INTRODUCTION

Since the original work due to G. I. Taylor (1934) numerous attempts have been made to explain the work hardening phenomena in terms of dislocation mechanisms. Much progress has been made on the experimental side in determining the nature of the stress–strain curves of various materials as a function of various parameters such as temperature, strain rate, grain size, crystal orientation (in the case of single crystals), alloy composition etc. The development of methods for the direct observation of dislocations, mainly by transmission electron microscopy and etch-pitting techniques, has made it possible to study in detail the internal distribution of dislocations as a function of deformation, and the dislocation arrangement is now quite well established in a number of cases. Studies of slip lines have also yielded valuable information on the scale on which the slip processes take place. Dislocation theory has seen much development and a number of mechanisms and dislocation interactions important in work hardening have been established. However, in spite of a spate of theories during the last few years designed to explain the work hardening of single crystals, the phenomena are still not well understood. The aim of any work hardening theory is to explain the stress–strain curve, and its dependence on temperature, strain-rate, etc. This involves usually the assumption of a model of the dislocated state, which is characterised by a flow stress which depends on one or more parameters of the dislocation distribution, and the variation of these parameters with strain.

ATOMIC FORCES AND THE STRENGTH OF SOLIDS

Professor Sir Nevill Mott was one of the first to realise, at the beginning of the 1930s, that, with the arrival of quantum mechanics, the road at last lay open for the creation of a general theory of bulk matter in terms of atomic and electronic structure and properties. The great programme of solid state physics was thus begun.

In parts of this programme, for example in the theory of metallic conductivity, a quantum theory of solids could be developed directly by finding suitable, if approximate, solutions of Schrodinger's equation. In others, however, the physical situation proved too complex for this approach. The strength and plastic properties of solids, for example, are not determined by the average behaviour of the atoms but by the exceptional behaviour of those relatively few atoms situated at lattice irregularities. The theories of such properties could not be developed without providing some intermediate concepts, to by-pass the mathematically formidable and to some extent physically irrelevant problem of solving Schrodinger's equation for lattice irregularities, and to enable the theory to work directly in terms of the atoms and their movements. One of the most useful of these has been the representation of the cohesion of a solid in terms of ‘atomic bonds’, i.e. force–displacement relations between pairs of atoms such as that in fig. 6.1.

For covalent solids this representation can be justified quantummechanically, but for metals it is highly artificial and must be used with care.

INTRODUCTION

In this chapter we shall mainly be concerned with point defects (vacancies and interstitials) in pure crystals, though substitutional and interstitial impurities will receive some attention, particularly as ideas originally developed to deal with them have subsequently been applied to vacancies and self-interstitials.

Section 1.2 is concerned with the formation of defects by thermal excitation, cold work and irradiation. Part of section 1.3 (on defect mobility) may appear a little out of proportion. However, it has seemed to the writer that many solid state and metallurgical texts introduce the standard jump frequency formula with much less discussion than they devote to related matters, and that therefore a slightly extended treatment at an intermediate level might be useful.

Section 1.4 (Physical properties of defects) is concerned with some of the basic physical properties (energy and volume of formation, effect on electrical resistance) of defects and their measurement. Since the presence of defects has usually to be inferrred from their influence on bulk properties this section could equally well be entitled ‘Effect of defects on physical properties’. It is perhaps best to confine this latter phrase to secondary effects consequent on the basic effects mentioned above. Important among them is the interaction between point defects and dislocations. Section 1.5 (‘Interaction energies of point defects’) serves as a link with other chapters.

By convention a Festschrift is a collection of pretty ‘essays’ by a miscellany of scholars, presented to an older and greater scholar at some significant milestone of his life. The present work conforms to this convention only in that it was conceived as a tribute to Professor Sir Nevill Mott upon the occasion of his 60th birthday.

The prime intention was to write a modern version of ‘Mott and Jones’ – a comprehensive and up-to-date account of the Theory of Metals and Alloys. But much has been discovered in 30 years, and this would have stretched to many thousands of pages. We therefore decided to concentrate on the two major topics where our interests mainly lie – the electronic structure and electrical properties of metals, on the one hand, and the mechanical properties of solids on the other. In the end, each topic grew and diverged into a separate book.

The first volume, Electrons, published in 1969, gave an account of current understanding of the electron theory of metals, with particular reference to band structure, Fermi surfaces and transport properties. The present, second, volume deals with lattice defects and the mechanical properties of metals and alloys. Although the titles of the various chapters (Point defects, Crystal dislocations, Observations of defects in metals by electron microscopy, Solution and precipitation hardening, Work hardening, and Fracture) suggest a systematic coverage of a very large field in Physics and Metallurgy, the book is certainly not intended to be a comprehensive review of the subject.

INTRODUCTION

This article is concerned with the hardening of solids, mainly metals, by various arrays of solute atoms. Random solid solutions, coherent and incoherent precipitates are considered. In addition, some attention is given to alloys containing long-range order, which have recently been studied in detail. The general theory of dislocation locking by atmospheres of solute atoms is omitted, but some recent developments are described. A study of the effect of the displacement of atoms of the matrix to produce vacant lattice sites or interstitial ions should logically form part of the present work. Largely on account of its importance in nuclear reactor technology, a great deal is known about this topic, and the reader is referred to the available books and conference reports (Dienes and Vineyard, 1957; IAEA, 1962–3; Strumane et al, 1964; Chadderton, 1965).

