A study of crystalline interfaces by means of electron diffraction and transmission electron microscopy

AmUDY OF CRYSTALLINE INTERFACES BY MEANS

OF ELECTRON DIFFRACTION ANDTRAN^ISSION

A STUDY OF CRYSTALLINE INTERFACES BY MEANS

OF ELECTRON DIFFRACTION AND TRANSMISSION

ELECTRON MICROSCOPY

Proefschrift

ter verkrijging van de graad van doctor in de

technische wetenschappen

aan deTechnische Hogeschool Delft

op gezag van de rector magnificus

prof. dr. ir. F. J.Kievits

vooreen commissie aangewezen

door het college van dekanen te verdedigen op

woensdag 14 november 1979 te 14.00 uur

door

Johannes Gustavus Erlings

metaalkundig ingenieur

geboren te Dordrecht

^^

o» o

Delft University Press /1979

0 00 b S ^ !^

Prof. dr. ir. S. Radelaar

en de

co-promotor

Dr. F.W.Schapink

t o reproduce i n t h i s t h e s i s p a p e r s , which have been p u b l i s h e d or a r e accepted for p u b l i c a t i o n i n t h e i r j o u r n a l s .

Akademie Verlag (Physica S t a t u s S o l i d i ) Pergamon P r e s s , I n c . ( S c r i p t a M e t a l l u r g i c a )

Chapman & Hall Ltd. (Journal of Materials Science) (Thin S o l i d Films)

General introduction 1

EARLY GRAIN BOUNDARY MODELS 3 The amorphous cement theory 3 The transition lattice model 4

The island model 4 Smoluchowski's model 5 Friedel et al.'s model 5

Gifkins' model 5

MODERN GRAIN BOUNDARY MODELS 7 Small-angle grain boundaries 7

The dislocation model 7

Energy of small-angle dislocation boundaries 9

High-angle boundaries 12

Introduction 12

Coincidence site lattice model and 0-lattiae concept 13

CSL model and 0-lattiae concept 13

The 0-lattioe theory and (111) twist grain boundaries 19

11.2.2.2.1. The O-lattice theory 19 11.2.2.2.2. Application of the O-lattice to (111) twist grain boundaries 22

11.2.2.2.3. Calculation of CSL's and DSC lattices from O-lattices at

(111) twist grain boundaries 24

11.2.3. Planar matching model ' 2 9

11.2.4. Near-coincidence model 30

11.2.5. Structural unit model 31

III. EXPERIMENTAL TECHNIQUES 35 111.1. Single and bicrystal preparation 35

111.2. UHV vacuum deposition of thin solid films 41

I . I . l . 1.2. 1 . 3 . I.A. 1 . 5 . 1.6. I I . I I . 1 I I . 1 I I . 1 I I . 2 I I . 2 I I . 2 I I . 2 I I . 2 1. 2 . 1. 2 . 2.1 2 . 2

IV.1. Electron diffraction from unrelaxed bicrystals 47

IV.2. Relaxation of small-angle boundaries 48 IV.3. Generation of a DSC dislocation network in a coherent

twin boundary 54 IV.4. Dislocation networks in twin boundaries 57

IV.5. Electron diffraction from near-coincidence twist boundaries 62 IV.6. A note on electron diffraction from near-coincidence twist

boundaries 66 IV.7. Electron diffraction from periodic <111> twist grain

boundary structures 70

V. PHASE BOUNDARIES 81 V.1. Epitaxial layers 81

V.1.1. The O-lattice and epitaxial layers 81

V.1.2. The interface structure of Pd/(lll)Au epitaxial films 85

V.2. General phase boundaries 94

V.2.1. Introduction and review 94

V.2.2. O-lattice and CSL for (111) twist phase boundaries 97

V.2.3. Electron diffraction from relaxed (111) twist boundaries 109

V.2.4. On the in-situ relaxation of interphase interfaces 113

V.2.5. Analysis of electron diffraction patterns from bicrystals

containing (111) twist phase boundaries 123

V.2.5.1. Experimental 123

V.2.5.2. Interpretation of diffraction patterns in terms of double

diffraction 126

V.2.5.3. Experimental results and discussion 126

References 131

Summary 134

General introduction

All metallic materials applied in engineering are in a polycrystalline form. Diverse phenomena of these materials such as yielding, corrosion, high

temperature creep, superplasticity, recrystallization, segregation, brittle fracture all depend strongly on effects at grain boundaries. Phase

transformations such as precipitation and spinodal decomposition are also influenced by the interphase interface structure and properties. Materials containing well oriented grain boundaries with specific properties are applied nowadays in thin film techniques, lasers, semiconductors, etc. The growth of large single crystals on a technical scale, which becomes more and more important, also demands knowledge of processes occurring at grain boundaries. From these examples it will be evident that the importance of interface structures and properties has been generally recognised,

Much effort has been put in fundamental and experimental investigations on interfaces, resulting in numerous publications on this subject in the last decade.

From both theoretical and experimental research grain boundary structure models have been proposed. In general these models have to describe several

features. First, they must describe the change from one orientation to another in terms that permit atoms to be positioned according to the same laws of force that bind them together in the perfect crystal. Second, the models should yield the correct magnitude and orientation dependence of the grain boundary free energy. Third, they must give a width for the grain boundary that is no more than the order of three atom diameters. Finally, the models should be a basis for describing properties associated with interfaces in a quantitative manner. At present such models do not exist. However few models have been shown to give a reasonable description of one or more of these points.

Most of the information about structures in metallic interfaces is obtained with transmission electron microscopy (TEM) and electron diffraction (ED).

Since m o d e m electron microscopes can be extended to complete analytical systems they are suitable instruments for such investigations. Except TEM, field-ion microscopy and x-ray diffraction techniques are applied on a small scale. Most of the present boundary structure information comes from thinned bulk specimens and oriented bicrystals, prepared by welding together thin vacuum deposited single crystal layers. In this thesis oriented bicrystals which are prepared by placing on top of one another thin gel-grown crystals and/or vacuum deposited single crystal layers, are investigated. Initially

both crystals are only locally in atomic contact in the interface. Boundaries are formed in situ during annealing. This has great advantages since electron images and diffraction patterns of unrelaxed and relaxed boundaries can now be compared, which gives additional information about the relaxation process. With such specimens it will be shown that structures observed in several boundary types can be explained with a geometrical model, which has been described in detail by Bollmann.

