figure 5. One must devise representations of the material’s structure and its response to time-dependent loading states, all within some context of numerical frameworks that may be used to perform design simulations. Figure 6 also shows examples of microstructural variables that can be important to performance variations across a population of turbine airfoils of a constant design and manufacturing process. These arise from both the complexity of the superalloys themselves and the methods of their manufacture (Pollock and Tin 2006). Figure 6b–e shows examples of misoriented or low-angle grain boundaries, the dendritic microstructure that results from chemical segregation during casting solidification and, a mixture of eutectic microconstituent, carbides and pores, respectively. Freckle grains may lead to locally high stresses since these are local polycrystalline regions. The low-angle boundaries are a generally accepted feature of the otherwise single-crystal materials; however, the superalloys exhibit severe crystal orientation sensitivity to their creep behavior (MacKay and Maier 1982). Thus, design should be able to assess the stress states relative to these features when the airfoils are configured. While much of the dendritic structure is annealed away during heat treatment, the homogenization is never complete and studies show that internal stresses develop at the scale of the dendrite spacing (Epishin et al. 2004). Pore and eutectic microconstituents (including carbides) tend to be locally soft or hard relative to the matrix, thus concentrating strain under load and leading to fatigue crack initiation (Yi et al. 2007; Liu et al. 2008). Thus, each of these features affect internal stresses but is not taken into account within current design methods. In fact, even when design practice extends beyond treatments of the single-crystal superalloy as an elastic solid, these microstructural features are not directly included, perhaps accounting for some of the variability between performance and design.
4.2.2 Deficiencies in the Processing–Properties–Design Paradigm The standard practice for including creep or fatigue response into a design falls back on the previously described processing–properties-design paradigm for materials engineering. The material is represented by a database of design minima curves from testing and basic feature configurations derived from experience. Nondestructive inspection methods are used to assure that cast blade crystal orientations are within the bounds set by the design curves and to selectively inspect for other defects. To establish those limitations, one might produce cast bars (having net sections much larger than the airfoils themselves) from which test specimens are prepared to evaluate creep and fatigue properties at a macroscopic scale. While a single primary crystallographic orientation and the secondary dendrite arm spacing may be evaluated for those bars, efforts rarely track/control other aspects of
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microstructure. Thus, the processing–properties–design paradigm employed in this case implicitly assumes that: (1) the cast microstructure of those specimens represents the same microstructures found in the turbine blade configurations, (2) the stochastic variations of properties within populations of tested specimens encompasses the property variations occurring within turbine blades and, (3) perhaps most importantly, that each of the microstructural details controlling properties is a homogeneous or equal-likelihood-of-occurrence entity over the configuration of the turbine blade. Within this paradigm, the “local continuum” (local) approximation is implicitly invoked well above the scale of key microstructural features, simply by the choice of specimens and scales used for determining material behavior. Nowhere within the processing–properties–design paradigm does one explicitly assure that the correlation lengths (de-correlation lengths) for the stochastic variations of microstructural features are exceeded, even though the local approximation cannot hold below such correlation lengths. Said differently, for any statistically varying aspect (descriptor) of microstructure, there is a minimum volume of material or number of cases of that aspect of the material that must be included in the tested volume, so that statistical fluctuations have no significant influence on the behavior of the volume. For sample sizes that equal or exceed the correlation length, any random sample is expected to behave similarly and independently of any other sample (they are de-correlated). Further, nowhere during these procedures, except perhaps by de-rating material performance capability (design minima) does one tailor the design process for the fact that some microstructure attributes cannot be homogenized. As examples, at the scale of the turbine blade feature sizes, freckle defects, and low-angle grain structures occur having only one to three features through the wall thickness. These are not present in sufficient numbers to be statistically or homogeneously represented over the airfoil. Coarse features such as these do not have a correlation length within the context of the turbine blade and, may not have one even at the larger scale of the tested specimens. The notion of establishing the de-correlation length for controlling microstructural features is an important one for setting foundations of microstructure–property relationship simulations. Such correlation lengths exist in a hierarchical way across microstructural scales and descriptors. That notion is pervasive and may lead to better foundations from which to build quantitative MSE tools and techniques. For example, chapters of this book suggest that fundamental scientific questions remain open regarding the viability of establishing representative volume elements (RVE) for evolving path-dependent plastic properties (e.g., see “Representation of Materials Constitutive Responses in Finite Element-Based Design Codes” by Choi and Brockman, “Accounting for Microstructure in Large Deformation Models of Polycrystalline Metallic Materials” by Bronkhorst et al.). At the scale of dislocations and substructure evolution, no quantitative theory exists and empirical approaches have not developed much beyond the scalar “dislocation density” and associated hypotheses. At the scales of grain structure, basic questions centered on establishing the correlation length for grain–grain interaction effects in a 3d elasticviscoplastic zone or for 3d plastic front propagation, remain relatively unaddressed except for highly idealized cases (in “Representation of Materials Constitutive
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Responses in Finite Element-Based Design Codes” by Choi and Brockman, Simonovski et al. 2004). From a multiscale science and physics perspective such absences make the prospects for accurate predictive simulations of failure properties, such as crack initiation, rather remote since the kinematical driving forces would not be known even if the atomic processes could be adequately represented. For such properties, one engineering challenge is to establish a protocol, using accessible methods for microstructure RVE construction, to assess the validity and inaccuracies of property simulations and to gain insights into the weakest volume elements that inevitably control mechanical behavior. Consequently, aspects of material property variability and design minima are perhaps reflections of the underdeveloped materials–design interface in that there are no sufficient methods to treat unsolved materials science issues within engineering design. These may also illustrate a growth area needed within microstructure–properties based materials engineering and computational methods for microstructure–property relationship.
4.2.3 A Look Forward Current research is exploring the use of concepts developed by Ghosh et al. (2001, 2007, 2008; Swaminathan et al. 2006; Swaminathan and Ghosh 2006) for 2d simulations of long-fiber composites, to build a framework for microstructure– property–design simulations of turbine blades (Groeber et al. 2009). At the scale of the entire blade, many analysis iterations are needed and a substantial number of volumetric analysis nodes are demanded simply from the spatial variation and complexity of the blade features. Therefore, the structure representations and analysis methods must be computationally quick or fast acting. Following Figs. 5 and 6, there are at least four steps to representing turbine airfoils at this scale. First, one needs to define a structure representation having sufficient fidelity to represent both the engineering design geometry and the microstructural features too large or heterogeneous to be represented within a single local continuum entity. One may also choose to represent distinct defect structures as identifiable features at this scale. Second, validated constitutive descriptions are needed for the property response of selected interest. These flow rules are assigned to the discrete continua of the structure representation. However, a key open aspect of these flow rules is the level of finer-scale microstructure and/or failure criteria that they represent (MacLachlan et al. 2001; Harrison et al. 2004; Ma et al. 2008; Choi et al. 2009). Usually, an anisotropic elastic-viscoplastic yield function or statevariable model, with or without crystallography and/or a damage model, would be the highest level of complexity that could be carried at this scale. Third, steps one and two need to be established within a numerical framework that is self-consistent with those selections. Within Ghosh’s scheme, a concurrent adaptive finite element method is preferred since such methods permit natural strain or damage localization during the strain evolution and couples them to lower length-scale aspects of the microstructure. Finally, a formal engineering protocol requires that a parameterization and validation testing methodology be established at the same dimensional scale as the microstructural discreteness selected for the blade representation. For this
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example, that engineering requirement implies isolating, sectioning and testing various feature and specimen sizes from actual airfoils, rather than from separate test bars as current procedures employ. The first three steps just described constitute an adaptation of what is termed a “Level 0” (L0) or part domain analysis within Ghosh’s scheme. However, one still needs to rigorously tie these to lower scale microstructural features and micromechanisms of behavior as deformation and damage evolve during analysis. To achieve this, Ghosh’s method defines a “Level 1” (L1) analysis domain at a lower scale that is a homogenized material point, for two reasons. Within the L1 domain, a statistically equivalent representative volume element (SERVE) may be defined from microstructure characterization and descriptor set development coupled with standard asymptotic homogenization methods (Swaminathan et al. 2006; Swaminathan and Ghosh 2006). These SERVEs are used to numerically compute the anisotropic yield functions used for L0 analysis and may be iterated to include the intrinsic statistical variations of microstructure. In addition, as an L0 domain simulation run evolves, the coupling of far-field loads to geometric and coarse microstructural features begins to localize against pre-selected criteria. The concurrent adaptive numerical scheme permits a new L1 SERVE analysis domain to be inserted, as a periodic domain that introduces a higher level of microstructural fidelity that interacts with the now localizing (stress) fields. Methods for establishing these domains and selected criteria for tracking localization have been previously described and are treated by Ghosh, et al. (Valiveti and Ghosh 2007, in “Multiscale Characterization and Domain Partitioning for Multiscale Analysis of Heterogeneous Materials”). For turbine blade analysis, establishing the L1 analysis tools presents several challenges. By definition, the L1 representation can only comprise microstructural features whose correlation length is smaller than the SERVE for that domain. That is, the microstructural features must be amenable to computational homogenization at that scale. In this example, low-angle grain structures and freckle grains would not qualify; however, the dendritic structure may. Characterization methods are needed for the dendritic microstructure to ascertain how many dendrite features are sufficient to establish the internal dendritic stress state. As Fig. 7 shows, Shade (Shade 2008) has made some progress in this regard by showing that differences in flow stress between dendrite cores and interdendritic regions can be directly measured. Also, the L1 representation must carry most aspects of the ”–” 0 microconstituent and any variations of it that may occur at the scale of the dendrites. What remains unclear in this example of the L1 domain is how much of the interdendritic microconstituents (eutectics, carbides, and pores) can be or should be represented at the L1 scale. Hence, a challenge for the materials engineer developing computational methods for microstructure–property relationships is to establish not only the microstructural de-correlation lengths, testing methods that correspond to those lengths and rigorous statistical representations of the microstructure variability, but also to establish all of these within standardized protocols that are consistent with simulation capabilities.
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Fig. 7 Microcrystal compression sample machined from dendrite core structure of cast single crystal superalloy and tested (Shade 2008)
As Ghosh et al. describe, a quantitative partitioning of the computational domain will inevitably lead to identifiable microstructural features that cannot or should not be homogenized within a selected L1 domain (in “Multiscale Characterization and Domain Partitioning for Multiscale Analysis of Heterogeneous Materials” by Ghosh). For the turbine blade example, one may anticipate that the eutectic microconstituent or pores that exceed some size dimension would fall into that realm. These are defined as Level 2 (L2) features. Any such features need to be represented within their own micromechanics frameworks that account for plastic processes and damage accumulation within those entities. Even coarse-scale extrinsic defects may fall into a similar L2 domain since they cannot be homogenized via asymptotics. For the intrinsic microstructural variation, as simulations proceed at the L0 and L1 levels, again localization is expected that will exceed the bounds of pre-selected failure criteria established within failure or damage models. Once that occurs, the adaptive scheme inserts an L2 domain into localization fields and permits a still higher fidelity representation of behavior to evolve in simulations. For the turbine blade material example, one may envisage that L2 computational methods consist of crystal plasticity models, non-local formulations of damage, or crack initiation models of various forms. Clearly, the concurrent multiscale adaptive scheme described here can be explicitly tied to microstructural–dependent properties and heterogeneous materials, to the extent that its various parts can be build in a computationally viable way.
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4.3 Advanced Engineering Design: A Virtual Materials Systems Paradigm Previously, the notion of virtual materials systems was introduced as a parallel to virtual biological systems. One may expect that as virtual materials systems become closer to reality, the interface between design engineering and MSE will further dissolve. Figure 4 described what is becoming current-day practice within the ICME paradigm for materials, which already contains broad overlap between design and materials. Here, that view is contrasted with a futuristic view partially described in previous reports (Dimiduk et al. 2004a, b). The view outlined the potential of virtual materials systems or virtual processing–microstructure–property–design relationships to affect the engineering design practices, especially for developing an unknown or new material. Figure 8 describes a design environment wherein the usual suite of engineering tools for shape and product performance in a mechanical engineering sense (boxes A and B) are integrally coupled to a comparable suite of design tools for microstructure–property relationships. Boxes E, F, and G describe microstructure sensitive representations of the part. From this one may synthesize probability of part population behavior, then link these to system fleet probability of behavior via other probabilistic tools.
Fig. 8 Schematic depiction of the microstructure–property aspects of the virtual materials systems paradigm. This paradigm places significant emphasis on building and using a validated representation of the material, via a suite of integrated small-scale experimental and simulation methodologies, as an integral part of the system design process. Within such an approach, there are no real boundaries between engineering design and materials engineering
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One may envisage an environment (Fig. 8) for which process modeling includes spatial-temporal simulations of microstructure evolution, both at the level of primary constituent kinematics (grain, fiber or primary matrix constituent level) and at the lower levels of microconstituent and defect chemical and kinetics behavior (box C). Utilizing those tools results in a virtual description of the part domain. The figure shows that the process modeling procedures result in two key attributes for the remaining design system. First, as suggested by box D, experiments would be initiated that are defined from the results of process simulations and are specifically focused on evaluating critical microstructures or pedigreed materials for bounding models. In parallel to the experimental activity, box E suggests that the preliminary designed part may be partitioned into microstructural simulation domains. These can be defined from both the spatial-temporal variations of continuum state-variable fields and from expectations of extrinsic defect influences on behavior. For each of those domains, SERVE must be constructed to manage the intrinsic evolution of both the grain or primary-constituent kinematics, and the lower-scale, single-grain or microconstituent level kinetics under service loads and environmental conditions. Construction of the RVE suite involves both small-scale experimental measurements and a more-substantial set of simulation-based activities that includes building synthetic statistical instantiations of microstructures that include imposed extrinsic defect structures. Those activities are shown in boxes F1-G4 in Fig. 8. The activities depicted in box G5 represent the use of the small-scale SERVE suite within simulation frameworks that derive a statistically relevant set of material responses to include probabilities of performance. These result in larger-scale constitutive descriptions with damage mechanisms and “materials allowables” for part design. The synthetically derived materials behavior descriptions are then reconciled and adjusted using information from historical databases for similar material behavior (box H). From this point, the whole set of procedures may be iterated and updated toward some optimization criteria, until there is a converged quantitative view of the expected microstructure–property relationships over all domains of a part, consistent with the desired design performance criteria. Only after such reconciliation of design goals would one have to prepare full-scale test articles for full certification of the part. Clearly, within such a future paradigm, there is little separation between MSE and design engineering. For such a paradigm, design with materials becomes symbiotically fused with design of materials. This is done in such a way that simulated responses are fused with heuristically known responses resulting in both cost and performance risk reductions. For such a long-term view of microstructure–property–design engineering as depicted in Fig. 8 to become a reality, it is important that virtual materials systems build from technologies that are viable today, but remain extensible into the representations of tomorrow. As previously discussed, one area that already poses a present-day challenge, for which today’s decisions will impact the broader longer-term evolution of the field, is data types and management. For example, the last 5 years have seen step-wise growth in the techniques for both characterizing
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microstructures in 3d and simulating their behavior. Those successes brought about what some have a called a “data tsunami.” Both models and experiments now overwhelm data management, storage and, most importantly analysis capabilities. The very existence of such data, together with the interests in mining such data, calls for extensible data structures that carry the data pedigree throughout and, for automated, unsupervised analysis tools especially for microstructural analysis.
5 The Present Book No book on microstructure–property sciences and techniques can be comprehensive, nor was it the goal of the editors of this work to cover the topic in a comprehensive way. Rather, the purpose of this book is to provide insights into selected aspects of microstructure–property science and provide views of what is in the realm of the possible when a computation and simulation centric perspective is adopted. By doing this, the editors believe that a vision for the future of the field can be shown, specific advances in the field conveyed and, gaps in the computational methods highlighted. The structure of the book follows to great degree from the introductory context discussed previously. The view is that the broad goal of attaining virtual materials systems provides the guiding principles and, that the four domains of multiscale materials modeling (Fig. 5) together with microstructure–property science provide the more detailed structure. Thus, the book consists of four parts. Part I describes selected methods for attaining virtual materials structure and directly tying that information to the computational domain, beginning with methods for experimentally determining 3d microstructure. Part II of the book shifts the focus onto virtual material response within the simulation environment, which is directly tied to the constitutive descriptions and kinematical frameworks selected for simulation. In Part III, the book describes selected numerical techniques and simulation frameworks for treating some aspects of engineering challenges associated with time-dependent material behavior at microstructural scales. While the first three parts of this work develop many key aspects of microstructure–property science from a computational perspective, there is still a great deal missing. Within Part IV of the book, three selected and current broader-interest topics in the MSE community are presented. This part includes a chapter that describes the multiscale framework for mechanical behavior testing that is evolving in parallel to the ICME paradigm and, a view of computational stochastic methods for describing effects across length scales. The final chapter returns to a broader look at the design and MSE fields and provides another perspective on the challenges of bringing microstructure–property information into the systems engineering optimization domain. The editors of this work operate from the strong belief that there is a quiet revolution under way within microstructure–property science that is being driven by the continuing advances in computing capabilities. That revolution is allowing
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designers to explicitly include microstructure heterogeneity and location- or feature-specific behavior directly within the design process. Ultimately, this revolution will lead to the availability of virtual materials systems, to more portable materials that are no longer so closely tied to application-specific processes and descriptions and, to transformed approaches to materials data and data structures. While much of this book describes the advances from the perspectives of metals technologies, there is ample reason to believe that the structuring of the challenges and aspects of the techniques are general and applicable to broad classes of materials beyond metals. Nonetheless, even for metals many of the techniques require significant further developments to realize engineering gains. Our hope is that many will be swayed by the art of the desirable that is conveyed herein and find ways to transform that into the art of the possible while continuing to advance the field. Acknowledgements There are many incremental contributors to the viewpoints expressed both in this introduction and the throughout remainder of the book. Among those contributors, the author gratefully acknowledges important and formulating discussions with Profs. H.L. Fraser, S. Ghosh and J.C. Williams; and with Drs. R.E. Dutton, J.P. Simmons, C. Woodward, Dr. C. Hartley, M.G. Mendiratta, T.A. Parthasarathy, J.M. Larsen, L. Christodoulou, S. Wax, D. Backman, H.A. Lipsitt, M.J. Blackburn and Mr. J. Schirra. We also gratefully acknowledge financial support from the Air Force Office of Scientific Research, under the direction of Dr. C. Hartley, and the Defense Advanced Research Projects Agency, especially during early periods of this effort.
