The dislocation structure of twist boundaries and the influence of impurities

and the influence of impurities

and the influence of impurities

and the influence of impurities

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de

Rector Magnificus, prof.dr. J . M . Dirken,

in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Dekanen op dinsdag 10 maart 1987 te 16.00 uur

door

Johan Hamelinh

TR diss

1528

prof.dr.ir. S. Radelaar.

Dr. F.W. Schapink heeft in hoge mate bijgedragen aan de begeleiding bij het tot stand

komen van dit proefschrift en is als zodanig door het College van Dekanen aangewezen.

boundaries and the influence of impurities'.

1. Bij de interpretatie van de resultaten van transmissie-elektronenmikroskoop onderzoek aan een driehoekig

gedissociëerd dislokatienetwerk in een (111) kleine-hoek korrelgrens houden Scott en Goodhew ten onrechte geen rekening met de superpositie van spanningskontrast, veroorzaakt door dislokaties, en moirékontrast.

- Scott, R.F. and P.J. Goodhew, Phil. Mag. A, 4_4^ (1981), p. 373.

- dit proefschrift, paragraaf II.3.4.

2. Bij de verklaring van het ontstaan van zogenoemde Bardeen-Herring dislokatiespiralen is door sommige onderzoekers de mogelijkheid van het inklappen van holtes in een korrelgrens ten onrechte buiten beschouwing gelaten.

- Balluffi, R.W., Y. Komem and T. Schober, Surf. Sci., 3^, (1972), p. 68.

- Erlings, J.G., proefschrift T.H. Delft, (1979), p. 37-40. - dit proefschrift, paragraaf II.2.

3. De veronderstelling van Schapink en Mertens dat er in het door hen bestudeerde dubbelkristal één koherente tweelinggrens aanwezig is en dat deze samenvalt met het middenvlak is

twij felach tig.

- Schapink, F.W. and F.J.M. Mertens, Scripta Met., L5, (1981), p.611.

- Blom, N.S., proefschrift T.U. Delft, (1987), hoofstuk 5. 4. Bij toepassing in een optisch systeem van een halfgeleider­

laser, waarvan de aktieve laag uit één of meer kwantumputten bestaat, verdient het aanbeveling het optische veld in

laterale richting te beperken door verschillen in effektieve brekingsindex.

- Prince, F.C. et al., IEEE J. Quantum Electron., 2J_, (1985), p. 634.

5. Een kliefstap op de spiegel van een halfgeleider laser kan, indien deze een sterke degradatie van de laser tot gevolg heeft, reeds vóór het uitvoeren van een levensduurtest worden opgemerkt door middel van elektro-optische metingen.

6. Het zou de veiligheid van luchtsporten ten goede komen wanneer bijvoorbeeld parachutisten en modelvliegers hun sport buiten een vliegveld zouden beoefenen.

7. Het opnemen van de letter ij in een Nederlands woordenboek onder de letter i is niet in overeenstemming met de plaats van de letter ij in het alfabet.

garagesektor, maar levert slechts een kleine bijdrage tot de bevordering van de verkeersveiligheid.

10. De lengte van een file, zoals vermeld bij verkeersinformatie, is voor een verkeersdeelnemer nauwelijks interessant.'Het is nuttiger de door hem of haar op te lopen vertraging te weten.

Johan Hamelink 10 maart 1987

CONTENTS

Summary

Samenvatting page

INTRODUCTION 1

Chapter I. THE STRUCTURE OF GRAIN BOUNDARIES 3 1.1. Description of several models for grain boundary structure 3

1.1.1. the dislocation model 5 1.1.2. the coincidence model 15 1.1.3. the planar matching model 25 1.1.4. the polyhedral unit model 28 1.1.5. models relating the structure and the energy of

grain boundaries 32 1.2. Review of experiments on grain boundary structure 36

Chapter II. EXPERIMENTS ON THE STRUCTURE OF (111) TWIST BOUNDARIES 41

11.1. Specimen preparation 42 11.1.1. crystal growth in a gel 42

11.1.2. UHV deposition of thin films 45

II. 1.3. bicrystal preparation 49 11.2. Bubbles in a bicrystal 54 11.3. Results and discussion 63

11.3.1. imaging dislocation networks by TEM 64 11.3.2. Knowles' equation applied to 2=1 and Z=3 grain

boundaries 71 11.3.3. relaxation in a bicrystal 73

11.3.4. experiments on low-angle grain boundaries 76 11.3.5. experiments on near-coherent twin boundaries 81

Chapter III. THE STRUCTURE OF (111) TWIST BOUNDARIES CONTAINING IMPURITIES 95

111.1. Introduction 96 111.2. EDS microanalysis 103 111.3. Results and discussion 109

111.3.1. impurities deposited on a gold crystal 109 111.3.2. the influence of impurities on the structure of

twist boundaries 118 111.4. Summary and conclusions 139

Chapter IV. SOME EXPERIMENTS ON GRAIN BOUNDARY MIGRATION 141

IV.1. Introduction 141 IV.2. Results and discussion 149

IV.2.1. migration of a low-angle boundary 149 IV.2.2. migration of a high-angle boundary 155

IV.3. Summary and conclusion 159

References 161

SUMMARy

In this thesis some properties of grain boundaries are described. The aim of this investigation is to contribute to the study of the relation between the structure of interfaces and the orientation difference of the two crystals surrounding the interface. This programme is carried out for some particular cases. Furthermore, the influence of impurities on the grain boundary structure is studied. This is achieved for low-angle twist boundaries (E = I) and for near-coherent twin boundaries (E = 3 ) . All experiments are carried out on thin gold single crystals. Two such crystals were manipulated on top of each other in such a manner that their orientation could be pre-determined. This was

completely effectuated under ultra high vacuum conditions. A grain boundary is formed during in-situ annealing in TEM. In chapter II the structure of intrinsic dislocations in I = 1 and in I = 3 boundaries is analysed and compared to theoretical models described in the literature. These models are described in chapter I. Special attention is focussed on the interpretation of images of regular dislocation networks by TEM and on the superposition of Moiré fringes and dislocation contrast. In some cases the presence of dislocation networks can be shown by the appearance of additional reflections in the electron diffraction pattern. In general, an agreement was found between the interface structure obtained and results from the models. Especially Bollmann's O-lattice and Knowles' equation were found to be very well applicable for the misorientations investigated. With these models both the dislocation spacing and their Burgers vectors can be predicted while from the reciprocal of the unit cell of the dislocation network extra reflections in the diffraction pattern can be predicted. Bubbles which formed during specimen preparation, were found to collapse upon annealing. During collapsing, Bardeen-Herring type dislocations were formed. This is described in section II.2.

Five different elements were used for deposition on a gold crystal: Co, Cr, Mn, Sb and Ti. Cobalt was found to grow epitaxially onto: the (11!) surface of

gold crystals during deposition with a dilatation of 12.5% relative to the gold lattice. The other elements formed islands during deposition while some elements formed an intermetallic compound: B o- A uM n , AuTi-i and AuSbo. Evidence was found for a phase change of AuSbj to AuSb2 during further annealing.

The influence of impurities on the structure of grain boundaries was investigated on UHV-prepared bicrystals containing a few atomic layers of a

known impurity element in the interface region. For this purpose Co, Mn and Cr were used. It was found that formation of dislocation networks occurred at

temperatures of about 500 C which is clearly higher than the temperature of about 250 C required for relaxation in 'clean' grain boundaries. The relaxation process starts when the impurity layer thickness has decreased to about one atomic layer by diffusion of the impurity out of the interface. Another i:

difference compared to 'clean' grain boundaries, when cobalt or chromium were used as an impurity, is the irregular dislocation networks at the beginning of the relaxation process, which become more regular during further annealing.

