Nano and Microstructural Design of Advanced Materials

NAN0 AND MICROSTRUCTURAL DESIGN OF ADVANCED MATERIALS A Commemorative Volume on Professor G. Thomas’ Seventieth Birthda...

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NAN0 AND MICROSTRUCTURAL DESIGN OF ADVANCED MATERIALS A Commemorative Volume on Professor G. Thomas’ Seventieth Birthday

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NAN0 AND MICROSTRUCTURAL DESIGN OF ADVANCED MATERIALS A Commemorative Volume on Professor G. Thomas’ Seventieth Birthday Edited by M.A. MEYERS University of California, San Diego, USA R.O. RITCHIE University of California, Berkeley, USA and M. SARIKAYA University of Washington, USA

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Preface The importance of the nanoscale effects has been recognized in materials research for over fifty years. The understanding and control of the nanostructure has been, to a large extent, made possible by new atomistic analysis and characterization methods. Transmission electron microscopy revolutionized the investigation of materials. This volume focuses on the effective use of advanced analysis and characterization methods for the design of materials. The nanostructural and microstructural design for a set of targeted mechanicaVfunctiona1properties has become a recognized field in Materials Science and Engineering. This book contains a series of authoritative and up-to-date articles by a group of experts and leaders in this field. It is based on a three-day symposium held at the joint TMS-ASM meeting in Columbus, Ohio. The book is comprised of three parts: Characterization, Functional Materials, and Structural Materials. The book is dedicated to Gareth Thomas who has pioneered this approach to materials science and engineering area over a wide range of materials problems and applications. Professor Thomas’ lifetime in research has been devoted to understanding the fundamentals of structure-property relations in materials for which he has also pioneered the development and applications of electron microscopy and microanalysis. He established the first laboratory for high voltage electron microscopy, at the Lawrence Berkeley National Laboratory. His research has contributed to the development and nano/microstructural tailoring of materials from steels and aluminum alloys, to high temperature and functional ceramics and magnetic materials, for specific property performances, and has resulted in a dozen patents. Professor Thomas is a pioneer and world leader in the applications of electron microscopy to materials in general. Following his Ph.D. at Cambridge in 1955, as an ICI Fellow, he resolved the problem of intergranular embrittlement in the AVZn/Mg high strength alloys which failed in the three Comet aircraft crashes and became identified with Prof. Jack Nutting as the “PFZ’ -precipitate-free-zones, condition, now in wide general use to describe grainboundary morphologies leading to intergranular corrosion and mechanical failure. This work prompted Dr. Kent van Horne of Alcoa to invite him to spend the summer of 1959 in their research labs at New Kensington, Pa. From there and after a trans-USA lecture tour he was invited in 1960 to join the Berkeley faculty, (becoming a full professor in 1966), where he started a major research program within the newly formed “Inorganic Materials Research Division” of the (now) Lawrence Berkeley National Laboratory. It was there, after nine years’ effort, that he founded the National Center for Electron Microscopy, which opened in 1982 and which he directed until he resigned in 1993, to spend 1.5 years helping establish the University of Science & Technology in Hong Kong. There he also set up and directed the Technology Transfer Centre. He returned to Berkeley in 1994 to continue teaching and research, and in his career has over 100 graduates. With his students and colleagues he has over 500 publications, several books, including the first text on Electron Microscopy ofMetals (1962), and in 1979 -with M.J. Goringe, a widely used referenced text- Transmission Electron Microscopy of Materials which was also translated into Russian and Chinese. His academic career in Berkeley has included administrative services as Associate Dean, Graduate Division, Assistant and Acting Vice-Chancellor-Academic Affairs, in the turbulent years of student unrest (1966-72). He was the Chair faculty of the College of Engineering (1972/73), and Senior Faculty Scientist, LBNL-DOE, which sponsored most of his research V

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Preface

funding. In 1995 he received the Berkeley Citation for “Distinguished Achievement” at UC Berkeley. Professor Thomas was Associate Director, Institute for Mechanics and Materials, UC San Diego, from 1993 to 1996. In this capacity, he formulated new research directions and stimulated research at the interface of Mechanics and Materials. He is currently Professor in the Graduate School, UC Berkeley, Professor-on-Recall, UC San Diego, and VP R&D of a new company, MMFX Technologies, founded in 1999, to utilize steels for improved corrosion resistant concrete reinforcement. In the USA the infrastructure repair costs are in the trillion dollar range. In 2002 the company received the Pankow award (American Inst. of Civil Engineers) for innovation in Engineering, based on Prof. Thomas’ patents on nano microcomposite steels. Professor Thomas has also played an important role in promoting the profession. He was president of the Electron Microscopy Society of the US in 1974, and in 1974 he became Secretary General of the International Societies for Electron Microscopy for an unprecedented 12 years, and was president in 1986-90. He lectured extensively in foreign countries and helped promote microscopy and materials in developing countries, also serving as advisor in China, Taiwan, Korea, Singapore, Poland, Mexico, et al. He also served on many committees of the ASM and TMS, and the National Research Council. After reorganizing the editorial structure of Acta and Scripta Metallurgica (now Materialia), when in 1995 he took over as Editor-in-chief, he became Technical Director, Acta Mat. Inc. 1998 until April 2002. He was Chairman of the Board in 1982/84. In recognition of his many achievements, Professor Thomas has received numerous honors and awards, including, besides his Sc.D.-Cambridge University in 1969: Honorary Doctorates from Lehigh (1996) and Krakow (1999); The Acta Materialia Gold Medal (2003), The ASM Gold Medal (200 l), Sauveur Achievement Award (ASM- 199l), Honorary Professor, Beijing University of Sci. & Technology (1958), Honorary Memberships in Foreign Materials societies (Japan, Korea, India, etc.), E.O. Lawrence Award (US Dept. of Energy-l978), Rosenhain Medal (The Metals Soc-UK-1977), Guggenheim Fellow (1972), von Humboldt Senior Scientist awards (1996 & 1981), the I-R Award (R&D Magazine-1987), Sorby Award, (IMS- 1987) and the Distinguished Scientist Award (EMSA-1980). He received the Bradley Stoughton Teaching Award (ASM) in 1956, and the Grossman (ASM), and Curtis-Mcgraw (ASEE) research awards in 1966. He is a Fellow of numerous scientific societies. In recognition of these achievements, Professor Thomas was elected to both the National Academy of Sciences (1983) and the National Academy of Engineering (1982). Professor Thomas, born in South Wales, UK, is also a former rugby and cricket player (member, MCC), enjoys skiing and grand opera. The editors thank the speakers at the symposium and the authors of the scholarly contributions presented in this volume. A special gratitude is expressed to Prof. S. Suresh for having enabled the publication of this volume by Elsevier. All royalties from the sale of this book are being donated to the TMS/AIME and ASM societies for the establishment of an award recognizing excellence in Mechanical Behavior of Materials. November, 2003

Curriculum Vitae of Professor Thomas

Date and Place of Birth: 9 August 1932, Maesteg, Glamorgan, U.K. Academic Qualifications B.Sc. with First Class Honors in Metallurgy, University of Wales (Cardiff), 1952. Ph.D. University of Cambridge, 1955; Sc.D. University of Cambridge, 1969.

Career Details 1956-59 ICI and St. Catharine’s College Fellow, University of Cambridge 1960 Visiting Assistant Professor, University of California, Berkeley 1961-Present University of California, Berkeley: Full Professor (1966); Associate Dean, Graduate Division (1968-69); Assistant to the Chancellor (1969-72); Acting Vice Chancellor, Academic Affairs (1971-72); Chairman, Faculty of the College of Engineering (1972-73); Senior Faculty Scientist, Materials Sciences Division, Lawrence Berkeley Laboratory; Founder and Scientific Director, National Center for Electron Microscopy, Lawrence Berkeley Laboratory (198 1-93); on special leave as Director, Technology Transfer Centre, Hong Kong University of Science and Technology, Kowloon, Hong Kong (1993-94); Professor in the Graduate School, University of California, Berkeley (1995-present).

Awards and Honors 2003 2003 200 1

Silver Medal in honor of Prof. C. S. Barrett, ASM Intl. Rocky Mountain Chapter Acta Materialia Gold Medal First Albany Int. Distinguished Lecture in Mat. Sci. & Eng. (RPI). Vii

Curriculum vitae of Professor Thomas

Viii

200 1 1999 1998 1996 1996 1996 1996 1995 1994 1994 1991 1987 1987 1987

1985 1983 1983 1982 1981 1980 1979 1978 1977 1976 1976 1973 1971-72 1966 1966 1965 1964 1953

American Society for Materials International, Gold Medallist Doctorate honoris causa, University of Krakbw, Poland Honorary Member, Japan Institute of Materials Honorary D.Sc., Lehigh University, Bethlehem, PA, USA, 1996 Honorary Member, Indian Institute of Metals Honorary Member, Korean Institute of Metals and Materials Alexander von Humboldt Senior Scientist Award, IFW, Dresden, Germany The Berkeley Citation for Distinguished Achievement, U. C. Berkeley Honorary Member, Mat. Res. SOC.of India Medal of Academy of Mining and Metallurgy, Polish Acad. of Sciences, Krakow Albert Sauveur Achievement Award (ASM International) I-R 100 Award, Research and Development Magazine Elected, Fellow, Univ. Wales, Cardiff, UK Henry Clifton Sorby Award, International Metallographic Society Honorary Professorship-Beijing University of Science & Technology Confucius Memorial Teaching Award, Republic of China (Taiwan) Elected to the National Academy of Sciences, U.S.A. Elected to the National Academy of Engineering, U.S.A. Alexander von Humboldt Senior Scientist Award, Max Planck Institute, Stuttgart EMSA Distinguished Scientist Award for Physical Sciences Fellow, Metallurgical Society of AIME Ernest 0. Lawrence Award (US. Department of Energy) The Rosenhain Medal (The Metals Society, U.K.) Fellow, Royal Microscopical Society, U.K. Fellow, American Society for Metals Visiting Professor at Nagoya University, Japan Society for Promotion of Science Guggenheim Fellow; Visiting Fellow, Clare Hall, Cambridge University Curtis-McGraw Research Award (American Society for Engineering Education) Grossman Publication Award (American Society for Metals) for paper “Structure and Strength of Ausformed Steels”, Trans. ASM, 58,563 (1965) Bradley Stoughton Teaching Award, American Society for Metals Miller Research Professor, UC Berkeley National Undergraduate Student Prize, Institute of Metals (London)

Professional Activities 19981995-98 1992 1991-95 1986-90 1974-86 1991-94 1987-88 1982-85

1985-90

Managing Director, Acta Metallurgica, Inc. Board of Governors Editor in Chief, Acta Materialia and Scripta Materialia Founder Member, Editorial Board, NanoStructured Materials (Elsevier) Vice President, International Federation of Societies for Electron Microscopy President, International Federation of Societies for Electron Microscopy Secretary General, International Federation of Societies for Electron Microscopy Reappointed, Member, Board of Governors Acta Metallurgica, Inc. Member, US Department of Energy E. 0.Lawrence Award Selection Committee Chairman, Acta Metallurgica, Inc. Board of Governors Member, Acta Metallnrgica, Inc. Board of Governors


ix

1978-8 1 1975 1972-73 1961-present

TMS-AIME Board of Directors President, Electron Microscopy Society of America UC Convenio Program Visiting Professor, University of Chile, Santiago, Chile Served on many national and international committees including National Research Council (USA), International Federation of Electron Microscopy Societies, EMSA, ASM, TMS, University of California, editorial boards, etc. Served on science and technology boards (Taiwan, Singapore, Korea, South Africa and Mexico) as materials advisor. Publications Over 550 papers, 2 books, numerous book chapters. Selected Publications 1. “Structure-Property Relations: Impact on Electron Microscopy,” in Mechanics and Materials: Fundamentals and Linkages, Marc A. Meyers, Ronald W. Armstrong and Helmut Kirchner, eds. New York: J. Wiley & Sons, 1999, pp. 99-121; LBNL 40317. 2. “Nd Rich Nd-Fe-B Tailored for Maximum Coercivity,” Er. Girt, Kannan M. Krishnan, G. Thomas, C. J. Echer and Z. Altounian, Mat. Res. SOC.Symp. Proc. 577, Michael Coey etal., eds. Warrendale, PA: The Materials Research Society, 1999, pp. 247-252. 3. “Some Relaxation Processes in Nanostructures and Diffusion Gradients in Functional Materials,” G. Thomas, in Deformation-Induced Microstructures: Analysis and Relation to Properties (Proc. 20th Ris# International Symposium on Mat. Sci.,), J. B. Bilde-S#rensen, J. V. Carstensen, N. Hansen, D. Juul Jensen, T. Leffers, W. Pantleon, 0. B. Pedersen and G. Winther, eds., Ris# National Laboratory, Roskilde, Denmark, 1999, pp. 505-521. 4. “Origin of Giant Magnetoresistance in Conventional AlNiCo, Magnets,” A. Hiitten, G. Reiss, W. Saikaly and G. Thomas,Actu Muteriuliu 49, 827-835 (2001). 5. “Novel Joining of Dissimilar Ceramics in the Si3N4-Al2O3 System Using Polytypoid Functional Gradients,” Caroline S. Lee, Xiao Feng Zhang and Gareth Thomas, Acta Materialia vo1.49,3767-3773, & 3775-3780 (2001). See web-site (below) for more details: Internet: http://www.mse.berkeley.edu/faculty/thomas/thomas.html Patents Process for Improving Stress-Corrosion Resistance of Age-Hardenable Alloys, U.S. Patent 3,133,839 (1964). High Strength, High Ductility Low Carbon Steel (J. Koo and G. Thomas), U.S. Patent 4,067,756 (1978). High Strength, Tough Alloy Steels (G. Thomas andB. V. N. Rao), U.S. Patent4,170,497 (1979). Method of Making High Strength, Tough Alloy Steels (G. Thomas and B. V. N. Rao), U.S. Patent 4,170,499 (1979). High Strength, Low Carbon, Dual Phase Steel Rods and Wires and Process for Making Same (G. Thomas and A. Nakagawa). U.S. Patent 4,613,385 (1986).

X


Controlled Rolling Process for Dual Phase Steels and Applications to Rod, Wire, Sheet and Other Shapes (G. Thomas, J. H. Ahn, and N. J. Kim), U.S. Patent 4,619,714 (1986). Method of Forming High-Strength, Corrosion-Resistant Steel (G. Thomas, N. J. Kim, and R. Ramesh), U.S. Patent 4,671,827 (1987). Method of Producing a Dense Refractory Silicon Nitride (Si3N4) Compact with One or More Crystalline Intergranular Phases (G. Thomas, S. M. Johnson, and T. R. Dinger), U.S. Patent 4,830,800 (1989). High Energy Product Permanent Magnet Having Improved Intrinsic Coercivity and Method of Making Same (R. Ramesh and G. Thomas), U.S. Patent 4,968,347 (1990). Giant Magnetoresistive Heterogeneous Alloys and Method of Making Same (J. J. Bernardi, G. Thomas, and A. R. Huetten), U S . Patents 5,824,165 (1998) and 5,882,436 (1999).

Table of Contents

Preface

V

vii

Curriculum Vitae of Professor Thomas

Part 1: Characterization 3

Characterization: The Key to Materials

R. Gronsky

Nanochemical and Nanostructural Studies of the Brittle Failure of Alloys D.B. Williams,M. Watanabe, C. Li and V.J. Keast

11

Transmission Electron Microscopy Study of the Early-Stage Precipitates in Al-Mg-Si Alloys H.W. Zandbergen, J.H. Chen, C.D. Marioara and E. Olariu

23

Laser Surface Alloying of Carbon Steels with Tantalum, Silicon and Chromium J. Kusinski and A. Woldan

35

In-Situ TEM Observation of Alloying Process in Isolated Nanometer-Sized Particles H. Mori, J.-G. Lee and H. Yasuda

49

Characterization of MetaVGlass Interfaces in Bioactive Glass Coatings on Ti-6A1-4V and Co-Cr Alloys E. Saiz, S. Lopez-Esteban, S. Fujino, T. Oku, K. Suganuma and A.P. Tomsia

61

Development of Advanced Materials by Aqueous Metal Injection Molding S.K. Das, J.C. LaSulle, J.M. Goldenberg and J. Lu

69

Part 2: Functional Materials Microstructural Design of Nanomultilayers (From Steel to Magnetics) G.J. Kusinski and G. Thomas Effects of Topography on the Magnetic Properties of Nano-Structured Films Investigated with Lorentz Transmission Electron Microscopy J.Th.M. De Hosson and N.G. Chechenin

81

93

Slip Induced Stress Amplification in Thin Ligaments

109

Materials, Structures and Applications of Some Advanced MEMS Devices Sungho Jin

117

X. Markenscogand V.A.Lubarda

xi

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Table of contents

Microstructure-Property Evolution in Cold-Worked Equiatomic Fe-Pd During Isothermal Annealing a t 500 O A. Deshpande, A. Al-Ghaferi, H. Xu, H. Heinrich and J.M.K. Wiezorek

Part 3: Structural Materials Microstructure and Properties of In Situ Toughened Silicon Carbide L.C. De Jonghe, R.O. Ritchie and X.F. Zhang Microstructure Design of Advanced Materials Through Microelement Models: WC-Co Cermets and Their Novel Architectures K.S. Ravi Chandran and Z. Zak Fang

145

157

The Ideal Strength of Iron D.M. Clatterbuck, D.C. Chrzan and J. W.Morris Jr.

173

Microstructure-Property Relationships of Nanostructured Al-Fe-Cr-Ti Alloys L. Shaw, H. Luo, J. Villegas and D. Miracle

191

Microstructural Dependence of Mechanical Properties in Bulk Metallic Glasses and Their Composites U. Ramamurty, R. Raghavan, J. Basu and S. Ranganathan

199

The Bottom-Up Approach to Materials by Design W.W. Gerberich, J.M. Jungk and W.M.Mook

21 1

The Onset of Twinning in Plastic Deformation and Martensitic Transformations M.A. Meyers, M.S. Schneider and 0. Voehringer

221

Crystal Imperfections Seen by X-Ray Diffraction Topography R.W.Armstrong

233

Synthetic Multi-Functional Materials by Design Using Metallic-Intermetallic Laminate (MIL) Composites K.S. Vecchio

243

Taylor Hardening in Five Power Law Creep of Metals and Class M Alloys M.E. Kassner and K. Kyle

255

Microstructural Design of 7x50 Aluminum Alloys for Fracture and Fatigue F.D.S. Marquis

273

Elastic Constants of Disordered Ternary Cubic Alloys C.S. Hartley

287

Index

299

PART 1: CHARACTERIZATION

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Nan0 and Microstructural Design of Advanced Materials M.A. Meyers, R.O. Ritchie and M. Sarikaya (Editors) 02003 Elsevier Ltd. All rights reserved.

CHARACTERIZATION: THE KEY TO MATERIALS R. Gronsky Department of Materials Science & Engineering, University of California Berkeley, California 94720-1760 USA

ABSTRACT His seventieth birthday offers this special occasion to recall the many seminal contributions made by Professor Gareth Thomas to the field of materials science and engineering. A brief reckoning of his career, his dedication to the development of electron microscopy techniques, his applications of high precision characterization methods to numerous engineering materials systems, and his successes as both researcher and educator are recounted here.

INTRODUCTION The development of advanced materials is guided by assessment at appropriate levels of resolution. This has always been the preferred protocol, and hallmark, of materials science and engineering. Our discipline seeks to understand all of the links connecting the synthesis and processing of materials with the evaluation of their properties, with their performance in engineering applications, and with their internal structure and composition. However, as modem engineering progresses towards increased complexity and reduced dimensionality, our discipline places ever higher demands on the diffraction, spectroscopy, and microscopy techniques used for microstructural analysis. There was a time when “pearlite” was an acceptable designation for a microstructural constituent associated with certain mechanical properties of steels. Thirty years ago, it became essential to know the composition of both the ferrite and the cementite in “pearlite,” including whether or not there were any gradients in carbon concentration at their contiguous interfaces. And as this manuscript is being written, hundreds of scientists around the world are struggling to sort out carbon nanotubes as single-walled or multi-walled, spiral or concentric, vacant or filled, with what species, at which specific locations. Consequently, the levels of resolution appropriate for contemporary materials science and engineering are those that reveal individual atomicpositions in the spatial domain, and individual atomic identities in the temporal or energy domain. It is now generally accepted that atomic level characterization is the essential key to materials, old and new. Today’s symposium highlights many of the triumphs of advanced materials development based upon this singular tenet of microstructural design, which has been championed by Professor Gareth Thomas throughout his long and illustrious career. It was just over thirty (30) years ago that I came to Berkeley to begin my graduate studies in Professor Thomas’s group, and I’m honored to offer this contribution in celebration of his seventieth (70th)birthday.

3

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R. Gronslcy

BACKGROUND Gareth Thomas was born on August 9, 1932. He completed his Bachelor of Science degree with First Class Honors in Metallurgy from the University of Wales, Cardiff, in 1952. Three years later, in 1955, he obtained his Ph.D. from the University of Cambridge, where he stayed through 1959 as an ICUSt. Catherine’s College Fellow. In 1960 he arrived in Berkeley as a Visiting Assistant Professor and joined the ladder rank faculty as an Assistant Professor in 1961. During his first year on the faculty, when other assistant professors seeking tenure were buried in labs or libraries struggling to solidify their academic careers, Professor Thomas chose instead to organize an international conference. Securing a prime location on the Berkeley campus, he hosted “The Impact of Transmission Electron Microscopy on Theories of the Strength of Metals” in 1961, providing an aggressive examination of the Orowan and Petch equations as well as new insights into the mechanisms of strengthening by finely dispersed (TEM-sized) obstacles. Many of the luminaries in the fledgling field of transmission electron microscopy were there (Figure l), taking note of both the ambition and the dedication their colleague Gareth Thomas, who would continue this tradition of global congresses to advance the practice of electron microscopy in applications to engineering materials throughout his career.

Figure 1: A few of the attendees at the 1961 Berkeley conference on the “Impact of TEM on Theories of the Strength of Metals.” L to R first row, R.B. Nicholson, M.J. Whelan, G . Thomas, J. Washbum; L to R second row, K. Melton, A. Kelly, G. Rothman, P.R. Swann.

Also during his first year on the faculty, Professor Thomas found time to draft and edit a complete textbook, Transmission Electron Microscopy of Metals, published by Wiley only one year later, in 1962. This treatise was the first of its kind, a practical, pedagogical, “hands-on” treatment of the transmission electron microscopy technique, annotated with instructions on how to prepare representative samples worthy of scientific investigation. It served generations of students for the next 17 years, until his second edition, co-authored with M.J. Goringe, was released in 1979. Thomas’s early emphasis on high-resolution microstructural characterization of metals was born of his notable successes during his time at Cambridge. One of the most perplexing problems of the day was the catastrophic failure of the Comet aircraft, prompting many investigations into the relationship between the microstructure and deformation behavior of aluminum-based alloys. Thomas’s work [ 1,2] showed quite clearly (Figure 2) the occurrence of a precipitate-free zone (PFZ) adjacent to grain boundaries, and a coarser precipitate distribution adjacent to the PFZ, when compared to the surrounding matrix. Implicating such inhomogeneities in microstructure as the likely cause for inhomogeneities in mechanical response, the path forward was revealed through microstructural design. Subsequent development of thermomechanical processing cycles to eliminate the formation of PFZs and their attendant problems was facilitated by electron microscopy, the only technique with sufficient spatial resolution to verify success. Professor Thomas developed similar processing methodologies to protect age-hardening alloys against stresscorrosion cracking (Figure 3), resulting in his first patent [3], also issued within a few short years of his debut on the faculty.

5

Characterization: The key to materials

Figure 2: Heterogeneous precipitation and precipitate-free zones (PFZs) in AI-6Zn-3Mg, after reference [2].

BI

I

.s

n 3% r ; -

*li

Figure 3: Plot of average stress corrosion life (days) vs aging time (hours) for aluminum alloys subjected to step agmg process, after reference [3].

EARLY DEVELOPMENTS In his quest for precision during diffraction analysis, Professor Thomas became an early advocate for the technique of Kikuchi electron diffraction [4],which results from an inelastic scattering event that is subsequently elastically scattered. Thomas and co-workers released a series of publications in the 1960s explaining the method and demonstrating its superior advantages over conventional (spot) electron diffraction for precise determination of crystalline orientations. By painstakingly assembling photo collages combining hundreds of Kikuchi electron diffraction patterns, they also generated “Kikuchi maps” to assist investigators in navigating reciprocal space. Figure 4 shows one such map for the diamond cubic structure [ 5 ] , but others were published for both body-centered cubic [6] and hexagonal close-packed [7] structures. Diffraction also figured prominently in the analysis of spinodal decomposition, but there was no more convincing evidence of structural modulation that the images published by Thomas and co-workers [S], Figure 5(a). Coarsening of the spinodally-decomposed product resulted in a square wave compositional profile seen in Figure 5(b), which was much less obvious, and sometimes completely obscured, in diffraction results. Thomas was also first to point out that microstructures generated by spinodal decomposition were not

6

R. Gronslcy

susceptible to the formation of detrimental PFZs, and he proposed employing spinodal decomposition where possible in alloy systems with known miscibility gaps as another method of intelligent microstructural design.

Figure 4. Kikuchi map of the diamond-cubic structure (silicon) after reference [ 5 ] The top pole is readily identified by its four-fold symmetry as 001, the bottom center pole is 113, representing an angular range of 25 2” East-west extremes are 102 and 012 poles, at 36 9” apart

Figure 5 : Spinodally decomposed Cu-Ni-Fe alloy showing (a) early stage and (b) later stage product resulting from aging within the temary miscibility gap. The light phase i s Cu rich, the dark phase, Ni-Fe rich.

Yet another method of microstructural analysis pioneered about this time was the application of phase contrast ‘‘lattice’’ imaging to directly assess the local lattice parameter in close-packed metallic alloys. The resolution performance of transmission electron microscopes was limited thirty years ago to approximately 0.25 nm, consequently a two-beam “sideband imaging method was the only feasible option for extracting phase contrast, generating images of a single spatial frequency. Figure 6 shows how the technique yielded the modulation wavelength in a spinodally decomposed Au-Ni alloy, the first such demonsbation of its type. Thomas and co-workers continued to apply lattice imaging to a range of spinodal and ordered alloys during the late 1970s, coupled to the development of subsidiary analytical techniques such as optical microdiffraction [9]. As specimen preparation procedures for non-metallic materials also improved in Thomas’s laboratories, phase contrast methods yielded new insights into novel polytypoid formation in the non-oxide ceramics. The example shown in Figure 7 documents the substructure of a beryllium silicon nitride, BesSi3Nl0, as alternating stacking sequences of three layers of BeSiN2 followed by two layers of Be3N2.

I


24t

12

0

30

60 I

90

*

120 i

20 40 60 DISTANCE (NO OF FRINGES)

Figure 6 Lattice image (top) and plot of d-spacing vs distance in a spinodally-decomposed Au-Ni alloy The “average” modulation wavelength IS 2.9 nm, after reference [9].

INNOVATIONS These successes with a growing number of applications of electron microscopy in materials engineering were clearly noticed by the scientific community at large. Consequently Professor Thomas chose to convene another gathering of participants in 1976 for the purpose of addressing what had become a burning question for him and many others: “Should the US support a National Center for Electron Microscopy?’ The question originated in the understanding that electron microscopy had taken on the earmarks of “big” science, requiring multi-million dollar investments in order to construct, maintain, and run the high voltage electron microscopes that exhibited superior performance at the time. Attendees included eighteen (18) from Berkeley, forty-one (41) from elsewhere in the US, and seven (7) from abroad, and at the end of the workshop, all concurred that the time was right to seek a national, shareable, user resource in the model of the photon beam lines and es that had recently been funded by the federal government. The original estimate for this facility was a modest $5M. In rapid succession, the Energy Research and Development Administration (later DOE) held two national Materials Sciences Overview meetings, the proceedings of which were published as ERDA 77-76-1 and ERDA 77-76-2 . In these reports, the Office of Basic Energy Sciences identified a “critical need” for state-of-the-art fa in transmission electron microscopy. Thomas and collaborators submitted their proposal that year, and the Atomic Resolution Microscope (ARM) became a line item in the FY 1980 Congressional Budget at $4.3M [ I l l . The ARM was installed in 1982 and sustained the best imaging

8

R. Gronsky

performance of any transmission electron microscope in the world for the next decade. With a top operating voltage of 1 MeV, a biaxial tilt stage of*45’ range, and an instrumental resolution limit of 0.16 nm, it’s utility extended to many new materials engineering problems requinng microstructural assessment at the atomic level. Moreover, the technological innovations funded by the federal government dunng this project spawned a new generation of “medium voltage” instruments with enhanced performarxe and smaller footpnnt, so they could be placed in a “normal” laboratory setting, instead of the three-story silo architecture needed by the larger megavolt units.

Figure 7: Phase contrast image of Be9Si3NIo(left) and structural model (right) showing three layers of BeSiNl interspersed with two layers of Be,N2, after [lo].

One of the most widely publicized images from the ARM is shown in Figure 8, showing the atomic structure of the double-layer defect in the high Tc superconductor, YBCO.

Figure 8: Phase contrast image of YBa2Cu3O,.* (left) and structural model (right) showing double layer CuO defect running horizontally through center of micrograph, after [ 1 I]. Only cations are visible.

It is instructive to compare Figures 7 and 8 for their historical significance since they represent best practice in

“contemporary” transmission electron microscopy, published in the world’s premiere scientific journals, one decade apart. The legacy of innovation that has distinguished Professor Gareth Thomas’s career is clearly revealed in these images.


9

LEGACY But Professor Thomas’s legacy extends well beyond his contributions to the field of electron microscopy. His innovations in the development of novel materials and processing procedures have resulted in a dozen patents. The first, described above, was issued for a process to enhance resistance to stress corrosion cracking in A1 alloys. Six more patents cover his development of new steels, some high-strength, some dual phase [14], and some corrosion-resistant. Another patent was granted for a method to produce dense refractory ceramics [15]. And his four most recent patents are for magnetic materials, to enhance intrinsic coercivity and to enhance their giant magnetoresistive (GMR) response [16]. Professor Thomas’s contributions to the scientific literature number over five hundred (500) and counting. Even more impressive than this number is the range of topics on which he’s written. Metals and alloys, ceramics, semiconductors, superconductors, magnetic materials, composite materials, polymeric materials, and even organic materials appear in his manuscripts, along with a widely varied range of electron microscopy, diffraction, and spectrometry methodologies used for their characterization. One of very few individuals to have been elected to membership in both the National Academy of Engineering (1982) and the National Academy of Sciences (1983), Professor Thomas’s recent awards include the Gold Medal from ASM International (2001), a Doctorate Honoris Causa from the University of Krakow, Poland (1999), election as an Honorary Member of the Japan Institute of Metals (1998), an Honorary D.Sc. from Lehigh University (1996), election as an Honorary Member of the Indian Institute of Metals (1996), election as an Honorary Member of the Korean Institute of Metals and Materials (1996), a Humboldt Senior Scientist Award (1996), and the highest award given by his home campus, the Berkeley Citation for Distinguished Achievement (1995). Professor Thomas’s dossier of service is equally rich. He devoted four years as Editor in Chief of Acta Materialia and Scripta Materialia, currently continuing as a Technical Director (1998-), another four years as President of the International Federation of Societies of Electron Microscopy, four years as a Member of the Board of Governors of Acta Metallurgica, Inc., another four years as Chairman of the Acta Metallurgical, Inc., Board of Governors, and four more years as a member of the TMS-AIME Board of Directors, among other appointments of lesser duration, such as his one year (1993) term as Director of the technology Transfer Center at the Hong Kong University of Science and Technology, and one year (1975) reign as President of the Electron Microscopy Society of America. As he engages his seventy-first year, Professor Thomas is enjoying his honorable emeritus status on the faculty after having supervised more than one hundred (100) students through the pursuit of their graduate degrees. He has taught thousands more, undergraduate, graduate, and post graduate, in lectures and seminars at home and abroad. But, as expected, Gareth Thomas is not “retired.” He currently holds the position of Vice President of Research and Development for MMFX Steel Corporation of America, returning to one of his favorite metallurgical pastimes: enhancing the performance of steel. In an aggressive campaign to extend the lifetime of rebar used in concrete construction, Thomas has claimed another success through clever microstructural design. By replacing the ferrite/carbide microstructure common to low carbon rebar-stock steels with a “dualphase” microstructure (ferrite/martensite, or austenite/martensite) through simple adjustments in processing, an astoundingly superior corrosion resistance has been demonstrated, with high payoff potential for applications in marine environments. There can be little doubt that Professor Thomas’s legacy will continue to live on through these and other advances made by materials characterization in the Thomas tradition.

R. Gronsky

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SUMMARY

Accomplished in science, accomplished in engineering, and accomplished in academia, Gareth Thomas has certainly made his mark on the historical record. It is also clear that he leaves all of us a timeless message. It first appeared in the preface to his textbook Transmission Electron Microscopy of Metals, dated 1961. “Over the last twenty-five years electron microscopy has become an increasingly popular technique for examining materials. ...The tremendous advantage of the transmission technique is, of course, that the results obtained are visual and therefore convincing.” Over the intervening forty-one years, the message has remained the same. Advancing the state of the art demands results that are both visual and convincing. Making the case for new and improved materials requires evidence that is both visual and convincing. And, as he continues to demonstrate so effectively, the execution of his successful brand of microstructural design is stunningly visual and convincing. Happy birthday to very visual and convincing guy! ACKNOWLEDGEMENTS

I shall always be grateful to Gareth Thomas for accepting me into his group during the early summer of 1972. Thanks to my colleagues Prof. M.A. Meyers, Prof. R.O. Ritchie, and Prof. M. Sarikaya for their kind invitation to contribute to this commemorative volume. Thanks also to the stalwart program managers at OBES in DOE and ERDA before them who recognized the wisdom of microstructural design and funded this nation’s effort in electron microscopy through all of these years. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Thomas, G., and Nutting, J. (1959-60) J. Inst. Metals 88, 81. Nicholson, R.B., Thomas, G., andNutting, J. (1960) ActaMet. 8, 172. Thomas, G. (1964) “Process for Improving Strength and Corrosion Resistance of Aluminum Alloys,” U.S. Patent # 3,133,839. Kikuchi, S. (1928) Jupnn J. Phys. 5, 83. Levine, E., Bell, W.L., and Thomas, G. (1966) J. Appl. Phys. 37,2141. Okamoto, P.R., Levine, E., andThomas, G. (1967)J. Appl. Phys. 38,289. Okamoto, P.R., and Thomas, G. (1968) Phys. Stat. Sol. 25, 81. Butler, E.P., and Thomas, G. (1970) Acta Met. 18, 347. Sinclair, R., Gronsky, R., and Thomas, G. (1976) AcfuMet. 24,789. Shaw, T.M., and Thomas, G. (1978) Science 202,625. Gronsky, R., (1980) in 38th Annual Proc. Electron Microscopy SOC.Amer., G.W. Bailey (ed.), p 2. Gronsky, R., and Thomas, G. (1983) in 41st Annual Proc. Electron Microscopy SOC.Amer., G.W. Bailey (ed.), p. 310. Zandbergen, H., Wang, K., Gronsky, R., and Thomas, G. (1988) Nature 331,596. Thomas, G., and Nakagawa, A. (1 986) “High Strength, Low Carbon, Dual Phase Steel Rods and Wires and Process for Making Same,” U S . Patent # 4,613,385. Thomas, G., Johnson, S.M., and Dinger, T.R. (1989) “Method of Producing a Dense Refractory Silicon Nitride Compact with One or More Crystalline Intergranular Phases,” U.S. Patent # 4,830,800. Bemardi, J.J., Thomas, G., and Heutten, A.R. (1999) “Giant Magnetoresistive Heterogeneous Alloys and Method of Making Same.” U.S. Patent # 5,882,436.