I. EARLY GRAIN BOUNDARY MODELS [1-3]*

I.l. The amorphous cement theory

The early amorphous cement theory, which has been worked out in detail by Rosenhain et al. [4] supposes that crystals in a polycrystalline metal are surrounded and cemented together by a very thin layer of the same metal in the amorphous state. The theory was devised to explain experiments on grain boundary sliding at elevated temperatures. Rosenhain explained the sliding as being due to the difference in behaviour of crystalline and amorphous metal with rising temperature. In this model the orientation relationship between

the crystals forming the boundary does not influence the boundary configuration. Numerous later observations on grain boundary properties, however, showed

correlations between boundary properties (for example grain boundary diffusion and grain boundary energy) and the orientation relationship of the boundary forming crystals as well as boundary inclination. Because an undercooled liquid does not show any texture these observations are in contradiction with the model. Another reason why a grain boundary cannot be depicted as completely aperiodic can be obtained from thermodynamic properties of the boundary. From thermodynamic considerations and experimental results it follows that in the case of gold, for example, the thickness of an amorphous boundary region would be about 27 atom diameters. This boundary width is much larger as found by direct observations on grain boundaries by field-ion microscopy. From such experiments a boundary thickness of the order of a few atom diameters has been determined. In addition it is not very likely that atoms in the boundary region, which are in contact with or close to the crystals are free to form a structure

that is entirely uninfluenced by the periodic lattice structures of the neighbouring crystals. These arguments and observations have led to the conclusion that the amorphous cement theory does not present an adequate description of the atomic arrangement in the boundary.

1.2. The transition lattice model

The transition lattice model, which has been described in detail by Hargreaves and Hill [5], assumes a transition lattice between the crystals forming the boundary. This transition lattice consists of elastically bent crystal lattice planes. It is expected that the structure of the transition lattice depends on the orientation relation between the crystals, the orientation of the boundary plane and crystal structures. Apart from this difference with the amorphous cement theory, the boundary width in the transition lattice model was supposed to be much smaller than the boundary width in the amorphous cement theory. Several experiments on boundary free energy and interaction between the boundary and slip along glide planes of either crystal showed a dependence on the orientation relation of both crystals. However, to explain grain boundary sliding speculative arguments had to be used, while these mechanical effects could be explained well by the amorphous boundary theory. Experiments on specific surface energy and diffusion together with field-ion microscopy showed that the boundary cpntents cannot be described by the elastically bent lattice planes. For this reason this model is no longer applied nowadays.

1.3. The island model

The island model and related models to be described in the following sections are based on the assumption that a boundary consists of islands where the atomic matching is good, separated by regions of poor matching. In this model, which has first been put forward by Mott [6], the island dimensions, geometry and distribution are not related to the orientation relation of the crystals. Mott based his model on observations on grain boundary sliding and grain boundary motion. To explain the grain boundary sliding two assumptions had to be made. The first assumption was that there is negligible resistance to sliding in the islands of good fit. The second assumption is that the elementary act which allows slip to occur is the disordering or melting by thermal agitation of the atoms around an island, resulting in a local vanishing of the resistance to shear. The total resistance to sliding at such a boundary is thus equal to the resistance of the disordered areas located between the islands. Because no relation between the island distribution and the

orientation parameters is assumed, the grain boundary properties are isotropic in this model. It has already been indicated in the previous sections that this is not a realistic situation. A slightly modified island model was presented by Ké [7]. He concluded that a grain boundary consists of an assembly of

lattice defects. This conclusion was based on experimental results which indicated that the activation energies for grain boundary sliding and lattice self-diffusion were equal. However, later experiments resulted in different activation energies for sliding and self-difussion for some metals; thus the analogy between the structure of a grain boundary and the lattice defects responsible for self-diffusion cannot be pressed too far.

1.4. Smoluchowski's model

In order to account for the orientation dependence of grain boundary migration Smoluchowski [8] modified the island model by proposing that the free energy of disordering a group of several atoms depends on the interface energy of the boundary which is assumed to be a function of the orientation relationship.

Smoluchowski [9] also proposed a dislocation model for general tilt boundaries in order to explain certain experimental results concerning diffusion along grain boundaries. In this model the boundaries are thought to consist of dislocations separated by more or less undistorted regions. The ratio of the volumes of these different areas depends on the orientation relation of the grains. With increasing misfit groups of dislocations condense forming misfit regions or dislocations with large Burgers vectors. For high misorientations a continuous misfit region covers the whole interface area. An increased

diffusion coefficient, which was determined above certain misorientations, could be explained by the condensed groups of dislocations which were supposed to form wide channels along which diffusion was rapid.

1.5. Friedel et al.'s model

To relate measured and calculated grain boundary energies as a function of orientation difference Friedel et al. [10] proposed a relatively simple model. The interface energy was calculated for special orientations for which most of the atoms in the interface were supposed to occupy their original lattice positions. This was worked out from an assumed law relating energy to distance of separation of both boundary forming crystals, considering only nearest and next nearest neighbours. These calculated energies correspond to the measured energies within 10-50 percent.

I .6. Gifkins' model

Gifkins' model pictures a grain boundary to consist of islands of good fit, separated by channels of relaxed vacancies. The boundary islands are

identified with micro-facets (observed by field-ion microscopy), which tend to be planes containing specific crystallographic directions such that continuity of slip is possible across them. The channels of relaxed vacancies reside in the ledges which surround the facets. The dimensions of the facets are

estimated to be in the range of 5 to 50 atom diameters. Gifkins could explain several grain boundary properties (e.g. diffusion, sliding and internal friction) with this model.