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Liu L, Husseine NS, Torbet CJ, Kumah DP, Clarke R, Pollock TM and Jones J W (2008) In situ imaging of high cycle fatigue crack growth in single crystal nickel-base superalloys by synchrotron X-radiation. J Eng Mater Technol 130:021008–1–6. Liu Z-K, Chen L-Q and Rajan K (2006) Linking length scales via materials informatics. JOM 58(11):42–50. Ma A, Dye D and Reed RC (2008) A model for the creep deformation behaviour of single-crystals superalloy CMSX-4. Acta Mater 56:1657–1670. MacKay RA and Maier RD (1982) The influence of orientation on the stress rupture properties of nickel base superalloy single crystals. Metall Trans 13A:1747–1754. MacLachlan DW, Wright LW, Gunturi S and Knowles DM (2001) Constitutive modeling of anisotropic creep deformation in single crystal blade alloys SRR99 and CMSX-4. Int J Plast 17:441–467. Mandel J (1973) Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int J Sol Struct 9:725–740. McClung RC et al. (2008) Turbine Rotor Material Design, Phase 2 Final Report, Southwest Research Institute, University of Texas at San Antonio, Mustard Seed Software, GE Aviation, Honeywell, Pratt & Whitney, Rolls-Royce Corporation, FAA Grant 99-G-016, Federal Aviation Administration, Washington, DC. McDowell DL (2000) Modeling and experiments in plasticity. Int J Solid Struct 37:293–309. Meric L, Pourbanne P and Cailletaud G (1991) Single crystal modeling for structural calculations: part 1-model presentation. J Eng Mater Tech 113:162–170. Millwater HR and Osborn RW (2006) Probabilistic sensitivities for fatigue analysis on turbine engine disks. Int J Rotat Mach 28487:1–12. National Materials Advisory Board (NMAB) (2004) Committee on Accelerating Technology Transition: Accelerating technology transition: bridging the valley of death for materials and processes in defense systems. National Academies Press, Washington, DC. National Materials Advisory Board (NMAB) (2008) Committee on Integrated Computational Materials Engineering: integrated computational materials engineering a transformational discipline for improved competitiveness and national security. National Academies Press, Washington, DC. Olson GB (1997) Computational design of hierarchically structured materials. Science 277:1237–1242. Pollock TM and Tin S (2006) Nickel-based superalloys for advanced turbine engines: chemistry, microstructure, and properties. J Prop Power 22:361–374. ProCAST (2009) http://www.esi-group.com/products/casting/procast Reed RC (2006) The superalloys: fundamentals and applications, Cambridge University Press, Cambridge, UK. Shade PA (2008) Small scale mechanical testing techniques and application to evaluate a single crystal nickel superalloy. Ph.D. Thesis, The Ohio State University, 68–109. Shercliff HR and Ashby MF (1990) A process model for age hardening of Al alloys—part I. the model. Acta Metall 38:1789–1802. Simonovski I, Kovac M and Cizelj L (2004) Estimating the correlation length of inhomogeneities in a polycrystalline material. Mater Sci Eng A 381:273–280. Smith CS (1988) A history of metallography. MIT Press, Cambridge, MA. Swaminathan S and Ghosh S (2006) Statistically equivalent representative volume elements for composite microstructures, part II: with damage. J Compos Mater 7:605–621. Swaminathan S, Ghosh S and Pagano NJ (2006) Statistically equivalent representative volume elements for composite microstructures, part I: without damage. J Compos Mater 7:583–604. Tschopp MA, Spearot DE and McDowell DL (2008) Influence of grain boundary structure on dislocation nucleation in FCC metals. In: Hirth JP (ed) Dislocations in solids, Vol 14, Elsevier, Oxford, UK, 42–139. Uchic MD, Dimiduk DM, Wheeler R, Shade PA and Fraser HL (2006) Application of microsample testing to study fundamental aspects of plastic flow. Scripta Mater 54:759–764.
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Valiveti DM and Ghosh S (2007) Morphology based domain partitioning of multi-phase materials: a preprocessor for multi-scale modeling. Int J Num Meth Eng 69:1717–1754. van de Walle A, Asta M and Ceder G (2002) The alloy theoretic automated toolkit: a user guide. CALPHAD 26:539–553. Vitek V (1985) Effect of dislocation core structure on the plastic properties of metallic materials. In: Dislocations and properties of real materials, The Institute of Metals, London. Wikipedia Virtual Human (2009) http://en.wikipedia.org/wiki/Virtual Physiological Human, 31 March 2006. Yi JZ, Torbet CJ, Feng Q, Pollock TM and Jones JW (2007) Ultrasonic fatigue of single crystal Ni-base superalloy at 1000ı C. Mater Sci Eng A 443:142–149. Yip S (2005) Handbook of materials modeling, Springer, Heidelberg. Zhao JC (2006) Combinatorial approaches as effective tools in the study of phase diagrams and composition–structure–property relationships. Prog Mater Sci 51:557–631.
Serial Sectioning Methods for Generating 3D Characterization Data of Grainand Precipitate-Scale Microstructures Michael D. Uchic
Abstract This chapter provides an overview of the current state-of-the-art for experimental collection of microstructural data of grain assemblages and other features of similar scale in three dimensions (3D). The chapter focuses on the use of serial sectioning methods and associated instrumentation, as this is the most widely available and accessible technique for collecting such data for the foreseeable future. Specifically, the chapter describes the serial sectioning methodology in detail, focusing in particular on automated systems that can be used for such experiments, highlights possibilities for including crystallographic and chemical data, provides a concise discussion of the post-experiment handling of the data, and identifies current shortcomings and future development needs for this field.
1 Introduction In the previous chapter, the concept of integrated computational materials engineering (ICME) via microstructurally informed, multiscale simulations was introduced. For this type of endeavor, it is incumbent that the required microstructural information be on hand as either input or validation for these simulations to properly account for microstructural dependencies. Today, this information is most commonly found in the form of mean values for selected features, e.g., average grain size, average precipitate size or spacing, or in more advanced models, distributions of these microstructural descriptors are required. To provide as complete and unbiased description of microstructure as possible, the field of materials characterization is gradually developing and adopting methods that provide quantitative microstructural information in three-dimensions (3D). The desire for 3D microstructural data is relatively straightforward. Primarily, it is
M.D. Uchic () Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA e-mail:
[email protected]
S. Ghosh and D. Dimiduk (eds.), Computational Methods for Microstructure-Property Relationships, DOI 10.1007/978-1-4419-0643-4 2, c Springer Science+Business Media, LLC 2011
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because 3D data provides access to some very important geometric and topological quantities that cannot be determined a priori by classical stereological methods that utilize only 2D images (DeHoff 1983). These quantities include assessing the true size, shape, distribution of both individual features and that of their local neighborhoods, determining the connectivity between features or networks, and counting of the number of features per unit volume (De Hoff 1983; Wolfsdorf et al. 1997). Experimental methods that enable 3D characterization have undergone dramatic improvements in the past decade, due in large part to advances in both computing power and visualization and analysis software that have been enabling factors for both the collection and interpretation of these massive data sets. The 3D data collection process requires significantly more effort compared to conventional 2D analysis, which has spurred the development of fully automated instruments that are capable of collecting such information (Alkemper and Voorhees 2001; Spowart et al. 2003), as well as software programs that take in the raw data stack and provide as output reconstructions and analysis of the microstructural features in 3D [see for example, IMOD (Kremer et al. 1996)]. The diverse size range of microstructural features has resulted in the development of a suite of instruments to address the collection of 3D data at various size scales. This ranges from counting individual atoms in nanometer-sized needles (Miller and Forbes 2009) to interrogating features within manufactured components (MA Groeber, DM Dimiduk, MD Uchic, C Woodward 2009, unpublished research) – a difference of 7–9 orders of magnitude in scale – and cube of this value for volumetric coverage! The state-of-the-art for the field of 3D materials characterization has been the focus of recent collections of papers in a number of materials journals (Spanos 2006; Uchic 2006; Thornton and Poulsen 2008), and has also been the topic of a number of symposia at materials society meetings, for example, the 3D Materials Science symposia I to VI at the TMS national meetings. As an aside, a similar renaissance in 3D characterization methodologies has already occurred in the biological and medical sciences, with instruments that are more suited to either sectioning soft matter, or in some cases making use of instrumentation that cannot be directly applied to opaque materials such as confocal laser microscopy. Nevertheless, the significant overlap in problems of data handling, data segmentation and feature extraction, 3D visualization, and surface meshing has accelerated the maturation of this methodology for the structural materials community. This chapter focuses on one aspect of microstructural characterization with respect to the ICME field, which is to discuss the methodologies that can be used to quantify the 3D microstructure associated with grain ensembles or other features that are of similar scale such as second-phases, dendrites, precipitates, dispersoids, and voids. These are ubiquitous features found in most structural alloys, and these features as a whole range in size from multiple millimeters to tens-of-nanometers in scale. There are two main experimental pathways to collect information over this size range. The first is the use of X-rays, which are nondestructive and therefore allow for time-dependent studies that examine microstructural changes due to thermal or mechanical input, i.e., 4D experiments (Juul Jensen et al. 2006). There are a number of
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different techniques that can be used to provide image contrast in X-ray tomography experiments (Ice 2004). The most common method obtains information by reconstructing a suite of transmission (absorption) images taken at various projections. This technique is very sensitive to differences in atomic number and density, so that microstructural features which are quite different in these characteristics – such as porosity relative to the matrix – can be readily detected as shown in Fig. 1a. Other methods utilize diffraction contrast and either ray tracing methods (Schmidt et al. 2004; Juul Jensen et al. 2006; Ludwig et al. 2009) or other spatial localization methods (Larson et al. 2002; Ice 2006) to define features such as individual grains from grain aggregates. These diffraction-contrast methods have been greatly advanced in the past few years, and for selected techniques have been demonstrated to rapidly produce 3D characterization data of grain ensembles as shown in Fig. 1b. The primary disadvantage of these experiments is that they require the use of very high-intensity X-rays to produce data that has acceptable signal-to-noise levels, such as those produced by synchrotron sources (Ice 2004). This requirement severely restricts the general availability and applicability of these methods until there is a revolutionary change in the ability to produce high brilliance X-rays in a laboratory setting. The other method to acquire 3D characterization data at the macro-to-microscale is through serial sectioning experiments. Serial sectioning is much more accessible experimental methodology compared to synchrotron-based tomography, but this methodology has a significant disadvantage that the sample volume is inevitably consumed during the data collection process, which precludes any re-examination or re-use of the material after analysis. In spite of this drawback, serial sectioning
Fig. 1 Examples of microstructural data that can be obtained with synchrotron X-ray methods. (a) 3D reconstruction of the porosity in a cast single-crystal nickel base superalloy, CMSX-10, using transmission (absorption) X-ray tomography (Link et al. 2006). The dimensions of the reconstructed volume are 500 500 800 m. Figure is used with permission from Elsevier. (b) 3D reconstruction of the 3D grain structure of a tensile sample of “-21 titanium alloy (Ludwig et al. 2009). The reconstruction contains 1,008 grains, and was collected using X-ray differential contrast tomography. Figure is used with permission from the American Institute of Physics
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experiments are becoming an increasingly common procedure to characterize microstructure in 3D, especially in the past decade with respect to the development and usage of automated instruments to perform of this task. This chapter endeavors to provide an overview of this technology, and discuss the state-of-the-art with regards to characterizing grain and precipitate scale microstructural features in structural materials.
2 Serial Sectioning For opaque materials, serial sectioning has been the most widely used method to acquire raw 3D characterization data at the macro-to-microscale, and in fact the first application of this methodology to examine the microstructure of structural metals was published over 90 years ago (Forsman 1918). Tomographic serial sectioning experiments are conceptually simple, being composed of two steps that are iteratively repeated until completion of the experiment. The first is to prepare a nominally flat surface, which can be accomplished by a variety of methods – a noninclusive list includes cutting, polishing, ablating, etching, and sputtering – where ideally a constant depth of material removal has occurred between each section. The second step is to collect two-dimensional (2D) characterization data after each section has been prepared, although data could also be collected continually during material removal depending on the particular sectioning method that is employed. After collection of the series of 2D data files, computer software programs are used to construct a 3D array of the characterization data that can be subsequently rendered as an image or analyzed for morphological or topological parameters. The 2D characterization data collected during a serial sectioning experiment can be comprised of number of different types and/or quantities of information. For example, in the particular case of characterizing grain microstructures, this could consist of using optical microscopy to image the structure of etched grain boundaries, as well as using an SEM to collect electron backscatter diffraction maps on key sections to characterize the average grain orientation, which was recently demonstrated by Spanos, Lewis, Rowenhorst and co-workers at the Naval Research Laboratory (Lewis et al. 2006; Spanos et al. 2008), as shown in Fig. 2. From a practical perspective, the main criteria for determining whether to incorporate a particular image or data map into a serial sectioning experiment is whether the microstructural feature or features of interest can be readily classified from this information, especially via unsupervised computer segmentation processes. In the most commonly performed experiment, the characterization data consists of a single 2D image per section (Mangan et al. 1997; Kral and Spanos 1999; Lund and Voorhees 2002; Holzer et al. 2006). Other examples include multiple images that highlight different aspects of the microstructure (Jorgensen et al. 2009), crystallographic (Wall et al. 2001) or chemical maps (Kotula et al. 2006; Schaffer et al. 2007), or conceivably any other 2D spatial measurement that is of interest (such as local measurements of resistivity or elastic modulus, etc.). The process of sectioning and
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Fig. 2 3D reconstruction of the austenite phase in a commercial austenitic stainless steel alloy AL-6XN (Lewis and Geltmacher 2006). The data set was produced via manual serial sectioning that incorporated collection of both optical images and EBSD maps. The volume contains 138 grains, and the arrow represents the normal of the serial sectioning plane. Although not readily visible because of the gray-scale coloring of this printing, the color of each grain corresponds to the crystallographic orientation relative to the arrow, which was determined by EBSD. Figure is adapted with permission from Elsevier
data collection is repeated until the desired sample volume has been interrogated, or perhaps more realistically for manual implementations of this methodology, the motivation to continue collecting the data falls below a critical value. One of the key aspects in the design of a serial sectioning experiment is to determine the minimal spatial resolution required by the subsequent microstructural analysis. For example, a serial sectioning study that will quantify aspects of feature shape such as surface area will require a much greater spatial resolution than one that is simply counting the number of features per unit volume. A rule-of-thumb is that one should strive for a minimum of ten sections per feature, although this is simply an ad hoc estimate. A better approach is to perform a critical examination of the effect that spatial resolution has on the accuracy or bias of any of the quantitative measurements-of-interest on simple test objects prior to initiating the experiment (Wojnar et al. 2004). Ideally, one would like to section at the finest possible step size and also collect high-resolution 2D data to generate the highest fidelity 3D data structures as possible. In practice, this goal is tempered by a number of factors. First, the precision of the sectioning technique should be assessed. The typical serial sectioning experiment employs a section thickness where the variability between sections is a small fraction of the total section thickness (<10%), as historically most studies assume a constant section thickness and do not take this variation into account when reconstructing the data. Second, the spatial resolution of the 2D characterization technique should be considered as well.
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For example, an experiment that utilizes X-ray spectroscopy via electron-beam irradiation [i.e., energy-dispersive spectroscopy (EDS) or wavelength-dispersive spectroscopy (WDS)] will have a much larger interaction volume and consequently a much poorer spatial resolution compared to an experiment that uses secondary electron imaging. Thus, the spatial resolution of each characterization technique that is employed should set a minimum bound, to prevent collecting “empty magnification” in either the 2D imaging plane or the sectioning depth. Additional feasibility issues include the proportional increase in time needed to complete a serial sectioning experiment as the spatial resolution is increased in the sectioning direction or the imaging plane, as well as the concomitant increase in computational resources for data handling and storage. Like any experimental methodology, serial sectioning has both advantages and disadvantages. One advantage is that both sectioning equipment and 2D characterization instruments are commonly found in most materials laboratories, therefore, manual implementation of this methodology can readily initiated, with the only significant “cost” being that of instrument and personnel time to perform this repetitious experiment. A wide variety of materials characterization methods are optimized for analysis of planar surfaces, such as light microscopy, scanning electron or ion microscopy, scanning probe microscopy or its derivatives, surface analysis techniques (EDS, XPS, Auger, SIMS, RBS, etc.), so that a wide range of materials characteristics can be obtained. The type of data that is collected during a serial sectioning experiment can have a profound impact on the ease of identifying and classifying microstructural features, and so this selection should be carefully considered in the design of the experiment. The primary disadvantage associated with this technique is the destruction of the sample, which for some applications is unacceptable. Another potential but much-less common issue is that the volume that is analyzed is always adjacent to or at the free-surface, which might affect the characterization measurement in an undesirable way. Manual demonstrations of this experimental methodology can be found periodically throughout the 1960s to the early 2000s, and a review of these studies can be found in the papers by Kral and co-authors (Kral et al. 2000, 2004). While serial sectioning experiments can be performed in this manner, the repetitive nature of this experiment is ideally suited for automation using instruments that are designed specifically for this application. Automation clearly reduces the tedium associated with the experiment (DeHoff 1983; Kammer and Voorhees 2008), thereby providing significant gains in terms of the amount of data that can be collected. In addition, there are other potential benefits to instrument automation, such as reductions in data variability via machine inspection and metrology. For example, pattern recognition methods can be incorporated to increase the accuracy and precision of the serial section thickness (Groeber et al. 2006), or image analysis methods can be used to adjust instrument settings so that the intensity histogram or image sharpness remains unchanged throughout the experiment. At present, there are only three devices that are capable of automatically acquiring 3D grain-level data in structural materials via serial sectioning. The next section describes these devices in some detail. Two of the devices – the Alkemper
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and Voorhees micromiller and Robo-Met.3D – utilize optical microscopy as the sole characterization method. These devices have been constructed and are suitable for characterization of the microstructure of millimeter scale volumes with micron-level precision. The third device, the focused ion beam–scanning electron microscope (FIB-SEM), is not specifically designed for serial sectioning experiments, but has been adapted for this purpose through the use of machine control software scripts. This device has approximately 1–2 orders of magnitude improvement in imaging resolution as well as sectioning fidelity compared to the other serial sectioning instruments, so that micron and submicron features can be accurately characterized. Also, both crystallographic and chemical data can be collected as part of the serial sectioning experiment using commercially available detectors that can be installed on the FIB-SEM. However, this instrument cannot currently characterize larger volumes like the other two devices.
3 Automated Serial Sectioning Instrumentation 3.1 Alkemper–Voorhees Micromiller This serial sectioning instrument was developed at Northwestern University by Alkemper and Voorhees (A&V) around the year 2000 (Alkemper and Voorhees 2001), to augment previous efforts by the Voorhees group (Wolfsdorf et al. 1997) to quantify materials microstructure in 3D using microtome milling, i.e., physical cutting with a rotating diamond knife. Unlike biological microtomy studies in which the thin section prepared by the cutting process is of interest, here the microtome blade is used as an end-mill to prepare an optical-quality surface in soft ductile metals and alloys that do not react adversely with the diamond blade, such as Pb, Sn, Al and Cu alloys (Kammer and Voorhees 2008). Images of this tomographic instrument are shown in Fig. 3. At the heart of the system is a commercial microtome that is outfitted to a rotary micro-milling attachment. The microtome performs the sectioning operation by moving a sample underneath the rotating micro-milling head using a linear stage. The key improvements by A&V were to incorporate an optical microscope with a digital camera, an etching/washing/drying station, and a linear variable differential transformer (LVDT) sensor within the commercial micromiller system. The cleaning, etching, and drying station was located next to the cutting head to remove machining chips after sectioning, and also to reveal microstructural features that could be selectively attacked via chemical etching. The linear stage then translates the sample underneath the optical microscope where an image of the freshly prepared surface can be captured using a digital camera. These first two modifications eliminated the need to remove the sample from the micromiller to collect an image of the serial section surface, which resulted in significant reductions in the time needed to complete one cycle of the experiment, as well as
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Fig. 3 The automated serial sectioning device developed by Alkemper and Voorhees (Alkemper and Voorhees 2001). (a) Image of the commercial Reichert-Jung Polycut E micromiller system, with modifications by A&V for cleaning the sample after the micromilling operation. (b) View of the microscope attachment and LVDT installed by A&V to perform automated serial sectioning. Figure is adapted from (Alkemper and Voorhees 2001) with permission from Wiley
minimizing positioning errors due to removal and subsequent re-mounting of the sample that affected the accuracy of both the cutting and imaging operations. The third modification of incorporating the LDVT allowed for an independent measurement of the spatial position of the sample when placed underneath the optical microscope. As a result, the absolute lateral position of the sample was quantitatively known for each slice, and this information was used to correct for in-plane translational errors in each of the 2D images that comprised the 3D data stack without resorting to image matching/correlation methods, which is useful in preventing alignment errors associated with using only internal features for this task (Russ 2002). With any serial sectioning experimental methodology, one needs to quantify both the accuracy and precision of the sectioning process, as these variations lead to both systematic and random distortions in the sectioning direction of the 3D data stack. In particular, this issue is vitally important for instruments that do not monitor or measure this quantity during the experiment. The A&V micromiller utilizes a precision mechanical feed to advance the depth of the milling head over an adjustable range of 1–20 m, but this device as constructed does not provide closed-loop control over the sectioning depth. Alkemper and Voorhees solved this problem by developing a custom calibration standard, and subsequent analysis of the 3D reconstruction of the calibration standard allowed A&V to determine that both machine errors as well as thermal expansion associated with heating of the system during operation were shown to affect the accuracy of the sectioning thickness (Alkemper and Voorhees 2001). Furthermore, they used the custom standard
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to quantify the average section thickness during steady-state operation, as well as identified a minimum warm-up time for instrument operation prior to which there were significant variations in the sectioning thickness. Examples of some of the data sets that have been collected with this device are shown in Fig. 4, which have been used to systematically quantify the dendrite coarsening process in simple binary alloy systems. The micro-milling process is fast, and optical cameras can also quickly acquire image data. As a result, this device can provide upward of 20 sections per hour, resulting in 3D data sets that are comprised of hundreds of images that are prepared in less than a day. It is clear from Fig. 4 that this device is useful for the 3D characterization of materials at the millimeter size-scale,
Fig. 4 Application of the A&V micromilling serial sectioning device to study dendrite coarsening processes (Kammer et al. 2006) (a) 3D reconstruction of the dendrite structure of a Pb-Sn alloy after a 3-min coarsening time. The solid corresponds to the “-Sn solid-solution dendrites, while the voids in the reconstruction correspond to the Pb-Sn eutectic phase. (b) 3D reconstruction of an Al-Cu alloy after a 3-week coarsening experiment. The solid corresponds to Al dendrites, while the voids in the reconstruction correspond to the Al-Cu eutectic phase. Note that this reconstructed volume is approximately 6 5 43 mm in dimension. Figure is adapted with permission from Elsevier
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as long as two conditions are met: one, the materials are compatible with diamond blade sectioning and two, the microstructural features of interest can be selectively identified from machined surfaces using only optical images and chemical etching.