In chapter IV the migration process of grain boundaries is reported from a theoretical as well as from an experimental point of view. Migration of a low-angle boundary, a near-coherent twin boundary and a E = 57 high-low-angle boundary were observed. Not all migration phenomena could be explained from the reduction of grain boundary area.

In dit proefschrift worden enige eigenschappen van korrelgrenzen beschreven. Het doel van dit onderzoek is het bestuderen van het verband tussen de struktuur van korrelgrenzen en de oriëntatierelatie tussen de twee aangrenzende kristallen voor bepaalde gevallen. Daarnaast is ook de invloed van verontreinigingen op de korrelgrensstruktuur bestudeerd. Het genoemde onderzoek is uitgevoerd aan kleine-hoek torsiegrenzen (E = 1) en aan bijna-koherente tweelinggrenzen (E = 3 ) . Voor alle experimenten is van dunne goud-éénkristallen gebruik gemaakt. Dubbel-kristallen met een vooraf bepaalde misoriëntatie werden geheel in ultra-hoog-vakuum geprepareerd door twee éénkristallen op elkaar te manipuleren. Een korrelgrens wordt gevormd door verhitten van een preparaat. Dit is in-situ uitgevoerd in TEM.

In hoofdstuk II wordt de analyse van de struktuur van intrinsieke dislokaties in E = 1 en in Z = 3 korrelgrenzen beschreven en worden deze resultaten verge­

leken met de theoretische modellen uit de literatuur. Deze modellen zijn in hoofdstuk I beschreven.

Er is bijzondere aandacht besteed aan de interpretatie van met TEM gemaakte afbeeldingen van regelmatige dislokatienetwerken en ook aan de superpositie van moirélijnen en dislokatiekontrast. In bepaalde gevallen kan de aanwezigheid van dislokatienetwerken aangetoond worden aan de hand van additionele reflekties in het diffraktiepatroon. In het algemeen is er een overeenstemming gevonden tussen de gevonden korrelgrensstruktuur en de resultaten van modellen. In het bijzonder voldoen het O-rooster van Bollmann en de vergelijking van Knowles erg goed in het geval van de bestudeerde misoriëntaties. Met behulp van deze modellen kunnen zowel de dislokatiespatie als de burgersvektoren worden bepaald. Extra reflek­ ties in het diffraktiepatroon kunnen worden voorspeld met behulp van de recipro-ke eenheidscel van het dislokatienetwerk.

Tijdens het prepareren van een dubbelkristal worden holtes gevormd tussen de twee deelkristallen. Deze holtes verdwijnen weer tijdens verhitten van het pre­ paraat en laten daar Bardeen-Herring-type dislokaties achter. Dit wordt in para­ graaf II.2. beschreven.

Er zijn vijf verschillende elementen gebruikt voor het opdampen op een goud­ kristal: Co, Cr, Mn, Sb en Ti. Kobalt groeit tijdens het opdampen epitaxiaal op het (111) oppervlak van goud. De dilatatie blijkt 12,5% ten opzichte van het goud-rooster te zijn. De andere elementen vormden eilanden gedurende het opdam­ pen. Ook vormen sommige elementen tijdens verhitten een intermetallische

ver-binding: 8„-AuMn, AuTi, en AuSb.. Tijdens langer verhitten zijn aanwijzingen gevonden voor een verandering van de AuSb. fase naar AuSb..

De invloed van verontreinigingen op de korrelgrensstruktuur is onderzocht aan dubbelkristallen die in UHV zijn gemaakt. Deze preparaten bevatten enige atoom-lagen van een bekende verontreiniging in het korrelgrensgebied. Hiervoor zijn Co, Mn en Cr gebruikt. Het bleek dat dislokatienetwerken ontstaan bij een tem­ peratuur van ongeveer 500 C hetgeen aanzienlijk hoger is dan de ca. 250 C die benodigd is voor relaxatie in 'schone' korrelgrenzen. Dislokatienetwerken worden gevormd zodra de dikte van de verontreinigingslaag door korrelgrensdiffusie is afgenomen tot ongeveer één atoomlaag. Een ander verschil in vergelijking met 'schone' korrelgrenzen is dat met Co of Cr aanvankelijk onregelmatige disloka­ tienetwerken werden gevormd. Gedurende verder verhitten worden deze netwerken regelmatiger.

In hoofdstuk IV is het migreren van korrelgrenzen beschreven zowel vanuit een theoretische als vanuit een experimentele benadering. In het bijzonder is de migratie van een kleine-hoek grens, een bijna-koherente tweelinggrens en een

Z = 57 grote-hoek grens beschreven. Niet in alle gevallen kon de korrelgrens­

INTRODUCTION

Interfaces in polycrystalline materials have been the subject of numerous investigations since many years. The reason of this interest is the role which interfaces play in material properties such as mechanical behaviour (deformation, embrittlement, yield strength), electrical behaviour (recombination of electron-hole pairs, change of electrical resistivity) and chemical behaviour (segregation, diffusion). Interfaces often contain intrinsic dislocations due to a difference in lattice parameter or due to a misorientation of the crystals surrounding the grain boundary. The grain boundary structure can often be described in terms of intrinsic dislocations and is related to the relative orientation of both surrounding crystals. This relation has been investigated frequently by setting up theoretical models as well as by carrying out appropriate experiments. The important role of interfaces as well as interface dislocations in material properties has been demonstrated in many investigations.

The material properties mentioned above are strongly influenced by the presence of impurities in the interface regions. For example, impurities segregating in the interface significantly reduce the grain boundary mobility. In the literature the influence of impurities on material properties was

described from experiments relating directly the quantity of impurity elements present in the interface region or in the bulk material to properties such as yield strength, embrittlement and corrosion. Not much attention has been paid, however, to the direct influence of impurities on the grain boundary structure. In this thesis some novel experiments on this subject are described in chapter III.

Many interfaces described in the literature were studied by means of transmission electron microscopy (TEM) in thinned bulk materials or in thinned pre-oriented bicrystals which were welded together at high temperatures and/or pressures. Impurities present in engineering materials were often found to be segregated in the interface region or more specifically onto the intrinsic dislocations. A depth profile of the elements present in the interface can be obtained for example by means of Auger electron spectroscopy (AES).

In this investigation thin dislocation-free gold crystals were used to prepare bicrystals. The misorientation could be predetermined. A great advantage

of this method is that relaxation effects, accompanying the onset of bonding in interface, can be studied completely in TEM and by electron diffraction

during in-situ annealing. Because the bicrystals are prepared in ultra high vacuum (UHV), interfaces can be studied without any influence of impurities. On the other hand the influence of impurities can be investigated by a comparison of results from 'clean' grain boundaries with results from boundaries containing impurities. These impurities can be deposited in UHV during preparation of a bicrystal. This means that the type and thickness of the impurity layer are precisely known and its influence on the interface relaxation process can be studied in-situ in TEM.

CHAPTER I.

THE STRUCTURE Of GRAIN BOUNDARIES

In this chapter the structure of grain boundaries is reviewed from a theoretical as well as an experimental point of view. First dislocation models and purely geometric models, based on crystallography of the unrelaxed lattice are reviewed in section 1.1. This is followed by the description of models, based on minimisation of the energy of a grain boundary. In section 1.2. the experimental results from the literature are discussed, which contribute to a better understanding of grain boundary structure and the results of the experiments described in chapter II.

Some models, for example the coincidence model, will be used in chapters II and III to explain experimental results. However, other models (planar matching, polyhedral units, disclinations and delocalisation) will also be described. The reason for this Is to obtain a general overview of grain boundary models.