II. MODERN GRAIN BOUNDARY MODELS

II.1. Small-angle grain boundaries

II.1.1. The dislocation model [1, 12-14]

A grain boundary is described completely by the orientation relation of the two crystals containing the boundary and the orientation of the boundary plane. Three degrees of freedom are related to the relative orientation of both

crystals while the boundary plane orientation has two degrees of freedom. Thus a general grain boundary has five degrees of freedom. However symmetrical tilt and twist boundaries are described completely by one parameter, as is shown in the following example. Starting from a perfect single crystal fig. (Il.l-la) shows how a symmetrical tilt boundary, described by a rotation 9 about e.g. [010] can be obtained. First a boundary plane is introduced into the (100) plane and secondly both crystals are rotated 6/2 in opposite direction about [010]. The parameter that describes this boundary is the rotation about [010]. Symmetrical twist boundaries can be obtained in an analogous way, as is shown in fig. (Il.l-lb). In addition to these two boundary types also asymmetrical tilt and twist boundaries will be discussed in this section. In such boundaries the boundary plane is specified by two additional parameters.

Dislocation models of grain boundaries were first proposed by Bragg [12] and Burgers [13]. In these models grain boundaries are pictured as consisting of planar arrays of dislocations or dislocation networks. In the simplest case of specific symmetrical tilt boundaries, only one array of edge dislocations parallel to the tilt axis is needed. In the more general case of boundaries which have both tilt and twist components, the interface is formed by a network of dislocations of mixed screw and edge type. In the case of small-angle boundaries the distance between individual dislocations is much larger than the dislocation core diameter and therefore the properties of small-angle grain boundaries may be derived approximately from dislocation theory

crystall -©• [001] [001], ( [001]2 boundary crystal 2 [Olü] [100] [001] [010] crystal 1 O crystal 2 [100] [OIOJ2, crystal 2 crystall

Fig. II.1-1. Sahematiaal drawing of a symmetrical tilt boundary (a) and a symmetrical twist boundary (b)

based on l i n e a r e l a s t i c i t y . The e f f e c t i v e width of a d i s l o c a t i o n g r a i n boundary i s dependent on the d i s l o c a t i o n content of the i n t e r f a c e . This can be i l l u s t r a t e d by c o n s i d e r i n g a row of edge d i s l o c a t i o n s i n a symmetrical t i l t boundary. The s t r e s s along such a row a l t e r n a t e s from t e n s i l e to compressive and w i l l t h e r e f o r e not reach f u r t h e r i n t o the c r y s t a l than a d i s t a n c e of the order of the d i s l o c a t i o n spacing ( d _ ) . For small-angle t i l t and t w i s t

boundaries d i s given by:

T . e e 2 s i n j

( I I . 1 - 1 )

where 8 is the tilt or twist angle and b is the length of the Burgers vector. A symmetrical low-angle tilt grain boundary in gold with a misorientation of 8 can thus be considered as a single crystal, containing a row of dislocations spaced 2 nm, which lies in a stressed area about 4 nm wide. Numerous

observations on such boundaries, most of them formed during polygonisation processes, are described in the literature [15].

On the basis of calculations of atomic positions van der Merwe [16] also found the dislocation model to be an adequate description. He started from a boundary

at which the atoms of each crystal were pulled out of their equilibrium positions by the atoms of the other crystal. He assumed a sinusoidal law of force which relates the force needed to displace the atoms from their

equilibrium positions and the atom displacements. From this model he calculated the precise positions of the atoms in the boundary plane. The calculated atomic arrangement corresponds to rows of dislocations in both tilt and twist

boundaries. Thus the interfacial structure consists of local regions of misfit having the form of dislocations, which separate regions of good fit.

With increasing angle of misfit the dislocation spacing will decrease resulting in dislocation core overlap. With emphasis on the effects of the cores of the dislocations Li [17] has attempted to extend the dislocation model to high-angle tilt boundaries. He assumed each dislocation of an infinite array of edge dislocations to be associated with a cylinder such that the stress at the surface of the cylinder vanishes. This assumption, which is based on the calculation of the strain energy of a single dislocation [18], is made since the stress at the boundary has to be finite. In addition he assumed linear elasticity theory to be valid even for high-angle boundaries. With these rather severe assumptions he discussed the structure of a tilt boundary as a function of misorientation. For small tilt angles the boundary consists of single dislocations with unchanged structures. With increasing misfit angle the cylinders are predicted to increase in size and to change shape from circular to elliptical, as if the cylinders are attracting each other. After a further increase of misfit angle the cylinders will merge into a slab that can be regarded as the core of the boundary. Recent observations e.g. on grain boundary energy and migration did not confirm this model. The difference can be attributed to the breakdown of linear elasticity theory in this region. In any realistic calculation based on the dislocation model it should be kept in mind that in most parts of the boundary the strain is so large that Hooke's

law does not hold and consequently linear elasticity theory cannot be applied.

11.1,2.

Energy of small-angle dislocation boundaries

The energy of dislocation boundaries has been calculated from dislocation theory by Read and Shockley [19]. Starting from the energy of a single dislocation in a dislocation row their main results can be derived simply by multiplying this energy with the dislocation density. The energy of a single

edge dislocation in a symmetrical tilt boundary is given by:

E = j - % - ^ In ^ + B e 4Tr(l-v) b e

where:

G is the shear modulus

b is the dislocation Burgers vector V is the Poisson's ratio

r is the effective radius of the region of the elastic distortion produced by the dislocation and

B is the core energy of the edge dislocation

It was shown by Read and Shockley that the effective radius r is approximately equal to the dislocation spacing b/9. With 9/b dislocations per unit length the interface energy per unit area is now given by:

^"^ m i . ^ B

Eq, (II.1-2) also holds for twist boundaries provided

T^ O b J , s

E = TT- and A = — = —

° 2^ Gb^

where B is the core energy of a screw dislocation. s

From eq. (II.1-2) it can be seen that the energy as a function of 6 consists of a linear part and a logarithmic part. The linear part describes the contribution from the dislocation core energies while the logarithmic part is due to the energies of the overlapping elastically distorted regions, Starting from the assumed sinusoidal law of forces, van der Merwe [16] also calculated the interfacial energy for symmetrical tilt and twist boundaries as a function of misorientation. Comparing these calculated energies with the results of Read and Shockley good agreement is found for both symmetrical low-angle tilt and twist boundaries. Calculated energies for high-angle boundaries agree well for tilt boundaries, while a large difference between the predictions from the two theories is found in the case of twist boundaries. Since the extrapolated dislocation model loses its physical significance at higher misorientations the calculated energies based on this model are significant only for small-angle boundaries.