3.2 RoboMet.3D Another serial sectioning device, RoboMet.3D, was developed by Spowart and Mullens (Spowart et al. 2003; Spowart 2006), and an image of this system is shown in Fig. 5. This device is conceptually similar to the A&V micromiller in that there are three stations to the system – sectioning, etching/washing/drying, and optical imaging. However, the RoboMet.3D can be used to examine a much broader range of materials compared to the A&V micromiller because it uses a precision mechanical polishing system for material removal rather than micromilling. Mechanical polishing is the most commonly used method to prepare materials for metallographic analysis, and therefore the sectioning system is well suited to examining many structural materials. Each of the stations is a physically separate unit on the RoboMet.3D, and therefore a six-axis robot arm is employed to move the sample between the various stations, and also holds the sample while it is being washed, etched, and dried with forced air. Notably, the commercially available optical microscope that is used on these systems is a fully automated device in its own right, being able to perform tasks such as focusing, contrast adjustments, and the capture of large-area montages of the serial section surface without human intervention. The sample-of-interest is mounted on a custom holder that minimizes rotational movement of the sample between polishing and imaging operations, but small lateral
Fig. 5 The automated serial sectioning device RoboMet3.D (Spowart et al. 2003; Spowart 2006). From left-to-right in the image are the precision metallographic polisher, six-axis robot, etching/washing/drying station, and a motorized inverted optical microscope. Figure is used with permission from Elsevier
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translations of the sample that occur between consecutive imaging operations are not measured – these must be removed via image analysis methods as will be discussed later in this chapter. Also, RoboMet.3D does not use closed-loop control over the material removal process, but rather section-to-section consistency is maintained by keeping common variables in the polishing process fixed, such as the time of polishing, applied load, wheel speed, in addition to using fresh diamond lapping films as the polishing media. With this protocol, RoboMet.3D has been demonstrated to achieve very good control over material recession rates, with a repeatability of ˙0:03 m for a section thickness of 0:8 m (Spowart 2006). The device can prepare sections ranging from 0:1 to 10 m in thickness and complete the sectioning cycle up to 20 times per hour (Spowart 2006), where the cycle time is mostly dependent on the sectioning depth, the etching time needed to resolve the feature-of-interest, and the on the quantity, resolution, and number of images that are acquired per section. Figure 6 shows representative data from the RoboMet.3D that highlights the good volumetric coverage that can be obtained with this device, as well as the diversity of materials that can be examined with this instrument. Like the A&V micromiller, this device can be successfully employed if two conditions are met: one, that a single polishing step provides both adequate material removal rates and sufficient metallographic surface quality and two, the microstructural features of interest can be identified using optical imaging methods.
3.3 Focused Ion Beam–Scanning Electron Microscopes For smaller-scale grain and precipitate structures – those that are approximately 10 m in scale or smaller – FIB-SEM are well suited to characterize these features in 3D via serial sectioning (Dunn and Hull 1999; Inkson et al. 2001; Uchic et al. 2006, 2007). FIB columns are able to focus highly energetic ions (typically GaC ) to small spot sizes that are on the order of 5–20 nm. The interaction of these energetic ions with a target results in localized material removed via ion sputtering interactions (Orloff et al. 2003). FIB microscopes are well suited to perform serial sectioning via cross-section milling with extremely fine resolution, and at the extreme can provide average serial section thickness of approximately 10–15 nm using closed-loop control measures (Bansal et al. 2006; Holzer et al. 2006). This value is at least one order of magnitude finer than the section thickness that can be nominally achieved with traditional mechanical removal methods such as metallographic polishing. Using the appropriate software control scripts, a typical serial sectioning experiment will usually encompass material volumes that are larger than 1;000 m3 with voxel dimensions approaching tens-of-nanometers. This combination of spatial coverage and resolution cannot be achieved with any other tomographic instrument. FIB-SEM microscopes have other advantages relative to the task of serial sectioning. Cross-section ion milling is an almost universally applicable method for
42 Fig. 6 Examples of 3D data sets that have been produced using RoboMet3.D. (a) Iso-surface rendering of a structural carbon foam (Maruyama et al. 2006). Dimensions of the 3D reconstruction are 1;526 1;526 776 m, and the average serial section thickness is 3:5 m. Figure is used with permission from Elsevier. (b) 3D reconstruction of a powder-compacted Fe-Cu alloy, which utilized an average serial section thickness of 1:2 m (Spowart 2006). Figure is used with permission from Elsevier. (c) 3D reconstruction of the mushy zone during directional solidification of a commercial cast single crystal Ni-base superalloy Ren´e N4 (Madison et al. 2008), which utilized an average serial section thickness of 2:2 m. Figure is used with permission from Springer
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preparing planar surfaces, and has been successfully applied to metallic alloys, ceramics, polymers, electronic materials and biological materials, although for some systems both low beam currents and sample cooling are required to prevent alteration of the starting microstructure. In comparison with most polishing or cutting methods, ion sputtering is a relatively low damage process that not only preserves the details of hard-to-prepare microstructures like those composed of both soft and hard phases, or brittle materials that contain significant porosity, but the depth of the damage layer is small enough to permit the usage of surface-damage sensitive techniques like electron backscattered diffraction (EBSD) for selected metallic alloys (Groeber et al. 2006; Konrad et al. 2006; Zaefferer et al. 2008). One other significant advantage of FIB-SEM microscopes is ability to incorporate imaging and surface analysis methodologies that can greatly mitigate the difficulty in classifying various microstructural features like grains and precipitates. These characterization methods include high-resolution backscattered electron (BSE) images that exhibit atomic-number contrast to differentiate between multiple phases, ion-induced secondary electron (ISE) images that often exhibit channeling contrast which can differentiate individual grains in polycrystalline materials (Orloff et al. 2003), EBSD mapping for local crystallographic orientation measurements, and EDS (Kotula et al. 2006), WDS, or secondary ion mass spectroscopy (SIMS) (Dunn and Hull 1999) for local chemical spectra mapping. The information limits for these methods are often of the same order of magnitude as the FIB sectioning capabilities, thus a properly outfitted microscope can provide the user tremendous flexibility in selecting which types of structural, chemical, or crystallographic information are important for their particular characterization study. For example, if only structural information is required, then image data may suffice. Chemical or crystallographic analysis can be included to help with the identification of particular phases and grain orientations, respectively. The procedure for performing an automated FIB-SEM serial sectioning experiment is roughly similar to the methods that use bulk sectioning processes. First the sample volume-of-interest is initially prepared, as FIB serial sectioning experiments have some specific sample geometries that can improve the quality of data that is collected (Uchic et al. 2006, 2007). Next, software control scripts are used to move the sample between sectioning and characterization steps. In their most basic form, these scripts perform the primary functions of cross-section ion milling of the sample and collection of electron images, as shown in Fig. 7. For these experiments, the sample does not need to move if it is placed at a position where both electron and ion columns can simultaneously image the same region of the sample, which greatly simplifies both the experimental setup and the complexity of the control scripts required to automate of the experiment. Like the other serial sectioning instruments discussed previously, the use of machine control scripts allows for a more consistent serial slice thickness, reduces the time needed per acquisition cycle, and enables the microscopes to run unattended for periods of about 1–2 days. More advanced control scripts incorporate image recognition procedures to minimize the effects of electrical, thermal, or mechanical drift, as well as to incorporate a wider range of data signals like ISE imaging or EBSD mapping, which requires accurate sample
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Fig. 7 The standard sample geometry and its orientation within a FIB-SEM microscope for a typical serial sectioning experiment (Holzer et al. 2006; Uchic et al. 2007). (a) Schematic of the experimental set-up for an FIB-SEM serial sectioning experiment where cross-section ion milling is used to controllably remove material at the micro- to nanoscale, and electron imaging is used to characterize the freshly prepared surface. If only electron imaging or EDS is used to characterize the sample surface, then the sample does not need to be moved during the tomographic experiment. (b) SEM image of a sample volume prior to sectioning. The trenches that surround the sample volume allow the electron beam to image the serial-sectioned surface, and also help prevent sputtered and re-deposited material from obscuring the surface-of-interest. Figure is used with permission from the Materials Research Society
repositioning after complex five-axis stage movements (Groeber et al. 2006). Note that for the experiments that use image recognition methods, these microscopes provide closed-loop control over sectioning thickness (unlike the two previously described serial sectioning instruments), thus enabling the user to simply select the serial sectioning depth and eliminating the need for calibration runs. Commercial FIB-SEM instrumentation is installed in many laboratories worldwide, and therefore this system is the most widely used of the three automated serial sectioning systems discussed in this chapter. Figure. 8 shows representative examples of 3D data that have been collected with these instruments, which highlights the diversity of size scales that can be examined (Fig. 8a), the ability to characterize precipitate- and multiple-phase microstructures at the microscale (Figs. 8b, c), and the ability to examine grain-level microstructures that includes the automated collection of orientation information (Fig. 8d). This figure demonstrates that FIB-SEM microscopes epitomize a new breed of multimodal serial sectioning instruments, as they are currently capable of high-fidelity characterization of the morphology, crystallography, and chemistry of micron and submicron size features in 3D. One final note is that these experiments can require significant instrument time to complete an experiment depending on the size of the volume that is examined and the type of data that is collected. Although some experiments require only a few hours – like those where the milling step takes a couple of minutes to execute, and a single electron image is collected for each section – others that include chemical or crystallographic maps, or those that attempt to interrogate volumes that have submillimeter dimensions may require multiple days.
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Fig. 8 Representative examples of 3D reconstructions that have been collected using a dual beam FIB-SEM. (a) Five reconstructions of portland cement agglomerants where the average particle size is different in each reconstruction, ranging from 0.68 to 14:2 m (Holzer et al. 2006). The voxel edge length as well as the number of particles identified in each volume is listed next to each reconstruction. Figure is used with permission from Wiley. (b) Reconstruction of ’-laths from a Ti-6242 colony (Simmons et al. 2009). Figure is used with permission from the IOP. (c) Anode of a solid-oxide fuel cell, which is composed of three phases: Ni (light gray), pores (dark gray), and yttria-stabilized zirconia (translucent phase) (Wilson et al. 2006). Figure is used with permission from the Nature Publishing Group. (d) A 3D reconstruction of the grain structure from a powdermetallurgy Ni-based superalloy, IN100. The volume shown has dimension of 96 36 46 m with a voxel edge length of 250 nm, and EBSD data was used to form this reconstruction, which provided information on both the morphology and crystallography of the grain structure (Groeber et al. 2008). Figure is used with permission from Elsevier
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4 Data Processing and Segmentation After completion of the serial sectioning experiment using either manual or automated methods, the stack of 2D data files need to be combined and processed in such a way so that the microstructural features that are within the 3D data stack can be classified. Said differently, for each object of interest in the data, i.e., every grain, precipitate, dendrite, void, and so on, one has to identify all of the voxels that are associated with that object. This procedure is referred to as data segmentation. Once all of the objects and their voxels are identified and classified, the process of quantifying microstructural characteristics using computational methods can be readily performed; this topic is the subject of other chapters in this book, and therefore will not be covered here. While the concepts of segmentation and classification is straightforward – as humans we perform this task during every waking moment – in practice this can be the toughest and often rate-limiting step to transforming the serial sectioning data into a useful form. The difficulty with data segmentation is utterly dependent on the type, quantity, and quality of data from the serial sectioning experiment, as well as the complexity of the microstructure of the material being examined. This is especially true for serial sectioning experiments that only collect image data, because of the general difficulty in using semi- or fully automated computer segmentation algorithms to defining objects from visually complex images. Note that unsupervised computerized segmentation processes are a necessary component of tomographic experiments because of the sheer volume of information that is contained within the data sets, thus eliminating the possibility of human-assisted segmentation except when only a handful of features require classification. A necessary step (typically performed before image segmentation) is to ensure that the spatial registry of each 2D data file that collectively comprises the 3D data stack is accurate. Each data file may have to be translated, stretched, rotated, or possibly all three in combination to account for the fact that the sample may have been in different physical locations when the 2D characterization data was collected, to correct for systematic or random distortions that may have also occurred during data collection (for example, image foreshortening), or to overlay data that may have been acquired at different spatial resolutions. The need for data alignment and registration is extremely common for most tomographic experiments, although manually performed experiments generally have more section-to-section variability unless specific measures such as dedicated sample fixtures are employed. One solution is to have an independent record of the spatial position of the sample relative to the 2D characterization device during data collection, like the LVDT data used in the A&V micromilling device. However, most tomographic instruments do not have the capability to provide this information at the present time. Another method to perform data alignment is through the use of fiducial markers that are within or on the outside of the sample volume. A commonly used method is the use of multiple hardness indentations (Kral and Spanos 1999). As long as the indentations are relatively large so that their shape is not grossly changed between consecutive sections, these markers not only provide an independent reference to adjust for any in-plane affine transformation of the data, but also the relative change
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of the diameter of the indent from section-to-section provides a local gage of the sectioning depth that can be used to correct for the position of the data along the sectioning direction within the 3D stack. Other fiducial marker strategies have also been proposed, involving machining patterns directly onto the side of the sample (Spanos et al. 2008) or onto a chip that is glued to the sample (Wall et al. 2001) that also provide both in-plane and depth removal information. A third option is to simply use the internal features that are present in the data for registration, but this method can introduce systematic errors unless the microstructure is relatively isotropic (Russ 2002). There are a number of signal processing algorithms that can be used for spatial alignment. Cross-correlation or other convolution methods are very commonly used if the data that is being registered is very similar from slice to slice, for example, image data that has approximately the same intensity and brightness variations throughout the data stack (Gulsoy et al. 2008). There are newer techniques such as mutual information (Pluim et al. 2003) that can also be used for this purpose, which is especially helpful in registering different forms of data, such as combining images with chemical or crystallographic maps (Gulsoy et al. 2008). After correcting for spatial registry, another common operation is to use interpolation methods to adjust the x-y resolution of the 2D data files to match the average sectioning depth, to produce cubic voxels for the 3D array. Finally, one usually selects a subregion from the 3D array, to define a volume that eliminate areas that have poor data quality or other artifacts, and if needed, to minimize the size of the data volume to prevent problems with computer memory allocation. The next step after data alignment is data segmentation, which Gonzales and Woods define for an image as “subdividing an image into its constituent regions or objects” (Gonzales and Woods 2002). For example, let us consider a two-phase microstructure that consists of a matrix with precipitates, as shown in Fig. 9. If one can obtain images of this structure that display a large contrast difference between
Fig. 9 (a) Intensity histogram for the ISE image shown in the upper frame of (b), which is of ” 0 precipitates from a Ni-based superalloy. For this histogram, the threshold value that correctly separates the pixels associated with ” 0 from the matrix can be readily drawn by eye, or determined analytically by fitting the intensity histogram to two Gaussian peaks (c). The application of this threshold can be seen in the lower frame of (b). Figure is used with permission from the IOP (Simmons et al. 2009)
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the two phases and the contrast/brightness levels are maintained throughout the experiment, data segmentation might consist only of a single threshold operation for the entire 3D stack. Unfortunately, the complexity found in most microstructures and/or limitations of the experimental data often makes life much more difficult! The combined subject of signal processing of images and its relationship to image segmentation is far too complex to cover even in brief detail in this chapter, as there are textbooks written to examine this subject (Gonzales and Woods 2002; Russ 2002). Rather, the reader is encouraged to review these resources, and this discussion will restrict its comments to the following two recommendations to help ameliorate this process. The first recommendation is in spite of the fact that the algorithms for performing image segmentation are becoming ever more sophisticated, especially for problems encountered in materials characterization [see for example (Jorgensen et al. 2009; Simmons et al. 2009)], there is no substitution for optimizing the “quality” of image data (Russ 2002). In particular, the experimentalist should strive to find an imaging condition or perhaps multiple imaging conditions that enable simple signal processing operations to readily threshold the features-of-interest. A recent example of this strategy is highlighted in the paper by Wilson et al. (2009), who utilized energybiased low-kV backscattered images to readily define the various phases observed in solid-oxide fuel cell cathodes. The second recommendation is to consider incorporating crystallographic or chemical maps rather than only image data, even though the amount of time needed to complete the experiment is usually significantly greater when these techniques are included. For the particular problem of characterizing grain structures, one method that is very attractive is to collect orientation information during the tomographic experiment, such as EBSD maps. The commercial manufacturers that supply such instrumentation have already developed a segmentation and analysis methodology to convert the Kikuchi band pattern generated by the interaction of the electron beam with the sample into a crystallographic orientation. Thus, the “difficult” part of image segmentation has already been solved, and more importantly, the singular characteristic that defines a grain – common crystallographic orientation – is an inherent part of the data collected during the experiment. The studies of Groeber (Groeber et al. 2006, 2008) and Zaefferer (Konrad et al. 2006; Zaefferer et al. 2008) have demonstrated that grain structures can be readily defined using this methodology. Similar comments hold as well for chemical spectra image data, especially when used in conjunction with automated phase analysis software, such as that developed by Kotula et al. (2003).
5 Summary Comments: Future Developments and Needs The serial sectioning methodologies and new experimental instrumentation shown in this chapter demonstrate a direct pathway to collecting and providing 3D data for both grain and precipitate structures. However, these achievements have only
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partially completed the anticipated ICME need for fully autonomous systems that can provide a complete description of microstructural distributions from the microto-macroscale. The necessity for autonomous systems is driven by both the inherent statistical nature of microstructural arrangements and the accelerated pace of twenty-first century research programs. For serial sectioning devices that operate at the millimeter scale, current automated instrumentation allows one to readily acquire structural information of grain-level microstructures via optical imaging techniques and simple mechanical removal methods, but collection of other basic information such as crystallographic or chemical information requires manual intervention. At the microscale, there is a greater quantity of spectral information that can be currently integrated into automated FIB-SEM data collection processes, via either electron or ion imaging and their derivatives, or chemical or crystallographic mapping. This chapter concludes with a list of future instrumentation and other advancements from the author’s perspective that should make for significant improvements in performing robust serial sectioning experiments, and ultimately make a positive impact in the field of ICME: Automated serial sectioning instrumentation that provides both high-spatial reso-
lution (nanometer-to-submicrometer voxels) and multimodal data collection that is also capable of interrogating millimeter scale and larger volumes. One example of such instrumentation would be to integrate fine-scale macroscale sectioning methods with an SEM outfitted with the current state-of-the-art EBSD and silicon-drift-detector EDS systems. Extending mechanical or other macroscale sectioning methods like ultrafast laser-ablation (Echlin and Pollock 2008) to enable automated characterization of coarse microstructural features within full-scale engineering components. Further incorporation of machine inspection and metrology tools to improve both the repeatability and accuracy of the sectioning process and the task of image stack registration. Improved multimodal data collection. For example, the ability to simultaneously collect both chemical and crystallographic data for automated phase analysis has already been commercialized for 2D data collection using electron microscopes (e.g., the Pegasus and Trident systems by EDAX, Inc, or the INCASynergy system by Oxford Instruments). However, the capability to perform similar data acquisition outside a high-vacuum environment remains to be demonstrated. Continued improvement in data segmentation algorithms. Real-time analysis and feedback to the characterization subsystems with respect to feature segmentation and classification during data acquisition. To the author’s knowledge, almost every tomographic experiment is performed asynchronously, that is, the data collection process is performed independently of data segmentation and classification. However, real-time interaction between the analysis software and characterization instruments would be especially useful for destructive techniques such as serial sectioning. One can envision that this capability would help ensure that all features within the analysis area would be positively
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identified before proceeding to the next section. Also, experimental acquisition times should be significantly reduced if the tomographic instrument collected information only at the location and frequency with which it is needed.