Nan0 and Microstructural Design of Advanced Materials M.A. Meyers, R.O. Ritchie and M. Sarikaya (Editors) 02003 Published by Elsevier Ltd.

NANOCHEMICAL AND NANOSTRUCTURAL STUDIES OF THE BRITTLE FAILURE OF ALLOYS D.B. Williams’,M. Watanabe!, C. Li’ and V.J. Keast’ ‘Department of Materials Science and Engineering and The Materials Research Center, Lehigh University, Bethlehem PA 18015, USA ’Australian Key Centre for Microscopy and Microanalysis, Madsen Building University of Sydney, NSW 2006, Australia

ABSTRACT Controlling the brittle intergranular failure of metals and alloys requires understanding the structure and chemistry of grain boundaries at the nanometer level or below. Recent developments in the analytical electron microscope (AEM) permit such studies. It is now feasible to determine, in a single AEM specimen, the grain boundary chemistry (using X-ray mapping), crystallographic characteristics (using automated crystallographic analysis) and the localized bonding changes that may accompany segregation (using fine structure changes in the electron energy loss spectrum). Computerized mapping techniques permit such information to be gained from dozens of grain boundaries. Integration of this knowledge may permit the design of new alloys and new heat treatments to create materials inherently resistant to the brittle failure often caused by nanometer level grain boundary segregation of impurities and alloying elements.

INTRODUCTION Gareth Thomas is primarily responsible for the development of the transmission electron microscope (TEM) as the most versatile and integrated technique for the solution of materials problems. Throughout his long and distinguished career Gareth has always stressed the essential need to use the TEM as one of a range of techniques to solve the problem at hand, rather than selecting a problem simply to suit the TEM’s capabilities. Nevertheless, he has also pushed the development of the TEM to its fullest capabilities, particularly in the exploration of its high-resolution imaging limits, embodied in the Atomic Resolution Microscope at the National Center for Electron Microscopy at Berkeley. At Lehigh, we have taken a similar approach to attacking materials problems, but emphasized the analytical side of the TEM, particularly elemental analysis via X-ray and electron spectroscopy. So we can perhaps talk about “Microchemical Design of Advanced Materials” in this article, in line with the theme of this book. This paper will review our implementation of Gareth’s philosophy to the long-standingissue of brittle failure. Brittle failure of metals and alloys remains a serious limitation to the development of new technologies and the improvement of existing ones. The record of brittle failure studies starts in the 19‘h century [ l ] and has encompassed classic examples such as the Titanic’s rivets [2], the Boston Molasses Tank [3], the SS Schenectady Liberty Ship [4], the Hinckley Point power-generation turbine [5] and the United Airlines DClO

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D.B. Williamset al.

12

crash at Sioux City [6] Despite such a long and painful history, the problem of bnttle failure remains as current as ever in its societal effects For example, during 2001, the space shuttle fleet was grounded twice, first by the discovery of cracking in the liquid hydrogen flow liners and second by beanng cracks in the crawlers that transport the shuttles to the launch site Similarly, the high-speed Amtrak Acela trams were pulled out of service following the discovery of cracks and breaks in brackets on the wheel sets of at least 8 of the 18 trains Brittle failure in metals takes many forms, e g hydrogen embrittlement [7,8], temper embrittlement [9], environmental degradation [ 101 and associated stress-corrosion cracking [ 111, fatigue failure [ 121, irradiationinduced embnttlement [ 131, liquid-metal embnttlement [ 141 and, more recently, such new phenomena as quench embnttlement [ 151 Two key factors transcend this diverse array of failure phenomena, namely the role of the grain boundary and segregation of undesirable elements to the boundary, as epitomized in Figure 1 There is a long history of research relating the structure of the grain boundary to vanous properties, including the tendency for segregation e g [ 16,171 Some studies have shown correlations between individual grain-boundary misonentation and the local chemistry, or related aspects such as grain-boundary precipitation [18,19] Such correlations have been few and have rarely been carried out on undisturbed (I e non-fractured) grain boundanes or on enough grain boundanes to permit any statistical correlation to be inferred In general it has not proven possible to relate directly the properties of grain boundanes to their structure While structure-property correlations are very strong at the structural extremes of coherent twins (C = 3 ) and random high-angle boundaries (C > -29), intermediate special boundaries (e g C = 5 , 7, 9 etc ) do not always correlate well with properties Part of the reason for this is undoubtedly that the grain-boundary structure is not the pure elemental construction that is commonly assumed, but is modified senously by local changes in the grain-boundary chemistry The analytical EM (AEM) is uniquely configured to study these phenomena because it combines high-resolution imaging, diffraction and nanometer-scale analysis of the same specimen at the same time, permitting correlation of the grain-boundary structure, misonentation, chemistry and bonding - all at the nanometer or sub-nanometer scale No other technique is so versatile at such a high resolution At Lehigh, we have been using the AEM to correlate the chemistry, structure and bonding of embnttled grain boundaries by a) performing quantitative analysis of nanometer-scale segregation to many grain boundanes using X-ray mapping (XRM) via energy dispersive spectrometry (EDS), b), for those same grain boundanes, determining their crystallographic misonentation via the latest computenzed diffraction techniques and c) relating the occurrence of segregation to changes in the atomic bonding at the grain boundary via electron energy-loss spectrometry (EELS) This paper gives an overview of the results of our integrated AEM studies in model embnttling systems such as Cu Bi, Cu-Sb and Fe-P We will first introduce bnefly the techniques used r

Figure 1: SEM images of the fracture surface of a) pure Cu and b) Cu doped with 20 ppm Bi

Nanochemical and nunostructural studies of the brittle failure of alloys

13

EXPERIMENTAL TECHNIQUES X-ray Energy Dispersive Spectroscopy (XEDS) XEDS has been the most extensively used technique for measuring segregation in the AEM, starting with the pioneering studies of Dog and Flewitt [20], who first demonstrated that monolayer-level segregation was detectable in thin foils. Subsequent work has been performed, for example, by Wittig et al. [21,22], Brummauer et al. [10,11,13,23], Rtihle and co-workers [24-261 and the Lehigh group [27-301. Today, with a field emission gun scanning transmission electron microscope (FEG-STEM) the spatial resolution of XEDS can be < 2 nm and segregation can be quantified with a sensitivity approaching 0.01 monolayer [31]. In contrast with the more widespread surface- analysis techniques, in the AEM it is possible to study both brittle and ductile grain boundaries since the boundaries are contained within the thin foil and do not need to be fractured. It is of course, also possible to determine the grain boundary crystallography at the same time via routine TEM diffraction methods, as amplified in the following section. A common approach is to acquire a segregation profile by stepping the electron beam along a line perpendicular to the grain boundary. For equilibrium segregation, the width of this profile will be determined by the size of the electron probe. Two-dimensional X-ray mapping (XRM) of segregant distributions has, until recently, rarely been performed but the unique 300kV ultra-high vacuum, field emission gun VG HB 603 FEG STEM permits the acquisition of compositional maps at high spatial resolution (< 2 nm) and high sensitivity (< 0.1 monolayer) [31-331. XRM offers the advantage that any compositional variations along a grain boundary plane or other complex elemental distributions are easily observed. Many (>30) boundaries can be studied in a single smallgrained sample and it is now possible to directly relate the segregation to the grain-boundary crystallography (misorientationand plane) via computerized diffraction pattern indexing as discussed below.

Automated Crystallographyfor TEM (ACT) While significant progress has been made in XRM and related AEM methods, most studies of grain-boundary crystallography still use standard TEM methods of selected-area or convergent-beam diffraction (SAED or CBED) which are generally non-computerized, labor-intensive and rarely produce statistically valid data. Recently, however, there have been attempts to characterize grain orientations automatically in the TEM [34371. This was stimulated by the success of electron back-scatter diffraction (EBSD) in the scanning electron microscope (SEM) [38,39], which gives computerized orientation of hundreds or even thousands of grains, thus permitting full microtexture analysis and other applications [40-431. Much EBSD work has been done exploring the effects of grain-boundary misorientation (and thus local texture if sufficient grain boundaries are analyzed) on materials properties (e.g. see texts [42.43]). Unfortunately, EBSD cannot be combined with the study of grain-boundary chemistry, because it is not possible to measure grain-boundary chemistry in the SEM. Both the spatial resolution and analytical sensitivity of SEM-XEDS are too coarse [44] to detect and quantify monolayerlevel grain-boundary segregation. To overcome these limitations we have used ACT in which the beam is scanned across the specimen and, when it satisfies the Bragg condition for a given grain, the corresponding area in the dark-filed image appears bright. The intensity of each pixel is recorded as a fbction of beam tilt and rotation angle (i.e. a diMaction pattern). A grain-orientation map is constructed and the misorientation between adjacent grains is calculated from the diffraction patterns, as in the EBSD technique. Eleetron Energy Loss Spectroscopy (EELS) The ionization edges in the EEL spectrum are also used to identify and quantify segregating elements [45.46], although the technique has been less frequently applied than XEDS. However, the fine structure on the ionization edges (the energy-loss near-edge structure (ELNES)) contains information about the unoccupied density of states (DOS) and can thus probe the interatomic bonding which is possibly affected when segregants induce intergranular brittle failure or, conversely, induce ductile behavior in otherwise brittle materials (e.g. B segregation to grain boundaries in NijAI;). There have been several recent examples where ELNES has been used to elucidate such changes in the atomic bonding at grain boundaries produced by segregating elements [45,47-501. The reliability and interpretation of ELNES at grain boundaries remains controversial. The main

D.B. Williamset al.

14

Figure 2: XRM of Bi segregation to grain boundaries in Cu a) localization of Bi to one grain boundary out of a complex intersection of multiple grain boundaries (see corresponding TEM bright field image in b) The grain boundary with Bi is a C=9 while all the others a e C=3 coherent or incoherent twins. In c) there are detectable differences in Bi level at the two different facets of the high angle grain boundary, indicating a role for the grain boundary plane in determining the degree of segregation The upper facet has 10 6 2 1 atoms/nm2while the lower facet has 12 4 + 1 3 atoms /nm2 In d) the map clearly reveals the presence of Bi segregation at a level of 0 8 Bi atomsinm' (reproduced from refs [3 11 by permission of Elsevier Science) expenmental difficulties are the limited spatial resolution and/or statistics It is anticipated that the introduction of sphencal-aberration correctors in STEMS [51] will increase current densities by and order of magnitude and ELNES of interfaces will become considerably more reliable (as indeed will XRM) RESULTS & DISCUSSION

XRM of Brittle Failure in Cu-Bi and Low-alloy Steels Using XRM it is possible to discern numerous aspects of the behavior of segregants that are not routinely accessible via more traditional point and line analyses As summanzed in Figure 2, it is possible to discern a), b) the absence of segregation on certain low-Z boundaries, c) differences in segregation levels between different facets and d) the detection of levels of segregation far below the monolayer level A monolayer in Cu approximates to 18 atoms/nm2 on the grain boundary so the presence of < 1 atom of Bi/nm2 as shown in Fig 2d would probably not cause bnttle failure and, therefore, would not be detected by classical surface-analysis techniques Perhaps the most intriguing result from the use of XlZM for the study of segregant distnbution on dozens of grain boundanes was the discovery that, in a highly embnttled system such as Cu-Bi, significant numbers ofthe grain boundanes exhibited NO detectable segregation [31] (see the histogram in Figure 3) This

-


Quenched Tempered

Stress relieved

15

embrittled

Detectabesegregation

1 1.5 x Detectable segregation 1 2.0 x Detectable segregation 1 Figure 3: A histogram of grain boundary coverage of Bi in Cu (atoms/nm2).-30% of the grain boundaries have no detectable Bi segregation. (One monolayer corresponds to -18 atoms/nm2. Reproduced from [31] by permission of Elsevier Science.

Figure 4: Pie charts showing the amount and degree of P and Ni segregation in a low-alloy steel The grain boundary segregation behavior vanes as a function of the heat treatment and the P is never present at all grain boundaries (courtesy A.J Papworth) Modified from [54].

result contradicts much of the common wisdom on the distnbution of Bi, which has traditionally been thought to be present on all grain boundanes in embnttled Cu (e g [52]) This conclusion anses probably because of the prevalence of surface analytical data (e g Auger Electron Spectroscopy (AES)) which, by its nature, pre-selects embnttled grain boundanes for analysis Parallel work [53] showed that minimum detection limits in the VG HB 603 field emission gun (FEG) AEM approached the single atom level in ideal conditions and that the detection limit for mapping of grain-boundary segregants was < 0 1 monolayers [31] So the absence of detectable grain-boundarysegregant was not a limitation of the AEM technique The distribution of P at grain boundaries in low-alloy steels was then studied to see if similar behavior occurred The evolution of grain-boundary segregation of a range of elements, subject to heat treatments that give rise to temper embnttlement was studied Typical results are summarized in Fig 4, which shows pie charts indicating changes in the amount (gray level) and degree (number of grain boundanes (30 per full pie chart)) of segregation dunng the heat treatments [54] The results indicate that, in t h s temper-embnttled condition (when the alloy shows 100% intergranular bnttle failure), P is present on some grain boundanes but it is neither present on all grain boundanes, nor is it the primary segregant In fact, both the amount and degree of P segregation has decreased from the pnor (more ductile) condition This fact in isolation strongly confirms the data from Cu-Bi in Fig 3, that bnttle matenals need not have 100% grain-boundary segregation for failure to occur, but also highlights the complexity of bnttle failure in multi-element systems So there is now corroborating evidence that embntthng species are not necessanly present at all grain boundanes Therefore, grain-boundarycharacteristics may play a greater role than hitherto believed 111 controlling the distnbution of Bi in Cu and P in steel CombinedXRMand ACT of Sb Segregation in Cu It is well known that grain boundary misorientation plays a role in segregation, accounting in part for the vanation from boundary to boundary [55] A widely accepted view is that high-angle grain boundaries are more accommodating than low-angle grain boundaries and strong segregants are present at all high-angle gram

D.B. Williamset al.

16

(a) BF TEM ima!

(b) diffraction patterns (c) orientation map

I 400nm

H

Figure 5: (a) BF image in Cu-0.08 wt%Sb alloy. (b) Diffraction patterns reconstructed from the DF images (c) Reconstructed onentation map. boundanes In fcc materials, e g , segregation is suppressed only at C=3 (the most close-packed) grain boundanes (as shown in Figs 2b and c) A possible reason for our observed lack of segregation at other highangle gram boundanes in Cu-Bi and Fe-P is that crystallography plays an even more important role than hitherto suspected and some high-angle grain boundanes are inherently resistant to segregation, perhaps because of a low density of ledges, intnnsic dislocations, etc To prove this hypothesis will require measurement of nanometer-level chemistry and local misonentation from many gram boundanes and companson with segregation in controlled textures and misonentations However, to date, it has proven extremely challenging to measure both these charactenstics from significant numbers of grain boundanes in the same specimen In order to pursue the details of the relationship between the occurrence (or absence) of segregation and the gramboundary characteristics, it is necessary to map the segregation of an embrittling species at a range of grain boundanes whose misonentation is determined, e g via ACT Figure 5a is a TEM image of a Cu-0 08 wt% Sb alloy, analyzed by ACT The diffraction patterns and onentation map are shown in Figs 5 (b) and (c) and the onentation relationship between adjacent grains is shown in Table 1 XRM was performed on these grain boundanes and the Sb images are shown in Figure 6 Clearly the segregation varies, while Sb is detectable on most gain boundanes, it is not detectable on #3, 4,s and 10 (and it has been shown this is not simply due to factors such as tilt of the grain-boundary plane) Again the crucial point is that a major embnttling agent is not present on many grain boundanes, supporting our data from Cu-Bi and Fe-P From Table 1 only one grain boundary (#3) is close to a E=3 structure, the rest are random high-angle grain boundanes Thus it has been demonstrated via a combination of XRM and ACT that a significant fraction (> -30%) of highangle grain boundanes, in 3 different strongly-segregating systems, Cu-Bi, Fe-P and CuSb exhbit no detectable segregation Therefore, it is reasonable to conclude that alloys in which such segregant-free grain boundanes are more prevalent should show enhanced resistance to segregation and any associated bnttle failure One method by which an increased fraction of segregation-resistant grain boundanes could be produced is grainboundary engineenng (GBE) which produces textures with a majonty of low-C grain boundanes, Such grain boundanes would shifi the distnbution of gram-boundary chemistry m Figure 3 towards the left-hand end of the spectrum, thereby significantly reducing the number of gram boundanes to which segregation can occur The concept of selectively enhancing the number of low-C grain boundanes through GBE, in order to reduce embnttlement was first proposed by Watanabe [41] As well as the pioneering work of Watanabe, GBE has also been implemented by other groups including, e g Lehockey et a1 156 571 and Kumar et a1 [58 591 GBE has


17

Sk Nt%)

Figure 6: Sb composition maps. The segregation between different grain boundaries is clearly not homogeneous and several grain boundaries have no detectable segregation although they are high-angle grain boundaries. TABLE 1 Axidangle pairs calculated from the ACT results for the grain boundaries numbered in Fig. 6 Boundary 1 2 3 4 5

Angle-axis 52'@[-11 -5 61 45"@[4 -2 -31 59"@[8 9 -81 -60°@,[1 1-11 28'@[15-1 I ] 49"@[-7 -8 121

Boundary 6 7

Angle-axis 43"@[9 -8 51 47"@[-2 -1 -51

8

47'@[2 5 -11

9 10

29'@ 12 -3 -71 40"@[-17 1-61

revealed the strong effect of the dzstnbutzon of grain boundary structures on properties, including embnttlement, and shown how, via thermo-mechanical processing (TMP), it is possible to engineer the distnbution of low-C grain boundaries to improve greatly the mechanical and other (e g corrosion) properties Implicit in GBE is the concept that manipulating the gramn=boundary structure may result in manipulating the chemistry, given the long-understood relationship between the two Since GBE produces a large fraction of

D.B. Williams et al.

18

2=3 grain boundaries (which exhibit no segregation) the possibility that GBE techniques will produce an inherently segregation-resistanttexture are now ripe for study using a combination of XRM and ACT. Energy Loss Near-Edge Fine Structure (ELNES) Studies of Embrittled GBs In addition to the role of the boundary crystallography in governing segregation it is important to discern the role of the segregant when it reaches the boundary plane. As has been noted, while this article emphasizes the segregant's role in brittle failure, segregants can also act to improve the ductility of otherwise brittle materials such as B in Ni3Al. So the segregation itself is not a necessary prerequisite for brittle failure. Clearly the segregant must be changing the character of the bonding at the grain boundary to induce either brittle failure or, conversely, ductilization. To probe the bonding changes at the nanometer level is also possible in the AEM using energy-loss spectrometry fine structure studies. We have demonstrated a clear relationship between the presence of Bi at grain boundaries in Cu and a change in the ELNES. The presence of Bi changes the electronic structure of the Cu but only within 100 ksiJns [&lo]. (B) The multilayered structure contains no precipitates (carbides etc.) and hence there are few microgalvanic corrosion pitting centers. Thus there is a considerable gain in corrosion resistance especially in saline conditions. These benefits are now being utilized in new and repaired infrastructures in which corrosion limits the lifetime of steel in concrete structures (See the web site: [ 5 ] ) . It is an astounding fact that infrastructure repair costs are estimated to be approximately 4 trillions $ over the next decade. ColPt MAGNETIC MULTILAYERS Basis The realization in the past decade or so that many important magnetic properties are microstructure sensitive has led to rapid developments in the field of magnetic devices, involving atomically engineered thin films, nanostructures and multilayers. In these composites the structure of the interfaces between dissimilar materials or/and at the grain boundaries governs the novel properties (e.g. perpendicular magnetic anisotropy, giant magnetic resistance, etc.) [ 151. In particular, magnetic multilayers (MLs) composed of modulated ferromagnetic-nonmagnetic layers such as Copt MLs [ 16,171 with large perpendicular magnetic anisotropy (PMA) and high coercivity have recently been proposed as future magnetic media in Terabit/in* magnetic recording systems [18]. In addition, recent interest in Co/Pt was sparked by the discovery that the magnetic properties can he locally modified by ion-beam irradiation [ 19,201. Characterization of the microstructure and the magnetic domain structure in such films is important from a technological as well as a fundamental perspective [21-231. In this paper, the influence of growth temperature and multilayer thickness on the structure and magnetic properties of Co/Pt MLs are discussed. Other parameters which affect the magnetic performance of the MLs include magnetic patterning by ion irradiation which allows the separation of the vertical and in-plane magnetization. Further details can be found elsewhere [21,24-261. Microstructural characterization For the specific experiments discussed here, Co/Pt multilayers with representative structure: / 20 nm Pt seed / Nx(t nm Co/l nm Pt) / 1 nm Pt cap layer /, were fabricated by electron beam evaporation using a Torr base pressure deposition system [27,28]. Figure 5 shows a plan-view, bright field Transmission Electron Microscopy (TEM) image and Selected Area Diffraction (SAD) patterns of a Co/Pt multilayer grown at 250°C. The fine structure visible in some of the grains is attributed to Moirk fringes caused by the small lattice mismatch between Co and Pt. The plan-view SAD pattern shows the typical ring spacing associated with polycrystalline face-centered cubic fee) Pt except for ring splitting due to the presence of highly strained Co layers, which will be addressed later. The intensity distribution of the rings indicates a strong texture with only some grains oriented randomly and contributing to the (1 11) and (002) ring intensities.

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G.J. Kusinski and G. Thomas

Figure 5: (a) Plan-view TEM micrograph and (b) SAD pattern of a lOx(3 8, Co / 10 A Pt) multilayer grown at TG= 25O”C, (c) SAD with 30” sample tilt showing arcing of the rings. One of the questions to be resolved by careful characterization is whether the multilayers are alloyed or not, i.e. are the Co/Pt “pure” components. Careful examination of the SAD patterns revealed that all of the rings associated with thefcc structure were split, with separation Ag increasing with the diffraction vector g. For each doublet, the inner ring corresponding to a larger real space parameter is always more intense and the outer ring is weaker. This implies two distinctive lattice parameters through the thickness of the Copt multilayer stack. Figure 6 shows (a) the SAD pattern for the discussed Co/Pt multilayer and simulated ring diffraction patterns of (b) Ptf,, and Co Pt3, (c) Ptf,, and Cohcp,(d) Ptf,, and Cof,,. On all patterns, triangles label Pt rings: (11 l), (200), (220), (1 13) and (222). By comparing the simulated SADs to the experimental data shown in Figure 6 , it is clear that the layers must be “pure” Co-Pt both fcc. No evidence for alloying or ordered compound formation was detected. However, the ratio of the equilibrium Cofccto Ptf,, spacing in the simulated pattern is not correct. As shown, any two Co and Pt rings on the recorded diffraction pattern, for example Co(220) and Pt(220), are much closer to each other than on the simulated pattern (d). This implies that the Co layers are strained. Figure 6 (e) and shows simulated diffraction rings as a function of increasing tensile strain of the Co layers, with image (f) showing a good match. Hence, Starting from the seed Pt layer, the Pt(ll1) layers are stacked according to the equilibriumfcc stacking sequence observed in bulk Pt. The Co layers, which, in bulk are hcp, follow thefcc stacking sequence of underlaying Pt.

(a

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87

Figure 6: SAD pattern for the Co/Pt multilayer and simulated plan-view diffraction patterns for (b) Ptpc and Copt,, (c) PtP, and Cq,,,, and (d) Ptfccand Cof,, tnaiigles label Pt rings. (e) and (0 Simulated plan-view diffraction patterns of Ptrccand Cotccat various strains: (e)qo= 3.65 A, u= +2.85%, acJap, = 0.92 and (f) ac0= 3 85 A, u= +8.45%, aco/ap,= 0.98 Numbers label rings. 1) Pt(lll), 2) Co(1 ll), 3) Pt(002), 4) C0(002), 5) Pt(022), 6) C0(022), 7) Pt(l13), 8) Pt(222), 9) Co(113), 10) C0(222), (see text).

Figure 7 shows a lattice image with fcc multilayers stacking and good coherency across the CoiPt interfaces Ihis (1 11) stacking further implies a small volume anisotropy of Co Thus the perpendicular easy axis of magnetization is attributed to the interface anisotropy and in particular to strain contributions [29] d(,,,,,, - 2 265

A, d(,,,,c o =

2 047 A

Figure 7: Cross-sectional micrograph of a IOx(3 8, Co and 10 8, Pt) multilayer grown at T,

= 250°C.

88


Analysis of the multilayers as a function ofgrowth temperature The microstructural evolution of multilayers as a function of growth temperature (TG)is presented in Figure 8, which shows plan-view BF images, SAD patterns and DF-hollow cone images obtained using (111) rings, (1 11)DF, of ML's grown at T, = 190"C, T, = 250°C and T, = 390°C. As depicted in the BF images, Figure 8(a), (e) and (i), all of the investigated samples were polycrystalline. The average grain size was found to increase with increasing T,. As discussed above the SAD patterns, Figure 8(d), (h) and (1) showed typical ring spacing associated with the polycrystalline fcc Pt structure. For all samples, a strong out-of-plane 4 11> texture was measured, as seen from very weak (111) and (002) rings and an intense (022) ring. Only some grains oriented randomly contributed to (1 11) and (002) rings. The SAD patterns from larger areas showed a uniform intensity distribution around all rings, indicating random in-plane orientation and only out-of-plane texture.

Figure 8: Plan-view BF TEM images, SAD patterns and (1 1 1)DF images of the Copt ML's grown at TG= 190°C, TG= 250°C and TG= 390oC, displayed in rows. All SAD'S were collected with a 5 pm aperture ~41.

The (1 1I)DF images, Figure 8(b), ( f ) and (d), show the same areas as the respective BF images and display only the grains without the - 4 1 1> texture. Figure 8(c), (g) and (k) are lower magnification images showing the distribution of such grains. As presented, samples grown at the low temperature of 190"C, @) and (c), had a large number of grains lacking the < l l 1 > texture. The distribution of such grains was very uniform. With increasing T,, the number of such misoriented grains decreased. For the T, = 250°C sample, ( f ) and


89

(g), the majority of misoriented grains were smaller than the average grain size, although a few, large misoriented grains were also found. For samples grown at 390°C, a good < l l 1 > texture was observed, with only very few grains lacking the texture, as shown in (i) and (k). Moreover, such misoriented grains were much smaller than the textured grains. Hence, as clearly shown by the set of (1 1l)DF images the texture was found to improve with an increase in sample growth temperature. For samples grown at TGup to 300°C, all diffraction rings were split, indicating two distinctive lattice parameters for the Co and Pt layers, as discussed above. For samples grown at TG= 390°C above a critical transition temperature, T,,,,, the ring splitting attributed to the two separate Pt and Co parameters was not detectable, indicating a continuous gradient in lattice parameter between that of strained Co and that of Pt.

Structure as a function of Co thickness

The structure of Co/Pt multilayers was also investigated as a function of Co layer thickness (b).In agreement with previously published data [30], it was found that the stacking sequence in the Co layers changes with increasing Co thickness. At 0.2 nm - 0.6 nm Co thickness, the (Co/Pt) multilayer structure showed fcc stacking with 2.2 8, layer spacing. When the Co thickness increased beyond -0.6 nm, hcp stacking faults (SF) were observed and their density increased with increasing Co layer thickness. Above approximately tco = 15 A,the Co layer stacking is mainly hcp. Co/Pt multilayers with Co layer thickness below approximately 6 A were uniform and relatively defect free. The only indication of the Co vs. Pt layer in the HRTEM image was a slightly higher intensity in the Co layer due to preferential thinning of the Co layer during specimen preparation by ion-milling and a lower atomic number contrast. This intensity variation had a periodicity of the analyzed multilayers. With increasing tco, roughening of the MLs was observed, with increased number of defects (Figure 9). As shown, the measured intensity is much higher within the Co layer; however, the precise location of the CoiPt interface is not known. The (1 11) stacking sequence in some of the locations isfcc, (same vertical atom position for every third layer, ABCABC), but some hcp stacking (same vertical atom position for every second layer, ABAB) is also seen.

Figure 9: Cross-section HRTEM image of (1 11) 15 A, Co layer in the C o p t multilayer.


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Magnetic properties In addition to affecting the grain size and the texture, TG is also known to influence the magnetic properties, such as coercivity (Hc). Magnetic hysteresis loops of the Co/Pt multilayers grown at different growth temperatures are shown in Figure 10. As shown, the perpendicular HC of the C o p t ML's was found to increase with increasing TG [27,24]. The coercivity increases by almost a factor of two on increasing the TG from 200°C to 300°C. However, when TG is increased beyond a critical transition temperature, Tc,it, a decrease in perpendicular coercivity is observed, and the HC for samples grown at 390°C is reduced to 5.8 kOe. A further increase in multilayer growth temperature results in loss of perpendicular anisotropy.

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Figure 10: Mgnetic hysteresis vs. multilayer growth temperature. (a) TG= 19OoC,(b) Tc = 250°C, (c) TG= 300°C and (d) TG= 390°C. These results show that the magnetic properties of such multilayer films depend on microstructure. With an increase in Tc up to a certain critical temperature, Tcrit, an increase in grain size and an improved texture were found, both contributing to an increase in Hc. For ML's grown at TG = 390°C > Tait, a decrease in HC was measured, and small magnetically decoupled domains were observed. The size of these domains was similar to the grain size. This correlates well with Co depletion at the column grain boundaries and with the diffuse Co/Pt interfaces, measured as one set of Co-Pt rings. Such diffuse interfaces are known to reduce the PMA and, hence, reduce H,. Also with increasing Co layer thickness, a decrease in H, was observed. The coercivity is therefore a function of both the microstructure (grain size and texture) and interface quality, which is strongly influenced by Tc. Thus the magnetic properties can be engineered by appropriate choice of growth parameters and multilayer repetition. CONCLUSIONS

In this brief review we have shown that controlled multilayer structures provide superior properties such as mechanical and corrosion in austeniticllath martensitic steels, and magnetic properties in Co/Pt multilayers.


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ACKNOWLEDGEMENTS Work at NCEM/LBNL was supported by the U.S. Department of Energy under Contract No. DE-AC0376SF00098. The authors acknowledge contributions from K.M. Krishnan, E.C. Nelson and G. Denbeaux of Lawrence Berkeley National Laboratory and B.D. Terris, C.T. Rettner, A. Kellock and J.E.E. Bagglin of IBM Almaden Research Center.

REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Williams, D.B. and Carter, C.B. (1996). Transmission electron microscopy: a textbookfor materials science. Plenum Press, New York and London. Thomas, G. (1998). In: Boundaries and Interfaces in Materials: The David A. Smith Symposium, pp. 3-17, Pond, R.C., Clark, W.A.T., King, A.H. and Williams, D.B. (Eds). The Minerals, Metals and Materials Society, Warrendale. Walton, C.C., Thomas, G. and Cortright, J.B. (1998), ActaMaterialia, 46,3767. Thomas, G. (1999). In: 20th Ris0 Int. '1 Symposium on Materials Science, pp. 505-521, www.mmfxsteel.com Krauss, G. and Marder, A.R. (1971), Metallurgical Transactions A , 2,2243. McMahon, J.A. and Thomas, G. (1973). In: 3rd Int'l Conference on the Strength of Metals and Alloys, pp. 180-184, LBL-1490 Thomas, G. (1973), Iron and Steel International, 46,451. Thomas, G. (1978), Metallurgical Transactions A , 9A, 439. Rao, B.V.N. and Thomas, G. (1980), Metallurgical Transactions A , 11A, 441. Thomas, G . and Kusinski, G. (2000). In: ASMHeat Treat Conference, An International Symposium in Honor of Professor George Krauss, pp. 515-518, Koo, J.Y., Young, M.J. and Thomas, G. (1980), Metallurgical Transactions A , 11A, 852. Kim, N.J. and Thomas, G. (1981), Metallurgical Transactions A, 12A, 483. Kusinski, G. and Thomas, G. (1999). In: Proceedings of 1st Int. Con$ on Automotive Heat Treating, pp. 17-25, ASM International Ultrathin Magnetic Structures I: an introduction to the electronic, magnetic, and structural properties, edited by Bland, J.A.C. and Heinrich, B., Springer-Verlag,Berlin, Heidelberg (1994) Carcia, P.F. (1988), Journal ofApplied Physics, 63, 5066. Weller, D., Farrow, R.F.C., Marks, R.F., Harp, G.R., Notarys, H. and Gorman, G. (1993), Materials Research Society Symposium Proceedings, 313, 791. Wood, R. (2000), IEEE Transactions on Magnetics, 36,36. Chappert, C., Bernas, H., Ferre, J., Kottler, V., Jamet, J.-P., Chen, Y . ,Cambril, E., Devolder, T., Rousseaux, F., Mathet, V. and Launois, H. (1998), Science, 280, 1919. Weller, D., Baglin, J.E.E., Kellock, A.J., Hannibal, K.A., Toney, M.F., Kusinski, G.J., Lang, S., Folks, L., Best, M.E. and Terris, B.D. (2000), Journal OfAppliedPhysics, 87, 5768. Kusinski, G.J., Krishnan, K.M., Denbeaux, G., Thomas, G., Weller, D. and Terris, B.D. (2001), Applied Physics Letters, 79, 221 1. Thomas, G. and Hutten, A. (1997), NanoStructuredMateriab, 9, 271. Denbeaux, G., Fischer, P., Kusinski, G., Gros, M.L., Pearson, A. and Attwood, D. (2001), IEEE Transactions on Magnetics, 37,2764. Kusinski, G.J., Thomas, G., Denbeaux, G., Krishnan, K.M. and Terris, B.D. (2002), Journal of Applied Physics, 91, 7541. Kusinski, G.J. and Thomas, G. (2002), Microscopy and Microanalysis, 8, 319. Kusinski, G.J., Krishnan, K.M., Denbeaux, G. and Thomas, G. (in press), Scripta Materialia, Weller, D., Folks, L., Best, M., Fullerton, E.E., Terris, B.D., Kusinski, G.J., Krishnan, K.M. and Thomas, G. (2001), Journal of Applied Physics, 89, 7525. Kusinski, G.J. (2001). Ph.D. Thesis, University of California, Berkeley. Zhang, B., Krishnan, K.M., Lee, C.H. andFarrow, R.F.C. (1993), Journal OfAppliedPhysics, 73, 6198. Li, Z.G., Carcia, P.F. and Cheng, Y. (1993), Journal OfAppliedPhysics, 73, 2433.