|- = f(l-lrf) • (II. 1-3)

m m m

where E is the maximum energy which occurs at a misorientation angle 9 .

m m Since 9 depends on the core energy its value cannot be calculated. Hence 6

has to be determined experimentally. The value of 6 is determined by fitting the calculated and observed energy curves. Obviously with this procedure a good agreement between those two curves can be obtained, Eq. (II.1-3) has also been applied to high-angle boundaries. However for these boundaries the relation should be considered as empirical. Nevertheless a good fit between measured values and eq. (II.1-3) can be obtained, provided that appropriate values for 9 different from the low-angle region are used,

From the agreement between calculated and experimentally determined energies and from the results presented in section 11,1,1 it can be concluded that the dislocation model is a correct description of the structure of low-angle boundaries. At higher misorientations, however, the model looses its

significance since the dislocation spacing will decrease resulting in dislocation core interactions. In such boundaries the misfit can no longer be localized in well-separated dislocations with lattice Burgers vectors. Suitable high-angle grain boundary models which have been shown to be in agreement with experiments will be discussed in section II.2,

II. 2. High-angle grain boundaries

II.2.1. Introduction [2, 3, 20-22]

In this section a number of recent high-angle grain boundary models will be reviewed. These models all try to predict under certain circumstances, the structure of a grain boundary between crystals with known orientation relationship.

Two groups of models can be distinguished: a) geometrical models and

b) models which take into account the interatomic forces.

Several geometrical models have been proposed in the last decades; the coincidence model and the O-lattice model, the planar matching (PM) model and the near coincidence model being the most important.

The coincidence model is based on the assumption that the grain boundary energy is related to the dimensions of a three-dimensional coincidence site lattice (CSL), which occurs at certain specific misorientations of the boundary forming grains. In this model, the CSL, which is a superlattice of both crystal lattices, is continuous across the boundary plane. The volume ratio of the CSL unit cell and the crystal lattice unit cell is designated by E.

The O-lattice theory presents a geometrical procedure to derive dislocation structures in boundaries that deviate slightly from an exact coincidence orientation. The O-lattice is defined mathematically as the coincidence lattice of points which are in equivalent positions in the two crystals. The Burgers vectors of the interfacial dislocations are vectors of the displacement shift complete (DSC) lattice. This lattice is the coarsest lattice that contains, in the coincidence orientation, both crystal lattices as superlattices.

The planar matching theory considers the matching of one or more stacks of low-index lattice planes of the boundary forming grains. Mismatch of these lattice- planes can be accommodated in the boundary plane by the introduction of interfacial dislocations. The Burgers vectors of these dislocations are given by the spacings of the mismatched stacks of planes projected onto the boundary plane. When a three-dimensional matching of three sets of planes with different orientations occurs the PM model is very similar to the CSL model. For a slight mismatch of such a three-dimensional matching structures predicted with the PM model and the O-lattice theory may differ. In this case the Burgers vectors of the interfacial dislocations are misorientation dependent in the PM model and DSC vectors in the O-lattice model.

in the boundary plane. Deviations from such a two-dimensional CSL can be accoiranodated for by grain boundary dislocation networks, where the Burgers vectors are given by the two-dimensional DSC lattice.

When only coincidence cells with E s 25 occur it is interesting to note that the O-lattice model covers 11% of all possible boundaries [23]. With the PM model 28.3% of all grain boundaries are generated by {111} matching, 12% by

{200} matching and 19.4% by {220} matching [23]. Thus the PM model describes almost 60% of all possible grain boundaries.

The second group of models takes account of the interatomic forces by the introduction of a suitable potential function. These models have been developed only recently since they require large computers in order to perform the necessary numerical calculations. Calculation of stable atomic positions in any boundary is possible by simulating the atomic relaxations, which are in some computer programs subjected to the restriction of overall conservation of volume.

With none of these models, however an exact determination of the atomic

positions in a boundary region is possible. The general solution of this problem can only be obtained by solving the Schrödinger equation for all possible structures, the structure with minimum free energy being the stable structure. The models mentioned will be described in more detail below, together with a number of relevant experimental results in order to demonstrate their applicability.

II.2.2. Coincidence site lattice model and O-lattice concept

II.2.2.1. CSL model and O-lattice concept

In this section the coincidence site lattice model and the O-lattice concept will be discussed Using these geometrical models three boundary types can be distinguished:

a) exact coincidence boundaries;

b) small deviations from exact coincidence boundaries; c) random boundaries.

a) Exact coincidence boundaries

Friedel [24] showed that a superlattice of two crystal lattices at certain special relative orientations can be continuous across the boundary plane. Such a superlattice, referred to as coincidence site lattice is independent of the

^ • ^ • • ® • • • • • »' ® • * • * . * • ' • ' • ' • ' • • * ® * « * » * » * . * . . * ® * ® , ' (i) ' * * • • ® * r ^ * * * * • • ' , ® ' (-\ ' • * • • ® • ^ * • • _ • • • • • © • • • • • » ® ® * r , * » * • ' • ' • * ® ' r>' ' ' ' a (t) . . . ^ . ® . „ . . . . ^ . ® . _ ® • . . . ® • • • • • • ® » _ * • • • • • • • • ® ' f-~, ' ' ' ' * . ® . • / I ^ • . • ® . * ( i ) * . * . * . * » ' « » * ® • • • • • ® • „ • • • ® CSL • • • ® . • ® . ® . - ®

Fig. II.2-1. a. An asyrrmetric boundary in two coincidence related crystals. The crystals are rotated 28.21 about <111> which results in a coincidence cell with 1 = 7.

b. The boundary from fig. (a) now having a faceted interface.

orientation of the interfacial plane. According to Ranganathan [25] a three dimensional CSL can be generated in the cubic system by a rotation 9 about <hkl> if:

9 = 2tan"' (y/x)y^

2 2 2

where x and y are integers and N = h + k + 1 . Rotations about a high-index axis generally result in CSL's with a symmetry lower than crystal symmetry. E.g. rotation of f.c.c. crystals about <100>, <110> and <111> generate CSL unit cells with tetragonal, orthorhombic and rhombohedral or hexagonal symmetry, respectively. The volume ratio (E) of the CSL unit cell and the crystal lattice unit cell is given by: E = x + Ny [25] . An example is shown in fig. (II.2-la)for <hkl> = <111> and X = 2 and y = 1. This results in 9 = 38.21° and E = 7. The two crystals in fig.(II.2-1 a),which are tilted about an axis normal to the plane of the paper, are separated by an asymmetrical tilt grain boundary. It should be noted that the grain boundary plane can have any orientation,since the concept of coincidence sites is independent of the choice of the boundary plane. However all grain-boundary coincidence models assume a relation between the purely geometrical construction of the coincidence lattice and the physical properties of the grain boundary. According to e.g. Brandon [26], the best fit and hence lowest interfacial energy occurs when the boundary follows a plane containing a high density of coincidence sites. Brandon's assumption was made because it was believed that a boundary which lies in a plane with a high density of coincidence sites has a small width, negligible long range strain fields and therefore low energy. This implies that the energy of a boundary consists of two contributions: one that is due to the arrangement of atoms in the boundary layer and a part that is due to the corresponding strain field in the grains. For boundaries in a plane of high coincidence site density at least the first contribution (probably also the second) may be smaller than in any other position. If the boundary makes an angle with a higi density plane of the CSL it will tend to take up a stepped structure such that it has a maximum surface area in the high density CSL planes. Applying this to the boundary in fig.(II.2-la)a stepped interface like in fig,(II,2-lb)would result in a lower energy. Thus according to Brandon a grain boundary in the most general case will be faceted.

The relation between E and the interface energy has not yet been fully

understood. Recently.computer simulations [27] of relaxation processes in grain boundaries showed a decrease in energy when E increases. These results are contradicting the normally assumed effect, the energy increasing with E as the structures become more complex. In our opinion these results indicate that at

certain misorientations close to two exact coincidence orientations (e.g. a twist boundary with 9 = 16.53 about <111>, which is close to 9 = 15.18°, E = 31 and 9 = 17.89 , E = 43) relaxations in the highest E boundaries occur. This is not in agreement with experimental results (see section IV.7). The first experimental observations which confirmed the relation between boundary properties and a coincidence orientation relationship were made by Kronberg and Wilson [28].They found a relation between the grain-boundary migration rate and orientation in recrystallizing Cu. Numerous other

experimental results which confirm the coincidence model can be found in ref.29.

b) Small deviations from exact coincidence orientations

In section II.1 it was noticed that Bragg [12] and Burgers [13] suggested that the interface energies of low-angle boundaries can be reduced by the

introduction of arrays of dislocations with lattice Burgers vectors. The application of this idea to coincidence boundaries was worked out by Brandon et al, [26], They suggested that boundaries close to exact coincidence may reduce the energy of the boundary region by the introduction of special dislocation networks. The Burgers vectors of such dislocations should be characteristic for the CSL and can be obtained in a way analogous to the low-angle boundary dislocation Burgers vectors. In low-low-angle boundaries the dislocation Burgers vectors are crystal lattice translation vectors. Those translation vectors, which leave the structure of the CSL invariant are the Burgers vectors of the dislocations in boundaries close to exact coincidence. These vectors are designated displacement shift complete (DSC) vectors. The DSC lattice and CSL in the (111) plane are drawn in fig.(II.2-2)for a rotation of 38.21° about <111> (E = 7 ) .

Fig. II.2-2. CSL and DSC lattice unit vectors in the (111) plane for a 38.21° <111> twist boundary.

The intrinsic structure of a boundary which deviates slightly from exact coincidence consists,according to this model, of a network of dislocations with Burgers vectors given by the DSC lattice. Such networks are indicated as secondary grain boundary dislocation (SGBD) networks. The mismatch from the exact coincidence orientation is located in the dislocation lines, with regions in between where the CSL periodicity is more or less preserved. The deviation from exact coincidence is limited by the dislocation spacing which should at least have a magnitude of the order of the CSL periodicity. This intrinsic boundary structure is somewhat similar to the fit/misfit model of Mott [6], which has been developed more recently by Gifkins [11] (see chapter I). The model of Mott is, however, rather qualitative and does not predict special ordered structures at CSL misorientations.

The mathematically most detailed treatment of grain boundary structures arose from considerations by Bollmann [30]. Bollmann extended the idea of coincidence of lattice points to coincidence of points with equivalent positions in the two grains which lead to his "O-lattice" of coincident internal points. Therefore the coincidence site lattice is a special case of this approach. From the definition of the O-lattice, 0-points can be considered as centres of regions of good fit, with misfit regions in between. In low-angle boundaries it is anticipated that this misfit would be concentrated in primary dislocations, the locations of which coincide with the intersection of the Wigner-Seitz (W.S.) cell walls of the O-lattice with the boundary plane. The W.S. cell walls of the O-lattice are composed of planes which are the bisectors of the smallest

0-point unit vectors. For boundaries close to a coincidence orientation the O-lattice is replaced by an O-lattice of second order; the 0„-lattice, The derivation of the 0_-lattice and the relation between the O-lattice and the 0.-lattice is given in ref. 31, The structure of these boundaries can be described with the 0„-lattice analogous to the description of the low-angle boundary with the O-lattice. The general mathematical procedure to

calculate these lattices, DSC lattices and CSL's is given in section 11,2.2.2.

At increasing deviation from the coincidence orientation the dislocation spacing will decrease. Eventually the dislocation cores overlap and the dislocations lose their identities. The maximum deviation at which such dislocations still have physical meaning is not clear at present. The maximum value of E at which a CSL is still a useful concept is not known either. Now some direct evidence for the existence of SGBD networks will be reviewed. Most of the evidence for the existence of SGBD networks in boundaries with small deviations from the exact coincidence orientation originates from

electron microscope studies of bicrystals prepared with controlled

misorientation. Schober and Balluffi [32] showed the existence of SGBD networks in (100) Au twist boundaries with values of E up to 17. SGBD networks are also observed by Silcock et al. [33] in a near E = 9 [Oil] symmetrical tilt boundary in Al and by Loberg and Smith [34] in a boundary near E = 13 (rotation of 28.6 about [1.00, 0.817, 9.58])in Mg, the boundary plane being approximately (814). Ishida et al, [35] observed SGBD's in a near E = 33 boundary (rotation of 58.2 about [0.699, 0,019, 0.715]) in an AlMg alloy, the boundary plane being close to (545). Diffraction patterns, containing spots arising from diffraction at SGBD networks, present additional evidence for relaxed boundary structures. For example Forwood and Clarebrough [36] showed the presence of SGBD networks in a near E = 57 boundary in a CuSi alloy and Erlings and Schapink (see section IV.7) presented evidence for SGBD networks in (111) twist boundaries in gold with E S 43. A major difficulty in these experiments is the distinction between reflections arising from double diffraction and spots due to a network

(see section IV,5 and IV,6).