References Alkemper J, Voorhees PW (2001) Quantitative serial sectioning analysis. J Microsc 201:388–394 Bansal RK, Kubis A, Hull R, Fitz-Gerald JM (2006) High-resolution three-dimensional reconstruction: a combined scanning electron microscope and focused ion-beam approach. J Vac Sci Technol B 24(2):554–561 DeHoff RT (1983) Quantitiatve serial sectioning analysis: preview. J Microsc 131:259–263 Dunn DN, Hull R (1999) Reconstruction of three-dimensional chemistry and geometry using focused ion beam microscopy. Appl Phys Lett 75:3414–3416 Echlin M, Pollock T (2008) Femtosecond Laser Serial Sectioning: A New Tomographic Technique. WCCM8/ECCOMAS Forsman O (1918) Unders¨okning av rymdstrukturen hos ett kolst˚a av hypereutectoid sammans¨attning. Jernkontorets Ann 102:1–30 Gonzales RC, Woods RE (2002) Digital Image Processing, 2nd edn. Prentice Hall, Upper Saddle River, NJ Groeber MA, Haley BK, Uchic MD, Dimiduk DM, Ghosh S (2006) 3D reconstruction and characterization of polycrystalline microstructures using a FIB-SEM system. Mater Char 57:259–273 Groeber MA, Ghosh S, Uchic MD, Dimiduk DM (2008) A framework for automated analysis and simulation of 3D polycrystalline microstructures. Part I: statistical characterization. Acta Mater 56:1257–1273 Gulsoy EB, Simmons JP, De Graef M (2008) Application of joint histogram and mutual information to registration and data fusion problems in serial sectioning microstructure studies. Scripta Mater 60:381–384 Holzer L, Muench B, Wegmann M, Gasser P, Flatt R (2006) FIB-nanotomography of particulate systems—Part I: particle shape and topology of interfaces. J Am Ceram Soc 89:2577–2585 Ice GE (2004) X-ray microtomography. In: Vander Voort GF (ed) ASM Handbook, Vol. 9, Metallography and Microstructure, pp. 461–464. ASM International, Materials Park, OH Ice GE, Pang JWL, Barabash RI, Puzrev Y (2006) Characterization of three-dimensional crystallographic distributions using polychromatic X-ray microdiffraction. Scritpa Mater 55:57–62 Inkson BJ, Mulvihill M, M¨obus G (2001) 3D determination of grain shape in a FeAl-based nanocomposite by 3D FIB tomography. Scripta Mater 45:753–758 Jorgensen PS, Hansen KV, Larsen R, Bowen JR (2009) A framework for automated segmentation in three dimensions of microstructural tomography data. Ultramicroscopy. doi: 10.1016/j.ultramic.2009.11.013 Juul Jensen D, Lauridsen EM, Margulies L, Poulsen HF, Schmidt S, Sorensen HO, Vaughan GBM (2006) X-ray microscopy in four dimensions. Mater Tod 9:18–25 Kammer D, Mendoza R, Voorhees PW (2006) Cylindrical domain formation in topologically complex structures. Scripta Mater 55:17–22 Kammer D, Voorhees PW (2008) Serial sectioning and phase-field simulations. MRS Bull 33:603–610 Kotula PG, Keenan MR, Michael JR (2003) Automated analysis of SEM X-ray spectral images: a powerful new microanalysis tool. Microsc Microanal 9:1–17 Kotula PG, Keenan MR, Michael JR (2006) Tomographic spectral imaging with multivariate statistical analysis: comprehensive 3D microanalysis. Microsc Microanal 12(1):36–48 Konrad J, Zaefferer S, Raabe D (2006) Investigation of orientation gradients around a hard Laves particle in a warm-rolled Fe3 Al-based alloy using a 3D EBSD-FIB technique. Acta Mater 54:1369–1380
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Kral M, Spanos G (1999) Three-dimensional analysis of proeutectoid cementite precipitates. Acta Mater 47:711–724 Kral MV, Mangan MA, Rosenberg RO, Spanos G (2000) Three-dimensional analysis of microstructures. Mater Charact 45:17–23 Kral MV, Ice GE, Miller MK, Uchic MD, Rosenberg RO (2004) Three dimensional microscopy. In: Vander Voort GF (ed) ASM Handbook, Vol. 9, Metallography and Microstructure. ASM International, Materials Park, OH Kremer JR, Mastronarde DN, McIntosh JR (1996) Computer visualization of three-dimensional image data using IMOD. J Struct Biol 116:71–76 Larson BC, Yang W, Ice GE, Budai JD, Tischler JZ (2002) Three-dimensional X-ray structural microscopy with submicrometre resolution. Nature 415:887–890 Lewis AC, Bingert JF, Rowenhorst DJ, Gupta A, Geltmacher AB, Spanos G (2006) Two- and three-dimensional microstructural characterization of a super-austenitic stainless steel. Mater Sci Eng A 418:11–18 Lewis AC, Geltmacher AB (2006) Image-based modeling of the response of experimental 3D microstructures to mechanical loading. Scripta Mater 55:81–85 Link T, Zabler S, Epishin A, Haibel A, Bansal M, Thibault X (2006) Synchrotron tomography of porosity in single-crystal nickel base superalloys. Mat Sci Eng A 425:47–54 Ludwig W, Reischig P, King A, Herbig M, Lauridsen EM, Johnson G, Marrow TJ, Buffiere JY (2009) Three-dimensional grain mapping by x-ray diffraction contrast tomography and the use of Friedel pairs in diffraction data analysis. Rev Sci Instrum 80:033905 Lund AC, Voorhees PW (2002) The effect of elastic stress on microstructural development: the three-dimensional microstructure of a - 0 alloy. Acta Mater 50:2585–2598 Madison J, Spowart JE, Rowenhorst DJ, Pollock TM (2008) The three-dimensional reconstruction of the dendrite structure at the solid-liquid interface of a Ni-based single crystal. JOM 60(7): 26–30 Mangan MA, Lauren PD, Shiflet GJ (1997) Three-dimensional reconstruction of Widmanst¨atten plates in Fe-123Mn-08C. J Microsc 188:36–41 Maruyama B, Spowart JE, Hooper DJ, Mullins HM, Druma AM, Druma C, Alam MK (2006) A new technique for obtaining three-dimensional structures in pitch-based carbon foams. Scripta Mater 54:1709–1713 Miller MK, Forbes RG (2009) Atom probe tomography. Mater Charact 60:461–469 Orloff J, Utlaut M, Swanson L (2003) High Resolution Focused Ion Beams: FIB and Its Applications. Kluwer Academic/Plenum, New York Pluim JPW, Maintz JBA, Viergever MA (2003) Mutual information based registration of medical images: a survey. IEEE Trans Med Imaging 22:986–1004 Russ JC (2002) The Image Processing Handbook, 4th edn. CRC Press, Boca Raton, FL Schaffer M, Wagner J, Schaffer B, Schmied M, Mulders H (2007) Automated three-dimensional X-ray analysis using a dual-beam FIB. Ultramicroscopy 107:587–597 Schmidt S, Nielsen SF, Gundlach C, Margulies L, Huang X, Juul Jensen D (2004) Watching the growth of bulk grains during recrystallization of deformed metals. Science 305:229–232 Simmons JP, Chuang P, Comer M, Spowart JE, Uchic MD, De Graef M (2009) Application and further development of advanced image processing algorithms for automated analysis of serial section image data. Mod Sim Mater Sci Eng 17:025002–0250024 Spanos G (2006) Foreword: scripta materialia viewpoint set on 3D characterization and analysis of materials. Scripta Mater 55:3 Spanos G, Rowenhorst DJ, Lewis AC, Geltmacher AB (2008) Combining serial sectioning, EBSD analysis, and image-based finite element modeling. MRS Bull 33:597–602 Spowart JE (2006) Automated serial sectioning for 3-D analysis of microstructures. Scripta Mater 55:5–10 Spowart JE, Mullens HM, Puchala BT (2003) Collecting and analyzing microstructures in three dimensions: a fully automated approach. JOM 55:35–37 Thornton K, Poulsen HF (2008) Three-dimensional materials science: an intersection of threedimensional reconstructions and simulations. MRS Bull 33:587–595
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Uchic MD (2006) 3D microstructural characterization: methods, analysis, and applications. JOM 58:24 Uchic MD, Groeber MA, Dimiduk DM, Simmons JP (2006) 3D microstructural characterization of nickel superalloys via serial-sectioning using a dual beam FIB-SEM. Scripta Mater 55:23–28 Uchic MD, Holzer L, Inkson BJ, Principe EL, Munroe P (2007) Three-dimensional microstructural characterization using focused ion beam tomography. MRS Bull 32:408–416 Wall MA, Schwartz AJ, Nguyen L (2001) A high-resolution serial sectioning specimen preparation technique for application to electron backscatter diffraction. Ultramicroscopy 88:73–83 Wilson JR, Kobsiriphat W, Mendoza R, Chen HY, Hiller JM, Miller DJ, Thornton K, Voorhees PW, Adler SB, Barnett SA (2006) Three-dimensional reconstruction of a solid-oxide fuel-cell anode. Nat Mater 5:541–544 Wilson JR, Duong AT, Gameiro M, Chen HY, Thornton K, Mumm DR, Barnett SA (2009) Quantitative three-dimensional microstructure of a solid oxide fuel cell cathode. Electrochem Commun 11:1052–1056 Wojnar L, Kurzydlowski JK, Szala J (2004) Quantitative image analysis. In: Vander Voort GF (ed) ASM Handbook, Vol. 9, Metallography and Microstructure. ASM International, Materials Park, OH Wolfsdorf TL, Bender WH, Voorhees PW (1997) The morphology of high volume fraction solidliquid mixtures: an application of microstructural tomography. Acta Mater 45:2279–2295 Zaefferer S, Wright SI, Raabe D (2008) Three-dimensional orientation microscopy in a focused ion beam-scanning electron microscope: a new dimension of microstructural characterization. Metall Mater Trans A 39A:374–389
Digital Representation of Materials Grain Structure Michael A. Groeber
Abstract Recent initiatives to accelerate the insertion of materials and link the materials design and systems design processes have called for the advancement of microstructure–property relationships. To achieve these goals, the development of digital microstructure models in conjunction with computational methods for simulating material response is a necessity. There have been significant advancements in the collection and representation of microstructure, which coupled with computational power increases, has yielded microstructure models with increasing complexity and accuracy. It is the emphasis of this chapter to discuss the state-ofthe-art methods and current limitations in the field of microstructure representation. Specific focus will be paid to the areas of: experimental data collection, feature identification, mesh generation, quantitative characterization, and synthetic structure generation. In presenting the status of the field, the key links to other fields that must be developed will also be addressed wherever possible.
1 Introduction This book is motivated by the practical assessment that many questions remain to be answered regarding the effect of microstructure on materials response. Attempts to answer these questions have produced microstructure–property relationships with varying levels of detail and empiricism. The empiricism of such relationships has limited the integration of materials design in the systems design process. Instead, the systems design community tends to use databases developed through extensive experimental testing to select existing materials with a set of required performance properties. The last chapter of this book outlines the prospect of designing new materials to a desired set of properties and doing so in an accelerated fashion. One
M.A. Groeber () Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, USA e-mail:
[email protected]
S. Ghosh and D. Dimiduk (eds.), Computational Methods for Microstructure-Property Relationships, DOI 10.1007/978-1-4419-0643-4 3, c Springer Science+Business Media, LLC 2011
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factor limiting the advancement of microstructure–property relationships to allow for accelerated materials design is the disjointed treatment of microstructure by the systems design, materials processing and materials development communities. The current “industry standard” in the systems design community is to treat microstructure as a set of notes or ASTM specifications on a part drawing. As a result, the processing community uses these notes/specifications as guidelines and quality control metrics when producing materials. This approach to microstructure description is inconsistent with that used in most computational microstructure–property models. This chapter attempts to address these inconsistencies in microstructure description and outlines a process for digital representation of microstructure, which allows for the more accurate inclusion of microstructure in property models. It has become common practice to generate microstructure models for property prediction simulations with limited amounts of microstructural information included. Typically, simple geometric shapes or tessellations are used to represent microstructural features with little attention paid to accuracy beyond average values. Often though, the use of simple average quantities such as “grain size” is likely to be inadequate in some microstructure–property relationships; instead one may need to consider the possibility that the full three-dimensional (3D) microstructure is important. The complexity and interplay of features in the 3D structure dictates that calculations by hand are generally impractical. Fortunately, computational power has provided a new pathway for the materials scientist to investigate microstructure– property relationships through microstructure modeling. However, computational modeling of materials requires the ability to generate digital microstructures in which the most relevant features are sufficiently described. There are multiple ways to tackle the challenge of generating 3D digital representations of microstructure. The obvious approach is to explicitly represent structure by collecting sample volumes of microstructure experimentally, whether it involves reconstructing direct images from serial sectioning or by reconstructing diffraction information from various 3D tomography techniques. A second, less direct approach is to represent structure statistically by developing tools to generate synthetic structures with statistics equivalent to some desired set, likely obtained from some experimental observation technique(s). Both of these approaches will be discussed in detail in this chapter. With regard to the explicit representation of structure, this chapter focuses on the specific topics of experimental interrogation techniques, data processing, and mesh generation. Experimental techniques to characterize microstructure in 3D have undergone dramatic improvements in the past decade, and there now exists a host of methodologies that are capable of collecting 3D microstructural information that range from counting individual atoms (nm) to imaging macro-scale volumes (mm). However, it is important to note that there are still gaps that exist in structure collection; for example, dislocation structures cannot be readily imaged in 3D. The state-of-the-art for this field has been reviewed recently in a Viewpoint Set for Scripta Materialia (Spanos 2006) and thus will not be presented here. Currently, there are two main experimental interrogation pathways to collect information about microstructural features in 3D. Serial sectioning experiments
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are more commonplace but consume the sample as part of the experiment, while X-ray methods are nondestructive but typically require the use of high-intensity X-ray sources. For the X-ray methods, there are a handful of groups worldwide that are working toward spatially resolved crystallographic analysis of grain structures in 3D using high-intensity X-ray systems (Schmidt et al. 2004; Lauridsen et al. 2006; Budai et al. 2004, 2008; Lienert et al. 2007). These methods have a significant advantage in that the sample remains intact after analysis, allowing for the possibility of time-dependent studies of microstructural changes due to thermal or mechanical input. Nonetheless, these experiments require the high brilliance of a synchrotron source that puts significant restrictions on the general applicability of the methods. Therefore, this chapter will skew its focus toward the processing and analysis of grain-level data using more universally accessible serial sectioning experiments. Electron backscattered diffraction (EBSD) maps are often coupled with the sectioning process as an integral component of the characterization method. The incorporation of crystallographic maps enables a straightforward approach to define and segment the individual grains or precipitates that compose aggregate assemblies, and also allow for orientation-based data analysis. Most of the data-processing steps presented here are tailored for the EBSD-based section data, but all are generic in their end goal of reconstructing and segmenting features with limited experimental artifacts. Finally, the reconstructed and processed data must be translated into a meshed structure for simulation. Various methods for surface representation and mesh generation will be presented with focus paid to the complexity, quality and physical accuracy of their resultant meshes. The quantification of microstructure and generation of equivalent synthetic structures will be discussed in the context of the statistical representation of materials structure. The motivation for such an approach is the ability to generate many different representative microstructures that match statistically the measurements on the material of interest to a pre-determined degree of accuracy. The ability to generate synthetic structures is meant to limit the need for abundant data collection as well as supplement when direct 3D information is unavailable. The experimental methods discussed in the section on explicit representation of structure provide direct 3D information. Three-dimensional (3D) characterization methods are then required to quantify the structure so as to serve as the input information for the synthetic structure generation process. In contrast to the use of direct 3D statistics, the prospect of inferring 3D statistical descriptors from 2D observations is also a topic of interest. These stereological approaches are necessary to develop due to the fact that 3D experimental techniques remain unavailable to a large portion of the materials community. The development of computational materials models with specific focus on the accurate incorporation of microstructure is the ultimate goal of this chapter. Two approaches for representing structure, explicit and statistical, will be addressed. The major steps in each of these processes will be described. The explicit approach requires experimental investigation, data processing, and mesh generation. The statistical approach consists of statistical quantification of experimental observations and synthetic structure generation also requiring subsequent mesh generation.
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2 Challenges and Previous Work 2.1 Characterization Characterizing microstructure has been classically limited to two-dimensional (2D) measurements and simplistic extrapolations to three dimensions (3D). Many microstructural features can be estimated with reasonable accuracy in this manner, but arguably far more are inaccurate, if calculable at all. Limitations in these techniques have fueled a drive toward direct 3D data collection. Early attempts to collect direct 3D data (Hillert 1962; Hopkins and Kraft 1965; Rhines et al. 1976) were somewhat successful, but only recently have experimental and computational tools advanced to make 3D data collection readily feasible. Additionally, many of the early 3D experiments provided only visualizations and not quantitative microstructural descriptors. Equally important in the collection of microstructure are the current limits on the automation of microstructure feature identification and segmentation. Direct 3D data has proven to be exhaustive and tedious to reconstruct and segment, which has greatly hindered its applicability. Even 2D measurements have been slowed by the common inability to distinguish features of microstructure. Some microstructures display the ability to easily segment features using only either optical or scanning electron microscope (SEM) imaging, but this is not always the case, especially as microstructures grow in complexity. The development of orientation imaging microscopy (OIM) via EBSD has cleared a path to automated segmentation of grains and phases over restricted scales, but does increase data collection times. Finally, and arguably most important, the quantitative description of microstructure has, in some cases, been left incomplete and inconsistent. Often only average values are measured neglecting the distributions of features, leaving the microstructure incompletely described. The distributions of parameters are certainly more arduous to measure and this is usually the cause for their absence in many characterizations. However, recent developments in data collection and feature identification have allowed for these measurements to be made relatively easily for many microstructures. Additionally, parameters are generally measured independently, resulting in the inability to correlate relationships between parameters. This is a major concern when representing microstructure, because correlations may describe the tendency of features to cluster, which may have significant affects on properties. Inconsistencies in microstructure description may arise due to a number of descriptors that quantify the same microstructural feature or parameter, but vary in their meanings and dimensionalities [e.g., mean linear intercept, equivalent sphere diameter (ESD), true grain volume, and mean width]. The various descriptors, when scaled to be comparable, can deviate significantly, causing confusion in which to use.
2.2 Modeling The representation of microstructure has traditionally been significantly limited by computational power, which has led to the common practice of grossly simplifying
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microstructure in many computational models. It is quite a standard approach to represent microstructural features as simple geometrical shapes, often with no size distribution, because “size” is rarely a part of model physics. Even in cases where attempts are made to match quantified microstructural parameters; there are generally not more than one or two parameters represented and rarely a correlation between multiple parameters. For some properties (i.e., texture or yield stress), it is possible that detailed morphological structure is unnecessary. However, other properties (i.e., fracture or strain hardening) are likely to require not only initial detailed structure, but evolving structure as well. The assumptions made in the representation of microstructure may significantly hinder the ability of physical models to predict material response. That is, a model which is developed from physical understanding of a process may be applied to a nonphysical microstructure, and as a result the simulation is inherently constrained and biased. Further, when deviations between simulations and experiments are encountered, they become difficult to classify as errors in microstructure representation or errors in the physical model itself. The inability to decipher the cause of error forces incorporation of scaling constants and other relatively empirical factors to match results. Additionally, when explicit 3D structures are available, the inability to mesh the structures with a consistent quality has limited the ability to analyze the structures. The mesh generation issue remains a key concern for computation materials modeling. Finally, determination of the necessary detail to be incorporated into a simulated volume has proven complicated. Identification of the so-called representative volume element (RVE), which specifies/bounds the extent to which the structure needs to be described to accurately simulate the underlying structure, has become an entire area of study. The combination of detailed structure quantification, accurate digital representation, robust meshing techniques, properly developed constitutive laws and significant computational power is required to determine an RVE for a given property. Given that the RVE is likely to change for various properties and the significance of a simulation is strongly linked to the use of an RVE, it becomes immediately clear that the aforementioned tools are key in the development of computational materials modeling.