EFFECTS OF TOPOGRAPHY ON THE MAGNETIC PROPERTIES OF NANO-STRUCTURED FILMS INVESTIGATED WITH LORENTZ TRANSMISSION ELECTRON MICROSCOPY Jeff Th.M. De Hosson and Nicolai G. Chechenin 'Department of Applied Physics, Materials Science Centre and Netherlands Institute for Metals Research, University of Groningen, Nijenborgh 4,9747 AG Groningen, The Netherlands. Skobeltsyn Institute of Nuclear Physics, Moscow State University, 118899 Moscow, Russian Federation.

ABSTRACT This paper concentrates on a detailed analysis of Lorentz transmission electron microscopy (LTEM) observations in the study of the magnetic properties of soft magnetic films. The nano-crystalline (Fe99Zrl)l-,NX, films have been prepared by DC magnetron reactive sputtering with a thickness between 50 and 1000 nm. The grain size decreased monotonically with N content from typically 100nm in the case of N-free films to less than 10 nm for films containing 8 at% N. Besides ripple fringes in the LTEM image that are commonly observed in nanocrystalline soft magnetic films also a dotted contrast apears along the ripple fringes. A theory of LTEM images for films with 1D- and 2D periodical topographies in combination with the micro-magnetic oscillations of the magnetization is presented. The theory predicts the 2D-pattern in LTEM in agreement with experimental observations. We show that in contrast to the traditional point of view not only the direction of the magnetization vector in nano-crystalline films makes a correlated small-angle wiggling but also the magnitude of the magnetization modulus fluctuates. LTEM studies were performed using JEOL 201 OF transmission electron microscope equipped with a post-column energy filter that provides an additional magnification of around 20 at the plane of the CCD camera with respect to the maximal attainable magnification by using a low field objective minilens.

INTRODUCTION In the early sixties of last century Gareth Thomas stood on the threshold of the development of transmission electron microscopy [ 11 and indisputably he became an inspiring and enthusiastic world leader in the application of this technique to solve materials problems. Only a few of us have pioneered so much in the structure-property relationship and in so many different areas of materials science. His research covered an extraordinarily broad range of topics, but it was and still is, characterized by a deep penetration into each subject to provide a thorough understanding. His seminal work stimulated hundreds of investigations over the last 40 years, clarifying structural aspects and thereby permitting analysis of the structural properties. Through these studies our understanding of the relationship between microstructure and property became more satisfactory and new concepts have been developed further. Besides structural properties Gareth Thomas showed great interest in the application of transmission electron microscopy to the field of functional materials. For instance he focused on a connection between the microstructure of various nano-structured alloys and the Giant Magneto-Resistive (GMR) properties [2,3] that also stimulated later our own scientific work [4,5]. Along these lines of his own interests this contribution concentrates on the application of transmission electron microscopy to functional materials, such as ultra-soft magnetic films for high-frequency inductors, to reveal the structure-property relationship. There exists an increasing demand for further miniaturization in portable appliances (e.g. mobile phones, palmtops), that is to say in communication tools. To this end the use of

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high-frequencies (e.g. 10-1000 MHz) in combination with thin magnetic materials is desirable. The use of magnetic films allows the integration of transformers and inductors into silicon 1C circuitry. Soft-magneticfilms are also widely used in modem electromagnetic devices as a high-frequency (>I00 MHz) field-amplifying component, e.g. in read-write heads for magnetic disk memories for computers and as a magnetic shielding material, e.g. in turners. The main requirements for the film material are: a high saturation magnetization, combined with a low coercivity and a small but finite anisotropy field. In addition the material should have a reasonably high specific electrical resistivity to reduce eddy currents, and also appropriate mechanical properties. To obtain the desired properties (low coercivity, little strain and very small magnetostriction) the use materials with grain size of the order of 10 nm nanometers, like nano-crystalline iron, becomes attractive [6]. Knowledge of local magnetic properties is essential for the development of new magnetic nano-sized materials. One of technique that is suitable for the measurement of local magnetic structures is Lorentz-Fresnel (or defocused) imaging mode of transmission electron microscopy. This rather classical TEM technique [7,8] has several outstanding advantages: uncomplicated application to various parts of thin foil, possibility of dynamical studies and good spatial resolution. Nevertheless, to obtain quantitative information from Lorentz micrographs is relatively difficult due to an in-direct link between image contrast and spatial variation of magnetic induction, which is problematic in regions of abrupt magnetization changes [9]. In this paper the possibility of a quantitative analysis of the magnetic properties of nano-crystalline iron using transmission electron microscopy is presented. The goal is to delineate a more quantitative way to obtain information of the magnetic induction and local magnetization. In particular the latter physical quantities affect the functional properties of ultra-soft magnetic materials for high-frequency inductors. One of the magnetic features that can provide quantitative magnetic information are the so-called magnetization or magnetic ripples, caused by local variation of magnetic induction that deviate from the mean magnetization direction [lo]. Besides Lorentz Fresnel (LTEM) also (off-axis) electron holographic modes can be used to analyze the magnetic structures [ 1I , 12,13,14,15]. For quite some time it is known that in polycrystalline and in nanocrystalline soft magnetic films with an induced uni-axial anisotropy, the direction of the magnetization wiggles around the easy axis producing a socalled micromagnetic ripple (see for example [lo] and references therein). The local variation of the magnetic field in a thin film influences the out-of-focus image in the transmission electron microscopy (TEM) via variation of the Lorentz force that acts perpendicular to directions of the electron beam and to the magnetic induction. Consequently, in Lorentz transmission electron microscopy (LTEM) the quasi-periodic oscillation of the transversal component of the local magnetization leads to almost parallel (1D) fringes in under- or overfocused images. From the spacing between the fringes the wavelength and the amplitude of the angular spread of the magnetization direction can be estimated using a simplified relation [I61 0.059-

C

Bt3,

where & is in degrees, B represents the magnetic induction in Tesla, f and Ax are the thickness of the film and the periodicity of the ripple structure (in micrometers), respectively. C = 2AUZo is the contrast of the LTEM image, where l o is the average intensity and AZ is the rms variation of the intensity due to ripples. This relation follows from both a classical approach of the LTEM imaging of micromagnetic ripple developed by Fuller and Hale [8] and a diffraction approach developed by Wohlleben [17], based on the theory of Aharonov and Bohm [18]. In both approaches the film is assumed to be flat and only the transversal component of the magnetization oscillates in the longitudinal direction. Often more complicated, e.g. bending or branching, Fresnel patterns are observed around magnetic domain walls, film edges, or for films with grain sizes larger than -50 nm [8,10,16,17]. The influences of the grain size and inhomogenieties on the periodicity and dispersion angle of the magnetization have been discussed in a number of papers [ 10,16,19,20]. Several of the studies has come to a quantitative analysis of Fresnel images of more realistic micromagnetic structure than proposed in [17], e.g. for a review reference is made to [21]. Here we focus on the analysis of an interesting complication of the Fresnel image, namely the variation of the contrast along the ripple fringes, which is visible in many LTEM micrographs, e.g. [17,22,23] but this has so far not been discussed in great detail. As an interpretation of this 2D contrast we suggest a model where the micromagnetic oscillations of the magnetization compete with oscillations due to topography of the film. Note

Effects of topography on the magnetic properties of nano-structured filmr

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that the fluctuations or variations discussed here are only related to space modulations either of magnetization in the film or of intensity in the LTEM image. Any time dependence is neglected.

EXPERIMENTAL OBSERVATIONS Nano-crystalline Fe-Zr-N films have been prepared by DC magnetron reactive sputtering with a thickness between 50 and 1000nm. The presence of zirconium is to catch the nitrogen in the iron matrix. Iron was chosen because it is easy to prepare and cheaper than other soft magnetic materials, e.g. cobalt. The nitrogen is added to get a small (nano-sized) grain size. Pure (99.96 at%) Fe sheets partially covered with Zr wires were used as targets. The N and Zr contents were controlled by varying the sputter power andor the ArlN2 gas mixture. A 64 k N m magnetic field was applied in the plane of the samples during deposition to induce uni-axial anisotropy of up to 1.6 kA/m. More details on the film deposition can be found in [6,20]. The films have been deposited on a glass or silicon substrates at several temperatures between room temperature and 200 "C. The DC-sputtered samples were deposited on either a silicon substrate covered by a polymer, which was removed in acetone after sputtering, or on a silicon substrate covered by a Si3N4 layer. The former samples were extracted on copper TEM-grids for support, while the latter samples kept their substrate because the layers were very thin. The deposition conditions were chosen to obtain a composition (FessZrl)l.,N, ,where the concentration of nitrogen was in the range x I 2 5 at%. The best films as far as nano-size dimensions are concerned have been obtained for 8 at%<x y,(Ll,). The combined reaction (CR) transformed alloys exhibit significantly more heterogeneous microstructures than conventionally processed FePd alloys, which is expected to alter the magnetic hysteresis behavior. It has been proposed [7] that the increased coercivity of the CR transformed alloys relative to the conventionally processed FePd alloys originates from a combination of a domain wall drag effect resulting from the reduced grain size (grain size hardening) and domain wall pinning by planar faults [7]. The intriguing interplay of recrystallization processes, which are driven by the stored energy of cold-work, and the ordering processes, which, interestingly, have been shown to include both continuous and discontinuous mechanisms [6], promises the potential to generate new microstructural morphologies in intermetallic FePd alloys with enhanced technical magnetic properties. However, in order to optimize properties intelligently a better basic understanding of processing-microstructure-property relationships for CR transformed FePd alloys has to be developed. To date, microstructural studies of CR transformed microstructures have been rather limited and only included results from transmission electron microscopy (TEM) [6, 71. Hence, here a systematic investigation the microstructure-property evolution during annealing of Fe-Pd alloy after cold-deformation by equal-channel angular pressing (ECAP) has been undertaken. The advantages of using ECAP for the cold-deformation prior to annealing are the preservation of the cross-sectional dimension and the controlled accumulation of very large amounts of total effective strain larger than 1.0 [8]. The latter enables the exploration of a wider range of driving forces for recrystallization than is attainable with cold-rolling for instance and expands the matrix of processing

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parameters. The evolution of properties and microstructure is observed by experimental studies using magnetic hysteresis measurements together with a combination of microstructural characterization methods of x-ray diffraction (XRD), scanning electron microscopy (SEM) and TEM. The combination of complementary methods of XRD, SEM and TEM enables a more complete microstructural characterization of the transforming Fe-Pd alloys that bridges the length scales from mesoscopic to nanoscopic. In his paper preliminary results regarding the evolution of microstructure and magnetic properties of equiatomic FePd that has been cold-deformed by a single pass of ECAP during isothermal annealing at 500°C are reported. Implications for structure-property relationships and the mechanisms of microstructural transformation of cold-deformed Fe-Pd during isothermal annealing are discussed.

EXPERIMENTAL PROCEDURE The equiatomic Fe-Pd alloy used in this study has been prepared from high-purity elemental starting materials using vacuum arc melting in an atmosphere of purified argon gas. Sections from the as-cast buttons have been cold-rolled to about 50% reduction in thickness, followed by a homogenization treatment at 950°C (1223K) for 6 hours (21.6ks) and quenching into ice-brine. In the as-quenched state the material consisted of grains of the disordered FCC y-(Fe,Pd) solid solution with an average diameter of approximately 130 pm. Billets of the as-quenched material 36mm long with square cross-sections of 6x6 mm2 have been cold-deformed at room temperature by a single ECAP-pass, 120" inner-die angle. Isothermal anneals after ECAP have been performed for up to 50 hours (180ks) at 500°C (773K), well below the critical ordering temperature TC=650"C(923K), in order to induce the combined reaction of concomitant recrystallization and ordering. The evolution of the magnetic properties was then documented as a function of annealing time using a vibrating sample magnetometer (VSM) producing a maximum field of 15 kOe at room temperature. The evolution of the microstructure as a function of annealing time has been studied by XRD, SEM and TEM using a Philips X'pert XR-diffractometer, a field-emission gun equipped Philips XL30 SEM and a Jeol 2000 FX TEM, respectively. Thin foils for TEM have been prepared by twin-jet electropolishing using an electrolyte of 82% acetic acid, 9% perchloric acid and 9% ethanol, all by volume, at approximately 0°C (273K) and 30 V.

RESULTS Figure 1 shows SEM micrographs obtained in the backscatter electron mode (BSE) of the microstructure of the Fe-Pd material in the as-quenched and as-deformed states respectively. The contrast observed is not associated with differences in elemental composition but rather purely due to differences in crystallographic orientation. The compositional homogeneity of the nominally equiatomic material has been confirmed by energy dispersive X-ray spectroscopy (EDS) of selected as-cast and as-quenched specimens. In Figure l a grains, including many annealing twins, can be distinguished for the as-quenched state. The average grain size after homogenization and quenching has been determined by computerassisted image analysis as approximately (130?5)p. After ECAP the microstructure consisted of heavily deformed grains that exhibited deformation bands at various length scales, i.e. shear, micro- and transition bands. The local rotations of the crystal lattice resulted in fairly rapid orientation changes and a mosaic

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Figure 1: SEM BSE micrographs of a) as-quenched Fe-Pd, scale marker 200pm, and b) as-deformed Fe-Pd, scale marker 50 pm.

b

Figure 2: a) TEM bright-field multi-beam micrograph of the dense dislocation cell-structure in a grain in the Fe-Pd after ECAP and b) SADP of the grain with beam approximately parallel to [loo]. Note the diffuse L1,-structure superlattice spots and spot splitting.

structure or cell structure inside a given deformed grain, which is reflected by the much more complex contrast in SEM BSE micrographs (e g Figure lb). The TEM micrograph in Figure 2 presents a typical example of the dense defect structure developed in the deformed grains. Using diffraction contrast images the dislocation density, pdlal=,has been estimated as pdlrloc = 10” - 10”cm ’. The accompanying selected area diffraction pattern (SADP) in Figure 2b clearly indicates the presence of a mosaic structure associated with the frequent small orientation changes across the dense dislocation walls between cells of lower dislocation density (Figure 2a). The matrix of the as-deformed material has the FCC crystal structure of the disordered y-phase and diffuse intensities for superlattice spots (e g. 110) in SADP’s indicate the presence of L1,-type either short-range order (SRO) or of very small (< 2nm) coherent precipitates of L1,-ordered orientation variants (Figure 2b) VSM experiments have been performed with the material prior to and after annealing to monitor the development of the magnetic age hardening curve. Some representative data collected from the VSM runs is collated in Table 1. The coercivity, H,, of the as-quenched disordered y-(Fe,Pd) solid solution was 27 Oe.


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The deformation induced during the single ECAP pass enhanced the coercivity to 37 Oe. During annealing the coercivity, H,, increases from a minimal value of the as-deformed state (37 Oe) to a maximum (523 Oe) after 12 hours of annealing at 5 0 0 ° C (673K) before decreasing again at longer annealing times to a lower value (349 Oe) (Table 1). Interestingly, coercivities of about 180 to 350 Oe have been reported previously [6, 71 for undeformed L1,-ordered FePd after long-time annealing at 5 0 0 " C , which had developed polytwin microstructures typical of the continuous ordering reaction. The type of magnetic age hardening behavior documented in Table 1 is consistent with previous studies [6, 71, which included a different mode of cold-deformation and different annealing temperatures. Thus this behavior appears to be characteristic of cold-deformed FePd during annealing at T y,(Llo), and the stored strain energy of cold-deformation. This driving


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force diminishes with increasing time during annealing mainly because the continuous mode of ordering is operative in the unrecrystallized volume and to a lesser degree because recovery processes lower the stored energy of cold-deformation. Thus, once the LRO parameter reaches the equilibrium value the recrystallization process or the CR transformation essentially comes to a halt. Then the annealing phenomenon of grain growth becomes significant in the CR transformed fraction of the material. This scenario is consistent with the experimental results presented here.

SUMMARY AND CONCLUSION 1)

2) 3) 4)

Isothermal annealing of the cold-deformed FePd at 500°C produces complex, hetcrogencous microstructures that exhibit significantly enhancd coercivities with respect to conventionally processed FePd with the polytwin structure. For the processing condition utilized here, the maximum coercivity (523 Oe) is associated with a partially recrystallized (=36%) microstructure of fully ordered y,-FePd with the L1,-structure. The magnetic age hardening behavior of the combined reaction transformed FePd observed here has been attributed to a grain size hardening effect. The observed microstructural evolution during annealing of the cold-deformed FePd has been rationalized in terms of the competition between the combined reaction, which may be considered as a discontinuous ordering mode with kinetics enhanced by the stored energy of cold-deformation, and the continuous ordering mode.

ACKNOWLEDGEMENTS The authors gratefully acknowledge support for this work from the National Science Foundation, DMRMetals (NSF-0094213), with Dr. K. L. Murty as program manager. One of the authors (HH) also gratefully acknowledges support from the Department of Materials Science and Engineering, University of Pittsburgh, during his sabbatical stay in Pittsburgh during the summer 2002.

REFERENCES 1. 2. 3. 4. 5. 6.

Weller, D., and Moser, A. (1999) IEEE Trans. Magn. 35,4423. Magat, L.M., Yermolenko, AS., Ivanova, G.V., Makarova, G.M., and Shur, YAS. (1968) Fiz. Metal. Metalloved. 26,511. Zhang, B., Lelovic, M., and Soffa, W.A. (1991) Scripta met. mat. 25, 1577. Khachaturyan, A.G. (1983) In: Theory of Structural Transformations in Solids”, p. 368, John Wiley & Sons, New York. Yanar, C., Wiezorek, J.M.K., and Soffa, W.A. (2000) In: Phase Transformations and Evolution of Microstructure in Materials, pp. 39-54, Turchi, P. et a1 (Eds).TMS, Warrendale. Klemmer, T., and Soffa, W.A. (1994). In: Solid-Solid Phase Transformations, pp. 969-974, Johnson, W.C., Howe, J.M., Laughlin, D.E., and Soffa, W.S. (Eds). TMS, Warrendale.

142 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

A. Deshpande et a1

Klemmer, T., Hoydick, D., Okumura, H., Zahng, B., and Soffa, W.A. (1995) Scripta met. mat. 33, 1793. Segal, V.M. (1999) Mat. Sci. Eng. A 271, 322. Berger, A,, Willbrandt, P.-J., Emst, F., Klement, U., and Haasen, P. (1988) Progr. Mat. Sci. ??, 1. Willbrandt, P.-J., and Haasen, P. (1980) Z. Metallkde 273. Doherty, R.D., Hughes, D.A., Humphreys, F.J., Jonas, J.J., Jensen, D.J., Kassner, M.E., King, W.E., McNelley, T.R., McQueen, H.J., and Rollett, A.D. (1997) Mat. Sci. Eng. A 238, 219. Reed-Hill, R.E., and Abbaschian, R. (1992) In: Physical Metallurgy Principle, p 247, PWS Pub. Co., Boston. Gleiter, H. (1969) Acta met. 12,1421. Yanar, C., Radmilovic, V., Soffa, W.A., and Wiezorek, J.M.K. (2001) Intermetallics 9, 949. Yanar, C.,Wiezorek, J.M.K., Radmilovic, V., and Soffa, W.A., (2002) Met. Mat. Trans A 33, 2413. Greenberg, B.A., Volkov, A.Y., Kruglikov, N.A., Rodionova, L.A., Grokhovskaya, L.G., Gushchin, G.M., and Sakhanskaya, I.N. (2002) Phys. Met. Metallog. 92,167. Kulovits, A. (2002) Diploma thesis, University of Vienna. Peiler, W. (2002) priv. comm. at MRS Fall Mtg, Boston.

a,

PART 3: STRUCTURAL MATERIALS


Nan0 and Microstructural Design of Advanced Materials M.A. Meyers, R.O. Ritchie and M. Sarikaya (Editors) Published by Elsevier Ltd.

MICROSTRUCTURE AND PROPERTIES OF IN SITU TOUGHENED SILICON CARBIDE Lutgard C. De Jonghe

R. 0. Ritchie

and Xiao Feng Zhang

'

' Materials Sciences Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA

ABSTRACT A silicon carbide with a fracture toughness as high as 9.1 MPa.ml' has been developed by hot pressing pSic powder with aluminum, boron, and carbon additions (ABC-Sic). Central in this material development has been systematic transmission electron microscopy (TEM) and mechanical characterizations. In particular, atomic-resolution electron microscopy and nanoprobe composition quantification were combined in analyzing grain boundary structure and nanoscale structural features. Elongated Sic grains with 1 nmwide amorphous intergranular films were believed to be responsible for the in situ toughening of this material, specifically by mechanisms of crack deflection and grain bridging. Two methods were found to be effective in modifying microstructure and optimizing mechanical performance. First, prescribed postannealing treatments at temperatures between 1100 and 15OO0Cwere found to cause full crystallization of the amorphous intergranular films and to introduce uniformly dispersed nanoprecipitates within Sic matrix grains; in addition, lattice diffusion of aluminum at elevated temperatures was seen to alter grain boundary composition. Second, adjusting the nominal content of sintering additives was also observed to change the grain morphology, the grain boundary structure, and the phase composition of the ABC-Sic. In this regard, the roles of individual additives in developing microstructure were identified; this was demonstrated to be critical in optimizing the mechanical properties, including fracture toughness and fatigue resistance at ambient and elevated temperatures, flexural strength, wear resistance, and creep resistance. INTRODUCTION Silicon carbide (Sic) offers many intrinsic advantages as a structural ceramic, including a high melting temperature, low density, high elastic modulus and hardness, excellent wear resistance, and low creep rates at elevated temperatures. This remarkable combination of features make Sic one of the most promising advanced structural ceramic materials for a variety of advanced engineering technologies. An imperative in these structural applicationsis high fracture toughness, which for commercially available Sic is typically on the order of 3 MPa.m'", well below that of grain-elongated Si3N4 and yttria-stabilized ZrOz, for example. This low toughness clearly limits its utility. The toughening of ceramic materials can best be induced by crack bridging mechanisms (i.e., extrinsic toughening by crack-tip shielding), which can result from crack paths bridged by unbroken reinforcement fibers in composite ceramics (ex situ toughening) [l-31, or by self-reinforcement from elongated grains in monolithic ceramics (in situ toughening) [4]. The latter approach was exploited in the present work by hot pressing Sic with Al, B and C additives (ABC-Sic [ 5 ] ) . Two strategies, specifically post-annealing heat treatments and changes in the additive content, were used to optimize the microstructure.

145

146

L.C. De Jonghe, R.O. Ritchie and X.F. Zhang

In this paper, we review the microstructure and mechanical properties of in situ toughened Sic. In particular, atomic-resolution TEM in conjunction with nanoprobe chemical analyses were employed to determine the structure and chemistty of the grain boundaries. Such atomic-scale characterization was correlated with materials processing and mechanical testing with the objective of optimizing the structural performance of ABC-Sic.

It is perhaps fitting that this work is presented in a symposium dedicated to the career of Professor Gareth Thomas, who was a pioneer in the development of materials through the use of transmission electron microscopy and intelligent microstructure design - this paper is dedicated to him in recognition of his leading contributionsto this field of materials science [e.g., [6,7]].

EXPERIMENTAL PROCEDURES Submicron P-SiC powder was mixed with 3 w P ? aluminum metal powder, 0.6 wt% boron, and 2 wt% carbon sintering. The slurry was stir dried, sieved, and uniaxially pressed at 35 MPa. The green bodies were hot pressed at 50 MPa at 190OOC for 1 hr in an Ar atmosphere. The final product, which was produced as 99% dense (3.18 g/cm3),4 mm thick and 38 mm diameter disks of polycrystalline Sic, was compromise of predominantly 4H and 6H a-Sic phases, with a minor fraction of 3C p-Sic. Some as-hot-pressed ABCSic samples were further annealed in a tungsten mesh furnace under flowing Ar, at temperatures between 1000 and 15OO0C, for times typically ranging from 72 to 168 hr. Structural and mechanical characterizations were performed for both as-hot-pressed and annealed samples. Structural and chemical analyses were carried out in a 200 kV Philips CM200 transmission electron microscope equipped with a windowless detector and corresponding X-ray energy-dispersive spectroscopy (EDS) system. A spatial-difference methodology was developed to determine the concentration of the impurities in the Sic grain boundaries using a nanoprobe with a diameter varied between 3 and 20 nm [8,9]. Indentation hardness, four-point bending strength, creep resistance, abrasive wear properties, R-curve fracture toughness, and cyclic fatigue-crack growth behavior at ambient and elevated temperatures were all evaluated for ABC-Sic. Experimental procedures for each of these mechanical property assessments are described in detail in the respective references quoted in this paper.

RESULTS General Aspects Some typical mechanical properties for as-processed ABC-Sic are summarized in Table 1. Of particular note is the fracture toughness, which has been measured to be as high as 7 to 9 MPa.m’”; this is the highest 4,value recorded for Sic to date.

Sample

ABC-Sic

4-Point Bending Strength (MPa)

691+12

Hardness* (GPa)

561f13

Fracture Toughness (MPa.mIn)

6.8-9.1 (iO.4)

Microstructure and properties of in situ toughened silicon carbide

147

Using X-ray diffraction, 70 vol.% of the hot-pressed structure was identified as hexagonal 4H-Sic and the remaining 30% as cubic 3C-Sic. TEM studies revealed that the 3C-to4H phase transformation, which is presumed to have occurred during hot pressing, promoted anisotropic grain growth [lo], resulting in high area density of plate-like, elongated S ic grains (Figure la); the length and width of these grains ranged from 3 to 11 pm and 1 to 3 pm, respectively, with an aspect ratio for 90 % of them between 2 and 5. Interlocking between the elongated Sic grains was often observed, as shown in Figure lb. Equiaxed 3CSic grains with a submicron size were also found in ABC-Sic. It is believed that the unusually high toughness of ABC-Sic results from a combination of crack deflection and principally frictional and elastic bridging by the elongated Sic grains in the wake of the crack tip [5,11]. To optimize this toughening mechanism, an understanding of the grain boundary properties and structure is crucial; consequently, extensive TEM studies were focused on this feature of the microstructure.

Figure 1: (a) Bright-field TEM images showing elongated Sic grains in ABC-Sic. (b) Two elongated grains are interlocked

Figure 2: (a) High-resolution electron micrograph showing a typical intergranular film between two Si c matrix grains. An amorphous structure is observed with a width of about 1 nm. (b) Distribution of amorphous grain boundary widths determined by high-resolution electron microscopy. The grain boundary width ranges from 0.75 nm to 2.75 nm, with a mean of about 1 nm. Figure 2a shows a high-rcsolution TEM image of a typical grain boundary area in as-pressed ABC-Sic. An amorphous intergranular film is seen between two adjacent Sic grains, about 1 nm in width. Using highresolution electron microscopy, the statistical width of these films was determined; results are shown in Figure 2b. Specifically, the amorphous grain boundary films range in width from about 0.75 to 2.75 nm, with a mean of -1 nm. This width is consistent with the values found in other ceramic materials [12,13], and with theoretical explanations developed by Clarke [14]. In some particularly wide films (e.g. -2.75 nm), nanoscale crystallites were recognizable, indicating local ordering in the glassy films. Earlier work on Sic sintered with boron and carbon showed a solid-phase sintering procedure with no grain boundary films being formed [ 15,161. The formation of the current films was due to liquid-phase sintering promoted by the A1 additives. The liquid phase allowed for densification of the Sic at temperatures roughly 200°C lower than Al-free compositions, i.e., at -1900°C instead of -2100°C. The amorphous intergranular films provide the preferred crack path, which is an essential element for the development of crack bridging and hence toughening in ABC-Sic.

148

L.C. De Jonghe, R.O. Rztchie and X.F. Zhang

Nanoprobe EDS analyses revealed that the grain boundary films contained substantial Al, 0, Si, and C. The oxygen came mainly from the SO2 surface oxidation of the Sic starting powder, and is one of the features responsible for difference in the behavior of different starting powders. Boron was below the detectability limit at the grain boundaries and in the interior of the Sic grains. In fact, boron was largely involved in forming A1&C7 secondary phases. Other secondary phases identified in ABC-SIC include A120C-Sic, mullite, A1203, A14C3, A1404C,and Al-0-C-B crystalline phases. These secondary phases were submicron in dimension and were present as triple-junction particles [8,17]. A1 in solution in the Sic matrix grains was also detected. Similar to Sic studied here, A1 and 0 concentrations in grain boundaries and their effects on Si3N4 phase transformation were reported by Goto and Thomas [ 181. Effects of Post-annealing The most significant phenomenon that we observed in the efforts of modifying microstructure of ABC-Sic was crystallization of amorphous intergranular films in post-annealing. While no significant change in the overall, large-scale microstructure was recognized, annealing at 1000°C for only 5-30 hours was sufficient to transform about half number of the amorphous grain boundary films into more ordered structures. Apparently, 1000°C was about the threshold temperature for activation of grain boundary diffusion. Annealing at higher temperatures for prolonged hours fully crystallized the intergranular films [8]. For example, whereas about 90% of the grain boundaries were amorphous in as-hot-pressed samples, 86% of the intergranular films in the annealed material were found to be crystalline. Figure 3a shows a highresolution TEM image of a grain boundary film crystallized after annealing at 1200°C for 500 hrs. In this image, the crystallized grain boundary film is not readily distinguishable because the grain boundary phase was strictly epitaxial with the 6H-Sic grain on the left-hand side. However, corresponding EDS detected substantial AI-0-Si-C segregations between the two Sic grains, confirming the existence of the boundary phase. An enlarged image from the framed area is shown in Figure 3b for closer inspection. The two adjacent Sic grains in this image show very different orientations. Due to a large deviation from the [l 1 201 type zone-axes, only (0004) lattice fringes, with the lattice spacing of 0.25 nm, could be resolved for the 4H-Sic grain on the upper-right side, while a two-dimensional [llZO] zone-axis lattice can be recognized in the 6H-Sic grain on the other side of the grain boundary. Under the imaging conditions for Figure 3, the black dots in the image correspond to cationic columns along the incident electron beam direction. Some black dots in the 6H-Sic grain were marked with black circles. The zigzag stacking of the dots correspond to the ...ABCACBABCACB... hexagonal structure of the 6H-Sic. The closest layer spacing along the c-direction is 0.25 nm.

Figure 3: (a) High-resolution electron microscopy image for a grain boundary area in ABC-Sic annealed at 12OO0Cfor 500 hrs. Amorphous intergranular film is crystallized. (b) Enlarged image from the framed area in (a). Atomic stacking in 6H-Sic grain on the bottom-left side is marked by black circles. 2Hwurtzite atomic stacking in grain boundary film (GB) are marked by white circles. Note the epitaxial orientation relationships between the grain boundary film and the 6H-Sic matrix grain.


149

The grain boundary layer outlined in Figure 3b can be distinguished by the abrupt change in the atomic arrangement. Some lattice points in the boundary region are marked with white circles. The projection distance between adjacent lattice dots in the grain boundary layers is very close to that in the neighboring 6H-Sic grain, and the characteristic zigzag lattice arrangement is also apparent in the boundary layer. These observations would suggest a grain boundary film structure similar to 6H-Sic. However, instead of the ...ABCACBABCACB.. . periodic stacking for every six layers in the 6H-type structure, the stacking period in the grain boundary phase contains only two layers in one period, resulting in ...ABABAB... stacking with an 0.5 nm repeat length, within a grain boundary width of about 1.25 nm. In conjunction with the quantitative analysis of the grain boundary composition and computer image simulations, we concluded that one of the intergranular crystalline phases was aluminosilicate with a AIzOC-SiC solid solution composition and a 2H-wurtzite structure (hexagonal unit cell, a = 3.1 8, c = 5.0 A) [8,19]. The crystallized structure usually has an epitaxial structural relationship with Si c matrix grains when the (0001) habit plane is available. As shown in Figure 3, the typical grain boundary width after crystallization remains about 1 nm.

To further study the crystallization process in intergranular films at elevated temperatures, an ABC-Sic sample was heated in situ in a 300 kV JEOL 3010 transmission electron microscope equipped with a hot stage. Figure 4 shows high-resolution images €or the same intergranular film before and after in sifu heating, respectively. The amorphous film prior to heating crystallized discretely after 25 hrs at 1200°C, as indicated by arrowheads in Figure 4b. The crystallization tended to proceed epitaxially on the (0001) plane of the adjacent 6H-Sic matrix grain, with a 2H-wurtzite structure similar to that observed in ex situ annealed Sic samples. No discemable features could be identified as potential preferential sites for nucleation at the SiCifilm interface. Presumably, local compositional fluctuations in the intergranular films serve as nucleation sites.

Figure 4: (a) High-resolution image of an amorphous intergranular film in as-hot-pressed ABC-Sic. (b) The same film as in (a) but after in situ heating in a transmission electron microscope at 1200°C for 25 hrs. Arrowheads indicate the discrete, crystallized boundary segments. Another significant consequence of the thermal treatment was coherent nano-precipitation within Si c matrix grains, as seen in Figure 5a. Although Figure 5a was taken from a sample annealed at 14OO0C,the plate-like nanoprecipitates first formed at 130OOC. The precipitates were uniform in size and shape, and dispersed within the Sic matrix grains. The projected dimension of the precipitates after 1300°C annealing is -4 x 1 nmz with a volumetric number density of 5x1OzZ/m3. The precipitates coarsened with annealing temperature, accompanied by decrease in the number density. Detailed high-resolution electron microscopy characterization and nanoprobe EDS analysis determined an AlbCs-based structure and composition for the nanoprecipitates [20]. A comparison between 6H-Sic and A14C3 structures projected along the [OOOl] direction is shown in Figure 5b. The similarity between the two structures caused coherent precipitation with the (0001) habit planes. The formation and coarsening of the precipitates at 1300 to 1600°C was a


150

consequence of lattice-diffusion-controlled classic nucleation and growth [20]. The diffusion of Al-rich species through Sic lattice starting at 130OoCresulted in At-enrichment in grain boundary films, as revealed by EDS.

Figure 5: (a) A14C3-based nanoprecipitate formed in a 6H-Sic grain annealed at 1400'C. The viewing directions are marked. (b) Atomic models for 6H-Sic and Al&3 projected along the [OOOl] direction. The similarity between the two structures can be seen.