Information about the intrinsic boundary structure can also be obtained from the interaction of lattice dislocations with SGBD networks. Experimental results on this subject have been published by several authors [37-40] . For example Dingley and Pond [40] observed the dissociation of a matrix dislocation into dislocations with DSC Burgers vectors in a near E = 41 boundary (rotation of 10.8° about [0.99, 0.01, 0.10]) in Al, the boundary plane being (0.2164, 0.6947, 0.6820). The matrix dislocation dissociated in six partials according to:

a/2[lT0] •* a/82 [41 4 5] + 5 a/82 [0 9 T]

Recently the validity of the SGBD description of a boundary close to

coincidence and the core width of SGBD's have been discussed by several authors [2, 41-43]. Gleiter [2] concluded that the dislocation model is only significant for very small deviations from a coincidence orientation. He suggested that at higher misorientations the misfit located in the dislocation cores will be delocalized. Estimates of dislocation core energies showed that delocalized cores are in a lower energy state than normal dislocations. The influence of the atomic configuration near the boundary on the dislocation core structure has been noticed by Gleiter and Pumphrey [42] . So far the delocalization of dislocation cores has not been fully proved experimentally although some results [43, 44] indicate a behaviour of this kind.

c) Random boundaries

From the experiments discussed above it follows that only exact coincidence boundaries and boundaries that deviate slightly (less than approximately 3 ) from exact coincidence orientations with E <^ 60 are described by the coincidence model. All other boundaries are designated random boundaries,

11,2,2,2, The O-lattice theory and (111) twist grain boundaries

In this section first the mathematical derivation of the O-lattice will be presented. Next, as an example, the O-lattice conceptwill be used to predict the dislocation structure of a small-angle (111) twist boundary. The 0„-lattice is described and it will be indicated how this concept can be used to calculate SGBD networks. Finally it will be shown how a CSL and DSC lattice can be obtained from the O-lattice,

11.2,2.2.1, The O-lattice theory [30]

The basis of the O-lattice theory is the assumption that the atoms in an interface between any two crystals will take up positions such that a more or less optimum matching occurs across the boundary plane. In a number of boundaries this optimum matching can be accomplished by the introduction of a dislocation network. The Burgers vectors, spacings and line directions of the network dislocations are described completely with the O-lattice in the case of low-angle boundaries and with the 0„-lattice in the case of small deviations from exact coincidence orientations. However since the O-lattice theory is a purely geometrical description no predictions about the generation and

stability [45] of such dislocation networks can be made,

The O-lattice is defined mathematically as the coincidence lattice of points with equivalent positions in both boundary forming lattices. The relation between the O-lattice and both crystals can be derived in the following way. Starting from an arbitrary point x in lattice 1 with arbitrary external and

(2)

internal coordinates, the corresponding point x in lattice 2 is given by (see fig, II.2-3):

x ( 2 ) = A x ( ' > II.2-1

where A represents a linear homogeneous transformation relating both lattices. In the crystal coordinate system the external and internal coordinates of e.g. the point (1,2, 1.3, 2.0) are (1, 1, 2) and(0.2, 0.3, 0.0),respectively.

Fig. II.2-3. O-lattioe between two rotated (18°)

hexagonal lattices. In this oase A is a

rotation.

Points with equivalent coordinates in lattice 1 (e^'^) can be found by adding translation vectors b ' of lattice 1 to x^'^,

Hence:

e('> =x('> . b ( ^ >

II.2-2

If the points defined by eqs. (II.2-1) and (II.2-2) coincide, we designate such a point by x :

x^O) = x ( 2 ) ^ ^ ( 0 ^ ^ ( L )

Together with eq. (II.2-1) we obtain:

x(°) = A-'x(°' . b(^> II.2-3 (I A-')x(°) = T x<°) .(0) .(L) II.2-4

Provided 111 5« 0 the x^ ^ vectors, which describe 0-points, are solutions of eq.(II,2-4),which is the fundamental equation of the geometrical theory of crystalline interfaces. If | T | = 0 the rank of T can be 2. 1 or 0. The latter is a trivial case. If rank T is 2 the solution of eq, (II.2-4)is a set of lines and when rank T is 1 a set of 0-planes is the solution of this equation.

A general interface can be described as the boundary between two crystals of different structures with known lattice parameters, joined in a given relative

orientation. The T matrix for this case will now be derived. In an orthogonal coordinate system with a vector basis u, the translation vectors of a primitive unit cell of lattice 1 are defined by:

x^') = s('>u II.2-5

where the columns of S represent the unit vectors of lattice 1, expressed in the orthogonal coordinate system. Expressed in the crystal coordinate system (indicated by a subscript c) of crystal 1 we thus have for the vectors x :

4 ' ^ = (s('))->x('> 11,2-6

A d i f f e r e n t p r i m i t i v e u n i t c e l l i n t h e c r y s t a l c o o r d i n a t e system i s given by:

x ( ' ' > = U ^ ' ) x ^ ' > 1 1 , 2 - 7 —c = —c

where U is a unimodular transformation leaving the cell volume unchanged (det y = 1 ) . In orthogonal coordinates eq. (II.2-7) becomes:

,(i') =s(>)y(')(s('>)-V') =sJ'\J'K II.2-8

the matrix R. Hence in o r t h o g o n a l c o o r d i n a t e s :

x(2) = R s^'K I I . 2 - 9

This leads t o

, ( 2 ' ) ^ 3 ( 2 ) ^ ( 2 ) ^ 3 ( 2 ) ^ - 1 ^ ^ ( 2 ) ^ ^^^^_^^

By eliminating _u from eqs, (11,2-8) and (II.2-10) we obtain

,(2') ^ 3(2)^(2)^3(2)^-1^ s(2)(y('))-'(S^»)-'xC'> 11,2-11

and

and

T =c

y<')(s(2))-'R-'s(2)(y(2))->(s(2))-l3(l)