3 Explicit Representation of Structure The most straightforward and likely most accurate method for representation of 3D microstructure is to explicitly translate an experimentally collected volume into a computational model. The direct inclusion of experimentally obtained microstructural information requires few or even no assumptions about the microstructure itself; although the data collection technique may require some prior knowledge of the microstructure. There are many experimental techniques that are capable of yielding information about the 3D grain structure of a material. In general, these techniques tend to fall in one of the two categories: X-ray based methods and serial sectioning methods. Both of these general categories have advantages over the other,
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as well as inherent limitations. X-ray based techniques have the major advantage of being nondestructive, but are often limited by resolution and cost/availability. Serial sectioning, given the many tools capable of undertaking it, provides adequate resolution across a significantly wide range of length scales (i.e., hundreds of nanometers to tens of millimeters). However, the process is destructive and can be time-consuming and erratic. This chapter will only briefly mention some aspects about the collection processes themselves; rather the processing, analysis, and application of the resultant data will be the main focus. Further technical details can be found earlier in this book as well as elsewhere for the various X-ray (Schmidt et al. 2004; Lauridsen et al. 2006; Budai et al. 2004, 2008; Lienert et al. 2007) and serial sectioning (Rowenhorst, Groeber, Raabe, Voorhies) techniques. Serial sectioning experiments are comprised of two main tasks – sectioning and data collection – which are repeated until the desired volume of material has been interrogated. The sectioning process involves the removal of a known volume slice of material, usually determined primarily by the size-scale of the microstructural features that are to be examined. A typical rule-of-thumb is to acquire a minimum of ten sections per average feature if feature shape is to be determined. The section thickness is often a compromise between the desire for high-fidelity data and the constraints of both personnel and instrument time to collect the data sets. After sectioning, the surface-of-interest is characterized through standard imaging methods or other mapping methods, such as EBSD, electron dispersive spectroscopy (EDS), etc. The resultant data obtained from a serial sectioning experiment is thus a series of parallel images that likely contain some alignment errors, require some form of feature segmentation, and may need some smoothing to account for the digital (or pixilated) images of the microstructure. X-ray-based experiments involve exposing the sample of interest with an incident beam of X-rays and measuring various aspects of the interaction of the X-rays with the sample. The fluorescence, absorption and diffraction scattering are examples of measurable results. The group of techniques that provide a description of the position and topology of the internal features of the sample are generally referred to as tomographic techniques. Within the broad class of tomography, there are multiple methods for obtaining contrast between the features of interest; for example, phase contrast tomography. In addition to obtaining topological/morphological information, if diffraction scattering is analyzed, the crystallographic information of the features can be measured as well. The limit on resolution and size of the data collected is closely linked to the intensity of the incident X-ray beam. The resultant data obtained from the X-ray experiment is a set of 3D positions with associated values of absorption (or other measures) and possibly coupled with crystallographic information. Similar to serial sectioning, there are usually artifacts in the X-ray data as well, many of which require filtering to allow for feature segmentation. The important point to consider is that the data collection technique, while it may certainly dictate/demand some specific processing substeps, does not greatly change the major post-collection steps necessary to represent the experimental data as a 3D model of microstructure. Every 3D dataset requires a thorough application of a series of tools to align consecutive images, segment features of interest, and
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Fig. 1 Sample reconstructions of (left) a nickel-base superalloy obtained through FIB-based sectioning and (right) a titanium alloy obtained by manual mechanical polishing
represent feature boundaries. Each of these major issues will be discussed with more focus toward serial section data. Figure 1 shows examples of some reconstructed 3D volumes obtained by serial sectioning.
3.1 Reconstruction and Feature Identification 3.1.1 Image Alignment and Stacking It is often the case that misalignment between sections occurs during the sectioning experiment and can cause difficulties in the subsequent analysis of the dataset. These misalignments can be minimized through the use of fiducial marks coupled with manual and automated alignment tools during the data collection process, but can rarely be totally avoided. This section will address techniques used to adjust sections after the collection process has completed. First, the more general alignment solution for image data is discussed and then the more unique condition involving specialized data, such as EBSD, is presented. The most straightforward registration procedures involve applying simple translations to sections to improve section-to-section alignment. In the general case, this can be done with images and image processing techniques such as least-square difference fitting or image convolution. In these procedures, images are translated (generally at multiples of the pixel size) in the x and y directions of the image until either a minimum difference or maximum product between pixels in consecutive images is obtained. Subpixel alignment can also be obtained by moving images fractions of the pixel size and interpolating the values on the new grid. Rotational alignment can also be achieved through these image processing procedures if interpolation is used to generate a grid coincident with the reference image. It should be mentioned that for these techniques to function properly, consecutive images should have consistent conditions (i.e., brightness and contrast). Often, this requirement necessitates a certain amount of image processing prior to the alignment procedures. In many cases, the registration of separate serial sections requires more than simple translations to gain proper alignment of the image stack. This is most prevalent
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when combining data collected by different methods, such as micrographs from a light optical microscope and EBSD scans from an SEM, where there may be different coordinate systems, orientations, and scaling between pixels. More complex alignment procedures are also needed when the sectioning experiment lacks a fixed reference frame. If the sections are obtained by sectioning slabs through a sample rather than polishing away material, the slabs, when mounted separately, may be rotated, translated and tilted relative to one another. Alignment of these images begins by identifying a number of point features (examples would include the centers of grains, triple junctions or sample edges). One then must choose a reference frame for to which images are aligned, such as an optical image (here by noted as X 0 ; Y 0 ). The points (X; Y ) in a corresponding EBSD scan, for example, should then be brought into coincidence with the optical image reference frame. The transformation from the EBSD image to the reference optical image is given by: 3 2 3 X0 X 4Y 0 5 D T 4Y 5; 1 1 2
(1)
where T is a two-dimensional transformation matrix of the form: 2
t11 T D 4 t12 0
t21 t22 0
3 t31 t32 5 : 1
For an affine transform, where image rotation and separate translation, scaling and skewing are all allowed for the two directions, T is solved for all values without constraint, using the pseudo inverse matrix (a least squares fit to a system of linear equations). However, when allowing skewing of the images, it is not possible to independently determine the rigid rotation of the two coordinate frames, which is necessary to correct the measurement of the crystallographic orientations between each EBSD section. Therefore, a limited transformation matrix can be used that only allows independent scaling in the x-direction and y-direction, S , translations in the x-direction and y-direction, P , and finally an image rotation (of angle ) normal to the sectioning plane, R. Thus T D RPS;
(2)
where: 3 2 3 2 3 Sx 0 0 1 0 Px cos./ sin./ 0 R D 4 sin./ cos./ 0 5 I P D 4 0 1 Py 5 I S D 4 0 Sy 0 5 0 0 1 0 0 1 0 0 1 2
The terms in the transformation matrix are then determined by a least-squares optimization. As mentioned, the decoupling of the matrix T is only required to correct
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crystallographic data associated with EBSD and is not necessary to do for data without such information. Further, the z-components of the matrices can be included to correct tilt errors introduced by remounting or nonparallel polishing. As mentioned, alignment errors can be corrected in image space by processes such as least-square difference fitting, image convolutions, and Fourier transforms (Gulsoy et al. 2009; Simmons et al. 2009). However, these methods may not utilize all of the data in an image equally and image conditions can vary significantly between sections. Varying image conditions can create problems identifying features as the same feature in consecutive images. EBSD or other special data types enable the use of all data points equally and generally provide a more invariant parameter to follow between sections. This is generally true because while contrast in electron images (i.e., secondary or backscatter) does have some dependence on the structure of the individual microstructural constituents, there is not a unique probing of the structure like is the case with EBSD or chemical mapping, for example. In the case of EBSD, the misorientation between a data point and the corresponding data point on a neighboring section can be calculated for all points in a given section. A parameter (‰) can then be created to define the amount of misalignment between the two sections. The definition of ‰ is given by: ‰ .k/ D
y max x max X X
.i; j; k/;
(3)
j D0 i D0
where xmax and ymax are the total number of data-points in the corresponding directions and an example of .i; j; k/ is given by: ( .i; j; k/ D
M
ŒP .i; j; k/; P .i; j; k 1/
0
otherwise;
;
where M ŒP .i; j; k/; P .i; j; k 1/ is the misorientation between points i; j; k and i; j; k1. The calculation of misorientation is discussed further in the following section on feature segmentation. The section k can be translated, by multiples of the EBSD (or other data-type) step-size, in the x and y direction until a position with minimum ‰ is located. The parameter .i; j; k/ can be adjusted to handle criterion tailored to other data types. Subpixel alignment can also be obtained connecting triple points on consecutive sections and minimizing their alignment (Rollett and Manohar 2004).
3.1.2 Feature Segmentation and Clean-Up Segmentation of individual features is necessary to allow for the measurement of each separately. Additionally, identifying and separating microstructural constituents provides the ability to investigate features removed from their surroundings. Grain segmentation is greatly aided by the quantitative orientation information
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provided by the EBSD maps, as local orientation information is the most direct means to group voxels which reduces issues of image contrast, thresholding and feature identification. There have been other, image-based techniques shown to be applicable to feature segmentation, but few have been utilized to segment features in 3D without significant user intervention (Rowenhorst, DeGraef). Feature segmentation using EBSD allows for the complete automation of the segmentation process. During segmentation, grains are identified as groups of voxels that share a similar orientation. Algorithms for identifying these groups of voxels have been presented previously (Groeber 2007; Bhandari et al. 2007; Ghosh et al. 2008). Commercial analysis packages (TSL, HKL) likely use a similar approach to identifying grains. The major steps of the algorithm are outlined in this section. The initiation of each identified grain is the selection of a seed voxel. Generally, it is a useful idea to select a voxel that has been deemed to be of “good” quality. Quality is defined by the EBSD data collection software and refers to the sharpness of the pattern, which is often correlated with the confidence in the assigned orientation. Usually, the highest quality data tends to lie within the center of a grain, and these voxels serve as a reliable point to begin grain segmentation. A grain is then defined as the set of voxels contiguous to and with the same orientation as the seed voxel. The requirement of the voxels to have the same orientation (within a defined tolerance) is based on the fact that all regions within a grain should share a similar orientation. A list of voxels assigned to the grain is created, which initially contains only the seed voxel. The voxels that neighbor the seed voxel are checked to determine whether they have a similar orientation. The misorientation is measured to evaluate the orientation difference between the seed voxel and each of its neighbors. The value of misorientation is given by the following equation: ˇ ˇ 1 ˇ 1 tr.Oc gA gB Oc / 1 ˇˇ ˇ (4) D min ˇcos ˇ; 2 where Oc is the crystal symmetry operator and gA and gB are the rotation matrices of voxel A and B and are given by: gi D
cos '1 cos '2 sin '1 sin '2 cos ˆ sin '1 cos '2 C cos '1 sin '2 cos ˆ sin '2 sin ˆ cos '1 sin '2 sin '1 cos '2 cos ˆ sin '1 sin '2 C cos '1 cos '2 cos ˆ cos '2 sin ˆ sin '1 sin ˆ cos '1 sin ˆ cos ˆ
! ;
where ('1 ; ˆ; '2 ) are the Bunge Euler angles of the voxel. If the misorientation is less than the defined tolerance (i.e., 5ı ), then that voxel is added to the list of voxels of the grain being segmented. Each voxel on the list undergoes the same process of checking its neighboring voxels until no new voxels are added to the list. After the list is complete, a new seed point is generated and the voxel assignment is repeated for the next grain. An option during segmentation is to terminate the process when no unassigned voxels remain above a data quality tolerance. This ensures that no grains can be formed that include all low quality voxels. “Clean-up” routines can be implemented to handle low quality points either before or after assignment. Some examples of clean-up routines will be discussed later.
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The process of grouping voxels is not unique to data with orientation information. Similar algorithms can be used for image data or chemical data, where voxels of similar grayscale or chemistry are grouped together. In all three cases, it may be important to consider the possibility of gradients in the data. Rarely is the interface between neighboring features perfectly sharp in the collected data. The grouping criterion may allow for additive deviation from the original seed voxel if only immediate neighbors are checked. Possible solutions include comparing the new voxel to both its immediate neighbors and the original seed voxel or comparing the new voxel to an updated average value of all the previously grouped voxels. Clean-up routines are important to handle low-quality data as well as treat features that may be nonphysical or difficult for the simulation tools to handle. Clean-up should ideally be performed in 3D, to provide the most information during the clean-up process. In any experimental technique, there will be data points that are indexed incorrectly, due either to sample preparation or to collection error. These points will either be left unassigned during the feature segmentation process or they will be identified as a feature of their own. The latter case results in a large number of extremely small features that will be difficult for the simulation to handle. A minimum feature size criterion can be used to filter any extremely small features. A feature which contains less than a defined number voxels can be dissolved and its voxels reassigned to neighboring features. The motivation to remove extremely small grains is the inability to accurately characterize features made of so few voxels and the trouble obtaining results in the simulation at those regions. The minimum size should be carefully defined and not selected arbitrarily. One important factor in selecting the minimum size is the number of voxels needed to generate a reasonable description of feature shape. Another factor in selecting the minimum size is the fraction of features that will be removed by the filter. Generally, it is undesirable to remove a large percentage of features with any one filter. If more than a few percent of the features are removed due to the minimum size criteria, it may be an indication that the resolution or quality of the data is insufficient. Any remaining voxels, whether unassigned during or dissolved after the segmentation process, should still be assigned to neighboring features to create a fully dense structure. There are multiple options for deciding how to assign these remaining points, each of which has advantages and disadvantages. The remaining low-quality points can be assigned to the feature with which they share the most surface area, to the feature which owns the highest quality neighbor of the unassigned voxel, or to the largest feature which they neighbor. In addition to the assignment of problem voxels, there are a variety of additional data processing procedures that can be applied to the 3D EBSD data, many of which are simple extensions of the 2D filters supplied in commercial EBSD analysis programs. For example, the average orientation of each grain can be calculated and assigned to all the voxels that constitute the grain. The average orientation is the orientation that minimizes the total misorientation with all the voxels in the grain. The average orientation can be solved for numerically as shown by Barton and Dawson (2001). Generally, an adequate initial estimate of the average orientation can be obtained by transforming
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the orientation of each voxel into a single fundamental zone and finding the center of mass of the resultant point cloud in orientation space. Additionally, the dataset can be scanned for grains are that are fully contained within another grain and as a result have only one neighbor. These grains could potentially be small subgrains that are misoriented only slightly larger than the misorientation tolerance. Data sets may also contain grains with special boundaries (i.e., twin boundaries) that can be identified and omitted from certain analyses if desired. Grains sharing these special misorientations can be merged together to leave only general grain boundaries in the data set. These processing possibilities are provided as selected examples and not meant to comprise a full list of the options available to further process 3D EBSD data.
3.2 Feature Surface Representation and Mesh Generation The following section will discuss a set of mesh generation techniques. Each technique will be critiqued with focus on its surface representation/structure, mesh quality, difficulty of implementation, inherent physicality, and user bias. It is not the author’s intent to identify a “best” technique, but rather to highlight the strengths and weaknesses of each technique. All of these techniques are designed to accept as input, a discrete voxel-based structure that has been segmented and labeled as unique features. It is not necessary for the discrete data to be on a cubic grid, although the grid should be orthogonal to produce the best results in most cases. Additionally, the data need not be obtained experimentally; a synthetic structure builder can be used to generate the voxel-based structure, which will be discussed later in this chapter. Figure 2 shows a grain with different surface representations to illustrate the effect of the techniques discussed here.
Fig. 2 Surface representations of a grain using (left) voxel-based meshing (middle) marching cubes without smoothing as commonly used in direct image-based meshing and (right) marching cubes coupled with line tension-based smoothing
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3.2.1 Voxel-Based Mesh Many, if not all, experimental methods collect data on a regular, discrete gird. The resultant data can then be treated as either a set of pixels in 2D or a set of voxels in 3D. Pixels and voxels provide a useful construction for automated mesh generation without the need for a separate surface representation step. The pixels or voxels can be directly imported into a finite element (FE) or fast Fourier transform (FFT) analysis code as brick elements or integration points, respectively. Another advantage of the voxel-based mesh is that the voxels (or brick elements) have good quality metrics. Two common measures of a mesh’s quality are the distribution of dihedral angles of the elements and the size distribution of the elements. Voxels, even rectangular voxels, have all dihedral angles equal to 90ı , which is nearly optimal for FE analysis. In general, all dihedral angles between 20ı and 160ı are acceptable for most analyses. The size distribution of a voxel-based mesh is a delta function at the grid spacing of the experimental data. Thus, there is no distribution in size of the voxels, unless some nonhomogenous decimation process has been employed. With a single element size, there is no concern of tiny elements much smaller than the average element size. Tiny elements are problematic when simulating the structure using FE. However, the voxel-based mesh is not without issues to consider. Voxels create two major concerns that must be discussed to determine the suitability of this technique. First, the voxel-based construction produces a surface structure that is aliased or “stair-stepped”. The stepped nature of the surface is directly linked to the facts that the elements have all dihedral angles equal to 90ı and no surface representation step is used. The edges and corners of elements on the surface create sharp discontinuities between features and can give rise to mesh instabilities. These sharp features can behave as stress concentrations during deformation simulations, leading to artificial stress and strain localizations. Second, the lack of distribution in the size of elements often creates an abundant number of elements. If the microstructural features vary in size, it is practical to have elements that vary in size as well. Also, even if the microstructural features are of uniform size, the mesh size need not be uniform. The gradient in the property of interest can be used to grade the mesh size, where larger elements should be used where the gradient is small and smaller elements used where the gradient is large. In general, feature boundaries are areas of high gradient and the feature centers exhibit lower gradients. It is possible to decimate the voxel structure by combining neighboring voxels together in regions with small gradients in the property of interest, but not without some additional mesh compatibility steps that will not be discussed here. The voxel-based mesh technique is certainly the simplest to implement of the techniques to be presented here. The mesh quality is also very high. The other techniques can also produce high quality meshes, but none have the inherent quality of the voxel-based mesh. However, the voxel-based mesh has arguably the worst surface structure due mainly to its complete lack of physical meaning. The “stair-stepped” structure is solely an artifact of the data collection process and not any true physical structure. The user bias is eliminated by simply using the
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experimental data, which at least confines the error to only the data collection itself. Due to the relative simplicity of implementation, many investigators have used the voxel-based construct for both FE and FFT simulations.
3.2.2 CAD-Based Surface Fitting The voxel structure obtained from experimental data collection can be used as input into a surface-fitting algorithm. It is often practical to perform the surface fitting and subsequent surface reconstruction in a CAD environment. The goal of CAD-based surface fitting is to correct the “stair-stepped” nature of the voxel structure. The process consists of identifying the set of voxels that make up the boundary/interface between two microstructural features and fitting a polynomial or spline surface through them. After completing this step for every interface, each microstructural feature is defined as the region bounded by all of the fit surfaces corresponding to the interfaces between itself and its neighbors. The intricacies of this approach have been detailed previously (Bhandari et al. 2007; Ghosh et al. 2008) and are far too extensive for this chapter. The advantage of this technique is the smooth surface representation that results from the adjustment of the polynomial surfaces or splines. However, there are difficulties encountered during the fitting and reconstruction/bounding process. First, the surfaces can be adjusted to be increasing more complex by increasing the order of the polynomial or spline. Increasing the order of the surface generally improves the fit to the voxels, but also creates issues such as self intersection and local degenerate solutions. A practical limit for the surface order has been shown to be around three for a typical engineered microstructure (Bhandari et al. 2007; Ghosh et al. 2008). Second, creating the region bounded by a set of fit surfaces is not trivial. Each surface is fit independently and there can be incompatibilities at the areas of their intersection. Neighboring surfaces may not actually intersect or may intersect at a point significantly away from the true feature surface. This is due in large part to the lower number of data points at the edges of the surfaces. Also, data at the edges and corners of features tends to have a higher propensity for error. These issues combined with the high order fit surfaces can lead to spurious events at the areas where surfaces intersect. In addition, each feature is bounded separately and thus overlaps and gaps can be artificially created between neighboring features. These generally small artifacts are often the result of locally perturbing fit surfaces to ensure their intersection. Artifact cleaning routines can correct the overlaps and gaps by overlaying the original voxel data (Bhandari et al. 2007; Ghosh et al. 2008), but not without significant user intervention or advanced CAD programming. Following the bounding of each feature and artifact correction, a volume mesh must be generated. Each surface of the bounded region can be discretized, with local curvatures of the surface dictating the local density of points on the surface. The points on the surface serve as nodes for an FE mesh. Additional nodes are required in the interior of the feature and can be positioned with various goals, such as a graded size distribution toward the center of the feature, where gradients are often lower.