1.1. DESCRIPTION OF SEVERAL MODELS FOR GRAIN BOUNDARY STRUCTURE

The purpose of this section is to review current models of grain boundary structure and to define associated parameters in order to provide background for analysis of experimental results. The geometry of a general grain boundary can be described by two single crystals which are in atomic contact along the grain boundary plane. In general, for rigid crystals eight parameters are required to define this geometry, three of which specify the orientation relationship of the two crystals, two for the orientation of the boundary plane and three to describe the translation of one crystal relative to the second. For fee crystals a change of the position of the boundary plane is equivalent to a relative translation of the two crystals.

In most models the number of parameters is reduced to five, three for the orientation relationship of the crystals and two for the boundary orientation.

F i g . 1.1.Schematical drawing of a symmetrical t i l t boundary. The m l s o r i e n t a t i o n a x i s l i e s i n t h e boundary plane normal t o t h e plane of t h i s drawing [ 5 ] .

F i g . 1.2. For a t w i s t boundary t h e m l s o r i e n t a t i o n a x i s i s normal t o t h e boundary p l a n e . Open and c l o s e d c i r c l e s r e p r e s e n t atoms j u s t above and below t h e boundary p l a n e r e s p e c t i v e l y [ 5 ] .

For a tilt boundary the mlsorientation axis of the two crystals lies in the boundary plane, as schematically shovra in fig. 1.1. For a complete description both boundary plane and mlsorientation axis should be given, e.g. (h, k,1. )[hjkjlo] tilt boundary. In the case of a symmetrical tilt boundary the crystals are rotated over the same but opposite angle 0/2. For a twist boundary the mlsorientation axis is normal to the grain boundary plane (fig. 1.2). This will be indicated by (hkl) twist boundary, meaning that the boundary plane is (hkl) and the mlsorientation axis is [hkl]. When the mlsorientation axis lies neither in the boundary plane nor normal to this plane, a boundary with a mixed character is formed.

In the models, developed to describe grain boundaries, different criteria have been used to examine the stability of the boundary structure. Dislocation models relate the orientation relationship of the crystals to intrinsic dislocations in the interface, while in other models attention is focussed on the coincidence of atom positions, on matching of atomic planes crossing the boundary or on the existence of certain polyhedral units of atom sites. Intrinsic dislocations are those dislocations that are part of the equilibrium boundary structure, in contrast to extrinsic boundary dislocations which may have entered the grain boundary from the matrix and which are not part of the equilibrium boundary structure. In a third group of models the calculated grain boundary energy is minimised with respect to the exact atomic positions in the boundary region. Different criteria may lead to different metastable minima of the grain boundary energy. This minimization occurs during relaxation of the grain boundary, which means that atoms near the interface can change their position slightly to decrease the boundary energy. To calculate this energy, computer simulations have been performed in order to obtain a set of optimum atom positions in the boundary area.

1.1.1. The dislocation model

A dislocation model of low-angle grain boundaries has been proposed by Burgers [1,2,3] and Bragg [4]. The grain boundary is described by the orientation relation of the two crystals and the orientation of the boundary plane, which requires in total 5 parameters. In general the boundary structure consists of a network of dislocations. When the orientation relation of the

two crystals is expressed as a rotation of one crystal through an angle 9 around a common axis, the dislocation spacing d for both a low-angle t i l t and twist boundary follows directly from:

b b , , ,..

2sin(9/2) 9 K '

where b is the length of the Burgers vector (see for example Read [5]). In the case of a symmetrical low-angle tilt boundary for example, the boundary structure consists of a planar array of edge dislocations parallel to the misorientation axis. In other cases more than one set of dislocations is required to accomodate the misfit of the two crystals.

Energy of a low-angle grain boundary

Read and Shockley [6] have calculated the energy of a low-angle tilt undary starting from the e

a symmetrical tilt boundary:

boundary starting from the energy per unit length E of an edge dislocation in

Gb r

E = In - + B (1.2)

e Ait(l-u) b e

where G is the shear modulus u is Poisson's ratio

r is the effective radius of the region around the dislocation contributing to the energy

B is the core energy of an edge dislocation

The problem with equation (1.2) is the determination of the value of r. To overcome this problem, Read and Shockley showed that r can be reasonably estimated from the dislocation spacing d.

Now the energy of a symmetrical t i l t boundary per unit area Et can be obtained

by multiplying equation (1.2) with the dislocation density 1/d:

E = E 9(A - ln9) (1.3) t o Gb Au(l-u) where E = -.—^ r- and A = = B o 4 u ( l - u ) 2 e Ob

For a twist boundary, the derivation of the energy versus angle formula i s similar and equation (1.3) holds, provided that

Gb 2 l t Bs

E = -K— a n d A -,0 2 , , - u - ^

and B is the core energy of a screw dislocation.

High-angle grain boundaries

An extension of the dislocation model to include high-angle symmetric tilt boundaries was proposed by Read and Shockley [6,7] for (001) boundary

planes.In this model the dislocation spacing reduces to a small number of atomic spacings. The energy versus angle curve which they presented contains a number of cusps. These cusps correspond to orientations where the dislocation spacing is equal to an integral number of lattice constants. These orientations are equivalent to CSL orientations with a low value of 2 (see section 1.1.2. and eq. (1.9)).

The dislocation model for high-angle grain boundaries is based on the assumption that the linear elasticity theory is applicable (i.e. Hooke's law holds). This is the case in a low-angle boundary where the dislocation spacing is such that the dislocation cores are well separated. This assumption fails for high angles and dense dislocation networks, leading to differences between predictions from this theory and experiments. The problem of all models of high-angle grain boundaries is that the dislocation spacings are comparable with the dislocation core dimension and hence the dislocation core structure and core interaction effects may be significant.Here Hooke's law will not hold. The width of a grain boundary is assumed to be equal to the dislocation

spacing. To improve the dislocation model for high-angle tilt boundaries Li [8] suggested a dislocation core model. To calculate the strain field of a tilt boundary a cylinder with radius r is introduced to represent the dislocation core. This radius is chosen such that the stress at the surface of the core vanishes. Here the linear elasticity theory is also presumed to hold. For low-angles the dislocation core model is identical to the dislocation model of Burgers and Bragg. Upon increasing the misorientation, the core radius increases and the shape of the cores becomes elliptical because they tend to attract each other. From a misorientation of 37 onwards the cores will overlap and a continuous slab arises at the grain boundary (fig. 1.3).

20°

Fig. 1.3. The dislocation cores of a tilt boundary are drawn for a number of misorientations. Above 9

[8],

37l the cores overlap. Here r = b/2

This slab is supposed to behave as an amorphous layer and for these high angles the properties of the grain boundary are near enough independent of 6. The width of the boundary is again equal to the dislocation spacing when there is no core overlap. For high-angle boundaries the width changes with sin 9/2. With the aid of the dislocation core model Li was able to describe many properties of grain boundaries in terms of the interaction between the cores of the dislocations and the properties of the cores themselves, whereas earlier models were able to describe only a few particular properties in a limited angular range. Among the properties, explained in terms of the dislocation core model are: strain energy, intergranular facture, preferential melting of the grain boundaries, preferential diffusion, boundary migration

and segregation. The fact that for some properties, e.g. the energy dependence of the misorientation angle, the experimental results do not coincide with the predictions from the model is probably due to the fact that the linear elasticity theory is not applicable for high-angle boundaries. Because the properties of high-angle boundaries are found to be nearly independent of 9, cusps in the energy-misorientation curve at coincidence mlsorientations are not predicted. To improve some deficiencies of the continuum dislocation array model, Glicksman and Void [9] added a thermodynamic term to the core energy leading to the concept of a heterophase dislocation model with a second phase in the core. An earlier model of Frank [10] might be regarded as a special case of this heterophase dislocation model. He discussed the case of "hollow" dislocations. Glicksman and Void treated simple grain boundaries in crystals near the melting point as arrays of heterophase misfit dislocations with a liquid-like core phase similar to the ordinary liquid phase. The results of this model are in accordance with experimental results for low-angle boundaries in crystals not far from the melting point where liquid core dislocations should be prevalent, while for high-angle mlsorientations deviations from the predicted energy-misorientation curve are found. For mlsorientations of over 15° a transition is predicted. Here the dislocation cores will suddenly merge, forming a slab of core phase, the energy of which is then independent of the misorientation between the two crystals and a liquid boundary core will arise.