The A1 content in intergranular films as a function of annealing temperature was analyzed with EDS. The results are plotted in Figure 6. It is apparent that Al solution in Sic grains decreased at llOO°C and especially above 1300'C, consistent with TEM observations that Al solutions formed nano recipitates &) changed exsolved from the Sic lattice. Not surprisingly, the A1 site density in the grain boundaries (NAI as well upon annealing. Below 12OO0C,the composition of intergranular films was virtually invariant, even while the intergranular films crystallized. The N*yB value was doubled at 1300°C, which can be readily correlated with diffusion of the Al-rich chemical species into the grain boundary films. The Al content in intergranular films after annealing at 1300°C is in agreement with All lSi0 @ c , a solid solution between 2Hwurtzite AlzOC and Sic [19]. At even higher annealing temperatures up to 16OO0C, N A ? ~changed marginally taking the standard deviation into account.

- . ; i

0

.

.

.

.

.

.

.

.

.-

200 400 600 800 1000 1200 1400 1600 1800 Anneallng Temperature ( O C )

0

Figure 6: Plots of EDS-determined Al site density in grain boundary films @A?*) (AVSiC, wt??)as a function of annealing temperature.

and in Sic matrix grains

The structural evolutions in intergranular films during thermal treatment demonstrated a profound influence on mechanical properties. It was found that the high strength, cyclic fatigue resistance, and particularly the fracture toughness of A3C-Sic at ambient temperature were not severely compromised at elevated temperatures. For example, the fatigue-crack growth properties up to 130OoC were essentially identical to those at 25OC. Figure 7 illustrates the variation in fatigue-crack growth rates, du/dN, with applied stressintensity range, AK, at a load ratio (minimum to maximum load) of R = 0.1 for ABC-Sic under different test conditions. It can be seen that at both 2% and 13OO0C,crack-growth rates display a marked sensitivity to the stress intensity but little effect is from change of loading frequency over the range 3 to 1000 Hz [21-231. Mechanistically, the lack of a frequency effect in ABC-Sic is expected as crack advance occurs via predominantly intergranular cracking ahead of the tip, as shown in Figure 8. Grain bridging in the crack


151

wake is a common feature. However, the absence of a frequency effect at elevated temperatures is surprising, particularly since comparable materials, such as Si3N4, A1203 and silicide-matrix ceramics, show a marked sensitivity to frequency at above 1000°C [24-281. In these later materials, softening of the intergranular films, grain boundary cavitation and viscous-phase bridging are common. In contrast, TEM studies of regions in the immediate vicinity of the crack tip in ABC-Sic (e.g. Figure 8) provided direct confirmation of fracture mechanisms which were similar at ambient and elevated temperatures, with no evidence for grain boundary cavitation (creep) damage or viscous-phase bridging at temperatures as high as 1300°C. We conclude that the unique high-temperature mechanical characteristics of ABC-Sic appear to be a result of the thermal-induced crystallization of intergranular glassy films. vZ25Hz. R=0.1 1 0 25% o 850°C A 1200°C 0 13OO0C F 3 H z . R=0.1 U 850°C 6 A 1200’C % 1300°C F25Hz. R=0.5 @ @ 1300°C

i

8

2 3 4 5 6 stress intensity Range, AK (MPam”‘)

Figure 7: Cyclic fatigue-crack growth rates, da/dN, in ABC-SIC as a function of the applied stress-intensity range, AK,for the tests conducted at temperatures between 25 and 13OO0C,load ratio R = 0.1, and frequencies between 3 and 1000 Hz.

Figure 8: TEM images of the intergranular crack profiles at the crack tip region in ABC-Sic at 1300°C under (a) cyclic loading (25 Hz, R = O.l), and (b) static loading. Arrows indicate the general direction of crack propagation. No evidence of viscous grain boundary layers or creep damage. The structural evolutions in grain boundary and matrix grains at elevated temperatures also benefit other mechanical properties of ABC-Sic. For example, the steady-state creep rate at high temperature of 1500°C was still impressively low (5 x 10-9/sat 100 MPa), and the creep rates at 120OoC in ABC-Sic was about three orders of magnitude slower than in single-crystal Ni-base superalloy tested under the same conditions [29]. In addition, 80% strength loss at 1300°C was restored by post-annealing [20]. Abrasive wear resistance was improved as well by formation of nanoprecipitates and by structural and compositional changes in grain boundaries after annealing [30]. These structural and mechanical characterization results demonstrate that the prescribed annealing is an effective way in tuning the microstructure and in turn optimizing the mechanical properties of ABC-Sic.


152

Effects of Additive Content Parallel to the post-annealing, another effort in tailoring microstructure and mechanical properties of ABCSic was in adjusting the nominal contents of the Al, B, and C sintering additives. A series of samples were prepared by changing content of one of the three additives. For example, the A1 content was 3 w f ?in most ABC-Sic samples. This was then increased to 4 np to 7 wt%, while boron and carbon contents remained at 0.6 and 2 wt%, respectively. The samples were referred to as 3ABC-, 4ABC, up to 7ABC-SiC, according to the weight percentage of the A1 content. Structural characterizations showed that A1 variations between 3 and 7 wt% did not reduce the densification of Sic samples under the same processing conditions. However, changing the A1 content did alter the microstructure, as illustrated in Figure 9 (only images for the 3ABC5ABC-, and 7ABC-Sic are shown). Although all samples were composed of plate-like, elongated Sic grains and equiaxed Sic grains, the size, aspect ratio and area density of the elongated grains varied significantly with increasing A1 content. The length of the elongated grains was found to be at a maximum in the 5ABC-Sic, with the aspect ratio increasing almost linearly up to 6 wt% Al. Compared to 3ABC-SiC, the aspect ratios in the 4ABC- to the 7ABC-Sic are much higher, but the area density of these elongated grains continuously decreases. A distinct bimodal grain distribution is seen in the 5ABC-Sic, with elongated a-Sic grains and submicron-sized equiaxed p-Sic grains [3 11.

Figure 9: Morphologies of the 3ABC-, 5ABC-, and 7ABC-Sic are shown in images in (a) to (c), respectively. Note the significant changes in dimension and area density of the elongated Sic grains. Note also the bimodal grain systems with 5ABC-Sic.

Samples 3ABC-Sic 4ABC-Sic

I

Phase and Volume Fraction 70% 4H 30% 3 c 19% 6H, 23% 4H 58% 3 c 67% 3C 24% 6H 76% 3C 20% 6H

Toughness Hardness* (GPa) 6.8fO.3 24.1

4-Point Bending Strength (MPa) 691f12

I

561k13

(MPa.ml’*)

I

6.3f0.7

480f30

8.9f0.4

598f34

3.2f0.2

533f58

3.9f0.4

I

23.9

*Vickers indentation, Load = 10 kg. E = 430 GPa (for calculation). Changing of the A1 content by 1 wt% not only caused a considerable change in the grain morphology, but also induced different degrees of 3C-to-4H and/or 6H-to-4H Sic phase conversion during hot-pressing. Xray diffraction spectra were collected from polished surfaces of the 3ABC- to 7ABC-Sic samples and from the starting powder for comparison. The spectra were matched with standard 3C-, 4H-, 6H-, and 15R-Sic phase spectra; results are summarized in Table 2. It was found that the 3 wt% A1 addition in the 3ABC-Sic was sufficient to transform all of the 20% preexisting 6H-phase seeds in 3C-dominanted starting powder


153

into 4H-SiC, with more than 50% of the p-3C transforming into a-4H phase as well. However, a 1 wt% higher A1 content resulted in a completely different phase composition by retarding the 3C-to-4H transformation. Further increases in A1 monotonically decreased the extent of the 3C-to-4H transformation, and apparently inhibited the preexisting 6H phase to be transformed into 4H. These results imply that the 3C-to-4H transformation was mainly promoted by boron and carbon, and the transformation was retarded when boron and carbon were compensated by increasing the A1 content. The experimental data also suggest close relationships between the phase composition and grain morphology, which would be expected if the pto-a transformation promotes the grain elongation [ 101.

In addition to the grain morphology, a factor that may have a determining influence on mechanical properties of dense, polycrystalline ceramic materials is the grain boundary structure and composition, as noted above. High-resolution TEM showed that about 90% of the intergranular films examined in the 3ABC-Sic processed an amorphous structure. In contrast, this fraction dropped to about 50% in the 5ABCSic, and all grain boundary films examined in the 7ABC-Sic were crystalline in structure. Quantitative EDS analyses indicated that increasing the nominal A1 content enhanced A1 concentration in the intergranular films, as well as in the Sic grains bulk, but the concentrations saturated in 5ABC-Sic. More than 5 wt?? A1 resulted in precipitation of excessive free Al, as observed in the 6ABC-and 7ABC-Sic. It should be pointed out that crystalline grain boundaries were prevalent in as-hot-pressed SABC-, to 7ABCSic without any post-treatment. It is thus plausible that enhanced A1 facilitates formation of crystalline intergranular films. These changes in microstructure can be expected to affect mechanical properties. Table 2 lists various mechanical properties for the sample series. The mechanical data illustrate the tradeoff mechanical performance which is often encountered in developing advanced ceramic materials. While the highest strength was obtained in the 3ABC-SiC, the hardness was at maximum for the 6ABC-Sic. As for toughness, it is clear that 5 wt% A1 resulted in the best toughness whereas 6 wt?? or higher A1 additions significantly degraded the toughness so that the materials became extremely brittle. Cyclic fatigue tests revealed that the 3ABC- and particularly the 5ABC-Sic displayed excellent crack-growth resistance at both ambient (25T) and elevated (1300'C) temperatures. Again, the crack propagation in both 3ABC- and 5ABC-Sic was intergranular, and crack bridging was observed in the crack wake. No evidence of viscous grain boundary layers and creep damage, in the form of grain-boundary cavitation, was seen at temperatures up to 13OO0C. The substantially enhanced toughness in the SABC-Sic was associated with extensive crack bridging from both interlocking grains, as in 3ABC-SiC, and uncracked ligaments, which only occurred in 5ABC-Sic. No toughening by crack bridging was apparent in 7ABC-SiC, concomitant with a transition from intergranular to transgranular fracture [32]. Similar to the aluminum concentration variations, changes in boron and carbon contents also alter the microstructure and phase composition. Typical results are shown in Figure 10. In this case, the A1 content was fixed at 6 wt%, while either the B or the C concentrations were changed. The phase composition determined by X-ray diffraction is noted in each image. It is clear that at fixed A1 (6 wt??)and B(0.6 wt%) contents, 1 wt?hadditional carbon promoted the formation of more elongated grains and enhanced the 3Cto-4H transformation (most of the 6H phase should originate from starting powder). This effect of carbon on 3C-to-4H transformation was also reported before by Sakai et al [33]. Growth of the equiaxed grains was found to be limited. With the A1 and C contents kept constant, increasing the boron content from 0.6 wt?? to 0.9 wt% largely increased the number density of elongated grains, but reduced their aspect ratio. In addition, boron promoted the 3C-to-4H and 6H-to-4H phase transformations more effectively than carbon. The positive effect of boron on the 6H-to-4H transformation is consistent with the previous observations of Huang et a1 [34].

-


154

6 Wt% A1

0 6 wt% Boron

0.9 wt% Boron

2wt%C

1 3wt%C

Figure 10: Microstructural changes with fixed A1 (6 wPh) and adjusted boron or carbon contents. Phase compositions are marked in each image. Effects of boron and carbon additions in changing microstructure and phase composition are clear. The systematic processing and characterizations of a series of ABC-Sic described above also allowed for the determination of the roles of Al, B, and C additives in developing the microstructures of silicon carbide. The observed effects may be summarized as follows: in terms of developing phase composition, boron is more effective in promoting the p-to-a phase transformations than carbon. Aluminum retards the p-to-a phase transformation,but promotes the 6H-to-4H transformation. As for the developing grain morphology, aluminum and carbon both promote anisotropic grain growth, whereas boron tends to coarsen the volume fraction, but reduce the aspect ratio, of the elongated grains. It should be noted that during processing, the combined roles of the Al, B and C additives often override their individual roles. For example, B and C together favor of the P-to-a phase transformation associated with grain elongation; however, the final microstructure does not necessarily have strongly elongated grains as B and C have opposite effects on anisotropic grain growth. Actually, the B:C ratio determines the final grain configuration. Aluminum has a different effect: when the B:C ratio favors the anisotropic grain growth, Al-rich liquid phase accelerates such growth so that the aspect ratio is hrther increased. However, if the AI:B and AI:C ratios are reduced, less liquid phases are expected to be present after the Al-B-C reactions so that the effects of A1 are diminished. Further experiments have indicated that even at constant A1:B:C ratios, a change in total amount of additives can still alter the grain configuration and phase composition significantly. This emphasizes the fact that the optimization of the mechanical properties of many structural ceramics such as ABC-Sic is a strong function of the absolute and relative amounts of the sintering additives.

SUMMARY ABC-Sic ceramics with unprecedented toughness values as high as 9 MPa.m'" have been developed by hot pressing @-Sicpowder with additions of Al, B and C. Such high fracture toughnesses were attributed to an in situ toughening mechanism primarily involving crack bridging by interlocked and elongated Sic grains. The anisotropic growth of Sic grains, which promoted such toughening, was the result of the liquid-phase sintering in which A1 additives acted to form a liquid phase. The existence of this liquid phase also lowered the sintering temperature to 190OoC. The toughening mechanism also requires intergranular cracking which was aided by the presence of amorphous intergranular films (typically 1 nm in width) in the grain boundaries of the as-processed ABC-Sic. Two effective methods for modifying the grain boundary structure and chemistry, as well as the overall microstructure, were investigated using post-processing annealing and adjustments in the nominal contents of sintering additives. Prescribed post-annealing treatments at higher than 1OOO°C were found to activate grain boundary diffusion and consequently caused the crystallization of most of the glassy intergranular films (not simply the


155

“pockets” at the grain boundary triple points). This led to superior high temperature strength, creep and fatigue properties at elevated temperatures with no evidence of creep damage in the form of grain boundary cavitation until temperatures above 14OO0C were reached. Using atomic-resolution electron microscopy and quantitative nanoprobe EDS, one of the crystallized intergranular structures was identified as 2H-wurtzite ahminosilicate. Heating at 13OO0C or higher also resulted in uniformly dispersed nanoprecipitates as a result of lattice diffusion. In addition, the lattice diffusion almost doubled the segregation of A1 into the intergranular films. It is believed that this significant microstructural evolution, which occurs during postannealing and is not characteristically observed in other advanced ceramics such as Si3N4, and A1203, is the origin of the many outstanding mechanical property attributes of ABC-Sic at elevated temperatures. These include high resistance to crack growth at temperatures between ambient and 13OO0C,far superior steadystate creep resistance than in single-crystal Ni-base superalloys, enhanced resistance to abrasive wear, and high-temperature strength loss recovery. Changing the nominal contents of the sintering additives, Al, or B, or C, was also used as an effective means to modify the microstructure. In particular, the area density and dimensions of the elongated Sic grains, their phase composition, and the grain boundary structure were all found to be sensitive to small variations in additive content. Using this approach, a series of materials processing with systematic changes in sintering atom additions were used to develop optimal microstructures for ABC-Sic. Compositions with superior fracture toughness, or creep properties or abrasive wear resistance were all defined. Such methods, in conjunction with prescribed post-annealing heat treatments, permit the tailoring of microstructures in ABC-Sic to achieve and optimize a wide range of mechanical properties in a highly controlled manner. ACKNOWLEDGMENTS

This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering of the U.S. Department of Energy under Contract No. DE-AC0376SF0098. Part of this work was made possible through the use of the National Center for Electron Microscopy facility at the Lawrence Berkeley National Laboratory. Thanks are due to Da Chen, Mark E. Sixta, Qing Yang, Rong Yuan, Jay J. Kruzic, and Rowland Cannon for their assistance and discussion in this work. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Faber, K.T. and Evans, A.G. (1983) Acta Metall. 31,565. Faber, K.T. and Evans, A.G. (1983) Acta Metall. 31,577. Becher, P.F. (1991)J. Am. Ceram. Soc., 74,255. Becher, P.F., Sun, E.Y., Plucknett, K.P., Alexander, K.B., Husueh, C.-H., Lin, H.-T., Waters, S.B. and Westmoreland, C.G. (1998) J. Am. Ceram. Soc. 81,2821. Cao, J.J., MoberlyChan, W.J., De Jonghe, L.C., Gilbert, C.J. and. Ritchie, R.O. (1996)J. Am. Ceram. Soc. 79,461. Thomas, G. (1994) Ultramicroscopy 54,145. Thomas, G. (1996) J. Euro. Ceram. Soc. 16,323. Zhang, X.F., Sixta, M.E. and De Jonghe, L.C. (2000) J. Am. Cerum. Soc. 83,2813. Zhang, X.F., Yang, Q., De Jonghe, L.C. and Zhang, Z. (2002) J. Microsc. 207,58. MoberlyChan, W.J., Cao, J.J., Gilbert, C.J., Ritchie, R.O. and De Jonghe, L.C. (1998), In: Ceramic Microstructure: Control at the Atomic Level, pp. 177-190,Tomsia A.P. and Glaeser A. (Eds). New York Plenum Press. Gilbert, C.J., Cao, J.J., De Jonghe, L.C. and Ritchie, R.O. (1997) J. Am. Cerum. Soc. 80,2253. Kleebe, H.-J., Cinibulk, M.K., Cannon, R.M. andRuhle, M. (1993)J. Am. Ceram. Soc. 76, 1969. Chiang, Y.-M., Silverman, L.A., French, R.H. and Cannon, R.M. (1994) J. Am. Ceram. Soc., 77, 1143. Clarke, D.R. (1987) J. Am. Ceram. SOC.70, 15.

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15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

L.C. De Jonghe, R.O. Ritchie and X.F. Zhang

Hamminger, R., Grathwohl, G. and Thummler, F. (1983) J. Muter. Sci. 18,3154. Lane, J. E., Carter, C.H. and Davis, R. F. (1988)J. Am. Cerum. Soc., 71,281. Zhang, X.F., Sixta, M.E. and De Jonghe, L.C. (2001) J. Am. Cerum. SOC.84,813. Goto, Y. and Thomas, G. (1995) J. Muter. Sci. 30,2194. Cutler, I.B., Miller, P.D., Rafaniello, W., Park, H.K., Thompson, D.P. and Jack, K.H. (1978) Nature 275,434. Zhang, X.F., Sixta, M.E. and De Jonghe, L.C. (2001) J. Muter. Sci. 36,5447. Chen, D., Gilbert, C.J., Zhang, X.F. and Ritchie, R.O. (2000) Actu Muter, 48,659. Chen, D., Zhang, X.F. and Ritchie, R.O. (2000) J. Am. Cerum. SOC.83,2079. Chen, D., Sixta, M.E., Zhang, X.F., De Jonghe, L.C. and Ritchie, R.O. (2000) Actu Muter. 48,4599. Hansson, T., Miyashita, Y. and Mutoh, Y. (1996), In: Fracture Mechanics of Ceramics, Vol. 12, pp. 187-201, Bradt R.C. (Ed). Plenum Press, New York. Zhang, Y.H. and Edwards, L. (1998)Muter. Sci. Eng. A256, 144. Edwards, L. and Suresh, S. (1992) J. Muter. Sci. 27,5181. Liu S.Y., Chen, I.W. andTien, T.Y. (1994)J. Am. Cerum. SOC.77, 137. Ramamurty, U., Kim, A.S., Suresh, S. (1993) J. Am. Cerurn. SOC.76, 1953. Sixta, M.E., Zhang, X.F. and De Jonghe, L.C. (2001) J. Am. Cerum. SOC.84,2022. Zhang, X.F., Lee, G.Y., Chen, D., Ritchie, R.O. and De Jonghe, L.C. (2003) J. Am. Ceram. Soc., in press. Zhang, X.F., Yang, Q. and De Jonghe, L.C. (2003) Actu Muter.,in press. Yuan, R., Kruzic, J. J., Zhang, X. F., De Jonghe, L. C. and Ritchie, R. 0. (2003) Actu Muter., in review. Sakai, T. and Aikawa, T. (1988) J. Am. Cerum. SOC.71, C-7. Huang, J.-L., Hurford, A.C., Cutler, R.A. and Virkar, A.V. (1986) J. Muter. Sci. Lett. 21, 1448.

Nano and Microstructural Design of Advanced Materials M.A. Meyers, R.O. Ritche and M. Sarikaya (Editors) 02003 Elsevier Ltd. All rights reserved.

MICROSTRUCTURE DESIGN OF ADVANCED MATERIALS THROUGH MICROELEMENT MODELS: WC-COCERMETS AND THEIR NOVEL ARCHITECTURES K. S. Ravi Chandran’ and Z. Zak Fang Department of Metallurgical Engineering 135 South 1460 East, Room 412, The University of Utah, Salt Lake City, UT 841 12

ABSTRACT Design and development of advanced materials for superior strength and toughness is a perpetual effort in meeting the material needs for demanding applications. The WC-Co cermet is one of the truly advanced materials, due to its unique combination of properties. Although such cermets are widely used, further improvements require a good mechanistic understanding of the microstructural aspects that govem the mechanical properties. The broad goal of this research is to establish such a mechanistic basis that enables a sound explanation of their excellent properties. The origins of the unique mechanical properties of traditional and novel cermets are closely examined using some microstructure-based models. It is shown that the superior properties arise as a result of the spatial arrangement of WC grains within Co, the strength and stiffness of W C and the constrained plastic deformation behavior of ductile Co layer binding the WC grains. It is also shown that the high toughness of “functional” cermets with hierarchical microstructures can be understood on the basis of the microstructure-based mechanistic models. The microstructure-based models illustrate the key aspects in traditional WC-Co microstructures that make them unique as well as the pathways for designing advanced composites materials following the architecture of WC-Co cermets.

INTRODUCTION WC-Co cermets as a class are one of the truly advanced materials, with their processing, microstructure and mechanical properties having been optimized through several decades of research and development. Cermets such as WC-Co, WC-Ni and WC-Tic-Co are successfully produced commercially with a high degree of control and reliability in mechanical properties [l-51. Owing to the large industrial demands, their continued development is an active area of research [6-101. It may, to some extent, be surprising to note that although empirical correlations between microstructural parameters and properties exist, a complete mechanistic basis for their superior mechanical properties is yet to be developed. The WC-Co microstructure is made of angular and hard WC grains that are nearly completely surrounded by ductile Co binder formed either by liquid phase sintering (LPS) of WCiCo under vacuum or by LPS followed by a low pressure hot isostatic pressing (sinter-HIP). Figure 1 illustrates typical microstructures of WC-Co at two different volume fractions of Co. When a comparison is made between several hard materials listed in Table 1, the general superiority of WC-Co is clearly evident: the possession of high elastic modulus, high hardness and flexure strength and acceptable toughness that is unique to WC-Co cermets. The objective of this study is to demonstrate the micromechanisms behind the unique properties of WC-Co cermets by using the approach termed, ‘microelement-modeling’ that captures the effects of the relevant microstructural parameters on properties. Idealized geometrical models consisting of microelements and ’ Author for correspondence, ernail r.i\icu.inincsiuIi cdu. Ph 1-801-581-7197. Fax I-801-581-4937

157

K.S. Ruvz Chundran and Z. Zak Fang

158

simple relationships of stress and strain from the mechanics of materials have been employed for this purpose The goal IS to develop simple yet reasonably accurate analytical models that allow insight into the aspects of the microstructural factors pertaining to WC and Co that lead to the superior properties This work also illustrates the fracture toughness characteristics of hierarchically structured “functional” WC-Co cermets through modeling It is shown that the present modeling approach is very effective in illustrating the microstructural effects on strength, creep resistance and fracture toughness of WC-Co cermets and their derivatives The approach may be useful in designing the microstructures of advanced materials, in general

WC-G%Co

wc-1O%Co

Figure 1: Microstructures of WC-Co cermets

MICROELEMENT MODELS OF WC-CO

In WC-Co cermets, the microstructure morphology can be idealized as WC grains enveloped by Co binder forming a nearly continuous matnx In such an arrangement, the ductile Co binder sandwiched between strong and stiff WC grains is in a physically constrained state during deformation and any modeling attempt should capture this aspect accurately and consistently In this work, the WC-Co microstn~cture idealized as a periodic arrangement of cubic WC inclusions in a continuous Co matrix as illustrated in Fig 2(a) Fig 2(b) illustrates the three dimensional geometry of the WC-Co unit cell. This unit cell can be thought of built upon the parallel and series arrangements of WC and Co phases, identified generally as A and B in Figures 3(a) and (b) The modeling approach basically involves first dividing the unit cell into such parallel and senes microelements and deducing the composlte behavior from the behavior of the elements It is first essential to determine the appropnate volume fractions of the microelements at different levels of division The division of the WC-Co unit cell can be done as illustrated in Figure 3(a-d) First, the two microelements of volume fractions V3 (WC) and V4 (Co) in Fig 3(d) add up in senes to form a composite, which in turn forms an effective microelement of volume fraction V1 (WC+Co) in Fig 3(c) Then, microelements of volume fractions V2 (Co) and V1 in Fig 3(c) add up in a parallel arrangement to complete the unit cell Hence, if the deformation behavior of parallel and series loading arrangements of WC and Co

WC-Cocermets and their novel architectures

159

are known, the macroscopic properties of the unit cell and hence the cermet can be derived using the appropriate volume fractions of the microelements at different levels.

Figure 2: Schematics of the idealized unit cell of the WC-Co microstructure

4 A co

Figure 3: Schematics of the division of WC-Co unit cell into microelements; (a) the parallel arrangement, (b) the series arrangement, (c) the parallel arrangement of microelements 1 and 2, and (d) the series arrangement of microelements 3 and 4. First, from geometric considerations of Figure 2(b), a microstructure parameter “c” can be defined in terms of the bulk volume fraction of Co, V, e o , as.

This definition allows the determination of volume fractions of the microelements as:

v,=

volume of element 1 1 volume of the unit cell (1 + c)’

v, =

volume of element 2 volume of the unit cell

v, = volume of element 3

volume ofelement 1

v4 =

-

1--

1 (1 + c)’

- 1

(1 + c)

volume of element 4 c -volume ofelement 1 (1 + c )

160

K.S. Ravi Chandran and Z. Zak Fang

It is to be noted that VI, V2, V3 and V4 are defined such that VI+V2=1 and V3+V4=1 to enable consistent division of microelements at different levels. The applicable mechanics of materials equations for the parallel arrangement of a two phase material, with the loading parallel to the interface are

E,, = EAVA+ EBVR

(3)

where Ell is the elastic modulus, EA and Eg are the modulus of phases A and B respectively, oapp is the applied stress, OA and CSB are the average stresses in A and B, respectively, cappis the composite strain and EA and EB are the average strains in phases A and B. respectively. VA and VB are the volume fractions of the respective phases. A similar set of equations for the series arrangement of phases A and B are

where E,, is the elastic modulus of the series composite loaded with the applied stress normal to the interface. Equations (3) through (8) along with equation (2) were used to determine the elastic modulus of WC-Co cermets and the results have been published elsewhere [l 13. Here we focus on the determination of the strength and creep resistance of the WC-Co cermets.

STRENGTH OF WC-COCERMETS In order to determine the strength of WC-Co cermets incorporating the constitutive properties of WC and Co, the plastic deformation behavior of the unit cell needs to be constructed. The deformation behavior of Co matrix can be represented in the Ludwik form as m=Eco~e

for the elastic part

d =K,,(E~)"

for the plastic part

with the total strain in the plastic regime being given by E = + .sP where E,,, 8 and B are the elastic modulus of Co, elastic strain and plastic strain in Co, respectively. K,, and n are constants of power law fit to the plastic part of the stress-strain curve of Co. Since the WC grains are much stiffer and harder than Co, to a first approximation, they can be considered to be rigid and the elastic strain contribution from microelement 3 is neglected leading to c3= 0 . The calculation of elastic stress-strain characteristics of WCCo cermet is straightforward and this is discussed elsewhere [12]. With the WC-Co unit cell deforming in the plastic regime, the following relationships can be written for the microelements 1, 3 and 4:


161

where & I , EZ, ~3 and 6 4 are the average plastic strains in the respective microelements. The superscript “p” is omitted hereafter for brevity CT{and C T ~ are the A ow stresses in microelements 1 and 2 resulting from the strain compatibility and the partitioning of the applied stress between microelements 1 and 2 according to the relation:

Simplification of the set of equations (1 1) leads to

At this stage, the physical constraint of Co deforming between WC grams should be considered. In the microelement scheme, this means that the plastic deformation of microelement 4 will be influenced by the constraint of rigid WC particles forming microelement 3, located above and below microelement 4 in the actual cermet. As the raho (hc,,/dwc) of thickness of Co in the microelement 4, ‘hc:, to the size of WC forming element 3, Idw:, decreases, the constraint expenenced by the Co matrix during plastic deformation would increase [13,14]. Murray [15] noted that through this effect the in-situ flow stress of Co is several times higher than that is observed in the bulk Co specimen It was suggested that the theoretical shear strength of Co is influenced by the presence of WC particle boundaries limiting the mean-free-path of shear deformation in Co as illustrated in Figure 4(a). To accurately represent the WC-Co plastic deformation behavior, this constraint effect should be incorporated in the model

Figure 4: (a) Schematic o f a ductile layer deforming under rigid WC grains [ 151 and (b) the normalized flow stress as a function of d/h as predicted by the plasticity solutions of Unksov [16]. One way to introduce the constrained deformation of Co is to modify the expression for in equation (13) to give a lower effective plastic strain for a given stress (in this case, 5,)acting on this element (see ref 12 for details):


162

Equation (15) is an approximation for the constrained deformation of Co, based on the analogy of Co deforming between WC grains to that of a ductile metal layer deforming between two rigid platens, as illustrated in Figure 4(b). Since there should be strain compatibility between microelements 1 and 2 , ~ & ~= : ,E, ,, = E~ and from equations (1 1) and (14)

From equations (12), (14) and (16) the plastic strain in the cermet is given by

The flow stress of Co is related to the plastic strain in Co as

Combining the equations (17) and (18), the normalized flow stress of the WC-Co cermet is given by

A

9

8

21

f

17

L,

v

Eqn. (19)

tn tn

13

5

9

zE ii

E x p t . Data (0.02 e p ) E x p t . Data (0.002 e p )

-a

-.Bm

5

z

1

E

0

0.2

0.4

0.6

0.8

1

W C V o l u m e Fraction, V p

Figure 5 : Comparison of the predictions from equation (19) with the experimental data of flow stress of WC-Co cermets normalized with the flow stress of Co. The experimental data is from: H. Doi, Elastic and Plastic Properties of WC-Co Composite Alloys, Freund Publishing Inc., Israel, 1974 Figure 5 illustrates the comparison of the normalized flow stress of the WC-Co cermet as a function of the volume fraction of WC. Both the predictions according to the microelement model (lines) for different strain hardening exponents of Co as well as the experimental data on compression strength of a variety of cermets are presented. The increase in flow stress, or alternatively the strength, comes from the fact that as WC


163

content increases, the extensive plastic deformation must be generated in the Co binder, especially the constrained region of Co present in the form of microelement 4. Equation (19) captures most of the dominant microstructural parameters that determine the strength of WCCo cermets, including, the size of WC grains, d,,, the thickness of the Co binder h,,, the volume fraction of Co, Vf,,, and the strain hardening behavior of Co, n, as well as the geometric constraint effect due to WC grains through the parameter For a given volume fraction of WC grains, increasing the constraint by manipulating the ratio h,,/d,, or the strain hardening characteristics of Co should lead to stronger WC-Co cermets. In particular, exceptionally high strength levels can be achieved if the WC volume fraction is increased as much as possible while ensuring a continuous layer of Co around WC grains throughout the microstructure. It is worth noting that the above conclusions are general and can be applied to any hard particle-ductile matrix composite system to achieve very high strength levels.

CREEP DEFORMATION BEHAVIOR OF WC-CO While a large fraction of commercial interest is in near-room-temperature applications of WC-Co, there are also research and developmental efforts [ 17, 181 in the area of high temperature deformation resistance in order to extend the temperature range of application of WC-Co cermets. Therefore, it is also of interest to examine the high temperature or creep deformation characteristics of WC-Co cermets. The microelement model for the creeping cermet can be devised following the same approach outlined in the previous section. The essential difference is that the strain rate takes the place of strain in the microelements. The strain rate compatibility between microelements 1 and 2 is expressed as

Since the WC grains can be considered as rigid and non-creeping due to its excellent strength retention at high temperature, the only constitutive relationship that is relevant here is that of the Co matrix:

where k,, and ncoare the constants characterizing the steady-state creep behavior of bulk Co. The complete details as well as extensive comparisons with the experimental data on several composites are published elsewhere [19]. The steady-state creep rates of WC-Co cermets can be derived as

The creep rate of the cermet, normalized with respect to that of Co is then expressed as

+=kcol E CO

r (1+ c)"

1

1"co

q 5 c +(1+c)2 o ( -1 y f l c o


164

Figures 6(a) and (b) illustrate the predicted normalized creep rates as a function of the volume fraction of rigid particles or second phase, in the general sense. In Figure 6(a) the effect of not including versus including &o in equation (23) is illustrated. It can be seen that without the constraint (i$co=l), there is virtually no difference between the creep rates of composites having matrices of with different creep exponents. In contrast, when the constraint factor, i$,, is included in the model, the effects of matrix creep exponents are differentiated. The predicted data are also in good agreement with the numerical results obtained by the self-consistent method, as illustrated in Figure 6(b), attesting to the reliability of the microelement modeling approach. For detailed comparisons involving other composite systems, ref. 19 may be consulted.

W

Unit cell model

E

...--.... Self-consistent Model

-

.WU

0

0.4

0.2

0.6

1

0.8

Vf of Second Phase, V

(a) (b) Figure 6: The predicted creep rates from the microelement model (equation (23)). (a) illustration of the effect of the WC grain constraint on Co and @) comparison with the data from a numerical model 0

,

--

,

,

,

/

,

-CO(8Oo"C) - - w c - c o (WC V( =o 81)

~

,

,

,

,

0

. I

,

.

,

co (8OOT)

I

I

I

,

I

I

,

w c - c o (WC V f =o 9)

0

-

T=800"C -12

"

1

"

2

1

'

"

3

Log Stress, MPa

'

:...