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The volume node generation process becomes a balance between spacing nodes homogeneously to ensure the best possible dihedral angle distribution and grading the node density to yield a graded mesh structure. Once the surface and volume nodes are created, the nodes can be connected by a Delaunay triangulation process to generate a set of tetrahedrons that will be the elements for a FE analysis. The nature of the Delaunay process results in a near-optimal mesh for the given nodes. However, the tetrahedral elements generated by the Delaunay triangulation will likely still require some clean-up processes to ensure proper element quality. In the interest of brevity, these clean-uproutines will not be discussed here, but can be found elsewhere (Barry 1995; Amenzua et al. 1995; Parthasarathy and Kodiyalam 1991). The CAD-based technique is likely the most time intensive technique to implement. The mesh quality can be very high, provided the clean-up tools are designed properly, but it does not have the inherent quality of the voxel-based construct. The largest benefit of the CAD-based mesh is the smooth surface structure, which is arguably better than any of the other techniques. The “stair-stepped” structure is generally completely eliminated. However, the process for smoothing the surfaces is not inherently physical. For example, the surface fitting criteria are generally defined to smooth while minimizing deviation from the experimental data and not to approach some physical property such as minimum surface area or specified angle of surface intersection. The user bias can also be significant during the artifact correction steps, where overlaps and gaps must be assigned to neighboring features. Bhandari et al. (2007) developed come physically based tools to correct these artifacts with limited bias.
3.2.3 Direct Image-Based Meshing Direct image-based meshing is similar to the voxel-based approach, but with some attempt to smooth the surface structure. The central tool used in most direct imagebased approaches is the marching cubes algorithm first developed by Lorensen and Cline (1987) and modified by others (refs). The marching cubes algorithm breaks a voxel-based structure into tetrahedrons based on local voxel neighborhoods. The tetrahedrons are generally high quality elements because of their tendency to be right tetrahedrons with dihedral angles of 45ı and 90ı . The surface structure is also generally improved over that of the voxel-based representation. The voxels on the surfaces of features are broken in a manner that removes many of the abrupt 90ı angles that causes the “stair-stepped” surface structure discussed previously. The marching cubes algorithm creates an extremely large number of elements, because each voxel is broken into multiple tetrahedrons. The tetrahedrons tend to be close to uniform in size and thus, they are far from efficient in meshing the microstructural features. The tetrahedrons can be decimated to improve the mesh efficiency by reducing the number of elements. The decimation of the tetrahedral mesh is actually more straightforward than the decimation of the voxel-based structure directly. The surface representation, while improved, is still far from the smooth surfaces generated by the CAD-based surface-fitting approach. Some commercial
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programs designed to generate direct image-based meshes have smoothing tools built in, but it is not known to this author what, if any, physical criteria are used. It is also possible to combine techniques by running a marching cubes algorithm on the original voxel-based structure before passing the semi-smoothed structure on to a CAD-based approach. The semi-smooth structure is likely to improve the CADbased approach by supplying a smoother initial surface structure, which may allow for the use of lower order fit surfaces. The direct image-based approach has nearly the same pros and cons as the voxel-based approach, albeit the surface representation is improved. The ease of implementation, high-quality metrics, and limited user bias are key advantages, but the relatively poor surface representation and the lack of physical smoothing criteria are drawbacks.
3.2.4 Surface Area/Line Tension-Based Smoothing Methods The surface area/line tension smoothing technique is an attempt to combine the direct image-based method with a semi-physical smoothing process. The first step in this method is to run the marching cubes algorithm on the original voxel-based structure. After the voxel-based structure is converted into a tetrahedral mesh, a surface smoothing algorithm is applied to improve the surface representation. The difference between this smoothing algorithm and the CAD-based approach is that this algorithm has a physical driving force and the features are not treated independently. The physical driving force is related to the general belief that interfaces between features will tend to minimize their surface area in an attempt to minimize the energy of the structure. The smoothing process is presented in detail by Lee (ref) and will only be outlined here. The smoothing process begins with the identification of all quadruple points, which are points where four features meet. Each quadruple point is connected to another quadruple point (with three of the same four features) and these connections make up the triple line network. The triple lines are not made by connecting the points directly with a line segment; they are connected by following edges of the tetrahedrons in the directions of the original voxel edges. The triple lines are initially “jagged” from following along edges of the original voxels, but will be smoothed by balancing the minimization of both line length and deviation from the experimental data. The smoothing process involves starting at one end of the line and progressively skipping edge segments on the triple lines and monitoring the deviation of the new line from the original “jagged” line. If a user-defined tolerance is exceeded, the current segment is fixed and a new segment is initiated. The process continues until the other end of the triple line is reached. After all the triple lines have been smoothed, the surfaces must also be smoothed. The surfaces are treated as a series of lines connecting triple lines across the surface. These lines are smoothed using the same process as the triple lines. The end points of all the line segments on the triple lines and surface lines are the surface nodes, similar to the nodes placed on the fit surfaces in the CAD-based method. Volume nodes are added and connected using the same ideas discussed in the CAD-based section.
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The surface area/line tension smoothing approach is an attempt to smooth the data based on physical ideas, while still being true to the experimental data. The implementation is rather simple, especially when compared to the CAD-based approach. The quality of the elements is similar to the CAD-based approach, due mainly to the similar Delaunay triangulation method for element creation used in both methods. The physically based smoothing criterion makes for the most “realistic” surface representation of the methods presented here. The user bias, however, is still somewhat significant due to the user-defined tolerance for deviation from the experimental data.
4 Statistical Representation of Structure Representing structure in a statistical sense has multiple benefits to the computational material scientist. First, some material properties show a significant amount of scatter with the apparent cause linked to specific local feature interactions. Thus, an individual experimental volume may not accurately represent the variation in locally critical neighborhoods. Statistically equivalent instantiations of the experimental volume may highlight the likelihood of the occurrence of such neighborhoods and permit simulation of their influence on behavior. Additionally, statistically quantifying structure can offer insight into the representative nature of a structure in relation to a specific property of interest. That is, the convergence of a property and some set of statistical descriptors can be linked to determine a RVE for that property. Further, statistical-based structure builders can be used when the 3D structure is not available. If the 3D statistics of a structure can be inferred from 2D observations, which will be discussed later, then a 3D structure can be created where one was not available. Finally, the statistical description of structure, if coupled with rapid and robust instantiation tools, can allow for the compression of data from large experimental datasets to a list of statistical descriptors used to generate structures when needed. The following section will discuss the critical steps in generation of statistically based synthetic structures, specifically statistical quantification methods, structure generation processes, and metrics for the validation of equivalence.
4.1 Quantitative Description of Structure 4.1.1 Feature Size and Volume Feature volume can be calculated simply by summing the number of voxels that are assigned to each feature. Each voxel has an associated volume given by Vvoxel D ı"2 , where ı is the section thickness and " is the pixel/step-size of the 2D image or EBSD map. The feature volume is calculated by Vgrain D Nv Vvoxel , where Nv is the number of voxels in the grain. In addition to measuring the true feature volume, the
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Fig. 3 Plot of equivalent sphere radius (ESR) for a nickel-base superalloy along with three fit theoretical distributions
equivalent sphere radius (ESR) can be calculated (Groeber 2007). The distribution of ESRs is useful for comparison with classical descriptions of feature size (or radius/diameter), which often involved some extrapolation from 2D. Additionally, various theoretical distributions have been developed to fit the ESR distribution (Zhang et al. 2004; Feltham 1957; Hillert 1965; Louat 1974). Figure 3 shows a sample distribution of ESR for a nickel-base superalloy along with fit theoretical distributions. Another measure of feature size is the mean width of the feature. The mean width is given by: v 1 X "i ˇi ; (5) LD 2 i D1
where v is the number of edges on the feature, " is the length of the edge, and ˇ is the angle between the normals of the faces that meet at the edge. Mean width is a measure of the linear dimension of a feature and was shown by Hadwiger (1957) to exhibit the property of additivity. Mean width can be calculated analytically for all flat-faced polyhedra (MacPherson and Srolovitz 2007), which means that mean width can be easily computed for all grains after any of the surface mesh generation processes discussed previously. 4.1.2 Feature Shape The irregular geometries that are typical of features in some materials make feature shape a difficult parameter to unambiguously describe. This is especially true due to the lack of general shape descriptors, where most descriptions of shape involve
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combining groups of size parameters to generate a unitless value (Russ 1986). Examples of possible shape descriptors include: length/width (aspect ratio), area/ convex area (solidity), and length/fiber length (curl). A common practice is to fit an ellipsoid (or ellipse in 2D) to the feature (Saylor et al. 2004a; Brahme et al. 2006). A systematic method of generating ellipsoidal inclusions from voxel data obtained by serial-sectioning has also been developed (Li et al. 1999). In this method, the zeroth order moment (I0 ), first-order moments (Ix , Iy , Iz ), and second-order moments (Ixx ; Iyy ; Izz ) are first calculated for each feature by adding the contribution of each voxel belonging to an identified feature. The coordinates of the centroid for the best-fit ellipsoid are computed from the zeroth- and first-order moments as: xc D
Ix Iy Iz ; yc D ; zc D : I0 I0 I0
(6)
Next, the principal directions corresponding to the principal axes of the ellipsoid are calculated from the eigenvalues and eigenvectors of the second-order moments Iij ; i; j D 1; 2; 3. The major axis (2a), minor axis (2c), and intermediate axis (2b) of the ellipsoidal grain are solved from the relations of the principal second moments of inertia as: aD
A4 BC
1=10
;b D
B4 AC
1=10
;c D
C4 AB
1=10 ;
(7)
where A, B, and C are given by:
15 I1 C I2 I3 15 I1 C I3 I2 ;B D ; 4 2 4 2 15 I2 C I3 I1 C D : 4 2 AD
In some cases, the ellipsoidal representation of features is an oversimplification that should be quantified. Correspondingly, an accuracy indicator of the ellipsoidal representation can be developed (Groeber 2007). The best-fit ellipsoid based on the moment analysis is scaled slightly to be of the same volume as the grain with the same aspect ratios and orientation. The fraction of the grain’s voxels that lie within the best-fit ellipsoid is calculated. If the grain is perfectly ellipsoidal, then the value of this quantity would be very near to 1 (with the voxel size relative to the feature size controlling the nearness to 1). Decreasing values indicate more complex and likely concave shapes that are poorly represented by an ellipsoid. Figure 4 shows a sample plot of the distribution of grain aspect ratios for a nickel-base superalloy as well as the corresponding accuracy indicator plot. In the case that the ellipsoidal representation is not sufficient, higher-order moments may need to be considered. However, there is more information in the second-order moments, beyond the aspect ratios, that is rarely used in microstructure description. For the second-order moments, there are three moment invariants.
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Fig. 4 Plot of (left) the distribution of grain aspect ratios for a nickel-base superalloy, obtained from a best-fit ellipsoid and (right) the accuracy indicator, termed ellipsoidal misfit, that describes the closeness of the fit ellisoids
The third moment invariant, where the first two are essentially the aspect ratios, offers more detail on the shape of a feature (MacSleyne et al. 2008). While it is true that the three invariants of the second-order moments are not sufficient in describing the true shape of the feature, there are techniques that can be used to incorporate all three invariants and improve on the current description of shape. For example, if a class of shapes is selected (i.e., superellipsoids or truncated octahedrons), the third moment invariant can be linked to parameters that fully define these shapes. In the case of the superellipsoid, the third invariant, known as 3 , can be linked to the exponent in the equation that defines a “superellipsoid” (MacSleyne et al. 2008): x n y n z n C C D 1; (8) a b c where a, b, and c are the semi-axes of the superellipsoid. The relationship of 3 to n is given by: 3 D F 3 Œn; (9) where
3 1 C n1 n5 F Œn D 20 5= 3 n3 1 C n3
and Œx is the complete Gamma function. Figure 5 shows the relationship of 3 and n, along with some example feature shapes for selected values of n. It becomes obvious from the plot that utilizing 3 enables the description of shape variation beyond just aspect ratio changes.
4.1.3 Number of Neighbors In addition to the shape of features, the number of nearest neighbors is another example of a parameter that cannot be determined directly from 2D measurements
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Fig. 5 Plot of the relationship between 3 and n, where n is the exponent in the superellipsoid equation given by equation 8
but is easily determined in 3D. Here, the voxels of each feature are checked to see whether they neighbor another feature. If a neighboring voxel belongs to a different feature, the two features are neighbors. This process is exceedingly useful because it not only determines the number of neighbors, but it also identifies the feature connectivity in the structure, which allows for further automated investigation of any parameter involving neighbor interactions. Note that when checking neighboring voxels, only voxels that share a common face, not a common edge or corner, are considered. This requires the features share some actual area and avoids counting features that meet at only an edge or point. Figure 6 shows a sample plot of the number of neighbors distribution for a nickel-base superalloy.
4.1.4 Correlations between Parameters The correlation or mutual relationship between parameters may be important because it provides additional information regarding the morphology and spatial arrangement of features. The feature connectivity coupled with the parameters of each individual feature allows for the relationships between parameters as well as the clustering of critical features to be studied. It is desirable to quantify the degree of correlation between two selected parameters. A quantity called the correlation ratio, 2 , can be used for this purpose (Kenney and Keeping 1947). The correlation ratio is a preferred metric in comparison with the correlation coefficient, r, because it can be used with nonlinear relationships. The correlation ratio is the square of the correlation coefficient (r 2 ) if the relationship is linear. If the relationship is
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Fig. 6 Plot of the distribution of the number of neighbors for grains in a nickel-base superalloy
nonlinear, the magnitude of the correlation ratio is larger, but retains a value between 0 and 1. The formula for the correlation ratio is given by: PNb
2
D
N 2 bD0 nb .yNb y/ ; PNg N 2 i D0 .yi y/
(10)
where Ng is the total number of features, yi is the value of the i th feature, Nb is the number of bins, nb is the number of observations in a given bin b and, PNb
P nb yNb D
i D0 ybi
nb
;
bD0 yN D P Nb
nb yNb
bD0
nb
;
where yb is the value of the i th feature in bin b. The correlation ratio can be thought of as the percent of the total variance of the dependent variable accounted for by the variance between groups of the independent variable. Seen in the equation is that if there is a large difference between the mean of the whole data set and the means of the individual bins, then the correlation ratio is high. Figure 7 shows a correlation plot relating feature size to a number of morphological parameters for a nickel-base superalloy. The grains of the superalloy were separated by their ESD and the averages of other morphological parameters were calculated for grains of similar ESD. The averages for each size bin were then normalized by the average of the corresponding morphological parameter for all sizes. It becomes clear which parameters are strongly correlated with size when observing the trends in Fig. 7. Parameters that deviate from a value of 1, such as number of neighbors and surface-to-volume ratio, are correlated with the grain diameter.
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Fig. 7 Plot of the correlation between grain size (i.e., ESD D 2 ESR) and other morphological parameters. Bin average refers to the average of a parameter for all grains within a given size bin and sample average refers to the average of the parameter for all grains, regardless of size bin
4.1.5 Crystallographic Texture Many of the traditional analysis of 3D EBSD data are directly analogous to the traditional two-dimensional counter analysis. For example, the volume fraction of a particular phase and the orientation distribution function (ODF), since the area fraction is exactly equal to the volume fraction for a random section through a material. However, a 3D analysis can often add information to these analyses, for example, only through a 3D analysis can the morphological texture of the grains be correlated with the crystallographic texture measured in the ODF. Other measurements have 2D to 3D analogs that are related through stereological relationships that give a statistical equivalence of the properties. An example of this would be the distribution of misorientations across feature boundaries in a material. In the 2D section, one can easily measure the misorientation across the boundary, but the line-length of a given boundary is not directly related to that boundaries area. However, if enough boundaries are collected in 2D section with a similar misorientation, one can apply the stereological relation, SV D 4BA = where SV is the surface area per unit volume, and BA is the boundary length per unit area (Russ and DeHoff 1986). It should be noted a very large number of boundaries need to be present in the 2D section to obtain a statistically significant number of boundaries for each misorientation type for this analysis to be valid. In the case of a 3D reconstruction, the measurement of the misorientation distribution is relatively straightforward since, as discussed above, the feature’s nearest neighbor misorientations and their boundary areas can be directly calculated from the 3D
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Fig. 8 Plot of (top) pole figures for a nickel-base superalloy and (bottom) the misorientation distribution of the same nickel-base superalloy. Note the misorientation angle reported here is the minimum of the set of equivalent misorientation angles and is often termed as the disorientation
reconstruction, thus the misorientation distribution function (MoDF) can be directly measured without assumption. Figure 8 shows a set of pole figures and the misorientation distribution for a nickel-base superalloy. 4.1.6 Interface Character Distribution There are many analyses that can only be measured directly through 3D reconstruction of the material. In a very general sense, these analyses can be seen as the correlation of a specific 3D geometry with the crystallographic orientation. One of the most relevant applications of this is the determination of crystallographic interface orientations, which requires directly correlating the crystallographic orientation of an object with the local interface normal. One of the most common 3D visualization techniques is to form a surface mesh of the object so that the interface of the object is described as a 3D mesh of discrete interconnected triangles. The conversion from a regularly gridded 3D array of data (such as a stack of images) is most often accomplished using a fast-marching cubes algorithm that converts the volumetric regular array data form to a surface mesh (Lorenson and Cline 1987). Often it is necessary to apply some degree of surface smoothing to remove pixel-like artifacts from the surface mesh, as discussed in the section on surface representation and mesh generation. Once the surface is described in terms of a set of triangles, the properties of these triangles can be used to quantify
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the interfaces of the 3D reconstruction, particularly the local interface normal. The normal nO and the area A of each triangle in the surface mesh is given by: ˇ ˇ eE1 eE2 ˇ I A D ˇeE1 eE2 ˇ =2; nO D ˇ ˇeE1 eE2 ˇ
(11)
where eE1 and eE2 are the two edge vectors of the triangle. One powerful construction that can be formed from this type of data is the interface normal distribution (IND) (Kammer et al. 2006). The IND is constructed by first placing the collection of all the interface normal vectors on to a unit sphere. The normals are then binned (weighted by the surface areas of the triangles) according to orientation, then normalized by a random distribution of orientations. Therefore, an orientation on the spherical histogram that has a strong intensity corresponds to a large surface area that shares that orientation. This spherical histogram is then projected to a 2D plot using a stereographic projection (typically along the Cz direction, but this is a matter of how one wants to represent the data), preserving the angular relationships between the orientations in the plot producing the IND plot. It should be noted that if the binning of the data occurred on the projected data space of the stereoplot, rather than on the unit sphere, the intensities of the histogram would be altered by the nonequal-area nature of the stereographic projection. Because the binning of the data occurs on the unit sphere, this artifact is avoided and a randomly distributed shape (like a sphere) would have a flat intensity. For INDs that contain peaks, the exact shape of peaks on the stereoplot would still be slightly altered by the projection especially close to the center poles and edges of the stereoplot. The stereographic projection has extremely large distortions for orientations that have a –z component. To avoid this, often it is necessary to project each hemisphere separately, however this can be avoided if there is crystallographic symmetry, which allows the plot to be compressed to a stereographic triangle. The inclusion of the local crystallographic orientation along with the local interface normal means that the interface normals can be expressed in the crystallographic coordinate system as well, creating a crystallographic interface normal distribution (CIND). The only variation in the construction of the CIND is that before the normals are binned on the unit sphere, they are rotated to the crystallographic coordinate system using the Euler angles of the object in question. There are two analyses that especially lend themselves to this type construction, the examination of individual objects and examining the overall crystal interface texture in a sample, which will be briefly reviewed. Rowenhorst et al. (2006) used the CIND analysis to examine the facet planes on individual coarse martensite crystals, as shown in Fig. 9 (right). By identifying the peaks within the CIND, and fitting planes to the corresponding triangles the average crystallographic facet normal was determined. This article also introduced a unique visualization technique in which the interfaces in the 3D reconstruction of the martensite crystals were colored according to the local interface crystallographic normal (Fig. 9, middle). The coloring scheme is identical to that used to create an inverse pole figure (IPF) but unlike the IPF, where each point in the image is colored according the crystallographic direction that points along an arbitrary section plane
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Fig. 9 Reconstruction of two coarse martensite crystals in HSLA-100 steel (left). Color indicates the crystallographic direction that is parallel with the z-axis. The difference in color indicates that the martensite crystals represent separate martensite variants (middle). Color indicates the crystallographic orientation of the local interface normal at each local patch of the interface. Note that while the crystals represent different variants, the crystal facets have similar crystallographic normals (right). Crystallographic interface normal distribution (CIND) plots of the two martensite crystals, where the upper and lower CINDs correspond, respectively, to the purple and yellow crystals in the left image. The peaks in the CIND indicate the average orientation of the facet planes of the crystals; peak intensities indicate Multiples of a Random Distribution (MRD)
direction (Fig. 9, left), here the interface was colored according to the crystallographic direction that is parallel with the surface normal. This visualization contains essentially the same exact information as the CIND construction with the added advantage of being able to link crystallographic orientation to specific morphological features, but unlike the CIND, the information presented is not quantitative. Saylor et al. (2004b, c) significantly expanded on the CIND construction to include not only the interface distrpoibution of grain boundaries in polycrystalline MgO, but also to include the full five-parameter space of the grain boundaries, constructing the grain boundary character distribution (GBCD). By examining the full five parameter space, they were able to not only determine the interface texture for the system, but also the interface texture for particular grain misorientations. Since this initial study, the GBCD has been determined for many more materials systems including Al, Ni, Cu, Brass, and Ti-6–4 (Saylor et al. 2004b, c; Randle et al. 2005; Randle et al. 2008a, b).