Frank's formula

Frank [11] developed a procedure for finding the dislocation structure of a general grain boundary via a geometrical analysis. This procedure is carried out for a grain boundary having five degrees of freedom: the misorientation' axis u_, the angle of misorientation 9 and the grain boundary normal n_. The

method is applicable to any grain boundary and will be discussed here for the case of a general low-angle boundary, meaning that the dislocation spacing is large compared to the unit cell dimensions. The model predicts the orientation and spacing of the dislocations in the grain boundary. In general more than one possible dislocation configuration is found, while the one having the lowest grain boundary energy is supposed to be the correct one.

<y=u0

Fig. 1.4. a) A general grain boundary can be obtained by cutting a single crystal along two arbitrary planes, removing the wedge shaped material and joining the grains. The mlsorientation Is given by a rotation through an angle 9 about an axis given by u.

b) One part of the bicrystal is shown. The vector V_ cuts a number of dislocations in the boundary plane [11].

An arbitrary vector V_ is chosen in the grain boundary plane and a vector V_'

is constructed by rotating V through the relative rotation to = 9u of the two grains, fig. 1.4. In general _V_' does not lie In the boundary plane and V and V_' would be identical if the two grains would have the same orientation. Next a circuit is constructed similar to a Burgers circuit around a dislocation. From the end point of V_' the circuit goes through one crystal to pass the

grain boundary at the common origin of V and V' and continues through the second crystal to the end point of _V_. The closure failure of this circuit is S_ = V - V_ and would be zero in a perfect crystal. S is equal to the sum of

the Burgers vectors of the dislocations enclosed. It can be written as:

S_ = Z n ^ (1.4)

i

The dislocations enclosed by the circuit are all cut by the vector _V^. For small 9 the closure failure S_ can be written as the vector product of w and V:

S_ = u x V = (u x V)9 (1.5)

This equation is generally called Frank's formula. By chosing two different vectors V_ and calculating the corresponding vectors j5_, the dislocation structure can be obtained. It Is always possible to construct a dislocation network consisting of three sets of dislocations. Sometimes an additional fourth set may reduce the boundary energy. In other cases one or two dislocation sets are sufficient. For example in a symmetrical tilt boundary j>_ = 0 if V_ is chosen parallel to the tilt axis, which means that all misfit

dislocations are parallel to u_. Their spacing can be obtained by chosing a second vector V which is not parallel to u*

Bollmann [12] has discussed the formula of Frank from a different point of view. The unit vector u is chosen along the z-axis of an orthogonal coordinate system meaning that S_ can be written as:

S_ = (-ey,9x,0) (1.6)

where V is given by (x,y,z).

It follows from eq. (1.6) that only those ^ points, which lie in the (x,y) plane, fulfill eq. (1.5) (see fig. 1.5a). Furthermore the vector j>_ in eq. (1.5) is described as a continuous vector with respect to changes of the length of _V, while eq. (1.4) is satisfied only for discrete ^_ points. The end points of the corresponding vectors V form a regular pattern of lines parallel to the z-axis. Only for points on these lines, which are called x-lines, is eq. (1.4) exact. The x-lines are separated by Wigner-Seitz type cell walls, which are perpendicularly bisecting the connections between the x-lines, as Illustrated in fig. 1.5b. Bollmann constructs the dislocation network in a low-angle grain boundary by intersecting the grain boundary plane with the honeycomb structure of the Wigner-Seitz cell walls. At the points where the x-lines Intersect the boundary plane (x-points) there is an optimum fit between the crystals, while the intersections with the Wigner-Seitz cell walls are the locations of worst fit and hence give the locations of the dislocations. Bollmann stated that the Burgers vectors of the dislocations is equal to the

Fig. 1.5. a) Only discrete points S_ of the (x,y) plane have a vector V_ as an image. The vectors _V_ form lines parallel to _u>

b) This honeycomb structure represents the geometrical interpretation of Frank's formula. The dislocation network is found from the intersection of the grain boundary plane with the Wigner-Seitz cell walls.

difference of the two vectors S corresponding to the two x-points at both sides of the dislocation line. An important result of this description is that it can easily be seen that the projection of the dislocation network on the plane normal to u is always the same, independent of the orientation of the boundary (fig. 1.5b).

Delocalisation

In dislocation models it is assumed that the structure of grain boundary dislocations may be similar in principle to the structure of perfect lattice dislocations in the sense that grain boundary dislocations may be considered as line defects whose core widths are comparable to those of perfect lattice

dislocations. These models are of physical significance only if the misfit dislocations are localized. Gleiter [13,14] and Pumphrey et al. [15] have calculated the width w of dislocations in grain boundaries by minimizing the boundary energy with respect to the dislocation widths. The boundary energy is divided into two contributions: the elastic strain energy stored in the two crystals on either side of the boundary and the energy associated with the local change in boundary structure owing to the dislocations. Gleiter [13] has shown that for a symmetrical tilt-boundary w can be obtained from:

/5 Gb^ _(1.7)

E is the boundary energy G is the shear modulus

b is the length of the Burgers vector involved -=x- is the slope of the E(0) curve

do 2 ^ 5.0 I t-9 5 z o < u o _ J C O o (a) 25 45 90 135 180 o IT LU Z w 2.0

>

<

>

/

°

\ / \°

_{\ / \°}

Fig. 1.6. The width of dislocations in [110] tilt boundaries in A.1 (divided by 2b) as a function of the misorientation angle calculated using the energy misorientation dependence given in (b), [14].

separation is small. On the other hand a disclinatlon dipole is equivalent to a dislocation wall for a large separation.

Fig.

6 = 2 L C J / H (C)

1.7. The low-angle tilt boundary of (a) can be given as a set of edge dislocations (b) or as wedge disclinations (c), [16].

Now a general grain boundary can be made up of regions of low energy separated by disclinations. It should be noted that both the disclinatlon model and the dislocation model can only be applied to low-angle grain boundaries.

1.1.2. The coincidence model

CSL model

In the CSL model the structure of grain boundaries is described from the viewpoint of coincidence of atom positions in neighbouring crystals. It is a purely geometric model and is based on the crystallography of the unrelaxed

l a t t i c e s . The c o i n c i d e n c e s i t e s a r e t h o s e e q u i v a l e n t p o s i t i o n s where t h e l a t t i c e s of t h e two c r y s t a l s would c o i n c i d e i f they were extended i n t o one a n o t h e r . The r e s u l t i n g c o i n c i d e n c e s i t e l a t t i c e (CSL) i s a s u b l a t t i c e of t h e two c r y s t a l l a t t i c e s , being continuous a c r o s s t h e g r a i n boundary, as was f i r s t argued by F r i e d e l [ 1 7 ] . A s u b l a t t i c e i m p l i e s t h a t every point of t h i s l a t t i c e c o i n c i d e s with a point of one of t h e two c r y s t a l l a t t i c e s . By s t u d y i n g copper a f t e r secondary r e l a x a t i o n Kronberg and Wilson [18] observed i n many c a s e s an o r i e n t a t i o n d i f f e r e n c e between neighbouring g r a i n s of 22 and 38° around t h e common [111] a x i s . The CSL corresponding t o t h e s e m i s o r i e n t a t i o n s has a r e l a t i v e small u n i t c e l l , corresponding t o E = 21 and 7 r e s p e c t i v e l y , according t o eq. 1.9. The CSL model has been extended by Brandon et a l . [ 1 9 ] , Brandon [20] and Ranganathan [ 2 1 ] . Ranganathan showed t h a t for cubic c r y s t a l s a t h r e e dimensional CSL i s g e n e r a t e d by a r o t a t i o n through an a n g l e 9 about an a x i s [ h k l ] i f

t a n ( ^ 9 ) = 2 - / ( h2 + k2 + l2) ( 1 . 8 )

