Advances in Research and Applications (eBook)

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1996 | 1. Auflage
403 Seiten
Elsevier Science (Verlag)
978-0-08-086514-0 (ISBN)

Infos

Solid State Physics, Volume 50 continues the series' tradition of excellence by focusing on the optical and electronic properties and applications of semiconductors. All of the topics in this volume are at thecutting-edge of research in the semiconductor field and will be of great interest to the scientific community.

Inhalt

Front Cover 1
Solid State Physics, Volume 50 4
Copyright Page 5
Contents 6
Contributors 8
Preface 10
Chapter 1. Some Fundamentals of Grain Growth 14
I. Introduction 14
II. Ideal Grain Growth 16
III. Some Elementary Solutions 20
IV. The Motion of Boundary Intersections in 2D and 3D 23
V. The Law of Mullins and von Neumann 26
VI. Asymptotic Properties and the Fate of Small Cells 31
VII. Approximate Models 32
VIII. Three-Dimensional Simulation 37
IX. Analogous Systems 37
X. Separated Grains: The Lifshitz-Slyozov Theory 39
XI. Conclusions 48
Chapter 2. Nucleation Theory 50
I. Introduction 51
II. Classical Theory 52
III. Kinetic Approach 70
IV. Rate Equation 106
V. Fokker-Planck Equation 123
VI. Topics in Multicomponent Nucleation 148
Chapter 3. Driven Alloys 202
I. Introduction 202
II. Typical Examples of Phase Transformations in Driven Alloys 204
III. Phenomenological Theory 234
IV. Typical Predictions and Experimental Assessments 262
V. Conclusions 332
Chapter 4. Light Emission from Silicon 346
I. Overview 347
II. Intrinsic Luminescence in Silicon 352
III. Extrinsic Luminescence 355
IV. Silicon Alloys 373
V. Strain Fields 376
VI. Nanostructured Silicon 380
VII. Applications 386
VIII. Conclusions 393
Author Index 396
Subject Index 408

Textauszug

Some Fundamentals of Grain Growth

D. Weaire; S. McMurry    Department of Physics, Trinity College, Dublin 2, Ireland

I Introduction

The structure of many natural systems is essentially cellular:1 It consists closed cells with well-defined boundaries. In materials science the arrangement of grains in a polycrystalline solid is of such a character. The grain boundaries are associated with a positive surface energy. Accordingly, they migrate in such a way as to lower that energy. The grain structure evolves with time in the direction of increasing grain size.

The observation and description of grain growth has long been a favorite research topic for metallurgists and materials scientists. It continues to be the subject of regular international conferences. Even if we adopt a narrow theoretical focus on idealized models, as we shall do here, the literature is daunting in its extent and variety. Our limited goal will be to provide a coherent basis from which to approach this mountain of miscellaneous papers. We will examine some of its fundamentals and summarize the most reliable results of theory and computer simulation, in both two and three dimensions.

Other reviews undertaken in recent years include those of Atkinson,2 Stavans,3 Glazier and Weaire,4 and Fradkov and Udler,5 each with its own perspective. They all offer much more detail than is given here.

In the next section we shall define the canonical model for grain growth, in which the velocity of a boundary is proportional to its curvature. Some scaling solutions in two dimensions are then reviewed. We then describe the more general problem of the evolution of a random cellular structure. In two dimensions this is subject to a remarkable law due to Mullins, which is discussed in Section 5.

Section 6 covers the asymptotic properties of grain growth, whose description is often the main objective of theory. We will comment only briefly on the plethora of approximations which have been devised at various levels, as an alternative to a full simulation on the computer. There follows a discussion of the challenge of accurate simulation in three dimensions, still to be met.

The remaining sections deal with a number of generalizations or related problems, primarily that of the Lifshitz-Slyozov model, which represents the growth of separate particles rather than grains. This is also, in a sense, curvature-driven growth. While it appears at first to be an easier problem than grain growth, it has in practice proved controversial. We may not be able to dispel all of the controversy and confusion which surrounds it, but hope to contribute to doing so, for it is clearly unsatisfactory that a central model of materials science should be the subject of such uncertainty.

Other miscellaneous topics which are broadly analogous are mentioned. However, we will not stray far afield into the wide variety of cellular systems, among which may now be counted the large-scale structure of the universe. We will inevitably be drawn into discussion of coarsening of the ideal soap froth, since many authors have made no distinction between this and grain growth. As we shall see, this is not strictly correct.