11,2-13

From eq, (11,2-13) it can be seen that the O-lattice between two crystals, fixed in orientation, is dependent on the choice of the U matrices and therefore on the choice of the primitive unit cells. This means that several different O-lattices exist between two crystals in a given relative orientation. This is a serious limitation of the O-lattice theory arising, among other things, from the fact that no symmetry operations are involved in this geometrical model, Bollmann [46] dealt with this problem by postulating that the principle of conservation of structure calls for a transition which is as smooth as possible. This means geometrically that A in the surroundings of the origin should

correlate, as far as possible, the closest neighbour points in both lattices. This can be achieved by choosing U and U such that | T | is as small as possible.

II.2.2.2.2.Application of the O-lattice to (111) twist grain boundaries

As an example the O-lattice will be calculated for a small-angle (111) twist boundary between two f.c.c. crystals. For a small-angle grain boundary

y^'^ = y^^^ = I, S^'^ = S^^^ = S and T = (I - R ~ ' ) . For a rotation about [111] the simplest expression for the O-lattice is obtained in a coordinate system

(indicated by a subscript h ) , defined by:

'-1

Th e rotation matrix in this coordinate system, R, , is:

h

1 - cos9 + _{" T T ~ V 3 ~} sin9 -2sin9

^^^ = I 2sin9 , „ sin9 n I (0) T T -, w

-^ 1 — 7 3 - 1 - cos9 - - 7 j - O x^ ' II.2-14

Since the rank of T, is 2, x, describes, in this case, a set of parallel lines perpendicular to the interface. Identical results for the interface structure will thus be obtained in a two-dimensional (2D) and a three-dimensional (3D) description. The O-lattice in the (111) plane is given by:

cot (y) cot (j)

"273 7T

cot (-) J cot (j)

~73 2 "^ 273 ~

b,^^^ II. 2-15 —h

Since the K vectors in 2D are the unit vectors of the h coordinate system, the (0) -1

x/ vectors are the columns of T, . At small-angle boundaries the O-lattice is a coherence preserving lattice, thus O-points can be identified with points of minimum strain (MS points). This means that the misfit between the two crystals will be located at the intersection between the boundary plane and the cell walls of the Wigner Seitz cells of the O-lattice, which can be identified with a dislocation network (fig. II.2-4). The dislocation spacing (d ) is given by the length of the O-lattice unit vectors. In the case of a small-angle boundary d can be calculated from eq. (II.2-15) :

lh(^)|

The equation is identical to the relation for small-angle tilt and twist boundaries which is given in section II. 1. Thus the theory of small-angle grain boundaries is a special case of the general theory of crystalline interfaces.

The dislocation description of a small-angle boundary is valid for rotations up to '^10 . At higher misorientations the dislocation spacing becomes so small that the dislocations can no longer be considered as individual defects. In terms of the O-lattice this means that the 0-points are so close together that it is no longer reasonable to describe the boundary as coherent in the

Fig. II. 2-4. O-lattice and W.S. cell walls for two interpenetrated (111) planes, which are rotated relative to each other.

vicinity of the 0-points. However at higher rotation angles the procedure for calculation of the O-lattice is still very useful, since it can be employed to determine primitive CSL cells.

If the relative orientation of two joined crystals deviates only slightly from a coincidence orientation, the interface tends to conserve the CSL periodicity by localizing the misfit from the exact coincidence orientation in a network of secondary grain boundary dislocations. In order to calculate such a SGBD network, a formula analogous to the O-lattice basic equation

(eq. (II.2-4)) has been derived [31]:

(I - B-') x(02> ,DSC II.2-17

In eq. (II.2-17) B is a linear homogeneous transformation describing the deviation from exact coincidence, the x 2 vectors describe an O-lattice of

DSC

second order (0_-lattice) and the b vectors form the displacement shift complete lattice associated with the CSL. In eq.(II.2-17)transformation

(I - B ) describes an 0„-lattice between two DSC lattices. Comparing ~ ~ (0 ) DSC

eq.(II.2-4)and eq.(II.2-17)it follows that B, x^ 1' and b at boundaries close to coincidence have a meaning analogous to A, x and b^ at small-angle boundaries. SGBD networks in high-small-angle boundaries with a small deviation from a coincidence orientation, can be identified with the intersection between the boundary plane and the Wigner-Seitz cell walls of the O.-lattice.

II.2.2.2.3. Calculation of CSL's and DSC lattices from O-lattices at (111) twist grain boundaries

A procedure to calculate the CSL from the O-lattice for a given coincidence relation has been presented by Grimmer et al. [47]. Eq. (11,2-13) applied to a high-angle twist boundary between two identical crystals reduces to

T = I - y^'^s 'R 's(y^^-')"' II.2-18

When different y matrices are employed (i.e. different single crystal unit cells are used), different O-lattices are obtained. However all these 0-Iattices have to contain the same CSL as a superlattice. For this reason y and y can be chosen arbitrarily provided the rank of the matrix T is 3. Here U is chosen as I and U is chosen such that |T | ?^ 0. T now becomes:

T = I - y^'^S~'R~'s II.2-19

To obtain the CSL in crystal coordinates (CSL ) from x two properties of the CSL have to be used. Firstly we use the fact that a CSL is a coincidence

c c cell of lattice points (internal coordinates (0, 0, 0 ) ) , so every unit vector

of this lattice necessarily consists of integers in the crystal coordinate system. Secondly, since the CSL is a superlattice of the O-lattice the volume E of the CSL is n|T | where n is an integer. Starting from the

O-lattice defined by the eqs. (II.2-4) and (II.2-19)the CSL can be determined in -1 '^

two steps: first such operations are carried out on T that two columns consist of integers while the determinant remains unchanged. In the second step, the remaining column is made to consist of integers by multiplication with an integer n; the determinant becomes E. For certain cases n can be expressed as a product of several integers. For those cases x may be such that different column vectors have to be multiplied by different factors in order to obtain a primitive unit cell of the CSL . The column vectors of the

c matrix thus determined form a basis of the CSL .