Then

2 2 2 2 2

E = x + y . (h + k + 1 ) ( 1 . 9 )

where x and y are integers. If E is even it must be divided by two repeatedly until E is odd. The primitive unit cell of the CSL has a volume of E times that of the primitive unit cell of the crystal lattice while 1/E is the density of atoms on CSL positions. Brandon noted that boundaries between crystals in a CSL orientation are in good atomic fit. The grain boundary energy is relatively low for a low E CSL. The highest possible degree of coincidence and hence the lowest energy, apart from a perfect crystal, is a twin orientation, corresponding to E = 3. When a boundary does not lie in the most densely packed planes of the corresponding CSL it will take up a stepped structure to increase the density of CSL points within the boundary. As an example fig. 1.8 shows a stepped boundary, given by ABCD, in a E = 11 related bicrystal consisting of b.c.c. crystals. E varies dlscontinuously with the

misorientation of the two crystals. Geometrically the value of £ may increase to infinity although for high values of £ a CSL looses its physical meaning. A maximum is, however, not known, but is generally estimated between 25 [22] and 50. The CSL model is purely geometrical, while a relative translation destroys the coincidence of atom positions but does not change the translation symmetry.

Fig. 1.8. Facetting of a 50.5 [110] tilt boundary (£ = 11), along planes of high coincidence density, AB and CD. The filled symbols represent atoms on CSL positions, after Brandon et al. [19].

Bollmann's 0-lattice model

Starting from the coincidence site lattice model, Bollmann [23,24,12] has set up a more general mathematical model in which regions of good fit in a grain boundary are not solely identified by coincident lattice points, but also by coincidences of interior cell points. The 0-lattice is formed by all points having the same interior coordinates in two interpenetrating lattices

and changes continuously as the orientation relation of the two lattices is varied, whereas the CSL changes discontinuously. The 0-lattice model is used to discuss the dislocation content of grain boundaries as well as phase boundaries. The 0-lattice is the lattice of all the possible origins, from which a given lattice 2 can be produced out of a given lattice 1 by a linear homogeneous transformation, given by the transformation matrix A:

3^(2) = A ^( 1 ), |A| * 0 (1.10)

x^ ' and _x_^ ' are the position vectors of the same point given in the

coordinate system of crystal 1 and 2 respectively. The corresponding points have the same internal coordinates in both lattices but may have different external coordinates. When a point is given in crystal coordinates these coordinates are the sum of the external and internal coordinates, while the external coordinates are integers and the values of the internal coordinates vary between 0 and 1. The transformation matrix A is for example a rotation or an expansion. Next a vector _b_ is defined as the difference vector between a general vector x1, ' and its partner x^ ':

b = x (2> - x <1> (1.11)

With the aid of eq. (1.10) it can be written as:

b^=(A - I ) x( 1 ) =(I - A_ 1) x( 2 ) (1.12)

O) Crystal Space

b Space

Fig. 1.9. a) The lattices of crystals 1 and 2 showing the translation vectors of lattice 1 and given in the crystal space.

b) The bj '-lattice in the b-space pictured as a lattice of the

translation vectors of lattice 1 (taken from Bollmann [12]).

When b_ is a translation vector of lattice 1 (fig. 1.9), denoted by b (L) (where (2) _{and x (1)}

L signifies lattice) the vectors x_

the 0-lattice vectors x* ' defining the O-points

+ b (L) are identical. They are

x(2) .x( l )+ b( L ) = x( 0 )

(1.13)

Substitution of the inverse of eq. (10) gives:

A-1^0) + b(L) „ x(0)

The solutions of this equation define the 0-points when |I determine the 0-lattice:

* 0 and

t(0) . ( I _ A-l)-l b (L) _(1.15)

The rank of (I - A~ ) " can be 3, 2, 1 or 0. Then the 0-lattice respectively consists of 0-points, parallel 0-lines and parallel 0-planes, while the latter case is trivial. An example of an 0-lattice is given in fig. 1.10.

Fig. 1.10. The crystal lattices of crystals 1 and 2 and the 0-lattice are given for the case that the transformation is a rotation of 9. Each 0-point can be taken as an origin with respect to the transformation. For three 0-points it is indicated that a point of lattice 2 can be obtained from a point of lattice 1 by a rotation of 9 about an 0-point [23] (taken from Bollmann [12]).

The "O-points center the areas of best fit between the two lattices, while the worst fit is inbetween the O-points. This misfit can be accommodated by dislocations. The location of the dislocations is determined by the intersection of the grain boundary plane and the Wigner-Seitz cell walls, which are the planes perpendicularly bisecting the connecting lines between neighbouring O-points. The vectors of the _b_-lattice determine the possible Burgers vectors of the dislocations. For a low-angle boundary and an interphase boundary with a small change in lattice parameter the _b_-lattice is the translation lattice of crystal 1 and the dislocations are called primary dislocations. Here the O-lattice model is very similar to the dislocation model.

In the case of high-angle grain boundaries there exist orientations for which the grain boundary energy is a local minimum in the sense that a slight change in orientation increases the energy. These are the CSL orientations for which the atomic fit between the crystals is an optimum. For deviations from the CSL orientation of the two crystals, the boundary will preserve the minimum energy as far as possible by introducing secondary dislocations to accommodate the misfit. Bollmann has described geometrically the occurrence of secondary dislocation networks. Similar to eq. (1.14) Warrington and Bollmann [25] described the C^-lattice (a second-order O-lattice belonging to a CSL orientation) as solutions of:

b °S C - (I - B'1) ,<°2> (1.16)

where B is the transformation matrix describing the deviation from the CSL orientation, x. are the vectors determining the 02~lattice and the _b vectors describe the displacement-shift-complete (DSC) lattice. This DSC lattice consists of all possible sum vectors between crystals 1 and 2 in a CSL orientation. It is the coarsest lattice that contains both crystal lattices as sublattices. An example of a £ = 7 CSL is given in fig. 1.11.

During relaxation, atoms in the interface region may slightly alter their positions in order to occupy minimum energy sites. The structure of an arbitrary high-angle grain boundary can thus be described as a nearby CSL orientation with a superimposed secondary dislocation network to accommodate

the deviation from the exact CSL orientation. The secondary dislocation network can be constructed from the intersection of the grain boundary plane and the Wigner-Seitz cell walls of the (^-lattice and the possible Burgers vectors are the translation vectors of the DSC-lattice. This means that in high-angle grain boundaries dislocations may appear, which are partial dislocations in a perfect crystal.

0 0 = 38.21°

Z = 7

Fig. 1.11. The CSL and DSC lattice unit vectors in the (lll)-plane are given for a 38.21° [111] twist boundary, corresponding to £ = 7.

Bollmann [12] describes a bicrystal starting from two interpenetrating lattices, as is also the case in the CSL model. On one side of the grain boundary atoms are located on sites of lattice 1 and on the other side of the boundary they are on sites of lattice 2. The matching of the two lattices is restricted to the boundary region only, while the O-lattice is continuous across the boundary. The boundary will tend to adopt an orientation such that

it contains a maximum number of O-points (fig. 1.8). Thus, according to Bollmann, when the orientation of the boundary plane differs from the orientation of a plane with a high density of O-points the boundary will take up a stepped structure.