4.1.7 N-Point Statistics Two-point through N-point correlation functions are a useful set of descriptors that characterize the spatial arrangement of microstructural features. Generally, these correlation functions, most common of which is the two-point correlation function, are used to analyze spatial arrangement of particles or voids in materials such as: discontinuously reinforced composites, foams, geological samples, etc.
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(Torquato 2001). However, correlation functions can also be utilized to describe the spatial arrangement of grains with certain microstuctural parameters. For example, the spatial arrangement of grains with a given orientation in a polycrystal could be quantified using correlation functions. Two-point correlation functions can be determined by systematically placing line segments in the structure and noting where the endpoints lie. Therefore, for a microstructure with two entities, be they phases, orientations, sizes, shapes, etc., there are exactly four two-point correlation functions: hP11 hrii, hP12 hrii, hP21 hrii, hP22 hrii. hPnm hrii is the average probability that endpoints 1 and 2 of a randomly oriented line segment of length r lie in entity n and entity m, respectively (Tewari et al. 2006). Note that here the term entity is being used to mean a certain property and not a specific feature. Thus, for hP11 hrii, it is not a requirement that both endpoints lie within the same particle/feature with entity 1, just that both endpoints lie within a particle with entity 1. Two-point correlation functions can also be directionally dependent and thus, the values of each correlation function can be calculated as a function of the orientation of the line segment within the sample. The extension of two-point correlation functions to n-point correlation functions simply involves placing an n-cornered polygon, rather than a line segment, in the sample and noting the position of the endpoints. The number of functions increases with n and with the number of entities in the sample. N-point correlation functions could be used to describe the local neighborhood around a grain in a polycrystal. A specific property of a grain, its size, for example, could be chosen as entity 1 and all other sizes could be chosen as entity 2. Then line segments of varying lengths could be drawn from the centroids of all grains with size equal to (or near that of) entity 1. The resultant two-point correlation functions would describe the spatial arrangement of grains with size equal to (or near that of) entity 1. This same process could be carried out for the Schmid factor of a grain or its orientation, which would likely offer key insight into the clustering of critical orientations.
4.1.8 Limitations/Concerns when Using Statistical Descriptors The statistical descriptors presented here certainly offer a great deal of information about the microstructure, but are by no means a complete set. Other classical morphological descriptors, such as the number of faces, edges, and corners of a feature have not been presented here, but are readily available in the 3D volume. Mean width was explained to be the integral of a feature’s curvature and the IND analysis was shown to be the complete distribution of interface normals, but the curvatures of a grain’s individual boundaries have not been correlated with the grain’s size and shape or with the curvatures of neighboring boundaries. In general, the near-neighbor analyses have been limited to only contiguous neighbors and not next-nearest neighbors and beyond. These issues are not truly limitations of the descriptors themselves; rather they are an illustration of the abundant amount of information available in just one experimental volume. The limitation is actually in the ability to quantify every aspect of the structure, which has yet to be shown, is in
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fact necessary. It should be the goal of a concerted characterization-modeling effort to define to what extent microstructure needs to be quantified and represented for each property. Some areas of the current analysis remain incomplete. The correlation between descriptors is one of the areas most lacking. Especially when three or more descriptors are related, it becomes difficult to quantify their dependencies on one another. This is most frequently the result of an insufficient number of features in the interrogated volume. This can be corrected by collecting larger volumes, but limitations of experimental capabilities and availabilities may be preventative. Additionally, the statistical analyses have not yet advanced to the point of accurately informing the experimentalist of the proper number of features to collect for each type of analysis. Another area that is currently underinvestigated, at least in the context of materials microstructure, is the analysis of rare events. The extreme-value statistics of the microstructure is not well understood and has not been well addressed in the quantification and comparison of structures. The standard distributions (i.e., lognormal, beta, weibull) used to describe most descriptors provide a “good” fit over much of the descriptor’s range. However, when the extreme values are closely investigated, the experimental observations often appear to deviate systematically from the fit distribution in a manner that suggests more than simply sampling error. Until the extreme-value statistics is properly quantified and accounted for in digital microstructures, simulations of properties that are believed to be controlled by rare events or neighborhoods should be treated with some sense of skepticism.
4.2 Synthetic Structure Builders The synthetic builders that will be discussed here consist of two major steps. First, the features to be placed in the synthetic volume are generated. These features are generated by sampling the size, shape, and morphological and crystallographic orientation distributions of the features observed by some experimental technique. Second, the features are placed in the volume with specific focus on the local neighborhoods created by neighboring features. The sampling procedure for generating representative features as well as the constraints used to place the features in the volume will be described here. Figure 10 shows some example synthetic volumes generated by the different techniques discussed in this section. 4.2.1 Representative Feature Generation A representative feature generation process is responsible for the creation of a collection of idealized ellipsoidal (or alternative representation) features having distributions of size, shape, and morphological and crystallographic orientation equivalent to those observed in the experimental volume. In this representation, each grain is modeled as an ellipsoid as defined in the previous section on feature shape. The size corresponds to the volume of each ellipsoid, the shape corresponds to the
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Fig. 10 Example synthetic structures produced by (left) the sequential grain placement algorithm using ellipsoidal grains and (right) the sequential grain placement algorithm using superellipsoidal grains. Note that the grains generally appear less “idealized” in the right image, where the superellipsoid representation described in Sect. 4.1.2 was used
aspect ratios of the principal axes (b=a; c=a and c=b) and the morphological orientation corresponds to the orientation of the major principal axis (a W a b c) relative to the global coordinates. The first step in the process is to sample the experimental feature volume distribution, which is represented by the cumulative probability distribution function (CPDF) fit to the experimental data. Many investigations have shown the feature volume distribution to be best represented by a log-normal distribution (Zhang et al. 2004; Groeber et al. 2008a), whose CPDF is given by:
P .V / D
!# 8 " AVG V V 1 ˆ g ˆ ˆ p ! V < VgAVG 1 erf ˆ ˆ STD 2 ˆ Vg 2 < !# " ˆ ˆ AVG ˆ ˆ g ˆ 1 1 C erf V Vp ˆ ! V > VgAVG : 2 VgSTD 2
;
(12)
where V is the feature volume, P .V / is its cumulative probability, which has 0 and 1 as its limits. The average feature volume (VgAVG ) and the standard deviation (VgSTD ) are parameters that determine the precise shape of the distribution function. During volume assignment, a number within the limits of P .V / is randomly generated and the corresponding volume, given by equation 1, is assigned. This assignment process is continued until the total volume of all features generated equals a threshold defined as 110% of the volume of the synthetic microstructural model. The additional volume is needed because some features may lie partially outside the domain of the microstructural model or overlap other features. This issue will be discussed further in the next subsection. Subsequent to the volume assignment, feature shapes are assigned in conformity with CPDFs of the ellipsoid aspect ratios (b=a; c=a and c=b) that have been established a priori from the experimental data. The corresponding CPDFs can be represented in terms of a beta distribution, with the form: R b=a P .b=a/ D
0
t p1 .1 t/q1 dt ; B.p; q/
0 b=a 1:
(13)
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R1
In equation 13, B.p; q/ D 0 t p1 .1 t /q1 dt is the beta function, p and q are the shape parameters, and P .b=a/ is the cumulative probability. The statistical analysis establishes the correlation between the shape and the size of each grain, represented by the aspect ratios and volume of each ellipsoid. To establish this correlation for the synthetic ellipsoidal features, each volume is converted to an ESD using the relation: 1=3 3 V : (14) ESD D 2 4 The correlation is determined by assigning aspect ratios to different volumetric bins that are represented by ranges of ESD values. The two aspect ratios that define an ellipsoid (b=a and c=a) each has a CPDF in each volume bin. The sampling of the b=a and c=a CPDFs is identical to that of the feature volume CPDF. This process ensures appropriate correlation between the shape and size distributions. In addition to a correlation with volume, the aspect ratios b=a and c=a should also be mutually correlated. However, often there is an insufficient number of features in the experimental volume to determine this correlation table between the grain volume V , and both of the aspect ratios b=a and c=a. To overcome this shortcoming, individual correlation functions are generated between V and each of the aspect ratios separately. In this process, values of b=a and c=a are evaluated corresponding to randomly chosen P . ba / and P . ac / values from the P . ba / . ba / and P . ac / . ac / plots, respectively. The consequent aspect ratio c=b D .c=a/=.b=a/ is evaluated and its probability density p.c=b/ D .c=b/p1 .1 c=b/q1 =B.p; q/; 0 c=b 1 is ascertained from the experimentally observed distribution. The two individually generated aspect ratios are accepted with the same probability density of p.c=b/. The third variable, the morphological orientation of each ellipsoidal feature, is defined by a set of rotations (, , ) needed to transform the global coordinates (X; Y; Z) onto the principal axes of the ellipsoid (A; B; C ). The probability density function, f .g/ g D Ng =N is the probability of observing an orientation G in the interval g G g C g, where Ng is the number of orientations between g and g C g and N is the total number of experimentally observed ellipsoidal features. If N .i / is the number of observations in the i th orientation space element ranging from (; ; ) and . C ; C ; C /, then the density of orientations can be expressed as N .i / =N . To evaluate this density, the entire orientation space is defined as a finite cube with edge length (180ı) with the origin at one of its vertices. For the purpose of creating ranges in the orientation data, the orientation space is discretized into cubic bins of dimension =36 or 5ı . The morphological orientation density in each bin is calculated by dividing the number of orientations in the bin by the total number of orientations in the experimental data and normalizing by the size of the bin. Ellipsoidal orientations are created and assigned based on this probability density function. In summary, the output of this process is a set of representative ellipsoidal features having statistically equivalent volume, aspect ratio, and morphological orientations as the experimental reference data. However, this process does not arrange the features in their appropriate spatial locations, which is the function of the next
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process. A final step in this procedure can be included either before or after the feature placement routine. This step is the assignment of crystallographic orientations to the generated features. The process of assigning crystallographic orientations is exactly identical to that of the morphological orientation assignment process previous described. Here, the only difference is that the rotations assigned are those to transform the global coordinate axes to the coordinate system of the crystal, rather than the principal axes of the grain.
4.2.2 Feature Placement After generating a set of ellipsoids that is representative of the 3D features, the focus must be shifted to the placement of the ellipsoids in the volume. There are multiple issues to consider when packing the ellipsoids. The density of the objects represented by the ellipsoids is one of the largest factors in developing the packing algorithm. For example, ellipsoids representing particles of a low volume fraction phase will certainly be placed differently than ellipsoids representing grains in a fully dense polycrystalline material. In the fully dense grain example, care must be taken to pack the volume as densely as possible, but minimize overlap between ellipsoids to retain each ellipsoid’s prescribed shape. In both cases, the local neighborhood of the ellipsoid (i.e., neighboring ellipsoids) must also be addressed during placement. The low volume fraction particles should be spaced equivalently to the experimental/reference data and the densely packed grains should neighbor grains of similar sizes and shapes as seen in the experimental/reference data. Two inherently different, but viable options for ellipsoid packing will be discussed here. The first approach involves overpopulating the volume with a large number of ellipsoids. This approach is presented in greater detail elsewhere (Saylor et al. 2004a; Brahme et al. 2006). A large set of representative ellipsoids are placed into the model volume. The ellipsoids should have a total volume much larger than the volume to be filled and are allowed to overlap and extend outside of the volume. A simulated annealing procedure can then be employed to determine an “optimal” set of ellipsoids. An optimal set of ellipsoids would maximize space-filling, while minimizing overlap of ellipsoids. Saylor et al. (2004a) and Brahme et al. (2006) outline the development of a penalty function that promotes optimal space filling. It is easy to imagine the adjustment of this function to address the less dense packing of the distant particle case. At present, only the space-filling nature of the ellipsoids is addressed in this technique and not the local neighborhood of an ellipsoid. However, it is conceivable that penalty functions could be developed to encourage desired clustering or spacing of ellipsoids of given sizes or shapes. Once an active set of ellipsoids are selected by this method, a voxelized structure is generated using cellular automata (CA) where the centroid for each ellipsoid is a seed point in the simulation and voxels are added starting at this seed point until the entire structure is filled. The growth is constrained initially such that only those voxel locations that are located inside the ellipsoids are added. When each ellipsoid is completely filled, then the constraint is dropped and the remaining free volume is consumed.
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A second approach to the ellipsoid packing problem is to sequentially place the ellipsoids while using statistical descriptors from the experimental/reference data as constraints. One such implementation of this approach is presented by Groeber et al. (2008b). Here, the set of representative ellipsoids should have a total volume much nearer to the model volume than the first approach. The ellipsoids are allowed to extend outside the volume, similar to the first approach, and thus the total volume should be some small amount (i.e., 10%) above the model volume. Each ellipsoid is randomly placed in a sequential fashion and checked against a number of constraints to determine whether its current position is acceptable. Constraints can include, but are not limited to: overlap limits, number of neighboring ellipsoids, and size distributions of neighboring ellipsoids. This approach generally yields optimal space-filling through the overlap limits and produces realistic neighborhoods by constraining placement to locations that improve the surroundings of previously placed grains. Although this technique presents some advantages over the previous, there are complications that arise as well. In any sequential process, there should be concerns of a failure to locate a suitable position, especially near the end of the process. Generally, the process is more efficient and successful when the largest grains are placed first, when there is sufficient room left for their placement. Additionally, the number of constraints greatly affects the feasibility of locating a suitable position and thus should be optimized.
4.3 Measures of Goodness The “goodness” of the synthetic structure can be defined relative to two objectives, which may or may not be completely related. First, the statistical descriptors of the synthetic structure can be compared to those of the experimental structure. The similarity of these descriptors is certainly one measure of “goodness” and will be a focus of the following subsections. Second, the simulation results of the two structures can be compared. This definition of “goodness” is arguably a more practical one, in that the response of the material is often the overriding goal. However, depending on the property of interest, the two structures may be “equivalent” with respect to the statistics chosen, but still yield different simulation results. This is likely to be the case when the experimental structure is smaller than a RVE for the specific property or the statistical descriptors chosen are not directly linked to the response property. This section will not deal with the second definition; due mainly to a lack of simulation results, but it should be considered carefully in future structure–property relation investigations. 4.3.1 Size(s) Measuring the “size” of a grain may appear to be a simple matter at first sight. Even in 2D sections, however, computing the circle-equivalent diameter yields a (slightly) different result than the average linear intercept (Underwood 1970). In 3D, one must
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Fig. 11 Blasche diagram combining the properties that exhibit additivity in the three dimensions. The x-axis is the square root of surface area normalized by mean width and the y-axis is the volume normalized by mean width. The positions where some standard geometrical shapes reside on the plot are noted
be concerned with all three dimensions, of which measuring the volume and surface area of a grain is intuitively obvious. Less obvious is how best to measure the linear dimension of a grain, since there are so many possibilities (linear intercept, sphereequivalent radius, etc.). The recent publication by MacPherson and Srolovitz (2007) on the theory of grain growth has, however, pointed out to the materials science community that “mean width” is not only a useful measure of integral curvature of objects such as grains, but that it also is unique in its property of additivity. Hadwiger (1957) showed that there is only one measure in each dimension that has the property of additivity, which means that the volume/area/mean width of the union of two overlapping objects is the sum of the separate quantities, minus the volume/area/mean width of the overlapping region. This suggests that the distributions of the three basic quantities (volume, area, and mean width) should be part of the validation of a digital microstructure. Moreover, ratios between pairs of these quantities, as shown in Fig. 11, also provide basic information on the shape of objects. Fitting to distributions of such ratios may also be part of the development of feature geometry in 3D models. The definition of mean width was presented in Sect. 4.1.1. 4.3.2 Shape(s) A typical approach to quantifying the shape of grains is to fit an ellipsoid and report the aspect ratios (Groeber et al. 2008a; Saylor et al. 2004a). This approach, which was described in detail in Sect. 4.1.2, is useful in describing the distribution
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of the amount of elongation of the grains. However, aspect ratios are ambiguous in reference to many aspects of shape. For example, it is possible for an ellipsoid and a rectangular prism to have the same set of aspect ratios. The local curvatures of grain boundaries are often disregarded when a “simple” geometric feature is fit to represent a grain. It is this issue that makes shape one of the more complicated parameters to describe. MacSleyne et al. (2008) have presented a method for distinguishing shapes by utilizing all three of the second-order moment invariants. The moment invariant technique creates a three-dimensional ( 1 ; 2 ; 3 ) moment invariant space to represent a grain’s shape rather than the limited two-dimensional space defined by a pair of aspect ratios ( 1 and 2 ). In the 3D moment invariant space, shapes with similar aspect ratios lie on the same arc, but are separated along the third dimension, 3 . This third dimension can potentially add extra information on the general class of shapes present in the microstructure. Additionally, combinations of the calculated moments can yield interesting insights into the types of shapes present in the structure. An example of a moment invariant analysis is shown in Fig. 12. The analysis provides the distribution of aspect ratios, which appears roughly equivalent for the two structures. However, when the value of 3 is compared, there is a noticeable shift in the distribution between the two structures. Finally, the largest difference between the two structures can be seen in the comparison of the distribution of
Fig. 12 Example of results from a moment invariant analysis. The upper set of plots is from an experimentally collected volume (Groeber et al. 2008a). The lower set of plots is from a synthetic microstructure generated with the goal of matching the experimental volume’s statistics, using methods presented by Groeber et al. (2008a). The analysis highlights the need (and ability) to look past lower order descriptors like aspect ratios
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V =Vconv:Vconv is the volume of the convex hull of the grain and V is the volume of the grain itself. The ratio of these two volumes is bounded by 0 and 1 and compares the relative concavity of the grain. It should be clear that the aspect ratio comparison alone does not accurately highlight many of the differences between the shapes in the two structures. 4.3.3 Neighborhood(s) The local neighborhood of a grain can be a complicated aggregate of features that can be described by a number of different parameters. For example, the morphological descriptors of the neighboring features could be reported or their crystallographic relationship to the reference grain could be of more interest. Additionally, the approach to describing the local neighborhood of grain is likely to vary with the type of microstructure and data being investigated. A grain structure, which has been segmented, is likely to have a known connectivity of grains and contiguous neighbors can be characterized. In the case of low volume fraction second-phase particles, the nearest neighbors may not be known and a two-point statistics approach (Tewari et al. 2006; Torquato 2001) may be better suited. For describing the morphology and connectivity of a grain’s neighborhood, there are multiple distributions that can be created. First, the distribution of number of neighbors can be generated for all grains, as well as correlated with grain size by grouping grains of similar size. In addition to number of neighbors, the size distribution of the neighboring grains can also be considered. The size distribution of neighbors, when correlated with the size of the reference grain, offers insights into the tendency of grains of certain sizes to cluster together (i.e., Aboav–Wieve). The shapes of neighboring grains can be correlated with the reference grain’s shape to quantify the clustering of similar shaped grains, which may evolve during recrystallization or deformation. The crystallographic description of individual boundaries will be discussed in the next section, but there are other parameters that describe the crystallography of local grain aggregates to varying degrees. The MoDF can be calculated for the entire structure, which gives some insight into the local textures present in the material. However, the MoDF does not provide any knowledge of the spatial distribution of the misorientations in the MoDF. The known connectivity of the grains allows for the spatial description of the misorientations. For example, one could calculate the fraction of a grain’s neighbors that have a critical misorientation value, be it high, low, or special. This approach could then be expanded to include secondary neighbors (i.e., neighbors of neighbors) and would ultimately offer a more local estimate of the clustering of grains with similar orientation. Two-point statistics can also be employed to describe distributions of orientations as well. 4.3.4 Boundary Character(s) To generate a complete 3D microstructure, one must add grain (crystal lattice) orientations to the description. The current state-of-the-art is that the grain geometry
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is created first and then a set of orientations is optimized with respect to texture and grain boundary misorientation (Saylor et al. 2004a; Groeber et al. 2008b). The procedure relies on simulated annealing and is computationally straightforward on modern personal computers. This procedure has at least two significant limitations, however. The first is that it assumes that size and shape are uncorrelated with orientation. However, this is not always the case; Bozzolo et al. (2005) have demonstrated that in titanium that has been deformed and then recrystallized there are texture components that are more dominant in the small grains and vice versa. The second limitation is that it ignores the fact that grain boundary properties depend on the interface normal as well as the lattice misorientation across them. The full description of grain boundary character requires, in fact, five macroscopic parameters. Fitting orientations to include both texture and misorientation and interface normal distributions needs to be developed. Implementing such an algorithm in voxel-based representations requires some method to compute the local interface normal. Alternatively, interface normals are straightforward to compute in a surface or volumetric mesh representation of a microstructure, which has been discussed previously.