-

"

4

-12

"

1

'

I

2

"

"

3

T=800"C '

"

4

Log Stress, MPa

(a) (b) Figure 7: Comparisons of the predictions from equation (22) with the experimental data on steady state creep of WC-Co cermets Figures 7(a) and (b) illustrate the predicted steady-state creep rates compared with the experimental creep data of WC-Co cermets. The reasonable agreement is quite encouraging, given the idealizations employed in the microelement modeling. It is to be noted that according to the model, the creep exponent of WC-Co


165

cermet (equation (22)) is necessarily the same as that of Co. This is the result of the geometrical arrangement of Co as continuous matrix material nearly surrounding every WC grain in the cermet by a complex term that microstructure. The effect of WC grains is to modify the proportionality factor, to, includes most of the microstructural parameters, d,,, h,,, Vtca and $ca as in equation (22). As can be seen from Figure 7, the effect of WC grains is to decrease the value of proportionality factor, k, leading to a downward shift in the steady-state creep curve, relative to that of Co. FRACTURE TOUGHNESS OF WC-COCERMETS

Modeling of fracture toughness of WC-Co cermets is not only useful in developing an understanding of the microstructural factors that control the resistance to fracture, but also enables designing better cermets to mitigate cracking. This is of particular importance in cermets as a class, since the major constituent, WC, is nearly completely brittle and the cermet applications involve severe loading conditions, often leading to chipping. Fracture in WC-Co systems has been found [2,4,14] to occur mainly by the ductile rupture of Co through void nucleation and coalescence. Other fracture modes such as fracture along WC-Co interface and WCWC grain boundary decohesion as well as cleavage across WC grains were also noted [20,21]. These mechanisms occur especially at low volume fractions of Co binder in the composite at which the contiguity of WC grains begins to increase. The effect of the contiguity of WC skeleton on fracture toughness has also been demonstrated [22]. In a fracture toughness experiment, in a given crack plane, the crack propagation is easy along the relatively weak WC-WC boundaries and final fracture is primarily controlled by the area fraction of WC grains and Co regions intact across the crack plane ahead of the tip. Many studies have confirmed that the plastic deformation at the crack tip is restricted to a single binder layer in the crack plane located immediately ahead of the crack tip. Since the WC grains are nominally elastic under load, the deformation of Co at the crack tip will be highly constrained. Although there are a few fracture toughness models such as that developed by Nakamura and Gurland [23], these models do not explicitly consider this constraint effect in terms of microstructural parameters. More recently, a simple model to predict the fracture toughness of WC-Co cermets, incorporating the microstructural parameters was developed by one of the authors [24]. In that study, the total fracture energy of the WC-Co cermet is taken as the sum of the weighted fracture energies of WC and Co:

where Gwc is the critical strain energy release rate of WC phase and Vf, oeff,co and hco are respectively the volume fraction, the in-situ flow stress and the thickness of the Co phase. The parameter p represents the extent of plastic stretch of Co before fracture and is usually between 1 and 2. The first term accounts for the energy due to fracture along WC while the second incorporates the resistance arising from the plastic rupture of the ductile Co. Recalling

oeff,co

Kc,w,co

=

=

+

EW,,,

0.3(k)]

(1 - v/ )&,wc(l,2

(26)

where &,WCCO, Ewcco and vwcco are respectively the cermet fracture toughness, modulus and Poisson's ratio, and &.wc, Ewc and vwc are the fracture toughness, modulus and Poisson's ratio of WC respectively.


166

It can be seen that knowing Oeff,co, from equation (25), in addition to the properties and sizes of WC and Co phases, fracture toughness of the WC-Co cermet can be estimated. Equation (26) is not only considerably simpler, but also provides a direct correlation of fracture toughness to the microstructure parameters of the cermet.

rfiin

5

-

”

s

50

40

(II

Y

6

30

u)

0

9 C

c

20

0

0

10 20 30 40 50 M e a s u r e d Toughness, K c , e x p ,(MParn”2)

Figure 8: Comparison of the fracture toughness of WC-Co cermets calculated from equation (26) with the experimental data. The experimental data include over 100 data points from diverse studies, a listing of which may be found in ref. 24. The calculated fracture toughness according to equation (26) for a variety of WC-Co cermets reported in literature is plotted as a function of the experimentally measured fracture toughness data in Figure 8. An average value of 850 MPa for the bulk flow stress of Co in equation (25) and a value of p=2 in equation (26) were for these calculations. A study [25] confirms that the bulk flow stress of the Co may vary from 700 MPa to about 1100 MPa depending on the degree of dissolution of W and C as well as the nature and distribution of carbide precipitates in Co. Since the amount of W and C in solution would depend on a number of processing parameters such as that in hot pressing and heat treatment, the use of a more accurate value for the Co flow stress is not possible. Nevertheless, a reasonable agreement between the measured and the calculated toughness values, with only a few data points falling outside the +lo% confidence level, can be seen. The experimental data is from WC-Co microstructures having a wide variation of WC particle diameter from 0.66 pm to 7.8 pm and the mean free path in binder varying from 0.04 pm to 1.9 pm. The volume fraction of Co varied from 0.05 to 0.4 in these cermets. These represent a reasonably wide variation in the microstructural conditions of WC-Co cermets. Therefore, equation (26) may be taken as a fairly accurate representation of fracture toughness of WC-Co cermet class as a whole and may be used as a predictive tool in the microstructure design of two-phase cermets in general. DESIGNING CERMETS SUPERIOR TO WC-CO

Although WC-Co cermets occupy the largest share of tool applications, there have been considerable efforts in designing cermets with other hard materials and binders, but with a microstructural architecture similar to that of WC-Co. Noteworthy in this regard are the systems containing TaC or T i c or TiBz or A1203 as the hard component and Co, Ni, Fe or Cr as the binder component. With the increased availability of synthetic diamond, Co bonded diamond, with or without WC additions, has also become one of the contenders. It is of interest to examine where the level of fracture toughness that can be achieved in these cermet combinations lie relative to the WC-Co system. Equation (26) was used to calculate the fracture toughness values for the systems listed in Table 2 using the constitutive data also listed in the table. In the calculations, a hard particle size, d = 3 Fm and a binder volume fraction, Vf = 0.1 were assumed.


167

The calculated fracture toughness data is interesting for two reasons. First, all of the systems based on carbides and borides of Ti or Ta offer fracture toughness levels lower than that of WC-Co. Secondly, only the fracture toughness diamond-Co system was higher than that of the WC-Co system. The fracture toughness of the cermet is also influenced by the elastic modulus of the brittle phase, in addition to the volume fraction and properties of the binder. In fact, in table 2, the fracture toughness can be seen to increase in general with an increase in the modulus of the hard phase. This, although may seem to be unusual, is entirely consistent with fracture mechanics principles. The fracture toughness, K, is related to the critical strain energy released for an infinitesimal increase in crack length, G,, during unstable fracture as K,

=JG,E

and

where ys is the surface energy of the solid. The surface energy is indirectly related to the atomic cohesion in solids. It is known that the elastic modulus of solids in general increases with an increase in the melting point. The increase in melting point is generally thought to arise from an increased cohesion between in solids. Therefore, it is not unreasonable to expect that in principle, the surface energy should also be higher in a solid having a high stiffness, although this interesting aspect needs to be verified. Thus, the effect of increased elastic modulus should work to increase the fracture toughness directly through equation (27) and indirectly through equation (28).

TABLE 2. Calculated fracture toughness levels for different cermet systems

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.

.

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Figure 9: Fracture toughness of ceramics as a function of elastic modulus

i

168


In order to venfy the above hypothesis, the experimental fracture toughness data of a wide variety of ceramics are plotted as a function of elastic modulus in Figure 9 A reasonable correlation is obvious, with the fracture toughness increasing with an increase in elastic modulus Figure 9 may explain why only the diamond-Co system is supenor to WC-Co in terms of the calculated fracture toughness In equation (26), both the higher fracture toughness of diamond, together with its high value of modulus around 1000 MPa contribute to the increased value of the calculated toughness It also implies another remarkable fact to design cermets with fracture toughness levels higher than WC-Co, one should resort to hard constituents having elastic moduli higher than that of WC, with all other vanables in the cermet system being invariant HlERARCHlCALLY STRUCTURED “FUNCTIONAL” WC-COCERMETS In the past few years, a new approach that involves hierarchical structuring of microstructure constituents is used to design WC-Co cermets Figure 10 illustrates a “double cemented tungsten carbide (DC carbide)”, consisting of WC-Co granules (usually containing a very low cobalt content) embedded in a matnx of ductile Co Figure 11 illustrates an idealized schematic of this class of microstmctures This unique microstructure is functionally designed to boost the fracture toughness of the materiaIs without comprising its wear resistance Figure 12 illustrates the combination of superior toughness and abrasive wear resistance in the DC carbide The hierarchically structured DC carbide provides another degree of freedom that is not available in conventional WC-Co cermets One of the consequences of equation (26) is the fact that the fracture toughness of conventional WC-Co is derived nearly equally from WC as well as Co phases If the fracture resistance of WC can be increased by replacing it with a tougher WC-Co granule, the properties of DC carbide can then be augmented Additionally, the sizes of WC grains in the granules are much finer than that in the equivalent WC-Co cermet The contributions of the WC-Co granules to the overall properties of the DC carbide are two fold First, when compared to conventional WC-Co cermets, the fracture energy of the hard component in DC carbide, i e the term G,, in equation (24), is significantly higher The higher fracture energy of the hard component particles Contribute to increasing not only the overall fracture toughness, but also the micro-chipping resistance, which in turn is manifested in the form of good overall abrasive wear resistance Secondly, the WC-Co granules in DC carbide resist the tendency of being dug-out in an abrasive wear environment. Therefore, the excellent wear resistance of WC-Co can be preserved even though a significantly larger amount of Co is present in this matenal, relative to the conventional WC-Co cermet

Figure 10: Microstructures of a hierarchically structured “functional” WC-Co cermet. The microstructure contains a total of 27 wt. % Co. The fracture toughness of hierarchically structured DC carbide can be modeled using the same approach described in the previous section. This requires calculating first the fracture toughness of WC-Co granules, & grr according to equation (26) as:


169

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The fracture toughness of the DC carbide, &,Dc, can then be calculated as:

where Eoc is the elastic modulus of the DC carbide, V; is the volume fraction of Co in the matrix (not including the Co in granules) surrounding the granules, and cr>,co is the flow stress of Co in the matrix, hz,, is the thickness Co matrix and p=2. The

g&,co can

be expressed as

where oi,coand d,, are the unconstrained (bulk) flow stress of the Co matrix surrounding the WC-Co granules and the granule size The microstructures of two types of hierarchically structured DC carbides made in a recent study [26] are illustrated in Figures 13 and 14. Microstructures of DC carbides with a constant WC-Co granule size, but with a varying volume fraction of the matrix Co, are presented in Figure 13. Microstructures of DC carbides


170

1

having constant a volume fraction of Co in the matrix as well as in the WC-Co granules, but with varying sizes of granules are presented in Figure 14

I I

I.

Figure 13: Microstructures of hierarchically structured DC carbide cermets with varying Co volume fraction and constant WC-Co granule size. The matrix Co volume fractions from left to right are: lo%, 20% and 30%. In the granules, the WC grain size is 3 pm and the Co volume fraction is 10%.

Figure 14: Microstructures of hierarchically structured DC carbide cermets with constant Co volume fraction (30%) in the matrix and varying WC-Co granule size. The average granule sizes from left to right are. 130 pm, 90 pm and 60 pm. The size of WC grains in the granules is 3 pm,

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Figure 15 Comparisons of the predicted and experimental fracture toughness data for the hierarchically structured DC carbide cermets, as a function of (a) the volume fraction of matnx Co and @) the WC-Co granule size The effect of volume fraction of matrix Co on the fracture toughness of DC carbides with a constant WC-Co granule size is illustrated in Figure 15(a) The points are from expenments and the lines are the predictions


171

from equation (30). In these calculations, the value of C T : , ~ ~was taken as 87 MPa, based on the agreement between the experimental data and the theoretical predictions as illustrated in Figure 15(b). The effect of WC-Co granule size on the fracture toughness at a constant matrix Co volume fraction is illustrated in Figure 15(b). In this figure, the data predicted using equation (30) for four different o,',,, values is illustrated. It appears that the reasonable value of oi,,that can result in a good agreement between the experiment and theory is between 75 and 100 MPa. This choice of strength for the matrix Co surrounding WC-Co granules is not unreasonable, since a powder processing route was employed in the making of DC carbide cermets.

CONCLUDING REMARKS The present study has shown that the key aspects of the microstructure design of WC-Co cermets and its mechanistic basis can be understood by modeling with microelements. The microelements seem to capture important microstructure details such as the WC particle size, thickness and volume fraction of the Co binder, the constrained deformation behavior of Co as well as the fracture characteristics of WC and Co. The following conclusions may be drawn. 1. The strength of WC-Co cermets is largely controlled by the volume fraction of WC and is supplemented by the constrained plastic deformation of Co between the rigid WC grains. The expected functional form of this strength can be predicted using the microelement model that consistently incorporated the local average stresses and strains as well as the physical constraint experienced by the ductile Co during plastic deformation. 2. The creep or high temperature deformation behavior of Co is controlled primarily by the deformation characteristics of Co. The steady-state creep exponent of WC-Co is necessarily the same as that of Co. The effect of WC content is to shift the creep curves down in proportion to the WC volume fraction. An accurate prediction of creep deformation behavior requires consideration of the constrained deformation behavior of Co.

3. Fracture toughness of WC-Co cermets can be predicted quite accurately by considering fracture processes in WC and Co and is determined by the fracture properties of both phases. A large share of fracture toughness is contributed by the WC itself. Further, the constrained deformation behavior of Co during fracture, where the in-situ flow stress is several times larger than that of bulk Co significantly affects the fracture toughness. This latter aspect is responsible for the increase in the cermet toughness with an increase in Co content. Decreasing the WC particle size and increasing the strength of Co by alloying would help to increase the fracture toughness further.

4. A surprising finding of this study is that for designing cermets with toughness levels beyond WC-Co requires the use of hard materials with stiffness and fracture toughness levels higher than that of WC. The diamond-Co system is particularly attractive in this regard.

5 . Hierarchically structured WC-Co cermets with fracture toughness levels higher than that of the traditional WC-Co cermets are possible. The fracture toughness levels of such novel cermets can be predicted reasonably well using two-level models that consistently incorporate the microstructure details at different levels. These models explain well the effects of granule size and the Co volume fraction on fracture toughness.

6. The microelement models, having been validated by the extensive experimental data on WC-Co, suggest pathways for designing advanced composites with high strength and creep resistance and with reasonable fracture toughness. The simplest is to have a high volume fraction of hard and stiff constituent completely surrounded by a ductile and binding phase with good interface strength between them. Further, following the hierarchically structured WC-Co cermet design, different binder types at different levels of the structure may be chosen to enhance the combination of properties.


172

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

Pickens J.R., and Gurland, J. (1978) Muter. Sci. Eng. 33, 135. Viswanatham, R.K., Sun, T.S., Drake, E. F. and Peck, J.A. (1981) J. Mat. Sci. 16,1029. Lindau, L. (1977). In: Proc. 4th Int. Con$ Fracture, Fracture 1977, 2,215. Sigl, L.S. and Fischmeister, H.F. (1988) Acta Metall. 36, 887. Ravichandran, K. S. (1994) Acta Metall. Muter. 42, 143. Fang, Z. and Easton, J.W. (1995) Int. J. Refractory Metals and HardMaterials 13, 297. Fang, Z., Lockwood, G. and Griffo, A. (1999) Metall. Mater. Trans. 30A, 3231. Sommer, M., Schubert, W. and Warbichler, P. (2002) Int. J. of Refractory Metals and Hard Materials 20,41. Kinoshita, S., Kobayahsi, M. and Hayashi, K. (2002) J. Jpn. Soc. of Powder and Powder Metallurgy 49, 299. Carroll, D.F. and Conner, C.L. (1997). In: Advances in Powder Metallurgy and Particulate Materials, Proceedings of the I997 International Conference on Powder Metallurgy and Particulate Materials 2, 12-61, Metal Powder Industries Federation, Princeton, NJ, USA Ravichandran, K.S. (1994) J. Am. Ceram. Soc. 77, 1178. Ravichandran, K.S. (1994) Acta Metall. Mater. 42, 1113. Drucker, D.C. (1964). In: High Strength Materials, Zackay, V.F. (Ed), John Wiley & Sons, Inc., New York, NY. Evans, A.G. and Hirth, J.P. (1992) Scr. Metall. Mater. 26, 1675. Murray, M.J. (1977) Proc. Roy. Soc. Lond. 356A, 483. Unksov, A.P. (1961) An Engineering Theory of Plasticity, Buttenvorths, London. Mari, D., Ammann, J.J., Benoit, W. and Bonjour, C. (1988). In: Mechanical and Physical Behaviour of Metallic and Ceramic Composites, Proceedings of the 9th Riso International Symposium on Metallurgy and Materials Science, Published by Riso National Laboratory, Roskilde, Denmark, 433. Lee, I.C. and Sakuma, T. (1997) Metall. Mater. Trans. 28A, 1843. Ravichandran K.S. and Seetharaman, V. (1993) Acta Metall. Mater. 41,3351 Hong J. and Gurland, J. (1981). In: Science of Hard Materials, p. 649, Viswanatham, R.K., Rowcliffe D. J. and Gurland, J. (Eds) Plenum Press, New York. Slesar, M., Dusza, J. and Parilak, L. (1986). In: Science ofHard Materials, p. 657, Almond, E.A., Brookes, C. A. and Warren, R. (Eds), Inst. Phy. Conf. Series No. 75, Adam Higler Ltd., Bristol, UK. Chermant, J.L. and Osterstock, F. (1976) J. Mat. Sci. 11, 1939. Nakamura, M. and Gurland, J. (1980) Metall. Trans. 11A, 141. Ravichandran, K. S. (1994) Acta Metall. Mater. 42, 143. Roebuck, B., Almond, E. A. and Cottenden, A.M., (1984) Muter. Sci. Eng. 66,179. Deng, X., Patterson, B. R., Chawla, K.K., Koopman, M. C, Fang, Z., Lockwood, G. and Griffo, G, (2001) Int. J. Refiactory Metals and Hard Materials 19, 547.


THE IDEAL STRENGTH OF IRON D. M. Clatterbuck, D. C. Chrzan and J. W. Morris Jr. Department of Materials Science and Engineering, UC Berkeley and Materials Sciences Division, Lawrence Berkeley National Lab, Berkeley CA 94609 ABSTRACT

The ideal strength of a material can be defined as the stress that causes an infinite defect-free perfect crystal to become mechanically unstable. The ideal strength is of interest because it sets a firm upper bound on the mechanical strength the material can attain. It is also approached experimentally in situations where there are few mobile defects. The present paper is concerned with the ideal strength of iron. We specifically compute the ideal tensile strength of iron for tension along cool>, the weak direction, and the ideal shear strength for relaxed shear in the { 112) and 4 11>{110) systems. We also consider the influence of pressure on the strength. The computation is done ab initio using the Projector Augmented Wave Method within the framework of density functional theory and the generalized gradient approximation in order to account for the magnetism of the material. The fact that iron can have a high tensile strength is puzzling. Because Fe has a stable fcc phase (at least at moderate temperature) simple models suggest that it should be very weak in tension on . That it is not turns out to be due to the difference in magnetic character between the bcc and fcc phases, which has he consequence that iron behaves like a “typical” bcc metal in both its tensile and shear behavior. INTRODUCTION

For almost three millennia the element Fe and its alloys have provided the most commonly used structural metals, and remain the materials of choice for a large fraction of all engineering structures and devices. There are three basic reasons why this is true. First, Fe is common in the earth’s crust, and is relatively easy to extract from its ores. Second, Fe has reasonable inherent strength, and can be alloyed or processed so that it becomes even stronger. Third, Fe (or steel) is a very flexible material that can be made soft for forming, or hard for structural strength by fairly straightforward changes in the way it is processed. The strength of Fe is a prerequisite. If it did not lend itself to the creation of alloys that are “as strong as steel”, it would not be widely used. Advances in computational techniques and computing machines have recently made it possible to compute the ideal strength of a crystalline material - the stress that drives the crystal lattice itself unstable and sets an upper bound on the strength the material can possibly have [1,2]. When these techniques are applied to Fe, as was done for the first time in the work reported here [3,4], a couple of surprises emerge. First, Fe is not really all that strong. Second, it is surprising that Fe is as strong as it is. These statements can be based on a simple model that is derived from the dominant role of symmetry in determining the ideal strength, as is well established from first-principles calculations [2,5]. The energy of a deformed crystal depends on the six independent components 173

174

D.M. Clatterbuck, D.C. Chrzan and J.W. Morris Jr.

of the strain and is, hence, a 6-dimensional hypersurface in a 7-dimensional space. The local extrema and saddle points on this hypersurface almost always correspond to structures with high symmetry. The configurations of greatest interest are the saddle points on the energy surface that neighbor the initial state. For a given load geometry (stress state) the crystal deforms along a particular path on the strain-energy surface as the load is increased. The stress required to dnve the system is related to the local slope of the energy surface in the direction of travel. The ideal strength is determined by the steepest slope along the path between the initial state and the first saddle point. The steepest slope corresponds to the maximum stress; the crystal is elastically unstable as soon as the point of maximum stress is passed. To understand the ideal strength it is necessary to identify the relevant saddle point structures. Four such structures have been identified in studies of the ideal strengths of bcc metals: the fcc structure, the simple cubic (sc) structure, a body centered tetragonal (bct) structure, and a base centered orthorhombic structure [6-81. For example, relaxed shear of a bcc metal in the “easy” 4 1 1 > direction on the (112) or (110) plane generates a stress-free bct structure [8]. Because the {112} shear system is not symmetric in the sign of the shear direction, it is also possible to move from a bcc structure to a base centered orthorhombic structure by shearing in the “hard” direction. Uniaxial tension in the 4 1 1 > direction takes a bcc structure to a sc structure [6,8], while cool> tension moves it from bcc either to fcc (the Bain transformation in Fig. 1 [2]) or to bct (the same stress-free bct structure encountered in shear [7]). The competition between the two possible paths that can be reached by 4 0 1 1 tension has the result that, while most bcc crystals (Mo, W) are governed by the instability associated with the fcc structure and, ideally, fail in cIeavage, at least one (Nb) becomes unstable with respect to evolution toward the bct saddle point and fails in shear [7].

Figure 1: The bcc crystal structure becomes the fcc structure after elongation along the direction. Ab initio total energy calculations of the ideal tensile strengths of unconstrained bcc metals show that they are weakest when pulled in a direction [2] (unsurprisingly, {OOl} is the dominant cleavage plane in bcc metals). A constant volume tensile strain along converts the bcc structure into fcc at an engineering strain of about 0.26 (the ’Bain strain‘, Fig. 1). Since both structures are unstressed by symmetry, the tensile stress must pass through at least one maximum along the transformation path. If we follow Orowon [9] in assuming a single extremum (the solid line in Fig. 2), and fit the stress-strain curve with a sinusoid that has the correct modulus at low strain, the ideal tensile strength is approximately

in good agreement with ab initio calculations (for example, om = 30 GPa = 0.072 E for W [lo]). Since the modulus of Fe is significantly less than that of other transition metals, such as W, Mo and Ta, the upper bound on its strength is smaller as well.

The ideal strength of iron

0.00

0.10

0.20

Engineering Strain

0.30

0.00

175

0.10

0.20

Engineering Strain

0.30

Figure 2: The energy as a function of strain has an extremum at the fcc structure which can be a local maximum (solid line) or minimum (dotted line). Assuming sinusoidal form, the inflection point governing the ideal strength falls at a much lower strain in the latter case, and the ideal strength is significantly less. But a second problem intrudes when we extend this analysis to iron [3]. The fcc phase in Fe is known to have an energy only slightly above that of bcc and is at least metastable at low temperature. In fact, the thermomechanical treatments that are used to process structural steel (particularly including the many developed or explored by Gareth Thomas) rely on the ease of transforming it from bcc to fcc and back again. If we assume a metastable fcc phase connected by a continuous strain-energy curve (the dotted line in Fig. 2), the tensile instability intrudes at a much smaller strain, and the ideal strength should be only about 6 GPa (versus 12 GPa based on an unstable fcc). This number is too small to be credible. Since tensile stresses that are several times the yield strength are developed ahead of crack tips in elastic-plastic materials, steels with yield strengths much above 1 GPa would necessarily be brittle. In fact, steels with much larger yield strengths have high fracture toughness and considerable ductility. A possible resolution of this paradox is suggested by the work of Herper et al. [ 111. They computed the energies of Fe for various magnetic states and lattice strains. Their calculations suggest that the energy of ferromagnetic Fe increases monotonically if it is distorted toward an unstable, ferromagnetic fcc, which can be stabilized by transforming into a complex antiferromagnetic state. This has the consequence that the low energy antiferromagnetic fcc phase is a minimum rather than a saddle point on the strain-energy surface of Fe. While Herper, et al. [ 111 did not investigate the point (they had other interests), this suggests the possibility that ferromagnetic Fe may become mechanically unstable under 4 OO> tension before encountering the magnetic transition that converts it into the antiferromagnetic state that stabilizes fcc. If this is the case, Fe can have the mechanical behavior of a typical bcc metal, while still having a metastable fcc structure that is stabilized by a late magnetic transition. That is, it can be both strong and easy to process.

As we shall show below, this is the apparent explanation for the fact that Fe has both a useful tensile strength and an fcc phase that facilitates processing. Going further, we shall establish that Fe naturally fails in cleavage on (100) planes and compute its tensile and shear strengths. Finally, we consider how the strength of Fe is affected by superimposed hydrostatic stress as encountered, for example, at the tip of a sharp crack.

116


COMPUTATIONAL METHODS To compute the ideal strength of Fe we must include magnetic interactions. These are non-local, and considerably complicate the problem. In particular, the non-local magnetic interactions have the consequence that the local density approximation cannot be used. In fact, computations based on the local density approximation predict that the ground state of Fe should be a non-magnetic close packed structure rather than a bcc ferromagnet. The use of the generalized gradient approximation (GGA) has been shown to correct this problem [ 121. Full Potential Linearized Augmented Plane Wave (FLAPW) calculations which make no further approximations beyond the GGA (assuming convergence of the basis set, charge density representation and Brillouin Zone integration) are probably the most reliable [13]. Unfortunately, in order to efficiently relax the stresses orthogonal to the applied stress, the stresses on the unit cell must be directly computable, and there are no implementations of the FLAPW method known to us that can do this in a straightforward way. For this reason we selected the Projector Augmented Wave (PAW) method, originally developed by Blochl [ 141, for this work. PAW is an approximation to FLAPW that captures most of its important features and can be formulated to calculate local stresses by the Hellman-Feynman method. We also performed some calculations using the FLAPW method as a check on the accuracy of the results. The two methods are in reasonable agreement. The details of the computation are given in ref. [4]. The ideal strength was computed for uniaxial stress in tension or shear. The lattice vectors were incrementally deformed in the direction of the imposed stress, and at each step the structure was relaxed until the stresses orthogonal to the applied stress vanished, as indicated by those components of the Hellman-Feynman stresses being less than 0.15 GPa [15]. Because there is no unique measure of strain for a given finite deformation, we describe our deformations in terms of the engineering strain from the equilibrium structure. The initial set of lattice vectors ra (a=!,2,3) in an orthogonal coordinate system become the vectors ra' after homogeneous deformation by the transformation :r = rla + D, rJa . From this transformation, we define the strain to be e,=[D,,+DJ,]/2. As is customary, we redefine the shear strains to be yIJ=2elJfor i#j. While the engineering strain is convenient for describing the change in the lattice vectors from their original configuration, the Cauchy (true) stress cannot be calculated from the derivative of the free energy with respect to this strain measure. To compute the Cauchy stress we take the derivative of the free energy with respect to the incremental strain from a nearby reference state, yielding a stress that converges to the thermodynamic definition of the Cauchy stress in the limit of small incremental strain. It should be noted that the ideal strengths determined from these calculations are for quasi-static deformation at OK, and that other dynamic instabilities such as soft phonons may lower the ideal strength.

EQUILIBRIUM STRUCTURES We chose to compute the ideal strength with the Projector Augmented Wave (PAW) method because of its computational efficiency and its ability to treat lattice stress. To check the accuracy of the method, we computed the energy as a function of volume for several magnetic structures and compared the results to calculations done using the FLAPW method as well as

The ideal strength of iron

111

with available experimental data. The magnetic structures included the following: bcc ferromagnet (FM), fcc ferromagnet (FM), fcc antiferromagnet (AFM), fcc non-magnetic (NM). The results are plotted in Fig. 3. The results for the ferromagnetic bcc phase are tabulated in Table 1 .

%

a: E

I

45

-

40

.

35

.

30

-

25

.

+BCC-NM +BCC-FM +FCC-NM +-JFCC-FM -c- FCC-AFM * FCC-DAFM

20 .

60

P

64

68 72 76 80 Vdurneiatom (au')

84

88

Figure 3: The energy and magnetic moment per atom as functions of volume computed using the PAW method for Fe in the bcc (filled symbols) and fcc (open symbols) crystal structures for several magnetic states: (diamonds) non-magnetic (NM), (squares) ferromagnetic (FM), (circles) anti ferromagnetic (AFM), and (triangles) double period anti ferromagnetic (DAFM). The discontinuity in the fcc FM curve separates two distinct phases with different magnetic moments. In general the agreement between the two computational methods is good. Comparing the PAW and FLAPW methods, the equilibrium volumes of the various phases agree to within 1%. The elastic constants of the bcc FM phase agree to within 3% with the exception of c44, which has a discrepancy of 13%. Compared with experimental measurements of the bcc phase at 4 K, the PAW calculations predict a lattice parameter that is too small by 1% and elastic constants which are generally about 10% too large suggesting a slight over-binding (the only discrepancy being c44 which is 18% too small). From the computed elastic constants, the relaxed tensile modulus in the direction, E= l/sll, is found to be about 29% too large, while the relaxed shear modulus in the 4 11> direction, G= 3c44(c1I-c12)/(4c44+c~ I - C I ,~is) about 18% too large.


178

In regards to the magnetic properties, both the PAW and FLAPW methods correctly predict that the ground state is the bcc ferromagnetic phase with a magnetic moment of 2.20 pg and 2.15 p~ respectively as compared to the experimental value of 2.22 pg [. Both sets of calculations also predict that the ferromagnetic fcc phase undergoes a pressure-induced, first order phase transformation at a volume of 76-77 au3/atom from a low-volume, low-moment phase to a highvolume, high-moment phase. The groundstate magnetic structure of fcc Fe has been debated on theoretical grounds extensively in the literature. While bulk fcc Fe is difficult to achieve experimentally at low temperature, there is some probative experimental data on the magnetic state of nearly pure Fe in the fcc crystal structure. It is possible to stabilize fcc Fe by growth as a thin epitaxial film or as small precipitates in a copper matrix. Tsunoda [17,18] found that small fcc precipitates in Cu that are almost pure Fe have a spiral spin density wave (SSDW) ground state. Knopfle et al. [19] have recently published calculations using the modified augmented spherical wave method that show good agreement with this experimental data. Their minimum energy fcc structure has a spiral vector of q=(0.15,0,1) with an energy which lies m

Clea wage

; 2

-E rz

m

0.5

0

20

207

46

80

0.0 80

1o(E

Crystallinity. x (%)

Figure 8 Normalized impact toughness and transition in fracture behavior with increasing amount of crystallinity in La-based BMG.

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Annealing above T, leads to crystallization Partial crystallization increases strength of the BMGs However, controlling the shape, size and volume fraction of precipitated phase is necessary to optimize the mechanical performance of such systems It is generally observed that up to 30 vol.% of crystallization in the amorphous matrix, the strength increases whereas beyond this limit, a precipitous drop in the fracture strength of the alloy is observed as exemplified in Fig 7 [29] Also, the fracture morphology changes, from ductile vein type fracture to brittle intergranular cleavage type fracture Figure 8 represents the fracture behavior of a BMG with increasing amount of crystallinity Commensurate with the change in fracture morphology, marked changes in other physical properties such as viscosity and elastic modulus of the partially crystallized alloys are also observed, Fig 9 [29] This type of transition is attributed to the attainment of percolation threshold and networking of the crystalline phases At lower amount of crystallization, deformation takes place by viscous shear deformation D u n g this deformation, a considerable rise in temperature can be seen, which results in localized melting of the amorphous phase Residual amorphous phase accommodates most of the plastic deformation through the formation of shear bands and acts as a crack shield With increase in crystallization, shear band and cleavage dominate the nature of fracture and at higher levels o f crystallization, intergranular cleavage type fracture takes place [30] This change in the fracture behavior can also be explained on the basis of the viscoelastic response time As viscosity of the material increases with annealing and with the increase in crystallization, charactenstic response times also increase As a result of this tendency of the amorphous and partially amorphous alloy towards vlscous deformation decreases under the condition of same strain rate

56 54 52 50

48 46

0

20

40

Crystallinity,

60

x

(%)

80

100

Figure 9. Variation in viscosity and elastic modulus with crystallization for La-based BMG Hardness tests on these systems show evidence of plastic flow in the form of shear bands There is a drastic increase in hardness (and corresponding loss in ductility) upon heat treatment in the supercooled liquid region In the amorphous alloy shear bands can be seen around the indents and with the increase in annealing time and crystallization tendency for shear band formation decreases At large crystallizations, cracks emanate from the edges of the indent Indentation behavior of amorphous and partially crystallized Pd-NI-P

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and Zr-Cu-Ni-A1 BMG [31] has shown pile-ups at the edges of the indents. Examination of the subsurface deformation patterns (obtained using the bonded interface indentation technique) shows semicircular and radial shear bands at the interface. With increase in crystallinity density of semicircular shear bands decreases with a increase in radial shear bands. Cracks can be seen when the BMG is totally crystallized. In all the experiments constancy of the normalized deformed zone size can be seen. This observation proves the validity of the expanding cavity model and the pile-up at the edges substantiates the validity of the slip line field theory. Kim et al. [32] have reported the formation of nanocrystals by performing indentation tests on BMGs. The free volume theory is found to be very useful in explaining the mechanism of flow and fracture of metallic glasses. CONCLUDING REMARKS

Bulk metallic glasses are important materials, not only because of the unique combination of properties exhibited by them, but also because of the fact that they can be used as precursors for bulk nanocrystalline materials. From the limited studies conducted on the nanocrystalline materials, it can be concluded that their properties are better than those of bulk metallic glasses. The important point to note is that their properties are very sensitive to the microstructure. In order to precisely define a microstructure, the four basic parameters, i.e. phase, shape, size and volume fraction of precipitates are to be taken care of. While synthesizing ex situ composites, volume fraction, shape, size of the second phase particles can be controlled independently. In case of in situ composites, numerous choices exist. In-depth knowledge of phase selection sequence and selection of processing parameters is needed. There are a few studies in the growth behavior of the phases. In nanomaterials, interface to volume fraction is very large. The interface structure is the least studied and hence, understood in these systems. This, unless clearly understood, may lead to hindrance in our ability to design desired microstructures. While nanomaterials can be processed by powder compaction, their production using BMGs as precursors avoids problems (inherent to compaction) such as residual porosity and surface contamination. Nanocomposites can be considered to be more stable than their “parents” because the nanocrystals have very low grain growth rate and the residual amorphous phase becomes resistant to further crystallization. Mechanical behavior of the all these phases may be distinctly different and they should be treated separately. As has been observed from the mechanical behavior of BMGs and composites, they are highly sensitive to morphology and volume fraction of the precipitates. In order to design a microstructure with optimum and reproducible mechanical properties, proper understanding of the above parameters is needed. Thus, broadly speaking, the combination of amorphous regions and complex crystallites in the microstructure has the potential to revolutionize technological development. ACKNOWLEDGEMENTS

The authors would like to thank Profs. K. Chattopadhyay, V. Jayaram, B. S. Murty, and Drs. S. Banerjee, G. K. Dey and N. Nagendra for the fruitful interactions and discussions on various topics related to BMGs. They would also like to acknowledge the collaboration with Prof. A. Inoue and Drs. D. V. Louzguine and N. Nishiyaina of the Institute for Materials Research, Tohoku University, Sendai, Japan and Prof. Y . Li of National University of Singapore. Research funding from the Board of Research in Nuclear Sciences (DAEOIMMTISRGI105) is gratefully acknowledged.