Theory for grain growth has often been pursued without much contact or comparison with experiment. This may be deplorable in principle, but can be understood when one considers the multitude of effects which intrude in practice and are not in the ideal model. Among these, anisotropy of the surface energy and mobility must be paramount. Pinning of boundaries, which may arrest grain growth at low temperatures, is also common. Some of the true richness and difficulty of the practical side of the subject may be appreciated by consulting, for example, the book of Chadwick.6

The reader may also be disappointed by the repeated retreat into two dimensions, where both theory and simulation are much easier. The difficulty of the three-dimensional case is illustrated by a flurry of activity two decades ago. Rhines and Craig7 claimed to have significant new results of a general nature, for the analysis of 3D ideal grain growth. These were regarded as a major breakthrough.8 In due course various inconsistencies in that work were pointed out,9, 2 and it did not, after all, form the basis for further insights.

Only carefully selected experimental systems conform really well to the ideal model. These include plastic crystals such as CBr4. The favorite choice for more recent studies has been succonitrile.10

II Ideal Grain Growth

1 TWO DIMENSIONS

The standard model of idealized grain growth as shown in Fig. 1 may be defined as follows. It is to be applied to the boundaries of a network, enclosing cells which represent grains.

Fig. 1 Idealized grain growth in two dimensions. Each boundary moves with a local normal velocity proportional to its local curvature, while vertex angles remain fixed at 120°.

(a) The normal velocity υ of a boundary is proportional to its local curvature c (directed toward its concave side).

(b) Apart from configurations which occur only instantaneously, only threefold vertices are allowed, and the vertex angles are 120°. Fourfold vertices, once formed, dissociate.

Rule (a) is consistent with the motion of a boundary (which has no inertia) under the action of a force due to surface tension σ, opposed by a viscous force with the coefficient k, so that

=μc,whereμ=σk

  (2.1)

This is often called the Allen-Cahn equation. 11, 12 The constant μ is known as the grain boundary mobility. This is, of course, very much an ad hoc derivation. Allen and Cahn11 gave an explicit derivation from the Gins-burg-Landau equation in the context of phase separation. Eq. (2.1) emerges when the finite thickness of the boundaries is small compared with their radii.

The force/viscosity argument also implies rule (b). If we consider the forces acting on the elements of boundaries close to a vertex, the viscous force goes to zero as the extent of these is shrunk, and so the surface tension forces acting at the vertex must be in equilibrium with each other. It is possible to introduce a drag force on the point vertex (that is, a finite vertex mobility) and hence break the 120° rule,5, 13 but the physical significance of this is rather unclear.

The instability of a fourfold vertex in 2D is quite elementary: It is easy to see that there is always an infinitesimal displacement of the system which dissociates the vertex and lowers the energy of the system. It can therefore be expected to follow this course, if we assume that there are no additional physical effects which oppose such dissociation. Matters are not quite so elementary in 3D (next section).

If a fourfold vertex forms and dissociates as in Fig. 2, the two threefold vertices must initially move with infinite velocity (see Section 4). In a notation established by Weaire and Rivier1 this is known as a T1 process; it may also be called neighbor-switching.

Fig. 2 Three stages in the evolution of grain growth, in which a fourfold vertex is formed and dissociates.

A fully symmetric fourfold vertex would, of course, present a dilemma as to which way it should be dissociated. We will discount such pathological special cases, appealing to the fact that we are dealing with a disordered system.

The global consequence of these rules is coarsening, that is, the progressive elimination of cells. The disappearance of a cell is usually represented as in Fig. 3(a), which has been called the T2 process. However, one of the surprises of recent years has been the finding that, strictly speaking, this process never happens in the coarsening of a disordered grain structure in 2D. Instead, all such cells suffer a T1 change to form a two-sided grain, as in Fig. 3(b), which then vanishes. This might be called the T2’ process. For further details see Section 6.

Fig. 3 (a) A three-sided cell may appear to shrink to form a vertex, but it is now realized that this happens only in symmetric configurations, (b) In the absence of symmetry, all cells pass through a stage in which they are two-sided, if only at a very late stage in their elimination.

2 THREE DIMENSIONS

In 3D the curvature which determines boundary motion is the mean curvature (arithmetic mean of the two principal curvatures), not the Gaussian curvature (product of the principal curvatures).

Rule (a) is the same as in 2D, applied to every point on a cell surface. Rule (b) forces these surfaces to meet, three at a time, on edges at mutual angles of 120°. This in turn forces the vertices (which are the intersections of such edges) to have local tetrahedral symmetry (with angles taking the value −1−13).

The instability of higher vertices (where more than six surfaces meet) has been well known since the nineteenth century14 in the context of the equilibrium of soap films, but was only recently demonstrated with full mathematical rigor for that case. 15 Surprisingly, it was recently called...

Produkt-Details

• Erscheint lt. Verlag: 13.12.1996
• Mitarbeit: Herausgeber (Serie): Henry Ehrenreich, Frans Spaepen
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Festkörperphysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-086514-3 / 0080865143
• ISBN-13: 978-0-08-086514-0 / 9780080865140