The dislocation spacing d in a grain boundary, which is given by the length of the O-lattice unit vectors, is described by:

d ~ (1.17)

For low-angle grain boundaries this is the same result as eq. (1.1), while for a high-angle grain boundary with a slight deviation from an exact

CSL-orientation the same formula holds. Here 8 is the angular deviation from a nearby CSL orientation and b_ is the corresponding Burgers vector.

The O-lattice can describe grain boundaries corresponding to certain orientations between the two crystals. The possible orientations of these crystals are within the maximum angular deviation 9 from a CSL orientation:

(1-18) m d .

min

where d , i s t h e s m a l l e s t d i s l o c a t i o n spacing t h a t may occur. The r e l a t i o n between 8 and E has been d i s c u s s e d by Warrington and Grimmer [26] , Warrington

m

and Boon [27] and I s h i d a and Mclean [ 2 8 ] . I t has been proposed by r e f s . [ 2 7 ] and [28] r e s p e c t i v e l y t h a t :

) = 6 E l and 9 = 9 E~^ ( 1 . 1 9 )

m o m o

where 9 i s t h e maximum angular d e v i a t i o n from E = 1 f o r which a d i s l o c a t i o n network o c c u r s . As an example i t can be worked out t h a t for 9 = 8 ° and a l l o w i n g E v a l u e s up t o 25, t h e E dependent r e l a t i o n l e a d s t o a coverage of

11%,meaning t h a t 11% of a l l p o s s i b l e m i s o r i e n t a t i o n s can be d e s c i b e d ; 2% of the m i s o r i e n t a t i o n s a r e low-angle b o u n d a r i e s . Brandon [20] assumed t h a t 9 ~ 1 5 ° . In p r a c t i c e t h e coverage w i l l be h i g h e r , because low-energy b o u n d a r i e s w i l l a r i s e more o f t e n then high-energy b o u n d a r i e s .

The r e l a t i o n between F r a n k ' s formula ( s e c t i o n I . 1 . 1 ) and Bollmann's 0 -l a t t i c e ( s e c t i o n 1.1.2) has been s t u d i e d by Know-les [29] i n order t o determine the d i s l o c a t i o n content of a g e n e r a l i n t e r f a c e , i n c l u d i n g an i n t e r p h a s e boundary. Knowles derived an equation for t h e d i s l o c a t i o n d e n s i t y in a

boundary starting from the theory of a continuous distribution of dislocations of Bllby et al. [30,31] and Bullough and Bllby [32], In a manner similar to eq. (1.5), the sum of . the Burgers vectors (S_, which is a continuous vector

now) of all dislocations crossing a vector V in the interface is given as:

(I - A_ 1)V (1-20)

where A is the transformation matrix relating the lattices of the two crystals in the notation of Bilby [30] and Christian [33]. If the misfit is accommodated by a network of i sets of dislocations, each with Burgers vector b j , line vector |_ and array spacing d., eq. (1-20) can be written, analoguously to eq. (1.4), as:

l (Ü! ' I>ti " (Ï " f

)l (

)

where

1±

d

n_ i s the u n i t vector normal t o t h e boundary p l a n e .

The vector >L defines the 1 d i s l o c a t i o n s e t completely by i t s d i r e c t i o n , which i s normal t o § , and l i e s i n t h e boundary p l a n e , w h i l e t h e l e n g t h of NJ d e f i n e s t h e d i s l o c a t i o n s p a c i n g . I n the g e n e r a l case of t h r e e s e t s of d i s l o c a t i o n s and assuming d e t ( I - A ) £ 0 e q . ( 1 . 2 1 ) can be w r i t t e n a s :

( NrV ) ( I - A 1)~\l + (N2.V)(I-A 1) \ 2 + (N3.V)(I-A l) \ 3 = V ( 1 . 2 3 )

The v e c t o r s (I-A ) b can now be I d e n t i f i e d as t h e 0 - l a t t i c e

= =3 —i

( 0 )

< 2 . 1 - I > £ l( 0 ) + ( H2. V ) x2 ( 0 ) + ( N3. V ) x3 ( 0 ) = V (1.24)

Knowles has shown t h a t t h e v e c t o r s _Nj can be expressed as t h e p r o j e c t i o n s of TO) *

t h e r e c i p r o c a l O - l a t t i c e v e c t o r s (x_ ) onto t h e g r a i n boundary plane or t h e i n t e r p h a s e boundary p l a n e having u n i t normal n . The v e c t o r _N. can be given a s :

£i = ( *1 ( 0 )> * - { ( Ki ( (V . n } n (1.25)

where the reciprocal O-lattice vectors are defined by:

x ( 0 ) x x ( 0 )

, (OK* -2 -3 ,, ,,.

Upon rotating the subscripts this equation can also be applied to (x,*' ') and

{*jy ) • Now the relation between the line directions and array spacings of

the dislocations and the vectors N1 follow, using eq. (1.22), as:

(0) * 1

Li // (x, ) x n and d^ = ( m-5 (1-27)

(x^ ) x n

This i s t h e b a s i c equation for t h e d e t e r m i n a t i o n of t h e l i n e d i r e c t i o n s and a r r a y spacings of t h e d i s l o c a t i o n network, r e l i e v i n g t h e mismatch across a g r a i n boundary or an i n t e r p h a s e boundary for any matrix A r e l a t i n g the two l a t t i c e s on e i t h e r s i d e of t h e boundary. Knowles [29] has given some examples of t h e a p p l i c a t i o n of t h e derived e q u a t i o n s on some p a r t i c u l a r g e o m e t r i c a l problems r e l a t e d with i n t e r p h a s e b o u n d a r i e s .

1 . 1 . 3 . The p l a n a r matching model

The p l a n a r matching model was i n t r o d u c e d by Pumphrey [34] t o explain o b s e r v a t i o n s of p e r i o d i c s e t s of l i n e s i n h i g h - a n g l e g r a i n boundaries [ 3 5 , 3 6 ] , which could not b e e x p l a i n e d by o t h e r models. Pumphrey found t h a t t h e p e r i o d i c

With the aid of this formula Gleiter has calculated w as a function of the misorientation angle (fig. 1.6a) using the measured energy misorientation curve for symmetrical [110] tilt boundaries in aluminium (fig. 1.6b). It is seen that the misfit dislocations are only localized in a small angular range around cusps in the E-6 curve, corresponding to low energy orientations. Outside these areas dE/d9 decreases and the energy reduction obtained by retaining locally the low energy boundary structure is smaller than the strain energy stored in a network of localized misfit dislocations. Hence the concept of misfit dislocations, although it is geometrically correct, may lose its physical significance as the cores of the dislocations in these boundaries are delocalized, meaning that they are smeared out over the entire boundary area. In fact, the delocalisation model describes a transition to a continuous distribution of infinitesimal dislocations.

Extrinsic dislocations, which are not part of the equilibrium structure of the boundary and which may be formed by lattice dislocations entering the grain boundary, can appear in two forms, according to Gleiter [13,14]. In the first place the extrinsic dislocations can react with the localized intrinsic misfit dislocations in a manner analogous to the way they react with low-angle boundaries. Secondly their cores may spread out and thus they become delocalized dislocations.

To evaluate Gleiter's delocalisation model critically, one may consider the experimental observations from the literature. A number of results

[8,37,56,132-137] conform the delocalisation model, but other observations [138-144] appear to be controversial. In conclusion, this model Is not appropriate for general high-angle grain boundaries.