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REFERENCES 1. Johnson, W.L. (1999) MRS Bulletin 24,42. 2. Inoue, A. (2000) Acta Mater. 48,279. 3. Basu, J. and Ranganathan, S. (2002) Sadhana 28,l. 4. Inoue, A,, Fan, C., Saida, J. and Zhang, T. (2000) Sci. Tech. Adv. Mater. 1, 73. 5. Klement, W., Willen, R. and Duwez, P. (1960) Nature 187,869. 6. Drehman, A.J., Greer, A.L. andTumbul1, D. (1982)Appl. Phys. Lett. 41,716. 7. Inoue, A., Ohtera, K., Kita, K. and Masomoto, T. (1988) Jpn. J. Appl. Phys. 27, L2248. 8. Amiya, K. and Inoue, A. (2000) Mater. Trans. JIM 41,1460. 9. Kim, Y.J., Busch, R., Johnson, W,L., Rulison, A.J. and Rhim, W.K. (1994) Appl. Phys. Lett. 65,2136. 10. Inoue, A., Nakamura, T., Sugita, T., Zhang, T. and Masumoto, T. (1993) Mater. Trans. JIM 34, 351. 11. Zhang, T. and Inoue, A. (2002) Mater. Trans. JIM 43,708. 12. Nishiyama, N. and Inoue, A. (2002) Appl. Phys. Lett. 80,568. 13. Inoue, A. and Wang, X.M. (2000) Acta Mater. 48, 1383. 14. Choi-Yim, H. and Johnson, W.L. (1997) Appl. Phys. Lett. 71,3808. 15. Herold, U. and Koster, U. (1978) Rapidly Quenched Metals I11 1,281, (Ed. by B. Cantor) Metals Society, London. 16. Loffler, J.F., Johnson, W.L., Wagner, W. and Thiyagarajan, P. (2000) Mater. Sci. Forum 343-346, 179. 17. Kelton, K.F. (1993) J. Non-cryst. Solids 163,283. 18. Ranganathan, S. andHeimendah1, M.V. (1981) J. Mater. Sci. 16,2401. 19:Matsushita, M., Saida, J., Zhang, T., Inoue, A,, Chen, M.W. and Sakurai, T. (2000) Phil. Mag. Lett. 80, 79.

20. Xing, L.Q., Eckert, J., Loser, W. and Schultz, L. (1999) Appl. Phys. Lett. 74,664. 21. Inoue, A., Zhang, T., Saida, J., Matsushita, M., Chen, M.W. and Sakurai, T. (1999) Mater. Trans. JIM 40, 1181. 22. Murty, B.S., Ping, D.H. and Hono, K. (2000) Appl. Phys. Lett. 77, 1102. 23. Murty, B.S., Ping, D.H., Hono, K. and Inoue, A. (2000) Appl. Phys. Lett. 76, 55. 24. Eckert, J., Mattem, N., Zinkevitch, M. and Seidel, M. (1998) Mater. Trans. JIM 39, 623. 25. Louzguine, D.V., KO,M.S., Ranganathan, S. and Inoue, A. (2001) J. Nanosci. Nanotech. 1,185. 26. Li, C., Ranganathan, S. and Inoue, A. (2001) Acta Mater. 49, 1903. 27. Conner, R.D., Dandliker, R.B. and Johnson, W.L. (1998) Acta Mater. 46,6089. 28. Ramamurty, U., Lee, M.L., Basu, J. and Li, Y. (2002) Scr. Mater. 47, 107. 29. Basu, J., Nagendra, N., Ramamurty, U. and Li, Y. (2002) Phil. Mag. 83, 1747. 30. Nagendra, N., Ramamurty, U., Goh, T.T. and Li, Y. (2000) Acta Mater. 48,2603. 3 1. Jana, S., Ramamurty, U., Chattopadhyay, K. and Kawamura, Y. (2002) Mater. Sci. Engg. A (In press). 32. Kim, J.J., Choi, Y., Suresh, S. and Argon, AS. (2002) Science 295, 654. 33. Fan, C., Louzguine, D.V., Li, C. and Inoue, A. (1999) Appl. Phys. Lett. 75,340. 34. Bian, Z., He, G. and Chen, G.L. (2000) Scr. Mater. 43, 1003. 35. Heilmaier, M. and Eckert, J. (2000) JOM 52,43.


THE BOTTOM-UP APPROACH TO MATERIALS BY DESIGN W.W. Gerberich, J.M. Jungk and W.M. Mook Department of Chemical Engineering and Materials Science University of Minnesota Minneapolis, MN 55455

ABSTRACT Bottom-up approach implies understanding the building blocks and then assembling them into a useful structure. For nanoparticle and multilayer composites used in aggressive loading environments, this requires an understanding of the length scales controlling the strength and toughness of the blocks. Here, we examine both nanospheres of silicon and thin films of Au in the 30 to 300 nm regime. With mechanical probing by nanoindentation we show that length scales can be defined by volume to surface ratio with connectivity to dislocation evolution. These can predict to first order the variations in hardness that may be initially high due to an indentation size effect but then later dislocation harden due to a back stress mechanism. For small nanospheres of Si, hardnesses in the 15-30 GPa regime are measured, while for very thin Au films the apparent hardness can vary from 2 to 6 GPa.

INTRODUCTION In 1964, nearly four decades ago, the Proceedings of the 2nd Berkeley International Materials Conference addressed “High-Strength Materials” [ 11. The seeds of “materials by design” were strongly evident in at least half the papers with Friedel [2], Zackay and Parker [2], Honeycombe, et al. [2], Thomas, et al. [2], Davies [2],Westbrook [2], and Drucker [2] all discussing how to improve hardness and strength by refining the microstructure. These top-down approaches were typified by Drucker’s discussion of size effects associated with going from the macroscale to the microscale. There he suggested that a large size effect should be apparent in a WC-Co composite with a dislocation density of 10’4/m2if the Co layers were decreased from 1 pm to 0.1 pm but not if one went from 10 pm to 1 pm. Present-day discussions are not so different, with the main change being that we are expanding the size effect paradigm to include the nanoscale. Also, it’s not the size scale effect so much, but what the length scales are and how one measures and physically describes them. In two ongoing studies on thin films and on nanoparticles we have measured the length scale and shown it to be characterized by a volume to surface ratio (V/S). If you can now relate the mechanical property of interest to this length scale, then from a bottom-up approach one would be poised to produce materials by design. We briefly describe

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W.W. Gerberich, J.M. Jungk and W.M. Mook

212

the theoretical basis for why V/Smight represent an appropriate length scale for the hardness of films and nanospheres. We then present two sets of hardness data on silicon nanospheres and Au thin films.

THEORETICAL BASIS We originally became interested in length scale effects through very shallow depth nanoindentation of single crystals [4]. Here, the volumes of plasticity were extremely small. The motivation was to evaluate the indentation size effect (ISE) at the nanometer scale using spherical diamond tips. Following a suggestion by Baskes and Horstemeyer [5],we considered carefully the effect of surfaces at these small volumes. For shallow depths we discovered that the volume to surface ratio was constant for four single crystals of Au, Al, W and Fe-3wt%Si oriented in the . This led to a V/Slength scale concept that could be used to predict the ISE. With &= VIS,it was shown that [4]

where o,, was the bulk yeld stress, 6, was the penetration depth or tip displacement, and R was the indenter tip radius. Using various tip radii, Eqn. 1 predicted hardness vs. penetration depth for these four single crystals. Later this approach was applied to thin films [6] and nanoparticles in the form of single crystal nanospheres [7]. While we described the length scale in a similar way, no detailed comparison of the resulting mechanical response for the different geometries was conducted. As illustrated in Fig. 1, we schematically interrogate a bulk single crystal, a thin film of thickness, h, or a nanosphere of diameter 2r,. The schematics roughly indicate the contact diameter, Za, to be the same in all three cases. For a length scale in terms of V/S,we take the surface of contact to be between the diamond tip and the deforming crystal except for the nanosphere which involves a top and bottom contact. Alternatively, since the whole surface of the nanosphere is deformed one might use the entire spherical surface. This leads to two possible interpretations of length scale from V/S giving the two choices noted in Fig. 1. For consistency we will use the contact radius, a , and its surface area to illustrate the point. In Fig. I , the plastic zone diameter is 2c for bulk crystal and film but 2r for the

I

I

v

2c3

S

3a2

V S

-

c2h a'

Figure 1: Schematic of length scales for a conical diamond probing a bulk single crystal, a thin film or a nanosphere. Here, 2a and 2c are contact and plastic zone diameters, h is the film thickness and r is the nanosphere radius.

The bottom-up approach to materials by design

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nanosphere. Here we drop the subscript from r, because of constancy of volume. From previous studies it is known that cla 3 is fairly typical for many indentation conditions. Assume then that cia 3 for a 100 nm contact radius into a bulk single crystal, a 100 nm thick film or a 100 nm diameter nanosphere. The length scales corresponding to the V/Svalues indicated in Fig. 1 would be 1800 nm, 900 nm, and 8.33 nm, respectively. Not too surprising is the factor of two decrease from bulk to thin film. However, the two order of magnitude decrease for the nanosphere suggests a possible large change in behavior. One should be careful here as we are discussing single crystal behavior. The length scale appropriate to mechanical behavior for thin films may well be truncated by a nanocrystalline grain size length scale if these were sputter deposited for example. For small contacts this may not be too serious since the nanocrystalline grain size may approach the size of the plastic zone. It should be mentioned here that if the whole nanosphere surface were used, giving V/S= ri3, that the length scale would still only be 16.7 nm.

-

-

To assess the effect of these far-ranging length scales on mechanical behavior, we examine the variation in hardness as determined from monotonically increasing deformation of either a nanosphere or a thin film. While only two general results are examined in detail, the implications are considerable. EXPERIMENTAL PROCEDURES We choose not to present procedures in depth here but to direct the reader to several recent publications for the preparation of the single crystal nanosphere and the polycrystalline thin films used. Silicon single crystal nanospheres were grown by a hypersonic plasma particle deposition process [8]. These directed beams may be rastered to deposit lines of nanoparticles with relatively narrow particle size distributions in the range of 10 to 100 nm. With an atomic force microscopy based nanoindenter, these may then be measured as to their size and then deformed by crushing them between a sapphire substrate and a large radius diamond tip. For details see reference [7]. For determining hardness it is fairly simple since geometric contact areas can be used after plastic deformation. As will be shown, even silicon, with a brittle to ductile transition near 700°C in tension, deforms like a relatively ductile metal when compressing these nanospheres at room temperature. In a separate study of sputter deposited gold, fairly thin nanocrystalline films 30 nm thick were deposited onto silicon wafers. These were then nanoindented with a 1 pm diamond tip to a series of increasing depths. Contact radii were both measured and calculated while the plastic zone illustrated in Fig. 1 could be measured by AFM where the pile-up merges with the zero plane. See reference [9] for additional details. Length scales as determined from geometry, as in Fig. 1, were compared to thin film hardness as a function of depth. Results for the deformation of both nanospheres and thin films follow.

RESULTS AND DISCUSSION First consider the load-displacement response for a single nanosphere. In Fig. 2(a) we seen AFM height image of a 55 nm silicon nanosphere prior to deformation. It appears to be much wider since we are imaging the sphere with a very large 1 pm spherical diamond tip. Deconvolution shows this to be a 55 nm sphere as indicated by the height. The load displacement curve in Fig. 2(b) indicates considerable plastic deformation noting that a 27.5 nm displacement would be 50 percent strain taking the height to be the gage length. The unloading slope is quite steep with nearly all the displacement irreversible. Since the height was originally 55 nm and the apparent height of (55-50) 5 nm is too small, we conclude


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80 60

z a -50

0

1

40

2

3 80

Figure 2: Deformation of silicon nanospheres: (a) AFM cross-section of a 55 nm diameter nanosphere prior to deformation, (b) load-displacement curve for the particle in (a); (c) AFM crosssection of a 43 nm diameter nanosphere prior to deformation; (d) load-displacement curve for the particle in (c) that the silicon particle fractured with the tip sliding in between the vertical cavity formed by the two hemispheres moving laterally apart Since the large 1 pm diamond is still in contact with both halves, we simply overestimate the contact area and underestimate the hardness ( H A ) Given the shape of the curve, we believe this occurs in the vicinity of 6 = 25 nm A much more normal loading curve I S obtained for the smaller 43 nm sphere (see Fig 2(c)) and Fig 2(d) By determining the contact areas from the spherical geometry under load [7] we determined the hardness (Pha’) versus normalized displacement (6h) Note that the appropnate radius, Y, is for the nanosphere and not the diamond indenter since Y (( R. For the 55 nm diameter nanosphere data in Fig 3 we see that the hardness decreases continuously with increasing displacement. It immediately occurred to us that this was like an indentation size effect and employed the analysis that had worked well for shallow indentations of single crystals [4] For the initial deformation of the defect free single crystal nanosphere this IS a realistic expectation From the previous analysis, we found the appropriate length scale to be V/S and for a sphere this would be 4/3 xr3/4nr2or r/3 From Eqn 1 this would give the ISE [4] for a nanosphere to be


215

Analysis of length scales showing hardness versus normalized displacement for a 55 nm Figure silicon nanosphere. 113 with an appropriate yield strength for the deformation of the 55 nm particle. With o,, = 36 GPa, it is seen that this fits the initial portion of the loading curve at small displacements. The deviation near a Sir of 0.5 was initially thought to be associated with a dislocation back stress at a pile-up. The concept is that a dislocation pile-up as produced by prismatic punching at the sphereidiamond contact would provide a large back stress. Upon calculating this from Hirth and Loethe [ 101, we find this to be, HZ-

P a

n(1- v)x

-

-

P6 n(1- v)r

(3)

the latter resulting from the number of dislocations in the pile-up being 6ib. Using 66 GPa for the silicon shear modulus and a Poisson’s ratio of 0.218, this gives H 3 26.9 GPa 6/u. This intersects the ISE curve about where the data start deviating but the data do not follow the trend. Rather than the silicon particle hardening, we propose that it fractures. With the large release of dislocation loops, the local stress could exceed the fracture stress whereupon the particle halves split apart. This gives rise to extra displacement as the two halves slide apart. This resulted in the fairly large jump seen in Fig. 2@) near 20 nm. Beyond this the actual contact area would be less than experimentally determined from the measured displacement: This would result in the hardness data beyond the fracture region being clearly underestimated. A more realistic result was obtained for the 43 nm particle shown in Figs. 2(c) and = 36 GPa in Fig. 4(a). As is seen in 2(d). Hardness follows the ISE curve of Eqn. 2 using the same oYs Fig. 2(d), the load-displacement curve was much smoother with no displacement jumps. Even if the particle did fracture, it apparently stayed together. We did have to scale the contact area for S > r assuming a constancy of volume. This was accomplished by assuming a geometrical contact for displacements up to 6 = Y and then setting an equivalent right cylinder volume equal to the sphere volume for 6 > r. The same scaling had been used in Fig. 3. For the 43 nm nanosphere, we see in Fig.

216


n

m

c. I

Figure 4: Analysis of length scales showing hardness vs. normalized displacement for a 43 nm silicon nanosphere: (a) analysis for increasing hardness at deeper depths utilizing a dislocation back stress; (b) analysis for increasing hardness using a volumeisurface length scale. 4(a) that the hardness goes through a minimum near the point where Eqs. 2 and 3 intersect. Here, we believe that the calculated hardnesses are more representative and similar to the hardness observations seen recently as produced by repetitive work hardening [ 7 ] . Taking a closer look at the phenomenology, we note that the hardness follows a dislocation back stress argument more closely at higher displacements in Fig. 4(a). In a future paper under preparation [ll], we show that a work per unit fracture area approach can lead to a length scale given by

es =

PBYs

(4)

2 2 0 n(1-v ) YS

where E and cry. are modulus and yield strength, ys is the surface energy and p'is a constant which depends on the character of the dislocation structure created, One might expect such an approach to be more appropriate where the plastic work involved with dislocation loop evolution become dominant at larger displacements. Here, we use the second ViSdescription of k', in Fig. 1 and note that a* 6r for the nanosphere, giving

-

Since the hardness is some multiple of the yield strength H - p " ~ , , , w e combine constants,

P = P')3"''

and obtain from Eq. (4)


0

(a)

2 s

217

5 0 I-lm

10 0

7 5

(b)

Figure 5: Indentation into a 300 nm Au film supported on a silicon substrate image, (b) mid-plane cross-section

(a) AFM height

Since this is determined from a work per unit surface area concept, i t is not surprising this mimics the Griffith criterion except now the length scale is that associated with the defonnation process With a single fitting constant of = 6, this fits the 43 nm sphere data well in Fig 4(b) for the latter deformation stages Here, we use Eq. (6) for hardness with f, defined by the last relationship in Eqn. 5 allowing H to be determined as a function of S/r While it is tempting to suggest that the measured hardness curve is a simple superposition of these two hardening mechanisms, this is premature and awaits further material studies Given this description of nanosphere deformation in terms of VIS length scales, we next turn to thin films where 300 nm Au films had been sputter deposited on Si A typical conical diamond with a 1 pm spherical tip produces relatively constrained plastic deformation in very thin films Such a result is shown in Fig 5 with the pile-up and plastic zone size indicated in the cross-sectional AFM profile From a series of such indentations to different depths, measurements of contact diameter and plastic 5

a=15

[

.

ah2

=-

a

m

u

Figure 6: Analysis of normalized plastic zone (cia) as a function of normalized contact ( d h ) fur determining a length scale, i?,, in 300 nm Au films.

218


zone size indicated in Fig. 6 allowed a determination of the length scale. The data fit of cia vs. aih allowed a length scale, as defined by Fig. 1 for thin films, to be given by fj

ah a

=-

(7)

where h is the film thickness and a is the contact radius. As described elsewhere [6], this led to a fracture mechanics criteria for slow-crack growth representative of R-curve behavior. Here, for hardness we assume that initial deformation exhibits little if any ISE as these deposited Au films have a large inherent defect density. If the initial deformation is predominantly controlled by a dislocation back stress argument, then Eqn. 3 is appropriate as modified by the slip band length being the plastic zone size, c, and the fact that this is constrained plastic flow with H 3 9 , . With these modifications Eqn. 3 becomes HE-. 3P6 n(1- v)c

-

Comparing this to measured hardnesses in Fig. 7(a) gives a surprisingly good accountability of the hardness variations with increased normalized depth. This strongly suggests that the hardness of relatively ductile films on a rigid substrate need not take into consideration the strength contribution of the substrate as many composite approaches have suggested. In many ways this behavior is analogous to the silicon nanosphere being squeezed between two high stiffness, non-yielding platens. The hardening comes from the internal defect structure in the nanosphere. Similarly, the material trapped between the spherical diamond tip and the silicon substrate can harden due to the increased dislocation density being confined to decreasingly smaller spaces. It is significant that the plastic zone about the


219

indentation in those thin films increased as giving hardness proportional to 6”’from Eq. (8). Similarly, from Eqs. 5 and 6 it is seen that the hardness of the nanospheres would increase as Since the dislocation densities would increase in proportion to the displacement, this would give flow stress proportional to the square root of the dislocation density. This suggests a future area of investigation involving transmission electron microscopy. To assess whether our length scale argument held up for this thin film, we utilized the same Eqn. 6 with a tensile modulus of 80.8 GPa, a surface energy of 1.485 Jim’ and a Poisson’s ratio of 0.35 appropriate to gold. The length scale as given in Fig. 1 was used. For a reasonable fit as shown in Fig. 7(b), a considerably sharper increase in measured hardness with is observed than predicted by Eqn. 6. This may be partially a result of measurement difficulties for these extremely thin films but more likely a shortcoming of attempting to apply a model for single crystal thin films to nanocrystalline ones. The encouraging aspect is that for the data fit in Fig. 7(b), p was taken to be 15 whereas for the fit in Fig. 4(b) for the nanosphere it was 6. This difference could easily be the difference in constraint with the nanosphere exhibiting little and the thin film being fully constrained from the standpoint of contact mechanics. SUMMARY

This contribution purports to demonstrate the importance of length scales, their measurement, and their inclusion in design rules for materials performance. Here, we’ve emphasized two important aspects of volumeisurface and dislocation slip-band morphology in length scale interpretations. These are shown to be important in understanding the hardening mechanisms in both silicon nanospheres and thin gold films in the 30-300 nm regime. At very small increasing displacements into single crystal nanospheres, it is seen that hardness decreases with a (l/S)”’dependence. However, at larger displacements, it is seen that hardness can then increase with a(6)’” dependence due to dislocation hardening. The increase in film hardness with depth is predicted to give a similar(6)’” dependence when factoring in how the plastic zone increases with increasing depth of indentation. These length scales for nanospheres and thin films should eventually lead to design rules for incorporation into small volume structures. ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation under grant DMI-0103 169 and an NSFIGERT program through grant DGE-0114372. One of us (JMJ) would like to acknowledge support of Seagate Technology through the MINT program at the University of Minnesota. REFERENCES 1.

V. F. Zackay, High Strength Materials (New York, NY: J. Wiley and Sons, Inc., 1965).

2.

J. Friedel, “High Strength Materials,” (ibid.), 1-11; V. F. Zackay and E.R. Parker, “Some Fundamental Considerations in Design of High-Strength Metallic Materials,” (ibid.), 130-166; R. N. K. Honeyconlbe, H. J. Harding and J. J. Irani, “Strengthening Mechanism in Ferritic and Austenitic Steels” (ibid.), 213-250; G. Thomas, D. Schmatz and W. Gerberich, “Structure and Strength of Some Ausformed Steels,” (ibid.), 251-326; G. L. Davies, “The Growth of Fiber

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Structures from the Melt,” (ibid.), 603-650; J. H. Westbrook, “The Sources of Strength and Brittleness in Intermetallic Compounds,” (ibid.), 724-768; D.C. Drucker, “Engineering and Continuum Aspects of High Strength Materials,” (ibid.), 795-833. 3.

N. I. Tymiak, D. E. Kramer, D. F. Bahr and W. W. Gerberich, “Plastic Strain Gradients at Very Small Penetration Depths,” Acta Mater. 49 (2001), 1021-1034.

4.

W. W. Gerberich, N. I. Tymiak, J. C. Grunlan, M. F. Horstemeyer and M. I. Baskes, “Interpretations of Indentation Size Effects,” J. A ~ p lMech. . 62 (2002), 433-442.

5.

M. Baskes and M. Horstemeyer, private communication, Sandia National Labs, Livermore, CA (1999).

6.

W. W. Gerberich, J. M. Jungk, M. Li, A. A. Volinsky, J. W. Hoehn and K. Yoder, “Length Scales for the Fracture of Nano-structures,” Intern. J. of Fracture (2003), accepted.

7.

W. W. Gerberich, W. M. Mook, C.R. Perrey, C. B. Carter, M. I. Baskes, R. Mukherjee, A. Gidwani, J. Heberlein, P. H. McMurray and S. L. Girshick, “Superhard Silicon Nanospheres,” JMech. Phvs. Solids (2003), in press.

8.

F. DiFonzo, A. Gidwani, M. H. Fan, D. Neumann, D. J. Iordanoglu, J. V. R. Heberlein, P. H. McMuny, S. L. Girshick, N. Tymiak, W. W. Gerberich and N. P. Rao, Aml. Phvs. Lett. 77 (2000), 9 10-9 12.

9.

D. E. Kramer, H. Huang, M. Kriese, J. Robach, J. Nelson, A. Wright, D. Bahr and W. W. Gerberich, “Yield Strength Predictions from the Plastic Zone around Nanocontacts,” Acta Mater. 47 (1994), 333-343.

10.

J. P. Hirth and J. Loethe, Theory of Dislocations, 2nd ed. (New York, NY: John Wiley and Sons, 1982).

11.

J. M. Jungk, W. M. Mook and W. W. Gerberich, “Nanoindentation Length Scale Measures in Small Volumes,” in preparation.


THE ONSET OF TWINNING IN PLASTIC DEFORMATION AND MARTENSITIC TRANSFORMATIONS Marc AndrB Meyers', Matthew S. Schneider', and Otmar Voehringer* 'University of California-San Diego, Dept. of MAE, La Jolla, CA 92093 USA 'University of Karlsruhe, Inst. for Matls. Research., Karlsruhe Germany

ABSTRACT A quantitative constitutive description for the criterion postulated by Thomas [l-31 for the morphology of martensitic transformations is presented. Thomas observed that the temperature and strain-rate sensitivities of slip are much higher than those for twinning, rendering twinning a favored deformation mechanism at low temperatures and high strain rates. Constitutive relationships for slip and twinning are presented and applied to the martensitic transformation in steels: the lath to plate morphology change that is observed with increasing carbon content is successfully predicted by calculations incorporating the two modes of deformation. The Hall-Petch coefficient, for the inclusion of grain size effects is two times larger for twinning than slip. A simple calculation of the strain rates during martensitic transformation is also provided. For FCC metals, the constitutive description for the slip-twinning transition incorporates the effects of material (stacking-fault energy, grain size, composition) as well as external (temperature, strain rate) parameters successfully. It can also applied to the shock compression regime, where the shock front thickness (and, consequently, strain rate) is related to the peak pressure by the Swegle-Grady relationship. Predictions are compared to seminal shock loading work by Johari and Thomas [4] and Nolder and Thomas [5] demonstrating that there is a threshold pressure for twinning in copper and nickel.

INTRODUCTION It was shown by Thomas [ 1-31 that slip and twinning are competing deformation mechanisms and that they have a profound effect on the mechanical properties of martensitic steels and FCC metals like copper and nickel. The schematic representation by Thomas [2] is shown in Figure 1. This plot, although qualitative, provides deep insight into the mechanical response of metals and alloys, which can deform by slip, twinning, or martensitic transformations. Figure 1 shows that slip has substantially higher temperature dependence than twinning; hence, slip and twinning domains are established. Martensitic transformations are displacive, virtually diffusionless transformations with the thermodynamics and kinetics governed by the transformation strains. In steels, the martensite structure undergoes a drastic morphological transition as the carbon content reaches the 0.6-1 .O weight percent region. Early German literature classified the two regions into Schiebung and Umklapp; the current nomenclature is lath and plate where the basic difference resides in the deformation mode: the transition from slipped (lath) to twinned (plate) martensite. Figure 2 shows the change in Ms as well as the two modes as a function of increasing carbon content [6].Figure 3 shows transmission electron micrographs of the two morphologies in different compositions of steel. This paper provides a quantitative description of the transition from

221

M.A. Meyers, M.S. Schneider and 0. Voehringer

222

lath to plate martensite by applying the Zerrilli-Armstrong constitutive equation. It is a straightforward method of predicting the threshold carbon concentration for the nucleation of plate martensite as a function of strain rate, temperature, and grain size.

\

\

?

n

1

MARTENSITE SUBSTRUCTURE DISLOCATED

TEMPERATURE

4

Figure 1: Variation of the stress for slip and twinning as a function of temperature. Composition variations are expected to affect the slip-twin cross-over as suggested by the arrows. (Adapted from Thomas [2]). 'F

P 600t"

4m1

-

t

0 0

LATH

-l2zi2L MIXED

PLATE

0.2

0.4

0.6

0.8

1.0

WEIGHT PERCENT CARBON

1.2

1.4

1.6

Figure 2: Variation of Ms temperature with carbon content; notice transition from lath (slipped) to plate (twinned) morphology; from Marder and Krauss[l 13.

The onset of twinning in plastic deformation and martensitic transformations

(c)

223

a,

Figure 3: (a) Twinned plates of martensite and residual austenite in Fe-33Cr- 01C; (b) Twinned and dislocated martensite in Fe-28Ni-OtC; (c) Twinned plates of martensite in Fe-8Cr-1C; from Thomas [3]. CALCULATIONAL PROCEDURE The calculations require constitutive equations for slip and twinning that have the appropriate temperature and strain rate dependencies. These equations were implemented by Meyers et al. [7,8] into a sliptwinning transition criterion and will only be briefly described herein By considering slip and twinning as competing mechanisms, and equating the appropriate constitutive equations, one obtains the critical condition:

where 05 and OT are the slip and twinning stresses, respectively. Constitutive Description of Slip There are numerous equations that successfully incorporate the strain, strain rate and temperature effects and predict the mechanical response over a broad range of external parameters. The Zerilli-Armstrong [9] equation is used here to describe the lath to plate transition in martensite. It is modified to incorporate the solid solution hardening effects induced by carbon additions. There are two different forms of the equation applicable to FCC and BCC metals. The barrier size is quite different: dislocation forest dislocations are considered the primary barriers for FCC materials, whereas the Peierls-Nabarro stress is the principal obstacle for BCC materials. These differences are responsible for higher strain rate and temperature sensitivity for the BCC structure. When iron-based alloys undergo the martensitic transformation, the FCC structure transforms to BCC or BCT. This newly created structure has to undergo a complex deformation to accommodate the Bain and lattice invariant strains. The ZerilliArmstrong equation for the BCC structure has the form:

M.A. Meyers, M.S. Schneider and 0. Voehringer

224

The variables are strain, E, strain rate, E and temperature, T. The coefficients C,, Ct, Cr, Ch, and C, arc experimentally obtained parameters. The parameters, their physical meanings and values chosen are for pure iron [ 9 ] :

1033 MPa 0.00698 0.000415 266 MPa 0.289

CS: Stress Constant Ct : Thermal Softening Constant Cr : Strain Rate Constant c h : Strain Hardening Constant Cn. Strain Hardening Constant

The terms og and oc represent the athermal components of stress, which have minimal strain rate and temperature dependence. The grain size term, og represents the grain-size dependence, which is represented by a Hall-Petch relationship: og = k,d

-112

(3) I12

where ks for low carbon steels is found to vary between 15-18 MPdmm [lo]. An average value of 16.5 112 , MPdmm is used in the calculations. Figure 4 shows the Hall-Petch plots for both slip and twinning. The value matches well with experimental data by Marder and Krauss [I 11.

I100

-

(Armstrong and Worthington)

900

Kt data for Fe-C (Magee, et al.; McRickard and Chow)

m

2 700

A

v

v) v)

W

0

0

0

’** 0

.b ’

0

0

0

500

H 0.6T,) of pure metals and Class M (or Class I) alloys, that behave similarly to pure metals, rely on some aspect of the subgrain microstructure to describe the rate controlling mechanism. Many of the more recent theories rely on the details of the subgrain boundaries such as the spacing, d, of the dislocations that comprise the boundaries (related to the misorientation angle, 8, across boundaries) or the subgrain size, h [2-151. Subgrain boundaries have also been suggested to be the site of elevated long-range internal backstress [16-201 and the rate controlling process at the subgrain boundary has been associated with these stresses. Other work, however, placed this conclusion of long range internal-stresses in question [2 1-24]. The dislocations not associated with the subgrain boundaries, which are presumed to form a Frank network, are less commonly, especially recently, claimed to be associated with the rate controlling process of five power law creep. Dislocation network theories [ 1,25301 generally suppose that the creep behavior is explained in terms of network coarsening by the climb of the nodes and the activation of (critical-sized) links of the network. The acceptance of these models has been limited. This may be somewhat unjustified in view of some careful, and now well-established, experiments. For example, experiments under five-power-law conditions show that there is really no doubt that the elevated temperature flow stress of AISI 304 stainless steel (Class M alloy) is controlled by the density of dislocations, p, not associated with the subgrain boundaries [31]. Also, recent experiments have also shown that the flow properties of high purity aluminum and some (Class A) aluminum alloys under five-power-law creep conditions, appear independent of the subgrain size, A, or the nature (misorientation) of the subgrain boundaries [32-351. Traditionally, five-power-law creep theories have necessarily focused on the steady-state or secondary creep behavior. Of course, if a particular feature is associated with the rate controlling process for steady-state five-power-law creep, then the yield stress at a fixed temperature and strain-rate (within the five-power-law regime) under non-steady-state conditions would still be expected to be a function of the dimensions of this feature. This is basically equivalent to suggesting that non-steady-state behavior would be determined by the same microstructural features important for steady-state. Accordingly, it has been suggested that primary creep and creep transient conditions may obey a relationship ,

1 = A‘

[&I3

(s)” [D,,Gb/kT] ( o / G ) ~

where s is a substructural term, originally formulated by Sherby et al. [36] to be (l&J with p’ z 3. Sherby also suggested that N = 8 for aluminum in particular, and, perhaps, other metals as well. It was assumed that the activation energy for (non-steady-state) flow is equal to the activation energy for lattice self-diffusion over the five-power-law regime. Certainly, this equation has been illustrated to have some phenomenological merit, and fairly sophisticated phenomenological equations have been based on this relationship [36]. Sherby and coworkers suggested that the form of Eqn. 2 was reasonable since the established relationships [20] between the strengthening variables and the steady-state stress, e.g.,

!L= C, (l/hJ G

(3)

or

5= C, (p,,)p G

or

where p is E 0.5

(4)

Taylor hardening in five power law creep of metals and Class M alloys

251

when substituted into Eqn. 2, would yield the classic five-power-law equation (1). An important question is the nature of the “s” term in Eqn. 2 [i.e., p, d (or O), h]. There may be some problems with this logic, and N not being constant over the five-power-law regime may just be one [l]. Although outside the intended scope of this paper, another complication is that the activation energies in Eqns. 1 and 2 are not necessarily identical. Consistency with the (modified) Taylor equation [46] is expected if the influence of dislocation density on the flow stress is dominant, = oo+ aMGbp’”

where

(6)

is the applied stress at a given temperature and strain rate, G is the shear modulus, b is the

( T ~ , ~

Burgers’ vector, CL is a constant (typically 0.2-0.4 [31,38-421,although the Taylor factor, M, may not always be accurately known in the reported data leaving some uncertainty in this value), and oois the stress required to move a dislocation in the absence of other dislocations that can arise as a result of solutes, Peierls-type stresses, grain-size strengthening, etc. M can vary from 1 (pure shear) to about 3.67 for single crystals and is typically, in tension, 3.06 for polycrystals. This equation was shown to reasonably describe the 304 data within the five power law regime by assuming that oo was approximately equal to the yield stress of the annealed alloy. Furthermore, the value of CL for 304 was within the range observed in other metals at lower temperatures where dislocation hardening is confirmed. In principle, if Eqn. 6 is applicable, then the phenomenological relationship of (2) should reduce to the Taylor relationship of Eqn. 2. The first part of this work will demonstrate this same Taylor equation will.apply to pure aluminum (with a steady-state structure), having both a much higher stacking fault energy than stainless steel and an absence of substantial solute additions. Second, it will be shown that the microstructure and plastic flow characteristics of aluminum undergoing primary (Stage I) creep under either constant-stress or constant strain-rate power-law creep conditions are consistent with Taylor hardening. This latter point is important since it has long been suggested that because the total dislocation density decreases during primary creep under constunt-stress conditions, the “free” dislocations cannot rationalize hardening during this stage. ANALYSIS Steady-State Behavior Figure 1 shows earlier data [31] by the author where the elevated temperature flow stress of stainless steel is plotted as a function of the square-root of the (total) dislocation density in the subgrain interior. The data reflects steady-state structures as well as specially prepared specimens of stainless steels having various combinations of h and p microstructures produced by utilizing a variety of thermal and mechanical treatments. The specimens were mechanically tested at a given temperature and strain rate that nearly corresponded to the five-power-law creep range. It was found that the (Frank network) dislocation density not associated with subgrain boundaries dominated the strength, described by Eqn. 6. Furthermore, as just mentioned, CL = 0.28, which is consistent with observed values for Taylor hardening of about 0.2-0.4 at ambient temperature for pure metals. Some typical values of CL from the literature as well as this study are listed in Table 1. One complicating issue with Table 1 is that the value of CL is affected by the way p is calculated or reported. If p is measured as line length per unit volume, then the value of p is roughly twice that of p reported as intersections per unit area, thus affecting the constant a by a factor of about 1.4. The values by the author for Al and 304 are intersections per unit area, but the units of others of Table 1 are not known. Figure 7 utilizes line length per unit volume.