5 Inference of 3D Structure It is often the case that the true 3D structure of a material is not available to include directly in a computational model. This can be attributed to the cost, availability, and complexity of experimental tools. As a result, it is still a reality for many to infer 3D structure from 2D observations. As previously mentioned, if 3D statistical descriptors can be inferred from 2D observations, the synthetic structure builders discussed in the previous section can be used to generate 3D structures for simulation. The following section will discuss some potential methods for inferring 3D statistics from 2D observations.
5.1 Link Between 2D and 3D Structure The statistical reconstruction method described here is based on limited crosssectional information from a given material; it is essential, however, that crosssections are made on more than one sectioning plane, and preferably on three orthogonal planes. Statistical methods for reconstructing microstructures have been developed in a number of fields, especially for modeling geological materials (Fernandes 1996; Oren and Bakke 2002, 2003; Sundararaghavan and Zabaras 2005; Talukdar and Torsaeter 2002; Talukdar et al. 2002a, b, c). Saylor et al. proposed a method of constructing 3D models of polycrystalline materials based on the microstructural features observed in three orthogonal sections (Saylor et al. 2004a).
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In this report, the microstructural features of interest include size and shape of grains, misorientation distribution, orientation distribution, and the relative placement of grains with respect to size. The procedure outlined by Saylor, along with adaptations for elongated grain shape noted by Brahme et al. (2006), is the basis for one of the procedures described here. Groeber (2007) also offered a methodology for inferring 3D structure from 2D measurements. In Groeber’s study, the 2D sections were obtained by sectioning synthetically generated 3D structures, which provide a known set of 3D statistics to compare with the inferred statistics. There is a substantial literature on the general stereological problem of reconstructing 3D microstructures based on limited section information. When treating microstructures as collections of general particles whose size, shape, and orientation are to be reconstructed (without regard to their packing), the problem is known to lack a solution (Cruz-Orive 1976a, b). However, for particles that are monodisperse (in size and shape), this problem is well known and has semianalytical solutions for which the names Cahn and Saltykov are well known in the materials literature (Cahn and Fullman 1956; Saltykov 1958). For a historical overview, see Underwood (1970). In contrast to these more general cases, polycrystalline grain structures have an added constraint since grains are not independent particles (i.e., low volume fraction) because they fill space. This constraint enhances the ability to accurately reconstruct a 3D distribution from 2D observations (Przystupa 1997). This section will attempt to address the previously less investigated problem of space-filling particles (i.e., grains).
5.2 Probable Set Generation 5.2.1 Monte Carlo Histogram Fitting In this section, an example of generating a set of 3D (ellipsoidal) grains using statistical distributions calculated on three 2D, orthogonal, EBSD-based micrographs is given. The grain size distributions for the three orthogonal planes of a rolled aerospace aluminum alloy are shown in Fig. 13. Matching the statistics in the generated 3D structure to the measured 2D statistics is accomplished through a multistep process that includes: (1) generating representative ellipsoids (in terms of size and shape distributions), (2) placing those ellipsoids into a volume, (3) allowing that volume to be filled with voxels that are grouped as grains, and (4) modifying the grain structure by use of a (isotropic) Monte Carlo grain growth (Rollett and Manohar 2004). The first step in this process is the focus of this section, while the latter steps were previously discussed in the section on feature placement during synthetic structure generation. The transition from 2D to 3D is accomplished by assuming the grain shape is that of an ellipsoid and considering that the observations on 2D sections are only a portion of the true size and shape of the actual grain. The probability distribution
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Fig. 13 The resulting (top) PDFs and (bottom) CDFs for the OIM input data and ellipsoids generated to represent the data for the ND, RD, and TD directions
functions (PDFs) of the ellipse dimensions obtained from the sliced ellipsoids are given as f 0 .a/; f 0 .b/, where a and b are the semiaxes of an ellipse, and the prime accent indicates that it is from a section. In this case, the ellipse dimensions are assumed to be independent, and that a > b. References to the cumulative distribution function, CDF, utilize the same notation but with a capital “F .” The input data is used directly to create a CDF [i.e., F (TD)] of grain size where the range of the CDF is 0 to 1 and the domain is scaled in micrometers. The distribution for each direction is sampled and multiplied by a stereological constant to account for the fact that the CDF is generated from a 2D section. In this approach, the total number of ellipsoids generated is specified as input to the program and it proceeds to optimize the list such that they are a good match to the input data. The optimization is accomplished iteratively and the first step is to create an initial list of ellipsoids and then slice each of them many times and extract the 2D CDFs of grain size on each section. The RMS error between list and the input data is computed. The initial list is then modified by generating a new ellipsoid in the same manner as before and then randomly choosing an ellipsoid to replace from the list. The ellipsoid is replaced only if the new ellipsoid lowers the error of the system. The program completes when it has performed a number of user-specified iterations. The result of this method can be observed directly by comparing the PDFs and CDFs of the data and the simulated ellipsoids directly as can also be seen in Fig. 13. The distributions in both the ND and the TD directions are well matched to the input data. The discrepancy on the RD direction in this case is due to improper sampling of the grains in the RD direction (i.e., the majority of grains intersected the scan boundary).
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5.2.2 Domain Constraint The elemental assumption of this method is that the entire (and infinite) set of all possible ellipsoids can be bounded by observations of ellipses on experimentally collected, orthogonal 2D sections to leave a “most probable” set of ellipsoids. Factors such as the distributions of size, shape and orientation of the ellipses on the 2D sections are used to assign probabilities to groups of ellipsoids. This type of an approach is fundamentally different than the analytical developments made by Cruz-Orive (1976a, b) and DeHoff (1962) in that an exact solution is not the goal, rather a probable set is desired. An initial description of this technique is introduced by Groeber (2007). To assign probabilities to groups of ellipsoids, the infinite domain of ellipsoids must be initially truncated and discretized. A five dimensional space is created to define the ellipsoids. Three dimensions correspond to the orientation of the principal axes of the ellipsoid and are inherently bounded by the finite dimensions of Euler space, which describe the orientation of the ellipsoid (i.e., its principal axes). The other two dimensions correspond to the two aspect ratios of the ellipsoid. The aspect ratio dimensions are not inherently bounded, but can be truncated by using the 2D observations to make assumptions about reasonable upper and lower bounds. By definition, the upper limit of the two aspect ratio dimensions is 1 (i.e., a sphere) and the lower limit can be set to be the smallest aspect ratio observed in the 2D sections. In practicality, the accuracy of the estimated lower limit will be directly related to the number of observations, and thus, it may be prudent to reduce the minimum observed value by an additional 25–50%. The volume of the ellipsoids is treated only as a distribution within each discrete bin in the 5D space, not as its own dimension. This is because volume only scales the dimensions of an ellipsoid and any resultant elliptic section through it, which has no effect on the following process. Development of a probable set of ellipsoids is undertaken as an iterative process because of the inability to decouple the influences of ellipsoid shape (aspect ratios) and orientation on the resultant distribution of ellipses. The iterative process initiates by calculating a probable orientation distribution for the ellipsoids with an assumed uniform shape distribution. Then the shape distribution is updated using the calculated orientation distribution. Iteratively, each distribution is updated using the most recently calculated instance of the other distribution until a level of convergence is reached. Some details of the calculations are offered here as well as the process of extrapolation of the individual ellipses. Calculation of a probable orientation distribution requires sectioning a large number of ellipsoids within each discrete orientation bin and observing their resultant elliptic sections. Each ellipsoid is assigned a set of aspect ratios in accordance with the shape distribution, which is initially uniform. A two-dimensional histogram of resultant ellipse orientation and resultant ellipse aspect ratio is created for all ellipsoids in each of the orientation bins. An example histogram, shown as contour plots, is shown in Fig. 14. The histograms for each orientation bin are then compared to the same histogram made from the actual observations on the experimental 2D sections. The simulated histograms are fit to the experimental histogram by a least squares method to determine the probability of each orientation bin.
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Fig. 14 Plot of (left) density of ellipse principal axis orientation vs aspect ratio and (right) density of ellipse normalized size vs aspect ratio. The top third of each plot shows ellipses on the plane normal to the z-direction, the middle third shows ellipses on the plane normal to the y-direction, and the bottom third shows ellipses on the plane normal to the x-direction. In the left plot, the x-axis refers to the x-component of a unit vector oriented along the major axis of the ellipse and the y-axis is the aspect ratio (b=a) of the ellipse. In the right plot, the x-axis is the ellipse area divided by the average ellipse area and the y-axis is the aspect ratio (b=a) of the ellipse
Upon calculating the orientation probability distribution, the shape distribution can be updated by sectioning a large number of ellipsoids within each discrete shape bin, represented by a set of aspect ratios. Each ellipsoid is assigned an orientation in accordance with the previously calculated orientation distribution. A two-dimensional histogram of resultant ellipse aspect ratio and resultant ellipse normalized size is constructed for all ellipsoids in each of the shape bins. The normalized size is the area of the resultant ellipse divided by the average resultant ellipse area. An example histogram is shown in Fig. 14. The histograms for each shape bin are then compared to the same histogram made from the actual observations on the experimental 2D sections. The simulated histograms are fit to the experimental histogram by a least squares method to determine the probability of each shape bin. While sectioning the ellipsoids to determine a probable orientation and shape distribution, the distribution of fractional section size is constructed for each shape bin. The fractional section size is defined as the resultant ellipse area divided by the maximum possible resultant ellipse area for a given ellipsoid (i.e., when the ellipsoid is sectioned through the equatorial plane perpendicular to the minor axis). The fractional section size distribution is used to extrapolate the individual experimental ellipses. The shape of each experimental ellipse’s parent ellipsoid is predicted by the probability of an ellipsoid of a given shape producing the experimental ellipse (i.e., its normalized size and aspect ratio). Once the parent ellipsoid’s shape is assumed, the distribution of fractional section size for that shape can be used to convert the experimental section’s area to the maximum possible section area for the parent ellipsoid. With an assumed shape and maximum possible section size, everything necessary to fully define the parent ellipsoid is available. This process is carried
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Fig. 15 Results of the Observation-Based Domain Constraint method. The upper right image is the true 3D shape distribution (ellipsoid aspect ratios). The upper left image is the “probable” shape distribution calculated by the method. The lower image is a comparison of the true 3D grain size distribution (equivalent sphere radius) and the calculated “probable” size distribution
out for each experimental ellipse, resulting in a set of “probable” ellipsoids, whose statistics can be used to generate synthetic 3D volumes. Figure 15 shows the results of the observation-based domain constraint process for a sample microstructure.
5.3 Limitations and Possibilities Currently, there are a number of limitations that remain when inferring the true 3D microstructure of materials. First, the shape of the features being inferred has been limited to simple geometric shapes that can be easily described numerically, generally ellipsoids. This limitation need not be persistent, provided numerical descriptions of more complex shapes are developed. The moment invariant analysis mentioned previously may provide a key tool in developing this area. Additionally, the linkage between neighborhoods of grains in 3D and grains in 2D is not immediately clear to this author. The number of neighbors, as well as their size, shape, etc, is likely to be a function of the packing of the grains, which may not be known from even several 2D sections. In principle, it would be possible to section many different, yet known, microstructures and attempt to develop a “lookup” table that correlates parameters such as: number of 2D neighbors to number of 3D neighbors.
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6 Comments on Complex Microstructures The microstructures investigated in the works presented in this chapter are relatively idealized. The nickel-base superalloy used in the work of Groeber et al. was chosen for its small feature size, propensity to yield high-quality data, and microstructural homogeneity. The aluminum alloy used in the work by Saylor et al. and the steel used in the work by Rowenhorst et al. both are single-phase materials with standard boundary structures and limited heterogeneities. These properties enabled the investigation of microstructure with comparatively little difficulty. All three microstructures certainly have inherent difficulties as well. The nickel-base superalloy contains a second phase that is difficult to distinguish from the matrix as well as twin grains that are often too small to sample properly with the tools used. The rolled aluminum alloy has a grain size too large for the desired experimental techniques (i.e., EBSD). The steel required the precipitation of a second phase to identify boundaries easily, which was met with some difficulty. However, the complications encountered do not approach the level of some heavily engineered, more topologically complex microstructures, such as beta-processed titanium with a basketweave structure. Techniques are being expanded to treat such microstructures, but are currently in the developmental stages. It should not be a surprise that for the advancement of digital representation of grain-level microstructure to include more complex microstructures, there must be a coupled advancement in the ability to collect and quantify these microstructures. Additionally, clever techniques to homogenize or adaptively incorporate multiple microstructural scales will be an imperative to simulate microstructures that have critical features that exist of varying length scales. The utility of a microstructure representation can be limited greatly if the computational modeling community has no feasible method to simulate the structure. Again, this calls for a representation-modeling effort that defines the degree to which microstructure needs to be incorporated, both in the context of its effects on the property and on its ability to be simulated. For the example of beta-processed titanium, Venkatramani et al. (2008) have developed a homogenized grain representation that homogenizes the lamellar/lath structure of the titanium, which would be impossible to incorporate at the scale needed to include hundreds of prior beta grains. Thus, for properties that this homogenization is proper, the experimental techniques and representation tools can be tailored to identify and represent only the prior beta grains (and alpha colonies/variants) and omit the underlying lath-rib structure in an effort to increase simulation efficiency. Finally, the synthetic structure generation process should likely attempt to parallel the natural process that produced the microstructure of interest. That is, the application of an increasing number of generic constraints on the placement of features in the synthetic volume is likely to inhibit the generation process. However, if the generation process follows the natural process, then many constraints may become obsolete and unnecessary. Two examples of this idea are the inclusion of twin grains in a polycrystal and the generation of a colony (or basketweave) structure in a titanium alloy. In the case of twin grains, it is difficult (if not impossible) to treat
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the twin and its parent grain as independent features and ensure that they will be placed in proper relation in the synthetic structure. Rather, it is potentially a better strategy to remove the twin grains, by merging them with their parent grains, and characterize the simplified structure. Then, a synthetic structure without twins can be generated to match the simplified structure. If the statistical description of the twin structure (i.e., fraction of grains with twins, twin plate thickness, number of twins per grain, etc.) is measured during the merging, then twins could be inserted into the simplified synthetic structure, which would simulate the natural process of twin nucleation in a parent grain. In the case of the colony structure of a titanium alloy, if the colonies themselves are treated as the features to be placed, it is a complicated process to place colonies into neighborhoods that have the proper orientation relationships (in accordance with the Burgers’ relationship) and the correct topological structure (imposed by the prior beta grain boundaries). One option to circumvent this complication is to generate a synthetic structure that consists only of prior beta grains, with the statistics to match the experimental beta grain structure. The prior beta grains can be identified by grouping the colonies that came from the same beta grain, which is known through the Burgers’ relationship. Then, the synthetic beta grains can be divided into colonies, where the orientations are already constrained by the beta orientation and the topology is already constrained by the beta grain boundaries. Similar to the twin example, the statistics of the “secondary” features (i.e., the twins or colonies) can be measured during grouping to locate the primary features (i.e., the parent grains or prior beta grains). Dividing the beta grains into colonies simulates the natural precipitation of the alpha phase in the prior beta grains. Linking the synthetic generation process to the natural process may simplify the representation of complex structures and it is also likely to enhance the physical significance of the generated structure.
7 Conclusions This chapter has attempted to discuss the key areas necessary for the development of digital representations of materials microstructure at the grain level. Much of the detail in the various areas is presented elsewhere and has been noted wherever possible. It was the goal of the author to highlight the current “state-of-the-art” techniques and discuss their strengths and weaknesses. This field is still in a state of relative infancy and requires the cooperation of a number of other fields to properly evolve. The experimental community has elevated the ability and precision with which it can investigate microstructure in the last decade. It is important for communication with this community to tailor experiments to the needs of given representation requirements. After the generation of a microstructure representation, mesh generation remains a barrier to the simulation of models (without artifacts). The general meshing community has not been introduced to the needs of the materials modeling problem. Quantification of mesh error relative to the digital representation is one key metric, as well as the quality of elements required for a given simulation. Finally
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and arguably the most critical, the development of a connection between simulation results and statistical descriptors is imperative. Such an association is a necessary part of the pathway to the determination of a RVE for all materials properties. The improvement of constitutive relations requires knowledge of what descriptors influence properties and proper quantification of descriptors is key in defining their influence on properties. Digital representation of materials microstructure is an integral part in the determination of microstructure–property relationships, but cannot be treated as an independent step in the process. The full effect of developments presented here and those to come will only be realized when these collaborations have been cultivated. Acknowledgments The author would like to acknowledge his collaborators, all of whom contributed through detailed discussions and in many cases developed some of the tools and techniques presented in this chapter. All of the sections in this chapter were heavily influenced by them and in some instances their own words and terminology were used. The works of Profs. Somnath Ghosh, Tony Rollett, and Marc DeGraef; as well as Drs. David Rowenhorst, Dennis Dimiduk, Mike Uchic, Sukbin Lee, Jeremiah MacSleyne, and Mrs. Yash Bhandari and Steve Sintay and many others have greatly advanced this field and inspired this author.
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Multiscale Characterization and Domain Partitioning for Multiscale Analysis of Heterogeneous Materials Somnath Ghosh
Abstract This chapter discusses the development of a multiscale characterization methodology leading to microstructural morphology-based domain partitioning MDP methodology for materials with nonuniform heterogeneous microstructure. The comprehensive set of methods is intended to provide a concurrent multiscale analysis model with the initial computational domain that delineates regions of statistical homogeneity and inhomogeneity. The MDP methodology is intended as a preprocessor to multiscale analysis of mechanical behavior and damage of heterogeneous materials, e.g., cast aluminum alloys. It introduces a systematic three-step process that is based on geometric features of morphology. The first step simulates high-resolution microstructural information from low-resolution micrographs of the material and a limited number of high-resolution optical or scanning electron microscopy micrographs. The second step is quantitative characterization of the high-resolution images to create effective metrics that can relate microstructural descriptors to material behavior. The third step invokes a partitioning method to demarcate regions belonging to different length scales in a concurrent multiscale model. Partitioning criteria for domain partitioning are defined in terms of microstructural descriptors and their functions. The effectiveness of these metrics in differentiating microstructures of a 319-type cast aluminum alloy with different secondary dendrite arm spacings SDAS is demonstrated
