Geometrical Frustration and Glass Formation T. EGAMI, V. LEVASHOV, R. AGA, and J.R. MORRIS The effect of geometrical frustration in the atomic structure on the formability of bulk metallic glasses is discussed from a general point of view. It is pointed out that there are two distinct and complementing pathways to easy glass formation: stabilizing the glass itself and destabilizing the corresponding crystalline state. While the discussions in the field tend to focus on the first one, the second in fact is a more effective approach. Examples of both will be discussed using soft- sphere, rather than hard-sphere, packing concepts. DOI: 10.1007/s11661-008-9555-9 Ó The Minerals, Metals & Materials Society and ASM International 2008 I. INTRODUCTION EVER since the first metallic glass was obtained by rapid quenching from the melt, [1] glass-forming ability has been one of the central subjects of study in the field of metallic glasses. Because the effort to discover a new glass forming composition by experiment is a long and tedious one, there is a serious need for guiding principles for glass formation. The interest in this subject was particularly heightened by the recent discovery of bulk metallic glasses. [2,3] For instance, the group of Inoue at Tohoku University (Sendai, Japan) used the accumu- lated wisdom of the field to succeed in discovering a large number of bulk metallic glass compositions. [4] The three principles they used are as follows: (1) three or more elements should be used, (2) the difference in the atomic size should be greater than 12 pct, and (3) the heat of formation has to be negative. It has been known for a long time that the heat of formation and the atomic size difference are the key parameters for glass formation. [5,6] Recent studies fur- ther reinforce their importance. [7–9] The merit of prin- ciple 3 is relatively easy to understand, because if the heat of formation is positive, the system tends to phase separate, and alloying, necessary for glass formation, will be suppressed. It is also evident that these three principles are, in a sense, a converse of the Hume– Rothery rules [10] and the Darken plot [11] for solid solubility. However, they are not exactly a converse, because the Hume–Rothery rules and the Darken plot are addressing the questions in the equilibrium state, while glass formation is intrinsically a nonequilibrium phenomenon. As pointed out by Turnbull a long time ago, [12] glass formation is a kinetic process that can happen to any substance, as long as the substance is cooled sufficiently fast. Yet, there are close connections between solid solubility and glass formation. In this article, we will discuss the basis for principles 1 and 2 in some depth, in particular, in the context of the under- lying wider and more general issue of geometrical frustration in three dimensions. There have been discussions on glass formability in terms of secondary properties, such as the ratio of the glass transition temperature to the melting temperature or the crystallization temperature. [12,13] However, they are less insightful, because we do not understand these secondary properties very well, and are also less useful because they can be assessed only a posteriori. In this article, therefore, we focus on the geometrical issues related to the atomic size effect. We will first discuss the question of packing spheres in the space. This is an old and yet very difficult and unresolved question. While most of the discussions on this subject assume packing of hard and incompressive spheres, we assume the spheres to be soft and compres- sive, because atoms are compressive, and atomic defor- mation and elasticity is critically important in glass formation, as will be discussed subsequently. Also, in applying this discussion on the question of glass formability, we have to note that there are two aspects in the question, just as ‘‘a glass can be equally half- empty or half-full.’’ Glass stability can be improved either by lowering the energy of a glass or raising the energy of a competing crystal. When considering glass formability, many researchers tend to focus on the stability of a glass itself. For instance, the glass stability may be improved by more efficient packing of atoms, which would lower the free energy of the glass. [14] However, more often than not glass stability is a consequence of the instability of a competing crystalline phase. [6,7] We will consider both aspects with equal footing. T. EGAMI is with the Departments of Materials Science and Engineering and Physics and Astronomy, University of Tennessee, and with Oak Ridge National Laboratory. Contact e-mail: egami@utk.edu V. LEVASHOV is with the Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996. R. AGA is with Oak Ridge National Laboratory. J.R. MORRIS is with the Department of Materials Science and Engineering, University of Tennessee, Knox- ville, TN 37996 and Oak Ridge National Laboratory, Oak Ridge, TN 37831. This article is based on a presentation given in the symposium entitled ‘‘Bulk Metallic Glasses IV,’’ which occurred February 25–March 1, 2007 during the TMS Annual Meeting in Orlando, Florida under the auspices of the TMS/ASM Mechanical Behavior of Materials Committee. Article published online May 15, 2008 1786—VOLUME 39A, AUGUST 2008 METALLURGICAL AND MATERIALS TRANSACTIONS A