The steady-state flow stress is sometimes described by Eqn. 4:

258

M.E. Kussner and K. Kyle

120

m

h

$

v

80

4LL

2

5

0) S

9?

T~ (annealed)

c. v)

a,

F

40

304 stainless steel 750°C = 9.6 X l o 4 5.’ Compression

0

L

0

I

I

5

10

Figure 1: The elevated temperature yield strength of 304 stainless steel as a hnction of the square root of the dislocation density (not associated with subgrain boundaries) for specimens of a variety of subgrain sizes. (Approximately five-power-law temperature/strain-rate combination.) Based on [3 11.

where p is an exponent 1-2. Sometimes, p is chosen as 2 and

(where C is a constant) and the “classic” Taylor equation (e.g., Eqn. 6) has been suggested. However, this relationship between the steady state stress and the steady-state dislocation density is not for a fixed temperature and strain rate. Hence, it is not of a same type of equation as the classic Taylor equation. That is, this later equation tells us the dislocation density not associated with subgrain boundaries that can be expected for a given steady-state stress which varies with the temperature and strain-rate. However, according to Eqn. 2, if the strength is exclusively provided by pm t, then the strength is temperature dependent. Equation 7 is expected to be athermal [43,44]. Equation 6, however, contains a temperature dependent ooterm. A similar experiment illustrated in Figure 1 has also been performed on steady state structures of aluminum [36,45]. In one case [36], aluminum specimens were deformed to various steady state stresses at a given temperature by varying the applied strain-rate. The strain-rate was quickly changed to a common strain-rate after steady state was achieved and the new plastic flow stress (at a fixed temperature and strain-rate) was noted. The subgrain sizes were measured at each steady-state, so that the dependence of the flow stress at a


259

TABLE 1 TAYLOR EQUATION Cf VALUES FOR VARIOUS METALS

Metal

TIT,

304

0 57

cu

0 22

Fe

1

0 28

o,$0, polycrystal

[311

0.34

o,= 0, single crystal

~421

0 22

0.31

o,= 0, polycrystal

0.15

0 37

o,z 0.25-0.75 flow stress,

I: I 1: 1 TI

Ref.

1-6 slip crystal

-_

cu

Notes

6)

0 51-0 83 -

I

polycrystal

0 19-0.34

Stage 1 and I1 single crystal M = 1.78 - 1 Go f 0

031

oo= 0, polycrystal

0 20

oo# 0, polycrystal

0.23

oo# 0, polvcrystal

I

I

I

I I

I

[38] [391

[401 [411 This study

[381

Note: Q values of A1 and 304 stainless stress are based on dislocation densities of intersections per unit area. The units of the others is not known and these values would be adjusted lower by a factor of 1.4 if line-length per unit volume is utilized. specific temperature and strain-rate on the subgrain size could be determined. Equation 2 was basically formulated based on these experiments. In another case [45] three specimens were deformed at various temperatures and strain-rates, again, to steady-state. The specimens were quickly cooled to 300°C and redeformed at a fixed strain-rate. The new flow stress (again, at a fixed temperature and strain-rate) was also related to the measured (steady-state) subgrain sizes produced at the higher temperatures. The data in both cases suggests a phenomenological relationship between the flow stress at a fixed temperature and strain rate and the (steady-state) subgrain size (the network or the dislocation density within the subgrain was not considered):

where kl is a constant. It should be noted that the oo term is a substantial fraction of the steady-state flow curve (as illustrated subsequently) despite the high purity (also, see Fig. 6(a)). Thus a “friction stress” unrelated to dislocation hardening is still appropriate, just as with the stainless steels case. Again, the two phenomenological equations, Eqn. 2 and Eqn. 6, in principle, are equivalent at a fixed temperature and strain-rate. Equation 3 is based on steady-state deformation. Since the steady-state subgrain size is generally related to the steady-state dislocation density pss,

where, as pointed out earlier, p may vary from 1-2 [20]. Blum and coworkers’ [46] careful measurements suggest a value of about 1.6. Substituting Eqn. 9 into Eqn. 8 suggests that for steady state structures of aluminum (deforming under a non-steady-state “reference” temperature and strain-rate),


260

where oois roughly the yield stress of the annealed aluminum at the reference temperature and strain rate. This suggests that the same classic Taylor equation that can be used to describe elevated temperature dislocation hardening in stainless steel is applicable here, as well. An important additional question to assess the validity of the Taylor equation is to modify the dislocation density exponent to 0.5 in Eqn. 10 and assess the value of a. If both the phenomenological description of the influence of the strength of dislocations in high purity metals such as aluminum have the form of the Taylor equation and also have the expected values for the constants, then it would appear that the elevated temperature flow stress is provided by the Frank network rather than the subgrain walls. Figure 2 plots modulus-compensated steady-state stress versus diffusion-coefficient-compensated steadystate strain-rate. Figure 3 illustrates the well-established trend between the steady-state dislocation density (line length per unit volume) and the steady-state stress. The steady-state flow stress can be predicted at a reference strain rate (e.g., 5 x lo4, s-'), at a variety of temperatures, with an associated steady-state dislocation density from these two figures. If Eqn. 6 is valid, then the values for u could be calculated for each temperature, by assuming that the annealed dislocation density and the o0 values account for the annealed yield strength measured in this study and reported in Figure 4. 1oo

1o4

1o 8

1o-'*

10-l6

1o Z 0 1o - ~

1o

-~

1o 3

10'

osJG Figure 2: The compensated steady-state strain-rate versus the modulus compensated steady-state stress for 99.999 pure Al, based on [59].

Taylor hardening in five power law creep of metals and Class M alloys __

lo4

1014

.

9‘ I

1013

26 1

I

I

c!

E 10’2

n,’ lo1’

I

,

,

,

, u , d

, , , ,,,,

n

, , ,

, , , ,,,,,,

,,llii

, , ,

Lu

”%

n o d

On

o,JG Figure 3: The average steady-state subgrain intercept, h, density of dislocations not associated with subgrain walls, p, and the average separation of dislocations that comprise the subgrain boundaries for A1 [and A1-5 at%Zn that behaves, mechanically, essentially identical to Al, but is suggested to allow for a more accurate determination of p by TEM]. Based on [60]. Figure 5 reports the resulting a values. Figure 5 indicates, first, that typical values of a are within the range of those expected for Taylor strengthening. Said another way, strengthening of (steady-state) structures can be reasonably predicted based on a Taylor equation. The strength we predict, based only on the (network) dislocation density and completely independent of the heterogeneous (subgrain) dislocation substructure.

262

250

150

300

350

Temperature (C)

400

5w

450

550

Figure 4: The yield strength of 99.999% pure A1 as a function of temperature.

1

p=10”m-’

0.30

o p=2 5x10” m.’

025{

150

200

250

300

350

400

450

500

T (“C)

Figure 5: The values of the constant alpha in the Taylor equation (6) as a function of temperature. The alpha values depend somewhat on the assumed annealed dislocation density. Hollow dots, p = 2.5 x 10” m-2; solid, p = 10” m-*.


263

This point is consistent with the observation that the elevated-temperature yield strength of annealed, polycrystalline aluminum [high-angle boundaries (HABs) only] is essentially independent of the grain size [47]. It has further been established that for a fixed grain sizeisubgrain size, the flow stress is independent of large variations in the misorientation [33]. Furthermore, the values of a are completely consistent with the values of a in other metals (at high and low temperatures) in which dislocation hardening is established (see Table 1). The fact that the higher temperature a values of Al and 304 stainless steel are consistent with the ambient-temperature values is consistent with the athermal behavior of Figure 5. The non-near-zero annealed dislocation density observed experimentally may be consistent with Ardell et al. suggestion of network frustration creating a lower limit to the dislocation density. One point to note is that in Figure 5 the variation in a with temperature depends on the value selected for the annealed dislocation density. For a value of 2.5 x 10” m-*, the values of the alpha constant are nearly temperature independent, suggesting that the dislocation hardening is, in fact, theoretically palatable in that it is athermal. The annealed dislocation density for which athermal behavior is observed is that which is very close to the value observed by the author (Figure 6), and suggested by Blum [48]. The suggestion of athermal dislocation hardening is consistent with the model by Nes [44], where, as in the present case, the temperature dependence of the constant (or fixed dislocation substructure) structure flow stress is provided by the important temperature-dependent ooterm. It perhaps should be mentioned that if it is assumed both that oo= 0 and that the dislocation hardening is athermal (i,e., Eqn. 7 is “universally” valid) then a is about equal to 0.53, or about a factor of two larger than anticipated for dislocation hardening. Hence, aside from not including a oo term which allows temperature-dependence, the alpha term appears somewhat large. Primary Creep Behavior The trends in dislocation density during primary creep have been less completely investigated for the case of constant-strain-rate tests than constant-stress creep tests. Earlier work by the author [3 11 on 304 stainless steel found that at 0.57 T,,, (and the same strain-rates as Figure l), the increase in flow stress by a factor of three is associated with increases in dislocation density with strain that are consistent with the Taylor equation. That is, the p versus strain and stress versus strain give a o versus p that “falls” on the line of Figure 1 [49]. Similarly, the aluminum primary transient in Figure 6, where the dislocation density monotonically increases to the steady state value under constant strain-rate conditions, can also be shown consistent with the Taylor equation.

Challenges to the proposition of Taylor hardening for 5-power-law creep in metals and class M alloys include the microstructural observations during primary creep under constant-stress conditions. For example, it has nearly always been observed during primary creep of pure metals and Class M alloys that the density of dislocations not associated with subgrain boundaries increases from the annealed value to a peak value, but then gradually decreases to a steady-state value that is between the annealed and the peak density [50-551 (e.g., Figure 7). Typically, the peak dislocation density value, pp, measured at a strain level that is roughly one-fourth of the strain required to attain steady-state (cSs/4),is a factor of 1.5-4 higher than the steady-state pssvalue. It was believed, by some, difficult to rationalize hardening by network dislocations if the overall density is decreasing while the strain rate is decreasing. Therefore, an important question is whether the Taylor hardening, observed under constant strain-rate conditions, is consistent with this observation. This behavior could be interpreted as evidence that most of these dislocations have a dynamical role rather than a (Taylor) hardening role, since the initial strain-rates in constant-stress tests may require by the equation,

1 = (b/M) p, v

(1 1)

a high mobile (nonhardening) dislocation density, pm, that gives rise to high initial values of total density of dislocations not associated with subgrain boundaries, p, where v is the dislocation velocity. That is, of the


264

10

50

40

30

20

10 0

I

Al (99.999%) 371°C

= 5.04x lo-* s.'

c

!

_.

1

(4

....

Figure 6 : The work-hardening at a constant strain-rate creep transient for A1 illustrating the variation of h, p, d, and ex, over primary and secondary creep. The bracket refers to the range of steady-state dislocation density values observed at larger strains [e.g., see (b)]. From [33].


40 50

t

265

Al (99.999%) 371'C

0 ' ~

~

..

..i

.....

I

-1-

.....

L

i

..

1.50 1.oo

0.50

0.00

0

L-

0

2

4

6

8

10

12

........

14

1.

16

Strain,Y cb) Figure 6 (continued): The work-hardening at a constant strain-rate creep transient for A1 illustrating the variation of h, p, d, and Oh,,, over primary and secondary creep. The bracket refers to the range of steadystate dislocation density values observed at larger strains [e.g., see (b)]. From [33].


266 1o s

IU - 5 at% Zn

0

0.1

0.3

0.2 E

0.4

Figure 7: The constant-stressprimary creep transient in Al-Sat %Zn (essentially identical behavior to pure Al) illustrating the variation of the average subgrain intercept, h, density of dislocations not associated with subgrain walls, p, and the spacing, d, of dislocations that comprise the boundaries. The fraction of material occupied by subgrains is indicated by fsub.The subgrain size during primary creep reflects those regions of the grain where subgrain formation is observed. Based on [61].


261

total density of dislocations not associated with subgrain boundaries, at any instant, some are mobile (p,) while some are obstacles, perhaps as links of the Frank network (p - pm).As steady state is achieved and the strain rate decreases, so does pm and in turn, p. More specifically, Taylor hardening during primary constant-stress creep may be valid based on the following argument:

From Eqn. 11 B = p,,,vb/M. We assume [56] v = k, o'

and, therefore, for constant strain-rate tests,

The E~ (plastic strain) is small at the onset of yielding in a constant strain-rate test (E = E,, ), and there is only

minor hardening, and the mobile dislocation density is a fraction, f: , of the total density,

therefore, for aluminum (see Figure 6) pm(6,=n) = f:0.64 p,,

(based on p at E p = 0.03)

(14)

where f: is basically the fraction of dislocations in the annealed metal that are mobile at the yield stress (half the steady-state flow stress) in a constant strain-rate test. Also from Figure 6, oy/oss= 0.53. Therefore, at small strains,

is,= f: 0.64(0.53)[k1b/M] pssass (constant strain rate at E~

= 0.03)

At steady-state, o = oSsand pm =fkp,,, where fk is the fraction of the total dislocation density that is mobile at steady-state and

(constant strain rate at E~ > 0.20) By combining Eqns. 15 and 16 we find that f, at steady state is about 113 the fraction of mobile dislocations in the annealed polycrystals (0.34f: = f:). This suggests that during steady state only 113, or less, of the total dislocations (not associated with subgrain boundaries) are mobile and the remaining 213, or more, participate in hardening. The finding that a large fraction are immobile is consistent with the observation that increased dislocation density is associated with increased strength for steady-state deformation and constant strain-rate testing. Of course, there is the assumption that the stress acting on the dislocations as a finction of strain (microstructure) is proportional to the applied flow stress. This is sensible (and the fraction is probably unity) for a network model. Furthermore, we have presumed a 55% increase in p over primary creep with some uncertainty in the density measurements.

268


For the constant-stress case we again assume

t6p=o = f: [k,b/M]ppoSs

(constant stress)

where f i is the fraction of dislocations that are mobile at the peak (total) dislocation density of pp, the peak dislocation density, which will be assumed equal to the maximum dislocation density observed experimentally in a p-Eplot of a constant stress test. Since at steady-state from Eqn. 15, is, 0.34f: [ k , b / M ] p s , ~ , s by combining with Eqn. 17, t,p,o/~ss =[F)3pp/pss f",

(constant stress)

(18)

(fi/f:) is not known but if we assume that at macroscopic yielding, in a constant strain-rate test, for annealed metal, f: 2 1, then we might also expect at small strain levels and relatively high dislocation densities in a constant stress test, f i z 1. This would suggest that fractional decreases in t in a constant stress test are not equal to those of p. This apparent contradiction to purely dynamical theories (i,e., basted strictly on Eqn. 11) is reflected in experiments [50,51,53,55] where the kind of trend predicted in Eqn. 18 is in fact observed. Equation 18 and the observations of t against E in a constant stress test at the identical temperature can be used to roughly predict the expected constant-stress p-Ecurve in aluminum at 371°C and about 7.8 MPa; the same conditions as the constant strain-rate test of Figure 6 . If we use small plastic strain , p values have been measured in constant-stress tests), we can determine the levels, (e.g., E z ~ i 4 where ratio (e.g., iE=(E,,,4)/EE=E,,) in constant stress tests. This value seems to be roughly 6 at stresses and temperatures comparable to Figure 6 [11,50,51,57,58]. This ratio was applied to Eqn. 18 [assuming (fi/f:)z 11; the estimated p-E trends, in a constant-stress test in Al at 371"C, are shown in Figure 8. This estimate, which predicts a peak dislocation density of 2.0 pss,is consistent with the general observations discussed earlier for pure metals and Class M alloys, that pp is between 1.5 an 4 pss(1 5 2 . 0 for aluminum [50]). Thus, the peak-behavior observed in the dislocation density versus strain trends, which at first glance appear to impugn dislocation network hardening, is, actually, consistent, in terms of the observed p values, to Taylor hardening. Two particular imprecisions in the argument above are that it was assumed (based on some experimental work in the literature) that the stress exponent for the elevated temperature (low stress) dislocation velocity, v, is one. This exponent may not be well known and may be greater than 1. The ratio (pp/pss)is multiplied from a value of 3 in Eqn. 18 to higher values of 3[2"-'], where n is defined by v = d"'This means that the observed strain-rate "peaks" would predict smaller dislocation peaks or even an absence of peaks for the observed initial strain-rates in constant-stress tests. In a somewhat circular argument, the consistency between the predictions of Figure 8 and the experimental observations may suggest that the exponents of 1-2 may be reasonable. Also, the values of the peak dislocation densities and strain-rates are not unambiguous, and this creates additional uncertainty in the argument.

SUMMARY Previous work on aluminum and stainless steel show that the density of dislocations within the subgrain interior influences the flow stress for steady-state substructures and primary creep under constant strain-rate conditions. The hardening is consistent with the Taylor relation if a linear superposition of soluteilattice


269

~

1 E+11

0 00

015

0 30

0 45

0 60

stran

Figure 8: The predicted dislocation density (- - -) in the subgrain interior against strain for aluminum deforming under constant stress conditions is compared with that for constant strain-rate conditions (-). The predicted dislocation density is based on Eqn. 18 which assumes Taylor hardening. hardening (u0,or the stress necessary to cause dislocation motion in the absence of a dislocation substructure) and the dislocation hardening ( z uMGbp”2) is assumed. Here is assumed that the fraction of immobile dislocations (p - pm) is a constant fraction of the total dislocation density. It appears that the constant, a, is temperature independent and, thus, the dislocation hardening is athermal. Furthermore, it is shown that constant stress creep behavior where the total dislocation density (p) decreases during primary (hardening stage) creep, is actually consistent with Taylor hardening. The increase in total dislocation density simply reflects the high initial strain-rates in a constant-stress test. The obstacles for dislocation motion in this case are still the network dislocations. ACKNOWLEDGEMENTS This work was supported by Basic Energy Sciences, U.S. Department of Energy, under grant DE-FG0399ER45768. The mechanical testing by Dr. M.-Z. Wang is greatly appreciated. The comments to this manuscript by Prof. W. Blum are appreciated. REFERENCES 1. 2. 3. 4. 5.

Kassner, M.E. and Perez-Prado, M.T. (2000) Prog. Muter. Sci. 45, 1. Morrism, M.A. and Martin, J.L. (1984) ActaMetall. 32, 1609. Morris, M.A. and Martin, J.L. (1984) Acta Metall. 32, 549. Derby, B. and Ashby, M.F. (1987) Acta Metall. 35, 1349. Nix, W.D. and Ilschner, B. (1980). In: Strength of Metals and Alloys, pp. 1503-1530, Haasen, P., Gerold, V., and Kostorz, G. (Eds). Pergamon, Oxford.

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6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

41. 42. 43. 44. 45.

M.E. Kassner and K. Kyle Straub, S., Blum, W., Maier, H.J., Ungar, T., Borberly, A. and Renner, H. (1996) Acta Muter. 44, 4337. Argon, A.S. and Takeuchi, S. (1981) Acta Metall. 29, 1877. Gibeling, J.C. and Nix, W.D. (1980) Acta Metall. 29, 1769. Blum, W., Cegielska, A,, Rosen, A., and Martin, J.L. (1989) Acta Metall. 37, 2439. Hasegawa, T., Ikeuchi, Y., and Karashima, S., (1972) Metal. Sci. 6,78. Ginter, T.J. and Mohamed, F.A. (1982) J. Muter. Sci. Eng. 17,2007. Barrett, C.R., Nix, W.D., and Sherby, O.D. (1966) Trans. ASM59,3. Blum, W., Hausselt, J. and Konig, G. (1976) Acta Metall. 24,293. Weertman, J. (1 984). In Creep and Fracture of Engineering Materials and Structures, p. 1, Wilshire, B. (Ed). Pineridge, Swansea. Maruyama, K., Karachima, S., and Oikawa, H. (1983) Res. Mechanica 7,21. Mughrabi, H. (1983) Acta Metall. 31, 1367. Borbely, A., Blum, W., and Ungar, T. (2000) Muter. Sci. Eng. 276, 186. Borbely, A., Hoffinann, G., Aemoudt, E., and Ungar, T. (1997) Acta Muter. 45, 89. Lepinoux, J. and Kubin, L.P. (1985) Phil. Mag. A 57, 675. Mughrabi, H. and Ungar, T. (in press). In: Dislocations in Solids, Nabarro, F.R.N. (Ed). North Holland. Sleeswyk, A.W., James, M.R., Plantinga, D.H., and Maathuis, W.S.T. (1978)Acta Metall. 126, 1265. Kassner, M.E., PCrez-Prado, M.-T., Vecchio, K.S., and Wall, M.A. (2000) Acta Muter. 48,4247. Orowan, E. (1959). In: Internal Stress and Fatigue in Metals, p. 59. General Motors Symposium, Elsevier, Amsterdam. Gaal, I. (1984). In: Proc. 5th Inter. Riso Symp., pp. 249-254, Andersen, N.H., Eldrup, M., Hansen, N., Juul Jensen, D., Leffers, T., Lilholt, H., Pedersen, O.B., and Singh, B.N. (Eds). Riso National Lab., Roskilde, DK. Ostrom, P. and Lagneborg, R. (1980) Res Mechanica 1, 59. Kassner, M.E., Perez-Prado, M.T., Long, M., and Vecchio, K.S. (2002) Metall. and Muter. Trans 33A, 311. Ardell, A.J. and Przstupa, M.A. (1984) Mech. Muter. 3 , 3 19. Evans, H.E. and Knowles, G. (1977) Acta. Metall. 25, 963. McLean, D. (1968) Trans. AIME 22,1193. Przystupa, M.A. and Ardell, A.J. (2002) Metall. andMater. Trans. 33A, 231. Kassner, M.E. (1990) J. Muter. Sci. 25, 1997. Kassner, M.E. (1 993) Muter. Sci. and Eng. 166, 8 1. Kassner, M.E. and McMahon, M.E. (1987) Metall. Trans. 18A, 835. Doherty, R.D., Hughes, D.A., Humphreys, F.J., Jonas, J.J., Juul Jensen, D., Kassner, M.E., King, W.E., McNelley, T.R., McQueen, H.J., and Rollett, A.D. (1997) Muter. Sci. and Eng. A238,2 19. McQueen, H.J., Evangelista, E., and Kassner, M.E. (1991) 2. Metall. 82, 336. Young, C.M., Robinson, S.L., and Sherby, O.D. (1975) Acta Metall. 23, 633. Miller, A.K. (1 987). Constitutive Equations for Creep and Plasticity, Elsevier Applied Science, Essex, U.K. Widersich, H. (1963) J Metals, 423. Jones, R.L. and Conrad, H. (1969) TMS-AIME245,779. Levinstein, H.J. and Robinson, W.H. (January 1963). “The Relations between Structure and the Mechanical Properties of Metal.” In: Symp. at the National Physical Lab, p. 180. Her Majesty’s Stationery Office. From Weertman, J. and Weertman, J.L. (1983). In: Physical Metallurgy, p. 1259, Cahn, R.W. and Hassen, P. (Eds). Elsevier. Bailey, J.E. and Hirsch, P.B. (1960) PhilMag. 5,485. Taylor, G.I. (1934) Proc. Royal SOC.A145,362. Weertman, J. (1999). Mechanics and Materials Interlinkage, J. Wiley, New York. Nes, E. (1997) Prog. Muter. Sci. 41, 129. Kuchi, S.K. and Yamaghuchi, A. (1985). In: Strength of Metals and Alloys, p. 899, McQueen, H.J., Baillon, J.-P., Dickson, J.I., Jonas, J.J., and Akben, M.G. (Eds). Pergamon, Oxford.

Taylor hardening in five power law creep of metals and Class M alloys 46. 47. 48.

49. SO.

51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61.

27 1

Livingston, J.D. (1962) Actu Metall. 10, 229. Kassner, M.E. and Li, X. (1991) Scriptu Met. et Muter. 25,2833. Blum, W. (2002) private communication. Kassner, M.E., Miller, A.K., and Sherby, O.D. (1982) Metall. Trans. 13A, 1977. Blum, W., Absenger, A., and Feilhauer, R. (1980). In: Strength ofMetuls and Alloys, p. 265, Haasen, P., Gerold, V., and Kostorz, G. (Eds). Pergamon, Oxford. Daily, S. and Ahlquist, C.N. (1972) Scriptu Metall. 6, 95. Sikka, V.K., Nahm, H., and Moteff, J. (1975) Muter. Sci.and Eng. 20,95. Orlova, A,, Pahutova, M., and Cadek, J., (1972) Phil. Mag. 25, 865. Stang, R.G., Nix, W.D., and Barrett, C.R. (1971) Metall. Trans. 2, 1233. Clauer, A.H., Wilcox, B.A., and Hifih, J.P., (1970) ActaMetall. 18, 381. Gorman, J.A.,Wood, D.S., and Vreeland, T. (1969) J. App. Phys. 40,833. Parker, J.D. and Wilshire, B. (1980)Mater. Sci. and Eng. 43 271. Raymond, L. and Dorn, J.E. (1964) Trans. AIME 230, 560. Straub, S. and Blum, W. (1990) Scriptu Metall. et Mater. 24 1837. Blum, W. (1993). In: Materials Science and Technology, Vol. 6, p. 339, Cahn, R.W., Haasen, P., Kramer, E.S. and Mughrabi, H. (Eds). Wienheim, Verlag Chemie. Blum, W. (1991). In: Hot Deformation ofAluminumAlloys, p. 181, Langdon, T.G., Merchant, H.D., Morris, J.G., and Zaidi, M.A. (Eds). TMS.



MICROSTRUCTURAL DESIGN OF 7x50 ALUMINUM ALLOYS FOR FRACTURE AND FATIGUE

F. D. S. Marquis Department of Materials and Metallurgical Engineering College of Materials Science & Engineering South Dakota School of Mines and Technology, Rapid City, SD 57701

ABSTRACT This paper focuses on the microstructural design of the 7050 and 7150 aluminum alloys for the control of recrystallization, grain and subgrain morphologies, fracture toughness, fatigue crack initiation and fatigue crack growth of these alloys. An investigation of the evolution of the microstructure during primary, secondary and intermediate thermo-mechanical processing has been carried out. The study of the recrystallization behavior, the grain morphology and the sub-grain morphology has been carried out. Other microstructural features such as the morphology of the constituent particles, as the formation of primary hydrogen porosity and its evolution during intermediate thermo-mechanical processing have also been studied. The paper discusses the effect of these microstructural parameters on the fracture and fatigue behavior of these alloys. INTRODUCTION The high strength aluminum-zinc-magnesium-copper alloys gain strength by solution heat treatment, quenching and artificial aging through the formation of a complex sequence of intermediate microstructures, such as the Mg (Zn, Cu, Al)2 phase, that finally, in equilibrium, produce stable precipitates such as MgZn2 and Mg3Zn3A12. For higher strength, up to 3 wt% copper can be added. The copper content must be limited if the weldability and general corrosion resistance are necessary. Small amounts of manganese, chromium (7075, 7178), and zirconium (7049, 7050, 7150) are added to control the recrystallization and develop the highly directional wrought grain structure in wrought alloys. This structure is beneficial for stress corrosion cracking (SCC) resistance if the load is applied in the longitudinal rolling direction, because it is extremely difficult to propagate intergranular SCC cracks perpendicular to the highly elongated grain structure. For the same reasons this microstructure becomes very vulnerable if the load is applied in the short transverse (ST) direction, with consequent intergranular crack propagation in the longitudinal rolling (LR) or long transverse (LT) directions. The short transverse SCC susceptibility of these alloys is considerable in the underaged (W or T4) and peak aged (T6) tempers and is minimized in certain overaged (T7) tempers.

213

F.D.S. Marquis

214

These age-hardened high-strength alloys have been used successfully as structural materials due to their unique combination of low density, high strength, and high corrosion resistance. In addition their incorporation in airframe structures (aircraft and space vehicles) and light weight armored carriers has been critical to vehicle performance and cost due to their high strength-to-weight ratio, high specific stiffness, high durability, good machinability and formability, and low cost. In recent years, however, the more stringent demands established by the newer generation of aircraft and spacecraft have reactivated considerable scientific and technological interest in the development of improved high-strength alumipum alloys. This has been reinforced by the slow development of reproducible and reliable data (appropriate for design incorporation) on the fracture toughness and fatigue resistance of carbon fiber reinforced epoxy composites. These and other alloys are also being used in the design and manufacturing of advanced aluminum based metal matrix composites. 100

80

1

I

I

I

I

1

I

707 7178-T651

0

~

I

MD11 7150-T6151

0

4 400

LlOIl 7075T7651

DC-3 0 2024-T3

4

Junkers F-13

2o 10

I

01 1910

-I

2017-T4

Commerclat and mllitary alrcraff

I

1920

I

1930

I

1940

I

1950

1

757-767 0 0 OC-17

829 0 7470 7075-T651 7075T651

0

I

I

1960

I

1970

I

1980

I

1990

200 100

I

2000

Year first used in airplane

Figure 1 : Upper wing skin plate alloy and temper chronology In order to meet the demands of the aerospace industry, Alcoa in cooperation with Boeing developed a new aluminum alloy 7150 having improved combinations of strength and corrosion resistance. This alloy is a modified composition, modified processing, high purity version of the 7050 alloy, which was developed by J.T. Staley and co-workers at Alcoa (1, 2). In the T6 temper 7150 plates and extrusions developed high strength and adequate fracture toughness and fatigue resistance. However, in this temper the enhanced strength resulted in the degradation of the resistance to exfoliation corrosion, especially in the short transverse direction. In order to improve the resistance to exfoliation corrosion in the short transverse direction, without sacrificing strength, Alcoa developed a new temper, T77. This led to the incorporation of the 7150-T7751 in both plate and extrusions in the upper wing structures of the C-17 military transport plane and the 7150-T6151 in the upper wing structures of the European Airbus A310, McDonnell Douglas MD-11 and Boeing 757 and 767, as shown in figure 1 (3). The major portion of commercial 7x50 alloys are produced through conventional ingot casts typically 200"-190" x 60"-50" x 20"-16", which are subsequently processed into plates typically 2" to 12" thick, and then machined or formed into structural components. These components are subjected in service to multidirectional stresses and must possess the best combination of strength, ductility, fiacture toughness, and resistance to fatigue and stress corrosion cracking. For most

Microstructural design of 7x50 aluminum alloysfor fracture and fatigue

215

aerospace applications in recent years, the improvement of fracture toughness and fatigue resistance, especially in the short transverse direction, has become of crucial importance. The objective of this investigation was to evaluate the relative magnitude of various parameters such as design, microstructure, manufacturing, that significantly influence the above mechanical properties. MATERIALS AND EXPERIMENTAL METHODS Typical chemical compositions of the materials used in this investigation are presented in table 1.

TABLE 1 TYPICALCHEMICAL COMPOSITIONS (WT%) OF 7x50 ALLOYS Zn

Mg

cu

Zr

Cr

MU

Ti

Si Fe

6.316

2.135

2.25

0.102

0.002

0.01

0.029

0.027

V

BI

Be

Ni

Na

Ca

Pb

Cr-Eqv

Li

0.003

0.002

0.0004

0.004

0.0003

0.0004

0.003

0.037

---

Zr-2Ti

FeISi

Fe+Si

CuIMg Fe+Mu

0.078

Al

0.160 2.826 0.106 1.05 0.089 89.022 Typical values of Fe+ Si are approximately 0.1 in 7150 and 0.3 in 7050 The design of sampling of specimens for fracture toughness (Klc), fatigue crack initiation (FCI), and fatigue crack propagation (FCP) is discussed elsewhere (4, 5, 6). The rational for this design is based on macro and micro segregation and porosity distribution studies carried out by the author (7) and on the need to understand the correspondent scale up effects. Thus sections A and B are transverse (width x thickness) plates, taken at approximately 8" and 27" from the top (north) of the ingot. These sections have the benefit of representing microstructural gradients as function of the thickness, width and size of the ingots. Sections C and D are longitudinal (rolling direction x width) plates taken at T/4 and T2/2 respectively. T is the thickness of the ingot, which is 16" in this investigation. Since these plates have a typical thickness between 1 and 1 1/2", each of them has the benefit of representing relatively homogeneous microstructures, but quite different microstructures with different solute contents, from plate to plate. This design would thus allow an evaluation of the scale up effects from laboratory to commercial size ingots. This is so since the development of microstructural features such as micro and macro segregation, volume fraction and morphology of second phase particles and of the hydrogen induced porosity are microstructural variables that are ingot scale dependent. In addition, the microstructural evolution during advanced thermomechanical processing is very much dependent on the prior cast microstructure. Thus, in order to investigate the evolution of microstructures during industrial processing, four thermomechanical processes (A to D) were carried. Figure 2 shows processes A and B.