Disclination model

As an alternative to models based on dislocations, Li [16] has proposed a model to describe high-angle boundaries in terms of disclinations. The reason for this proposal is that a disclination is a rotational defect as is a grain boundary, whereas a dislocation is a translational defect. A dl sclination can be made by rotating the two surfaces of a cut with respect to each other instead of translating them in the case of a dislocation. In fig. 1.7 wedge disclinations and edge dislocations are compared. When comparing disclinations with dislocations it should be noted that a disclination dipole, consisting of two opposite disclinations, is equivalent to a dislocation, when the

lines were a result of the mismatch of families of atomic planes with traces in the grain boundary that are slightly mismatched, fig. 1.12. These traces result from rows of atoms where atomic planes end on the grain boundary. The plane matching boundaries preserve a repeat order only in one direction in the interface, rather than in two directions for a CSL boundary. Balluffi and Schober [38] have shown that plane matching boundaries can be treated as special cases of a CSL boundary where E approaches infinity. They are also termed coincident axial direction boundaries. In most cases one set of low index atomic planes can be found, from which the traces are slightly rotated and the spacings are different. In a similar manner as Moiré fringes, which

(a) (b)

Fig. 1.12 a) Two sets of planes (A and B) are mismatched across the grain

boundary (C).

b) After relaxation, an array of edge dislocations i s present to

accomodate the planar mismatch [40].

result from optical diffraction from two overlapping gratings, the two s e t s of

traces in the grain boundary give r i s e to lines of high atomic density,

fig. 1.13. Pumphrey suggested t h a t atomic relaxation occurs in the v i c i n i t y of

these Moiré-type fringes, where the atomic f i t is worst, leading to s t r a i n

fields that are imaged as as p a r a l l e l set of lines in the electron microscope.

When the trace spacings s, and S2 are known, the angle 9 between the traces

and the spacing d of the interference lines can be calculated.

The line features in the grain boundary are related to misfit dislocations and, according to Pumphrey, are only observed when a set of sufficiently densely packed planes (i.e. {ill}, {200} and {220} planes) are slightly mismatched. In general one set of dislocations is found to accommodate the misfit between the two crystals. The problem is to find the set of planes which

Fig. 1.13. Two sets of traces in the boundary of lattice planes with spacings Si and Sn are mismatched over an angle 8 . In this drawing s, and s~

are unequal. Moiré-type fringes appear with a spacing d [38].

corresponds to experimental observations. Sometimes more then one set is found. For example in a low-angle (100) twist boundary, where all sets of atomic planes are slightly mismatched, any of the densely packed planes will result in a different dislocation network. The experimentally observed square network of a/2 [Oil] and a/2[011] screw dislocations can, however, only be interpreted by considering the mismatch of {ill} planes. Another feature is that with the planar matching model only sets of continuous lines can be predicted, while more complicated dislocation structures, e.g. hexagonal, cannot be found. This means that the use of this model is strongly limited, mainly because the choice of the right set of planes is not established and only simple dislocation structures can be interpreted.

Balluffi and Schober [38] concluded that the planar matching model can only predict the grain boundary structure in certain cases, while the dislocation model is more general. They gave some examples where the planar matching model

leads to incorrect conclusions. Martikainen and Lindroos [39] used -several models to explain the structure of a grain boundary corresponding to 8 misorientation from a E = 7 CSL. The planar matching model gives a good explanation in this case. Pumphrey [22] has calculated the percentage of all possible grain boundaries that can be covered by the planar matching model. He used the criterion of a maximum angular deviation, similar to that of the CSL model (eq. 1.18). It was found that this model can be applied to the following percentages of grain boundaries in fee metals:

28,3% for {ill} matching 12,0% for {200} matching 19,4% for (220} matching

Thus the planar matching model was found to explain a larger percentage of grain boundaries than the CSL model. The planar matching model has been discussed extensively by Ralph et al. [40] and Schindler et al. [41] .

1.1.4. The polyhedral unit model

In the polyhedral unit model the structure of grain boundaries and their properties are related to a two-dimensional array of atomic configurations, which are called polyhedral or structural units or atomic clusters. Brandon et al. [19,20] have described a grain boundary as a two-dimensional array of atoms and noted that a low-energy boundary will pass through the most dense plane of CSL points, see fig. 1.8. The array of atoms consist of a coincidence atom surrounded by a number of atoms, the positions of which deviate slightly from those in a perfect crystal. Bishop and Chalmers [42] found an equivalent description of the structure of a grain boundary in terms of:

(i) a two-dimensional array of coincidence atoms in the boundary plane, (ii) atomic ledges and

(iii) a dislocation array.

These descriptions were applied to a symmetrical [100] tilt boundary. In order ro reduce the strain energy of atoms in the vicinity of a coincident atom, a relaxation process takes place, resulting in periodic groups of atoms of good fit in the boundary region.

boundary (c). The CSL points in the boundary form a regular pattern (d). In (e) the ledges form an alternating ledge sructure. It is seen that {530} planes are continuous near the boundary (f). A similar schematic representation is given when an additional tilt of 1.3° is supposed, (g), (h), (i), (j), (k) and (1). Now a second set of ledges is introduced [42].

Bishop and Chalmers found four different structural units for the orientation differences of the two crystals making a symmetrical [100] tilt boundary, corresponding to CSL's with E = 13, E = 17 and twice 2 = 5 . The boundary planes are then {320}, {530}, {210} and {310} respectively, see fig. 1.14. In the description of atomic ledges of the crystal surfaces (fig. 1.14 b ) , it was found that the ledges of the two crystals at the grain boundary interconnect exactly, as shown in fig. 1.14 c. In some cases it was concluded that the best geometrical fit between the two crystals is obtained when one crystal is translated relative to the second crystal, (fig. 1.14 e ) . In the case of such a rigid body translation the coincidence of atom positions is destroyed. This was also observed by Weins et al. [43,44,45]. By computer simulations of grain boundaries in fee metals, characteristic groups of atoms were found in the (111) plane where a central atom was surrounded by five, six or seven atoms, e.g. fig. 1.15. Here a two-dimensional cross-section of the bicrystal is made. According to Weins et al. [44] a general grain boundary can always be built up by a combination of several types of these polyhedral units. Alternating fivefold and sevenfold atomic clusters were seen to built up an 18° tilt boundary between crystals, formed by hexagonal arrays of atoms by Gleiter and Lissowski [46], as shown in fig. 1.15.

Fig. 1.15. In a two dimensional model of an 18° tilt boundary between two hexagonal arrays of atoms five-sided (A) and seven-sided (B) polyhedral units alternate in the boundary plane [46].

Some authors have compared the atomic arrangement of polyhedral units with the atomic arrangement in an amorphous material. Aaron and Boiling for example [47,48] developed a concept based on the free volume associated with grain boundaries and applied it as a criterion for grain boundary models. From calculations of the free volume for dislocation boundaries and boundary structures formed by random close-packed units it was concluded that the free (excess) volume of a grain boundary is the best criterion for the evaluation of a grain boundary model. The method of describing the structure of a grain boundary by a stacking of eight basic deltahedra was developed by Ashby et al. [49] and Frost et al. [50] . This was obtained for boundaries in coincidence orientation. The stacking of the deltahedra corresponds with the bicrystal symmetry (Pond et al. [51]).

Most investigations confirming the polyhedral unit model are based on computer simulations. These simulations are based either on the statics type model or on the molecular-dynamics model, (Balluffi et al. [89]). For the first type a static equilibrium is calculated for the sum of all potential energies of the atoms near the grain boundary. The molecular-dynamics model also includes the kinetic energy of the atoms in addition to the potential energy. This enables the possibility to study for example grain boundary migration. Due to excessively long computing time, the latter method is not very frequently used. The limitations of computer simulation are:

i) the boundary conditions associated with the limited number of atoms, involved in the calculations,

ii) the use of an oversimplified interatomic potential and the choice of a cutoff radius [90] and

iii) to avoid that the result is a metastable energy minimum [91].