TYPES OF LATTICE DEFECT

The presence of any imperfections in the lattice hinders the motion of dislocations through a crystal. Intrinsic point defects, vacant lattice sites and interstitial ions, are inevitably present in thermal equilibrium (see chapter 1). They may aggregate to form divacancies, larger aggregates, or possibly large spongy regions or voids. In face-centred cubic metals, the concentration of vacancies near the melting point may reach almost 1 in 103 (Simmons and Balluffi, 1960a); irradiation introduces vacancies and interstitial ions in large and essentially equal numbers. In solid solutions, solute atoms may either substitute for atoms of the matrix or lie in interstitial positions.

Liquid state physics no longer has the luxury status of an intellectual plaything – a kind of purgatory between gas and solid, a statistical mechanical jungle populated only by the foolhardy and/or academics. Pressing demands are increasingly being made upon the subject by adjacent branches of physics, and many people are being unwittingly drawn into the field from more conventional and immediately rewarding routes in physics. A subject which has not, as yet, satisfactorily accounted for the solid-fluid transition, nor which has yielded more than three further hard sphere virial coefficients in the 75 years since Boltzmann's original calculations, is evidently beset with problems: but there lies the source of the fascination.

This book is meant to be a guide to the uninitiated, and perhaps to broaden the outlook of those already there. It represents a very personal account of my own journey into the subject, and in consequence the content and approach might be considered in some respects somewhat idiosyncratic. I have made some attempt to present the material objectively, although inevitably one's own particular interests and viewpoints should and do assert themselves. Nonetheless, I felt the time was ripe to include separate chapters on the liquid surface and on the machine simulation methods, and to expand the by now routine statistical mechanical developments of the pair distribution associated with the names of Kirkwood, Born and coworkers, Bogolyubov, Percus and Yevick, and others. Of necessity some selection is inevitable, and I have arbitrarily restricted the discussion to the non-critical domain of the so-called ‘simple’ liquids, including the quantum liquids to a certain extent, although liquid water does receive a brief mention.

Introduction

We saw in the previous chapter how in the limit of low densities we were able to obtain a convergent expansion for the configurational partition function, and thereby determine all the thermodynamic functions of the system including the virial equation of state. However, what we did not have was a detailed knowledge, or indeed any knowledge, of the structure of the system.

Clearly, at liquid densities the series expansions will fail to converge(1), and anyway the computational labour in evaluating the multi-dimensional high-order cluster integrals which would develop in the highly connected fluid would be overwhelming, even in the economical Ree-Hoover formalism.

So, we are forced to adopt a new mathematical approach-the formalism of molecular distribution functions, or correlation functions. Instead of trying to evaluate the N-body configurational integral directly, the theory describes the probability of configurational groupings of two, three, and more particles. Further, it may be shown that we can still obtain the same amount of information concerning the system as is obtained from the study of the statistical integral itself. Moreover, in this way we obtain direct information on the molecular structure of the system being studied. Of course the method of correlation functions applies equally well to gases and solids, but we would not generally adopt that approach by choice since other characteristics of these phases suggest a more direct route to the partition function. The formalism is adopted, however, in the case of amorphous solids.

We shall find that the one- and two-body distribution functions, generally denoted by g(1) and g(2) respectively, will be of central importance in the equilibrium theory of liquids.

Introduction

In the previous chapters approximate integral equations were developed for the pair distribution. From this function we were able to establish virtually all the thermodynamic functions of interest, although it was clear that certain approximations had, of necessity, to be invoked either on the grounds of mathematical expediency as in the KBGY class of equations, or on more physical grounds in the case of the PY-HNC cluster expansions. The development of approximate integral equations was, of course, enforced by the impossibility of obtaining a direct evaluation of the N-body partition function for dense fluids, and as yet there appears to be no general method of calculating this quantity. It will be recalled that the difficulty was mathematical rather than conceptual, originating in the collective nature of the total potential ΦN even in the pairwise additive approximation. Powerful numerical techniques have been brought to bear upon the various integral equations, and these are generally evaluated by iterative techniques. The question arises as to whether these complex calculational techniques should not be used directly in rigorous variants of the theory. An ideal case would, of course, be the possibility of calculating directly the configuration integral of a system of a large number of particles.

It appears that with the advent of large electronic computers we have at our disposal a means of calculating an ensemble average in terms of the accessibility of states of the system. The ‘states’ of the system may be purely configurational and determinate as in the molecular dynamics approach, where the representative point evolves in accordance with a classical Liouville equation, subject to constraints of microscopic reversibility and ergodicity.

Introduction

Numerical evaluation and comparison of the various integrodifferential equations developed in chapter 2 may be conveniently divided into three sections. First the low density solutions will be discussed. In this case comparison is not made in terms of the form of the pair distribution function but rather by evaluating the equation of state, based on the pressure and compressibility relations, and comparing the virial coefficients so determined with the exact results of chapter I. Of course, if the theory were self-consistent the equation of state and virial coefficients would be independent of the method of approach-the pressure and compressibility equations would yield the same results. Some effort has gone into forcing self-consistency by choosing a form for c(r) which ensures identical results regardless of the approach (the self-consistent approximation: SCA). This device of enforcing thermodynamic consistency cannot be regarded as an advance of physical understanding; nevertheless, the results are excellent at least to the sixth virial coefficient for hard spheres.

At liquid densities direct comparison of the radial distribution function may be made. It will be seen that all the theories discussed in the previous chapter agree in the qualitative form of the pair distribution, but the quantitative discrepancy is large. Again, appeal to the equation of state is made. The extreme sensitivity of the pressure equation to the precise form of g(2)(r) (and, indeed, to the assumed form of the pair potential) provides a severe test of the theory. It is computationally convenient to work in terms of an idealized potential such as the hard sphere or square-well interactions- these models have the important advantage that the resulting equation of state may be compared directly with machine simulations.