In addition and in order to investigate the effect of strain on the degree of static and dynamic recrystallization and grain morphology, three deformation processes (1 to 3), each with five or six

F.D.S. Marquis

216

pass hot rolling breaking sequences, were carried out, as shown in table 2. Both the thermomechanical processes (A to D) and the deformation Processes (1 to 3) were carried out on sections A to D. In order to determine the degree of recrystallization an HN03 etching technique was found to be preferable to the conventionally used Keller's etch. Specimens were etched 5 minutes in a 30% HN03-70% water at 85 "C. After this treatment, subgrain and grain boundaries were visible. Hardness measurements (five on each sample) were measured by using a Vickers Diamond Pyramid Hardness Tester at room temperature with a load of 20 kilograms. Specimens for

PROCESS E

Figure 2: Typical simulation of industrial type thermo-mechanical processing TABLE 2

AMOUNT OF DEFORMATION (%) FOR INDUSTRIAL AND LABORATORY INGOT PROCESSING

Pass Number

Industrial Simulation

Laboratory Processing Process 1 Process 2 Process 3

13 15 18 22

20 25 22 14

13 15 14 16

28 ---

16

18 23

___

13 15 14 16 18

_-_

Transmission Electron Microscopy (TEM) were prepared by a jet polishing technique from the materials in the thermomechanically processed and aged conditions. Scanning Electron Microscopy (SEM) was used to examine laboratory induced monotonic fracture surfaces and fatigue fracture surfaces. Plane strain fracture toughness tests (KIc) were carried out in compact

Microstructural design of 7x50 alurninum alloysfor fracture and fatigue

211

tension specimens, under ASTM E399-83 standards. Fatigue crack growth tests were carried out in compact tension specimens, under ASTM E647-86 standards. Both tests were conducted on a Materials Test System (MTS) Closed-loop Electrohydraulic Testing System. The pre-cracking and the crack growth testing were performed at a frequency of 16 and 7.5 hertz respectively. Crack lengths were measured visually using a traveling microscope and on microstructural sections.

In order to investigate and fully characterize typical microstructures Analytical Electron Microscopy (AEM), quantitative x-ray microanalysis using Energy Dispersive Spectroscopy (EDS) techniques, electron and x-ray diffraction techniques were used. Techniques of conventional electron diffraction with Selected Area Diffraction Patterns (SADPs) and Convergent Beam Electron Diffraction Patterns (CBEDPs) were used. In analytical electron microscopy the probe size was controlled by the standard objective apertures and by the use of convergent beams of appropriate size for the characterization of very fine coherent precipitate particles. Quantitative optical, transmission, and scanning electron metallography with elemental x-ray mapping were carried out directly or through camera scanning of direct micrographs on a Quantitative Image Analysis System. RESULTS AND DISCUSSION In this investigation, various parameters were observed to influence significantly the fracture toughness, fatigue crack initiation and fatigue crack propagation of these materials, as shown in figure 3. As stated earlier, the scope of this investigation does not include the detailed study of

Ingot

Type and Volume Fraction of Porosity

Chemical Design

Processing Parameters

Fracture Toughness

Fntigue Crack Propagation

Figure 3: Flow diagram of parameters that influence significantly the fracture toughness and fatigue resistance of 7x50 alloys

27 8

F.D.S. Marquis

the individual contributions of each and every of these parameters, but only evaluates the relative magnitude of their effects and establishes combinations that could optimize the mechanical behavior of these materials. In order to achieve this, various thermomechanical processes were designed, as shown in figure 2 and table 3. Process A achieved a completely recrystallized TABLE 3 EFFECT OF THERMOMECHANICAL PROCESSING ON MICROSTRUCTURE

Industrial Processing Recrystallized VolumeFraction % Grain

Morphology

Laboratory Processing Process 1 Process 2 Process 3

.55

.45

Elongated 70

.50

Elongated 50

Equiaxed

50

.20 Equiaxed

I0

microstructure as a result of the low temperature of deformation. In this process the large amount of strain energy introduced into the ingot at the relatively low deformation temperature (final pass at 270 'C) provides a large driving force for the recrystallization to occur during later heat treatments. Process A applied to section A generates a similar microstructure but with considerable amount of intergranular precipitation of both second phase particles and hydrogen induced porosity. This is explained by the proximity of the top (north) of the large ingot. Process B generated a most unrecrystallized microstructure, and process C generated a partially recrystallized microstructure. Process D achieved a partially recrystallized microstructure with significant coarse intergranular precipitation of second phase particles. The effect of strain, at a constant entry temperature of 425 "C, on the degree of recrystallization and grain morphology is shown in table 3. This temperature was selected since a significant increase in the hardness was observed during rolling within a temperature range, which includes 425 "C (4). This is attributed to the precipitation of very fine Cu and Zn bearing particles, which formed during rolling, and, together with metastable ZrA13 dispersoids, inhibited significantly the recrystallization process. TABLE 4 EFFECT OF THERMO-MECHANICAL PROCESSING ON KIC IN T-L ORIENTATION

Process

A

B

C

D

KIC, Ksi(in)lR

22

29

25

24

K,c for specimensfrom section A and process A was 18 Ksi(in)1I2 Typical plane strain fracture toughness results, in the T-L orientation, are shown in table 4. The high fracture toughness of process B (unrecrystallized microstructure) is attributed to both the strength of the grains and grain boundaries. Typical subgrain morphologies and substructures are represented in figures 4, 5 and 6. These subgrains are elongated in the rolling direction and contain multidispersions of coherent and semicoherent precipitates. Homogeneous distributions

Microstructural design of 7 x 0 aluminum alloysfor fracture and fatigue

279

Figure 4: Partially recrystallized morphologies: (a) columnar and (b) equiaxed

F

S

f

?

. -

' - .

t

f

t

I urn

Figure 5: Subgrains elongated in the rolling direction, with semicoherent precipitates at low angle boundaries and multidispersions of coherent precipitates inside the subgrains.

280

F.D.S. Marquis

I

I

I

Figure 6 : Semicoherent precipitates at low angle boundaries and multi-dispersions of coherent precipitates inside the subgrains. of very fine coherent precipitates are observed inside the subgrains, and the semicoherent precipitates are observed both at the sub-boundaries and inside the subgrains. A detailed investigation of the nature of these particles was carried out in different areas of many specimens. Analytical electron microscopy, with quantitative x-ray microanalysis using energy dispersive spectroscopy techniques showed that the semicoherent precipitate particles nucleated at the subboundaries are rich in Cu and to a less extent in Zn. These Cu and Zn rich precipitates explain the capability of these 7x50 materials to develop high strength during aging at high temperatures, with simultaneous improvement of the fracture toughness and fatigue resistance. Representative selected area diffraction pattern, micro diffraction and convergent beam electron diffraction pattern analysis were carried out and discussed elsewhere (6). The analysis show that most of the semicoherent precipitates can be indexed as intermediate modifications of the q' phase and that their composition is consistent with the Mg (Zn, Cu, A1)2 formula. This phase could be indexed as a face centered orthorhombic crystal structure. The results of this microstructural analysis agree with very recent work (8, 9, 10) and are discussed in detail elsewhere (1 1). The large number of fine subgrains and the high density of dislocations acted as sites for the precipitation of fine q' phase (6), as shown in figure 7a. These partially coherent precipitates were observed to promote homogeneous slip by forcing dislocations to bow around the precipitate particles through Orowan type by-pass mechanisms, leaving fine dislocation loops around these precipitates (Fig. 7b). In addition, the growth of incipient slip bands was impeded by sub-boundaries and the dislocation substructures within the grains, which prevent the formation of dislocation pile-ups at the grain boundaries rendering crack initiation, by the Zener precipitates and the formation of wide precipitate fiee zones, which retarded crack initiation and decreased the rate of fatigue crack propagation (da/dN) by prevention of interface decohesion. This explains the fact that these specimens were observed to fracture mostly by the transgranular dimple rupture mode, were both deep and shallow dimples have been observed associated with subgrain


28 1

Figure 7: (a) Pined screw dislocations and (b) dislocation loops around q' precipitate particles. structures. This suggests that this type of advanced processing influences beneficially crack initiation and decreases the crack growth rate at both low and high AK. The low fracture toughness of process A (completely recrystallized microstructure) is attributed mostly to the large proportion of intergranular fracture due to strain concentrations in the grain boundanes caused by alterations in the dislocations, precipitates and grain boundary structures The high angle grain boundaries were observed to be preferred sites for thc formation of coarse intergranular precipitates and hydrogen gas porosity (fig 8) Both features promoted fracture at or near grain boundanes with low dissipation of elastic strain energy Although other morphologies for the porosity and second phase particles were observed, thc intergranular oncs werc observed to be the least desirable The partially recrystallized microstructures generated by processes C and D exhibit intermediate values of plane strain fracture toughness and mixed Eracture consisting of both transgranular and intergranular modes. This is in general agreement with the work of Alarcon, Nazar and Montciro (12) although significant diffcrcnccs in thc proccssing and microstructures were observed, such as the subgrain morphology and substructure, which lead to higher fracture toughness values in the present investigation. Typical data for the resistance to fatigue crack growth, plotted as the crack growth rate (da/dN) versus the stress intensity factor (AK) is presented in figures 9 left and 9 right In all cases these curves followed a sigmoidal shape, the second stage of which could be well descnbed by a Paris type of equation da/dN = A ((UOP, where p is the slope of the curve and A is the value of the crack growth for AK=l . Within the low stress intensity range and for recrystallized microstructures, the lowest crack growth rates were observed in microstructures developed by process A In this range the microstructure played a very important role as shown by the data obtained by process A, section A and B (fig. 9 left) and processes A and B (fig. 9 right) The lowest value of all microstructures was observed in process B (fig 9). Within the intermediate stress intensity range, process B exhibited the lowest values and the lowest slope of fatigue crack growth, although processes A and B exhibited significant overlapping

F.D.S. Marquis

282

Figure 8: Typical morphologies of Hydrogen Gas Porosity: (a) and (c) thermomechanical processed, (b) and (d) recrystallized.

D-3

z" 9 4 10-5 5

I0

l5

S T R E S S INTENSITY FALTOR. AqM?UET)

27

5 10 l5 irl STRESS INTENSITY FACTOR, A!+fPa?fEiii)

Figure 9: Fatigue crack growth rate dependence on AK and microstructure These results are significantly different from those reported by Zaiken and Ritchie (13, 14) This is explained by the difference in processing and microstructures developed prior to aging The very fine textured subgrains developed in the present work rendered the aged microstructures with higher resistance to fatigue crack propagation Within the high stress intensity range the microstructures developed by process B continued to exhibit crack propagation at much higher stress intensity values Within this range of stress intensity the crack growth rate was considerably influenced by the microstructure. The main inicrostructural features that were observed to contnbute significantly to the differences in the fatigue crack growth of these specimens were. (a) the type and volume fraction of porosity, (b) the type and volume fraction of constituent particles, and (c) the grain structure, as shown in figure 3 Type I1 and 111 crack paths


283

were the most predominant in microstructures developed by process B. Type I was most frequently observed in recrystallized microstructures. A typical example is shown in figure 10. Typical data for the resistance to fatigue crack initiation is shown in table 5. This is in general agreement with the work of Sanders and Starke (15 ) .

Figure 10: Fatigue crack growth (2,700 cycles) in microstructures developed by process C. A tortuous crack path with crack with significant arrest are observed in unrecrystallized areas TABLE 5 RESISTANCE TO FATIGUE CRACK INITIATION MEASURED AS THE NUMBER OF KILOCYCLES TO PRODUCE A PRE-CRACK OF 0.4 MM UNDER THE SAME LOADING CONDITION

Recystallized high fpo 46

Recystallized high @a 59

Recystallized

Recysta1lized-k Unrecystallized

Unrecystallized

low fpo & fpa

low fpo & fpa

low fpo & fpa

71.8

Note: fpo = volumefraction ofporosi@, a n d f i a

61 = volumefraction

58

of secondphaseparticles.

The very fine recrystallized microstructures with low volume fracture of porosity and low volume fraction of large particles exhibited the highest resistance. This is attnbuted to the randomly oriented fine grains and the effectiveness of their high angle grain boundaries in preventing the transfer of plasticity to the adjacent grains. This inhibits the formation of long-range persistent slip bands. This resistance dropped considerably when either the porosity and or the particle volume fractions were increased. All other factors being equal, the increment in either the volume fraction or size of the Hydrogen porosity was observed to have the most deleterious effect. Primary hydrogen induced porosity was observed to play the most deleterious role in providing nucleation sites (Fig. 1l a and b) and propagation paths (Fig. 1l c and d) for cracks. The mechanisms of formation of this porosity are discussed elsewhere (18). The very fine intermediate semicoherent dispersoids of the type ZrAllj played a beneficial role in refining the grain and subgrain structure under all the processing conditions. They were indexed as a A3B-type superlattice structure. Both the crystallographic and the grain refining results are in agreement with recent work of Yan, Chunzhi and Minggao (10).

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F.D.S. Marquis

I

. a

.

Figure 11: Effect of hydrogen porosity on faligue crack initiation (a) and (b) and on crack propagation: (c) and (d)

These metastable ZrA13 dispersoids were not observed to provide direct crack initiation sites or crack propagation paths under fatigue conditions. This role was observed only when they were associated with hydrogen and hydrogen induced porosity. Sub-boundaries were not observed to provide sites for crack initiation or paths for crack propagation in either theunaged or aged conditions. These results differ significantly with those of Karashima, Oikawa and Ogura (16) which reported that subgrain boundaries were preferred paths for fatigue crack propagation. A possible explanation may reside in the different processing parameters and consequent different substructures. Upon aging it was observed that the subgrain structures containing very fine dispersions of coherent and semicoherent precipitates were often associated with ductile fracture with characteristic deep an shallow dimples. In low purity materials large Fe, Cu and Si rich second phase particles were observed to influence significantly, especially in low purity materials (7050), the fracture toughness and the resistance to fatigue crack initiation and fatigue crack propagation and to enhance intergranular fracture. These results are in general agreement with models for the mechanics and kinetics of fracture processes discussed recently by Srivatsan (17). However in high purity materials (7150) this role of second phase particles was considerably reduced, the major role being played by the hydrogen induced porosity, which is discussed in more detail in a separate publication (18). It is important to notice that the effects of second phase particles and porosity change quantitatively and qualitatively during scaling up from laboratory castings to commercial size ingots. From the point of view of fracture toughness, resistance to fatigue crack initiation, and resistance to fatigue crack propagation, a good optimization of the microstructure consisted of finc unrecrystallized grains, (10 to 20 pm) with very fine subgrains (0.1 to 0.5 pm) containing a multidispersion of semicoherent and coherent precipitates and a considerable dislocation substructure. These microstructures could be achieved, representatively, through the type of advanced processing suggested in this work.

Microstructural design of 7x50 aluminum alloysfor fracture and fatigue

285

CONCLUSIONS In low purity materials recrystallized microstructures exhibited the lowest fracture toughness and intergranular fracture mode. In high purity materials, without intergranular porosity or intergranular coarse second phase particles, recrystallized microstructures, exhibited improved resistance to fatigue crack initiation. Unrecrystallized microstructures exhibited the highest fracture toughness and transgranular fracture mode. Subgrain structures developed during advanced processing containing fine dispersions of coherent and semicoherent precipitates, developed during double aging, showed with fracture. These microstructures exhibited improved fracture toughness and resistance to fatigue crack propagation. Hydrogen induced porosity was observed to play a considerable role in providing nucleation sites and propagation paths for cracks. In the manufacturing of thick structural plates through the casting of large ingots, special procedures must be implemented in order to decrease the size and volume fraction of hydrogen related porosity (both primary and secondary) and/or to change its morphology into a less deleterious one. In order to minimize the extent of recrystallization and to increase the resistance to fatigue crack initiation and fatigue crack propagation the following recommendations are made: (a) the amount of reduction per pass should not be higher than 24%, (b) the amount of reduction of the final pass should not be higher than 16%, (c) the hot rolling temperature should be adjusted in order to develop a very tine unrecrystallized microstructure.

ACKNOWLEDGEMENTS The author wants to acknowledge finding from the Institute for Mechanics and Materials at University of California-San Diego, the South Dakota School of Mines and Technology, and the Alcoa Foundation. In addition the author wants to thank Drs. J. Staley, D. Chakrabarti, and D. Granger for rewarding discussions.

REFERENCES 1.

Staley, J.T., Hunsicker, H.Y. and Schmidt, R. “New Aluminum Alloy 7050,”The Minerals, Metals, & Materials Society (1971).

2.

Staley, J.T. “Aging Kinetics of Aluminum Alloy 7050,” Met. Trans,, 5 (1974), 929.

3.

Staley, J.T. ”Advanced Aluminum Alloys,“ in Encyclopedia of Advanced Materials (Pergamon Press, 1994).

4.

Marquis, F.D.S. “Recrystallization Dynamics in 7050/7150 Aluminum Alloys”, Report to GOED, (1988).

5.

Marquis F.D.S, “Fatique Crack Propagation in 7050-7150 Aluminium Alloys,” Report to GOED, (1988).

6.

Marquis, F.D.S. “Design and Advanced Manufacturing for the Optimization of the Fracture Toughness and Fatigue Resistance of 7x50 Aluminum Alloys”, Institute for Mechanics and Materials, University of California, San Diego, Report 94-16, (1994).

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7.

Marquis, F.D.S., "Macro and Micro Segregation and Porosity Distribution in 7050/7150 Ingot Materials," Unpublished Work.

8.

Yan, J., Chunzhi, L. and Minggao, Y. "On the q' Precipitate Phase in 7050 Aluminum Alloy", Muter. Sci. & Eng., A, 141 (1991), 123.

9.

Brenner, S.S., Kowalik, J., and Ming-Jian, H. "FIM/Atom Probe Analysis of a Heat Treated 7150 Aluminum Alloy," Surface Sci. 246 (1991), 210.

10.

Yan, J., Chunzhi, L. and Minggao, Y. " Transmission Electron Microscopy on the Microstructure of 7050 Aluminum Alloy in the T74 Condition", J. Muter. Sci., 27 (1992), 197.

11.

Marquis, F.D.S. "Analytical Electron Microscopy of Advanced Processed 7x50 Aluminum Alloys," Unpublished Work.

12.

Alarcon, O.E., Nazar, A. M. M. and Monteiro, W. M. "The Effect of Microstructure on the Mechanical Behavior and Fracture Mechanism in a 7050-T76 Aluminum Alloy", Muter. Sci. & Eng., A138 (1991), 275.

13.

Zaiken, E. and Ritchie, R.O. "Effects of Microstructure on Fatigue Crack Propagation and Crack Closure Behavior in Aluminum Alloy 7150," Muter. Sci. & Eng., 70 (1985), 151.

14.

Zaiken, E. and Ritchie, R.O. "On The Development of Crack Closure and the Threshold Condition for Short and Long Fatigue Cracks in 7150 Aluminum Alloy," Met. Trans:, 16A (1985), 1467.

15.

Sanders, R.E. Jr. and Starke, E. A. Jr., "The Effect of Intermediate Thermomechanical Treatments on the Fatigue Properties of a 7050 Aluminum Alloy," Met. Trans:, 9A (1987), 1087.

16.

Karashima, S., Oikawa, H. and Ogura, T. "Studies on Substructures Around a Fatigue Crack in FCC Metals and Alloys," Trans. Japan Inst. Metals, 9, 3 (1968), 205.

17.

Srivatsan, T.S. "Microstructure, Tensile Properties and Fracture Behavior of Aluminum Alloy 7150," J. Muter. Sci:, 27 (1992), 4772.

18.

Marquis, F.D.S. "Mechanisms of Formation of Hydrogen Porosity in 7x50 and 2x24 Aluminum Alloys. Effects on Mechanical Behavior," in Gus Interactions in Nonferrous Metals Processing, ed. D. Saha (The Minerals, Metals & Materials Society, 1996), 43.

Nan0 and Microstructural Design of Advanced Materials M.A. Meyers, R.O. Ritchie and M. Sarikaya (Editors) Published by Elsevier Ltd.

ELASTIC CONSTANTS OF DISORDERED TERNARY CUBIC ALLOYS Craig S. Hartley Air Force Research Laboratory, A FO SW A Arlington, VA 22203-1954

ABSTRACT Single crystal elastic constants of disordered alloys having body-centered cubic and face-centered cubic Bravais lattices can be calculated as functions of composition by modelling the lattice as a virtual crystal. The technique is based on the method of long waves applied to the virtual crystal. The three independent elastic constants are related to four, axisymmetric force constants (ASFC) for first and second neighbors. The ASFC are defined in terms of the first and second derivatives of a three-parameter, virtual pair potential, which is determined from the corresponding pair potentials of like and unlike atom pairs in the crystal weighted by the probabilities of their existence in the first and second neighbor shells. This technique permits calculation of single crystal elastic constants of multicomponent disordered alloys using data obtained from elastic constants of terminal solid solutions of binary alloys and pure elements when the elements have the same crystal structure as the alloys. An illustration of the technique is given for ternary alloys of copper, aluminium and nickel.

INTRODUCTION Single crystal elastic constants of pure elements and stoichiometric compounds can be calculated ab initio with very good accuracy by calculating the quadratic variation in energy with strain of a computational cell as a function of the state of strain. Elastic constants of disordered alloys are not easily computed by these methods because of the difficulty of modelling structures for which the exact position and species of each atom is not known. Approximations to the disordered state include the Coherent Potential Approximation (CPA) for alloy potentials [ 11 and employing a large computational cell to perform calculations for several atomic configurations to simulate disorder. Alternatively, the semi-empirical Modified Embedded Atom Method (MEAM) [2] can be employed. For these techniques, the identity of atoms present on each lattice site of the computational cell must be specified and the energy of the computational cell minimized not only with respect to external boundary conditions but also with respect to the positions of all of the atoms in the computational cell. A recent development proposed by Hartley [3] models elastic constants of disordered alloy single crystals by assuming a homogeneous virtual crystal with interactions between first and second neighbors. Axisymmetric force constants for the first and second neighbor shells are derived from a virtual pair potential that is a sum of the pair potentials of the various types of atomic pairs weighted by the probability of existence of each type of pair. A similar approximation is employed for the nearest neighbor distance in the virtual crystal. Pair potentials for like and unlike pairs are assumed to depend only on the nature of the

287

C.S. Hurtley

288

atomic species and the position vector connecting the atom centers and not otherwise on the surroundings. Thus, to a first approximation, potentials for unlike pairs derived from elastic constant measurements on binary disordered alloys can be employed to construct virtual potentials in multicomponent alloys that contain the two atomic species. In the following sections, the construction of the virtual potential is summarized, followed by a review of the connection between ASFC and single crystal elastic constants for bcc and fcc lattices. These concepts are then applied to demonstrate the calculation of the composition dependence of the single crystal elastic constants of a representative disordered ternary alloy using data on the binary alloys and pure substances.

ELASTIC CONSTANTS AND THE VIRTUAL POTENTIAL General Considerations The internal energy per atom, U, is approximated as the sum of two terms: 1) the sum of painvise interaction energies between a limited number of neighbors, Up, and 2) a many body term, UV, that depends on the total volume of the crystal and accounts for the interaction of electrons with one another and with individual ions. If the painvise interaction term arises entirely from the Coulomb energy of positive point charges arranged on a lattice immersed in a uniform environment of negative charge, forces between neighboring ions are directed towards the centers of the atoms. However, if the deviation of electron energies from those of a free electron is included in the pair potential, these forces depend not only on the distance between ion centers but also on the crystallographicdirection connecting them. To determine Up, select any atom as the origin of a coordinate system with directions as the associated coordinate axes. Since interatomic forces are generally of short range, sum the painvise interaction term over only first and second neighbor atoms. Expanding Up in terms of atomic displacement about the origin gives* [4,5],

where s is the number of atoms interacting with the atom at the origin and the primes indicate the order of partial differentiation of the interatomic pair potential, , with respect to components of the position vectors connecting the origin to each neighboring atom, evaluated at the equilibrium spacing between the atoms. The term involving first derivatives corresponds to the total force exerted by neighboring atoms on the atom at the origin. To avoid imposition of the Cauchy condition on the elastic constants of the crystal it is customary to choose UV such that the first order term in its Taylor series expansion exactly cancels this force. The zerofhand higher order terms are subsumed into the corresponding terms in the expansion of 4 . The complete Taylor series expansion of U, therefore, contains contributions from both the pair potential and the many-body contributions represented by Uv. With the above restrictions, the internal energy can be represented as an appropriate sum over a net pair potential, cp, that contains many-body terms.

+

, The second derivatives form a force constant matrix such that the second derivatives, cp0”(’) = f,,@)represent the force exerted on the atom at the origin in the x, direction when an atom at the rh lattice point experiences a unit displacement in the x, direction. Independence of the order of differentiation requires that the force constant matrix be symmetric. Neglecting terms O(luf) results in the harmonic approximation for the total potential energy of the crystal at absolute zero. The quasi-harmonic approximation, in which elements of f,,@) are regarded as temperature-dependent material parameters, is a similar form that describes the potential energy at temperatures above zero K.

* Summation from 1 to 3 over repeated Latin suffixes

is implied unless otherwise indicated.

Elastic constants of disordered ternary cubic alloys

289

Applying conditions that insure that the force constant matrix possesses the symmetry of the cubic crystal system reduces the number of independent force constants to two each for the first and second neighbors [6]. The axisymmetric force constants (ASFCs) for each of the first and second neighbor shells are 1) a,,, corresponding to stretching bonds between nthneighbors (n = 1,2), and 2) Pa, corresponding to bending such bonds away from the direction joining the atom centers. In terms of appropriate derivatives of the pair potential:

where r(") is the position vector connecting an atom at the origin to an atom in the nth neighbor shell. Derivatives appearing in equation (1) can be expressed in terms of the ASFC:

where

= xi"' /Ir('')l. The relationship between the ASFC and the single crystal elastic moduli can then

be obtained by comparing appropriate terms in the equations of motion of the atom at the origin [7,8]. The magnitudes and signs of the ASFC give useful information on the dependence of the pair potential on interatomic distance. If the minimum in cp occurs between the first and second neighbors, a1 would be expected to be positive and PI, negative. The second neighbor stretching constant, a2, will be positive also unless the potential exhibits oscillations in the vicinity of this distance [9], and p 2 will be positive. The relative magnitudes of the ASFC will depend on the shape of the potential curve near the first and second neighbor distances. Since there are four independent ASFC but only three independent elastic constants for cubic crystals, either additional experimental information is required or some assumptions must be made about the potential to determine the force constants in terms of the elastic constants. If neutron scattering data are available, a fourth independent, equation can be derived to provide a unique solution for the four ASFC [lo]. Since such information is rare for alloys, it is convenient to make an assumption that links the ASFC through a potential that can be determined by fitting data to elastic constants alone. For cubic metals, the three independent elastic constants depend on the first and second derivatives of the pair potential. Consequently, a general cubic expression in Irl, cp(r) = Q~ + Q,r + Q z r z+ ~

~

r

~

,

(4)

suffices to provide the required number of independent parameters to construct the potential. The parameters to be determined are the coefficients of the first, second and third powers of the interatomic distance, since the constant term does not enter into the expressions for the ASFC. The potential so obtained cannot give accurate values for absolute energies since the constant term is not determined from the elastic constants. Coefficients of the cubic polynomial potential can be related to other mathematical forms, such as the Morse Potential Function (MF'F) [ 111, by expanding the latter in a Taylor series about the minimum in the potential to third order in the interatomic distance, then comparing coefficients of like powers of Irl [3]. Since the

290

C.S. Hurtley

MPF is a three-parameter function, this procedure makes the constant term in the polynomial a function of the other three coefficients. A similar process can be applied to express the polynomial potential in terms of other mathematical forms as long as there are only three adjustable parameters.

The Virtual Potential In order to develop expressions for the composition dependence of the elastic constants of an alloy it is necessary to construct a potential for the alloy based on the potentials for like and unlike atomic pairs. Consider a single-phase alloy single crystal to be a virtual crystal in which the mean interatomic spacing and virtual pair potential are obtained by a quasi-chemical approach using the potentials of like and unlike pairs. In the quasi-chemical approximation [12], the internal potential energy of a random solid solution is expressed as a sum over interatomic potentials of the several types of pairs in the alloy, weighted by the probability of existence of each pair. In an alloy of M components a virtual pair potential can be defined as:

where ppvrepresents the probability of the pair consisting of an atom of type p and one of type v, and qyPV)is are symmetric in p and v. the pair potential for the pv pair. Both pPvand qPV) The probability of a randomly chosen atomic site being occupied by an atom of species v is cv, where c is the atomic fraction of that species. For a disordered alloy having no short-range order the mean number of atoms of species p in the nth neighbor shell is z(")c;) where the coordination number of the nth neighbor shell is z(") and the concentration of species p in the same shell is c:).

Then the probability of finding an

atom of species p in the nthneighbor shell is c;' . For such an alloy the probability of finding a vp pair with one atom at the origin and the other in the nthneighbor shell is:

where':n is the number of atoms of species p in the nth neighbor shell. Then equation ( 5 ) gives for the virtual potential of a disordered, single-phase binary alloy of species A and B:

where the suffixes indicate the type of pair to which the potential applies. Writing the potential for each component pair as a cubic polynomial in the form of equation (4), inserting into equation (7) and collecting coefficients of like powers of r reveals that each in equation (4) depends on composition according to equation (7) with the appropriate coefficient substituted for the corresponding 'p(lrv). It is important to note that this quadratic composition dependence applies to the coefficients of the cubic polynomial, but not necessarily to the parameters appearing in other mathematical forms of the potential [3]. To determine the alloy ASFCs from equation (2), it is necessary to evaluate the virtual potential at appropriate values of the first and second neighbor distances. For this purpose, the mean nearest neighbor distance in the alloy can be expressed in terms of the corresponding spacings of the like and unlike pairs


29 1

present in the alloy [13,14]. The mean nearest neighbor distance in the disordered binary alloy considered above is

where the r,, refer to spacings of various kinds of atomic pairs in the alloy, which are assumed to be constant throughout the composition range of interest. The mean spacings of more distant neighbors are calculated from the geometry of the lattice. In the spirit of the quasi-chemical approximation, we assume that the nearest neighbor distances of atomic pairs depend only on the atomic species, but not otherwise on the surroundings of the pair.

Elastic Constants and Axisymmetric Force Constants Both face-centered (FCC) and body-centered cubic (BCC) crystals are characterized by a lattice parameter, a,, which is also the distance from an atom at the origin to its six, second neighbors, which lie along directions. The twelve nearest neighbors in FCC lie at a distance a0/d2 from the origin along 4 1 0 > directions, while the eight nearest neighbors in BCC are a0d3/2 from the origin along -411> directions. The relationship between single crystal elastic constants, referred to cube axes and expressed in reduced Voigt notation, and the first and second neighbor ASFCs can be expressed

where the order of the neighbor is indicated by the suffix on the ASFC and M is a 3 X 4 matrix that depends on the crystal structure. Values of M for both FCC and BCC crystals are given in reference [3]. In order to relate the ASFCs to the virtual potential, it is necessary to evaluate derivatives at mean neighbor distances appropriate to the alloy composition. These can be obtained by determining like and unlike pair spacings from data on the composition dependence of the lattice parameters in the phase field of interest using a least-squares tit to equation (S), which can then be employed to calculate the nearest neighbor spacing for any composition of interest in the single-phase field.

Composition Dependence of Elastic Constants The definitions of ASFC in terms of the virtual potential using equations (2) and (4) and the composition dependence of the pair spacings, equation (S), can be used to solve for the coefficients of the virtual potential in terms of experimentally determined elastic constants of alloy single crystals and their mean interatomic spacings. Equation (9) leads to:

for the coefficients of the polynomial form of the virtual potential. Values of N for FCC and BCC crystals are given in reference [3]. It is understood that the elastic constants and nearest neighbor spacing apply to the same alloy composition. Inserting the explicit composition dependence of the virtual potential

C.S. Hurtley

292

coefficients ofa binary alloy and inverting equation (10) leads to the expression

i; -

C, 1

for the explicit composition dependence of the elastic constants in terms of the polynomial coefficients and the mean nearest neighbor spacing corresponding to the composition, CA. This procedure has been demonstrated for several binary alloys having both face-centered cubic and body-centered cubic lattices and varying types of solubility conditions [3,15]. The procedure can easily be generalized to a multi-component, single-phase alloy by noting that for an M component alloy there are 3M(M+1)/2 independent Coefficients required to construct the polynomial potential and M(M+1)/2 independent pair spacings required for determination of the mean nearest neighbor distance. These parameters can be determined from the lattice parameter and elastic constants of the pure components when they have the same crystal structure as the alloy in question. Otherwise, it is necessary to obtain them from least squares fits to experimental data on elastic constants and lattice parameters of alloys in the single-phase field being studied. This process is facilitated by expressing the pair probabilities, the pair spacings and the Coefficients of the polynomial potential as multidimensional vector quantities. First, define the M (M+1)/2 dimensional pair probability vector, P(M)as

P MM

PM(M-I) + P(M-I)M where the elements of the matrix are the random probabilities of each type of pair. The top M terms are the probabilities of pairs of like species, while the bottom M(M-1)/2 terms are the probabilities of unlike pairs. In a similar manner, define the pair spacing vector, %MI, where the elements are the pair spacings of each type of pair in the alloy. Then the mean nearest neighbor spacing of the alloy in terms of these vectors becomes

where the superscript T indicates the transpose of the vector. In a completely analogous manner, the coefficients of the polynomial virtual potential can be expressed as functions of composition. Defining the vectors @{"' (i = 1,2,3) for the coefficients of the polynomial potential of the types of atomic pairs in the alloy permits writing the coefficients of the virtual potential as


293

(14)

is formed by stacking the transposes of the three vectors The 3 X M(M+1)/2 matrix, extension of equation (1 1) to M components can be written

Then the

which, with equation (13), gives the composition dependence of the elastic constants of alloys explicitly in terms of composition using parameters that describe the spacing and polynomial potential coefficients of each type of pair.. APPLICATION TO A TERNARY ALLOY SYSTEM

The procedure described in the previous section is illustrated in the following discussion by applying it to a calculation of the elastic constants of ternary alloys of copper, aluminium and nickel. All three of the components have the face-centered cubic crystal structure in the pure form and, although there is not complete miscibility of all three components in the solid state, there is a considerable single-phase field adjacent to the copper-nickel binary system extending well into the copper-rich and nickel-rich comers [16]. Lattice parameter data exist for binary alloys in sufficient quantity to determine the spacings of the like and unlike atomic pairs in the three binary systems using equation (13) [17]. Virtual potentials in the polynomial format have been constructed from data on single elastic constants and lattice parameters of a sufficient number of binary alloys to construct a virtual potential for the temary system [3,15] in the form of equation (14). Values of the relevant pair spacings and polynomial pair potential coefficients are given in Table 1.

Table 1. Pair Spacings and Polynomial Potential Coefficients

The data in Table 1 were used in Equation (15) with the value of N for face-centered cubic crystals, 1.414 4.121 -5.536

11.314 11.314 -11.314

36.728 22.607 -18.364

1

to obtain the room-temperature elastic constants of single crystals of ternary alloys in the range 0

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