To study the effect of the interatomic potential on the calculated grain boundary energy and structure, Wolf [92] applied six different interatomic potentials. Fig. 1.16 shows three non-oscillating potentials for aluminium. Wolf concluded that both the energy-misorientation relation and the calculated structure of a grain boundary vary greatly from one potential to another. Wolf did not find any uniform relation between the value of £ and the twist boundary energy, while no cusps in the E(S) curve were observed [93,94].

Models making use of polyhedral units can be interpreted as further developments of early island models. In these models grain boundaries are

supposed to consist of islands of good atomic fit separated by islands or channels of poor fit, Mott [52] and Kê [53]. The island models do not use any detailed atomic arrangements. Smoluchowski [54] was the first to suggest that the free energy of the islands of poor fit depends on the orientation relation between the crystals, whereas Mott and Kê did not suppose any orientation

o 2 0 I 0 0 > £ -oi ^ - 0 2 -0 3 - 0 4 06 08 10 12 14 16 r/0

Fig. 1.16.The interatomic potential V as a function of the interatomic distance r, divided by the lattice constant a. The Morse, Lennard-Jones and Rydberg potentials of aluminium are given, [92].

dependence. Gifkins [55] developed the island model more quantitatively by suggesting that the islands of good fit consist of facets of low-index directions. The islands are surrounded by channels of relaxed vacancies. In the polyhedral unit model the "islands of good fit" are supposed to be only a few atomic diameters, which is a smaller region than in the early island models.

1.1.5. Models relating the structure and the energy of a grain boundary

Many attempts have been made to describe unambigously the relationship between the grain boundary structure and its free energy. However, at present there exists no model that fulfils the requirement to describe the observations of the structure as well as the energetic and kinetic behaviour (e.g. grain boundary migration). In the case of a low-angle grain boundary

Read and Shockley [6] have calculated the misorlentation dependence of the grain boundary energy (eq.1.3), using the dislocation model to describe the grain boundary structure. Their assumption that linear elasticity theory is applicable limits the calculations to low-angle boundaries. For higher angles the dislocation cores will overlap and the dislocation energy exceeds that of the associated elastic strain fields. Read and Shockley extended the dislocation model to high-angle grain boundaries for energy calculations. The cusps they found in the E(9) curve correspond to low £ orientations (fig. 1.17). The deep cusp at 53 corresponds to a twin orientation. The depth of an energy cusp can be estimated from the energy of the most closely spaced dislocation array that is still likely to exist, as was shown by Pumphrey [56] . From these calculations it follows that cusp depths vary inversely with E. Although for some grain boundaries the cusp depths tend to decrease with l/£, this result does not agree with many other experimental results.

'e

Fig. 1.17.Energy versus misorlentation curve, showing cusps for certain misorientations, as given by Read and Shockley [6].

An elegant way of describing the structure and energy of a grain boundary according to the CSL model has been reported by Brokman and Balluffi [57]. They have described the relaxation process of a grain boundary in three steps: I Rigid body joining,

II Primary relaxation and III Secondary relaxation.

For the first step two rigid crystals are joined allowing a translation of crystal 1 relative to crystal 2. Then the energy of all pairs of atoms across

the boundary is summed and is minimized with respect to the translation. It was found that the calculated energies ER increase monotonically

with £ upto E = 50. Above £ = 50 the energy is independent of £. During step II, relaxation takes place around the 0-points. Consequently the lattices will fit better in these areas. The energy E related to primary relaxation varies continuously with 9, because the 0-lattice spacing varies continuously with 9. Above a critical angle 9 the energy E becomes independent of 9. 9 is estimated here about 20°. During the third step, relaxation occurs in the boundaries corresponding to a high E value and deviating from nearby low £ boundaries. A secondary dislocation network forms, while the energy E related to this step depends on the deviation A9 from the nearby low £ boundary and on the corresponding CSL. Fig. 1.18 shows schematicaly the result of the dependence of the grain boundary energy on the misorientation after the three relaxation steps for a [001] twist boundary. The total energy is now: E = ER + Ep + Eg ( 1 . 2 8 ) I = 25 13 17

I I

Y Y Y

Fig. 1.18. Schematic plot of the energy versus misorientation curve, showing the effect of rigid body joining for a number of £ values. Cusps due to secondary relaxation appear around these energy minima; after Brokman and Balluffi [57].

The continuous curve E(9) of fig. 1.18 due to primary relaxation shows minima for low £ orientations, corresponding with rigid body joining. The cusps

connecting these minima with the continuous E(9) curve are associated with the secondary relaxation.

1.2. REVIEW OF EXPERIMENTS ON GRAIN BOUNDARY STRUCTURE

Many experiments have been carried out to investigate the structure of grain boundaries. In this section a review will be given of experiments on grain boundaries, described in the literature, which are related to the subject of chapters I and II. In general it can be concluded that results obtained from low-angle grain boundaries support the dislocation model. For example the results on low-angle tilt and twist boundaries in gold from Schober and Balluffi [64,65,66] are in accordance with the dislocation model.

< o o HU 30 20 10 n

z

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_'

II'

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1 Sj

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Fig. 1.19. Dislocation spacing d of primary and secondary dislocation networks in (001) twist boundaries in gold. The filled symbols represent the measured data, while the open triangles denote the misorientations, where no dislocation networks were detected. The dashed curve is calculated, using eq. (1.17), [65].

The coincidence model may be used to explain Schober and Balluffi's results on low-angle boundaries as well as their observations in high-angle tilt and twist boundaries [65,67,68]. Twist boundaries were prepared by welding two (OOl)-oriented gold films together, after depostion of the films onto {OOlj cleaved faces of rocksalt. Dislocation networks were found in boundaries with a small deviation from a CSL orientation. Fig. 1.19 shows the

spacings of the secondary dislocation networks plotted as a function of the misorientation angle 8. The full circles represent the measured spacings, while the open triangles represent cases where no dislocations were detected. The dashed curves correspond to the calculated spacings from eq. (1.17). It is seen that for those misorientations where a dislocation network was found, the spacings are in agreement with the CSL theory. The angular range, within which relaxation effects can be observed, was shown to be larger when also electron diffraction from the intrinsic dislocation structures is observed, Sass et al.

[70] and Tan et al. [71]. The spacing of extra diffraction spots from a regular dislocation network is inversily related to the dislocation spacing and can be a useful means of detecting and measuring closely spaced dislocation networks.

The visibility of grain boundary dislocations has been discussed by Balluffi et al. [69,65]. They mentioned two limitations of resolving an intrinsic dislocation network. The contrast in the electron microscope of a dislocation network decreases for smaller Burgers vectors. This means that the contrast of secondary dislocation networks deteriorates when the corresponding value of £ increases, because then the magnitude of the Burgers vectors decreases. Moreover, for a certain angular deviation from a coincidence orientation, the dislocation spacing decreases for smaller Burgers vectors according to eq. (1.17), leading to a weaker strain contrast.

Rigid body translations of one crystal relative to the other crystal have been shown to occur by experimental observations [72-76] as well as by computer simulations [77-80]. For a given misorientation and grain boundary plane, several low-energy boundary structures may occur, differing only in rigid body translation [78]. The Burgers vector of a partial dislocation, separating two areas of the boundary with a different translation, is equal to the difference of the two translation vectors (Forwood and Clarebrough [81]).

A regular grid of intrinsic grain boundary dislocations acts as a two-dimensional diffraction grating for an electron beam. The extra diffraction spots in the diffraction pattern can be used to confirm the presence of the dislocation network and to analyse the spacings and line directions of the dislocation sets. This was proved to be a useful method by Balluffi et al. [69], Morton [82], Carter et al. [83], Sass et al. [70] and Tan et al. [71].