As an example we calculate the CSL(with E = 19)between two f.c.c. crystals, which is obtained by a rotation of 46.82 about [111]. T is obtained from eq. (II.2-19) with

-1 1 0 I and U*-'^ 0 1 0 0 0 1 1 0 0 1

3 - 3cos9 + /3 sine -2/3 sin9 cos9 + /3 sin9-l^ 2/3 sine 3 - 3cos9 - /3 sin9 cose-2 2/3 sine -3cos9 - /3 sin9 2cos9-2

Substitution of 8 = 46.82 results in:

14 -16

T = 1 ^ I 16 - 2 -4 1 with iT I = - - ^ and n = -8 =c 19 I I '=c' 19

16 -21 -4

In crystal coordinates the x unit vectors are the columns of T

•10

Multiplication of the first column with 4, the second column with -2 and rearrangement of the columns gives:

CSL = I 0 1 0 I and ICSL I = 19 5

This CSL is described on orthogonal coordinates by:

CSL = Y I 1 -1

Fora rotation of 9 + 120 = 166.82 , which is identical to the choice of another primitive unit cell thus a different U matrix, a different O-lattice will be obtained. However the same CSL should result from this O-lattice, as will be shown below. At a rotation of 166.82

35 -25 1 with |T^| = - "YI- and n = -50 16 • 3 •10 • 7

2±

rearrangement of the columns give:

1 - 5 • - •

CSL^ = I I 0 0 ° ° ' 0 land ICSL I = 19 4 - 1 / \ 4 5

We see that this CSL is identical to the CSL at 46.82°, while the O-lattice c c '

unit cells have different volumes.

With Grimmer's equation [48] the primitive DSC lattice can be calculated from the corresponding primitive CSL. Grimmer noted that the reciprocal of the DSC

D

lattice in real space (DSC ) is equal to the CSL in reciprocal space (CSL*),

R R

thus DSC = CSL*. It is obvious that we also have DSC* = CSL , which expresses that the DSC lattice in reciprocal space is equal to the reciprocal of the CSL in real space. Applying this to the example of the CSL with E = 19 we can calculate the DSC in real space from the CSL in reciprocal space. In

reciprocal space indicated by the index * T* is determined from eq. (II.2-19) with

/ -

^3 - 3cos9 + /3sin9 -2v'^sin9 0 cose + /3sine-l -2cose + 2 -3 cose - /3sine-l cose + /3sine-l 0

For the rotation of 46.82° about [111] T* is: =c

-16

4 -19 I with IT*! = T ^ and n = -4 =c' 19

6

and x ^ ° ^ * 4 ( 0 10 I4 ) = 1

By multiplication of the first and the last column with a factor two and minus two,respectively, CSL* is obtained: 0 0 4 16 10 0 16 14 0

crystal 1 crystal 2 boundary plane crystal 1 crystal 2 boundary plane edge dislocatiorTl

crystal 1 boundary plane rystal2

Fig. II.2-5. Schematic representation of the planar matching model;

(a) gives the situation when the normals to both plane stacks (x-. and £„j are precisely parallel to the axis of misorientation; (b) shows the effect of introducing a small asyrrmetric

tilt to the stacks. The result is that at ^he boundary only certain planes are brought into registry; (c) shows a possible relaxation patte-m, for the situation shown in fig. lb where the rrrismatch has been localized to form an intrinsic dislocation with Burgers vector d'.

2 5 3 - 3 - 2 5 13 15 10

-2 5 X /,

In the orthogonal coordinate system (CSL*) = (CSL* ) = DSC. The calculated DSC is:

DSC = 1 3

A number of CSL and DSC lattices are given in the references 47, 49 and 50.

II.2.3. Planar matching model

Pumphrey [51] was the first to suggest that boundary structures might be related to the match/mismatch of atomic planes across the boundary. This suggestion has been worked out in greater detail by Ralph et al. [52]. In this model the axis of misorientation, which is a common direction in the two grains, is usually taken as an axis which is normal or nearly normal to a low-index stack of planes ((111), (200) or (220)). If the axis of misorientation is precisely normal to such a set of planes the stacks on either side of the boundary will be accurately parallel. If the axis deviates from such a configuration these plane stacks are no longer presicely parallel and the resulting small deviations can be considered as the superposition of small twist components and/or small asymmetric tilt components. These small twist and tilt components bring the atomic planes on either side of the boundary into disregistry. Where this disregistry is small it can be accomodated by the formation of dislocations with Burgers vectors (parallel to the boundary plane), which are equal to the interplanar spacing (see fig. II.2-5) projected onto

the boundary plane. It is also possible to consider the intersection of more than one set of plane stacks at the boundary and as a result derive a complete description of the geometrical mismatch. Generally this mismatch will relax in three sets of intrinsic dislocations; this result can also be obtained with the Bollmann approach. The difference between the PM model and CSL model is that while the PM model requires that all grain boundaries created by rotations about a low index axis are of relatively low energy, the CSL model "predicts" that specific misorientations only have low energy. A PM boundary may be regarded formally [49] as the limiting case of a CSL boundary with E ->• <». The

PM theory has been criticized by Balluffi and Schober [53] on the grounds that it is yet another way of describing coincidence boundaries. It is obvious that this is only true if the boundary defects match planes belonging to more than one zone. A matching of planes in three dimensions must lead to matching of atoms, thus a CSL. Furthermore the PM theory does not predict Burgers vectors unambiguously. As dislocations in the CSL theory preserve coincident sites, they must also preserve plane matching. However the projected spacing in the boundary plane of different sets of planes is different. Depending upon which planes are supposed to match only some of the Burgers vectors according to the PM model are equal to the ones predicted by the CSL theory.

Electron microscopy has revealed many examples of defects of the PM type. Sets of lines, with directions and spacings independent of the deviations from coincidence orientations were first reported [54] in [100] tilt boundaries in Al bicrystals. These lines could be analyzed [55] in terms of a small twist component misorienting the (200) planes across the boundary. Similar dislocation lines have been reported by Pumphrey [56] in a 10.4 [110] tilt boundary in an AlMg alloy. These lines, which could be attributed to an additional 0.9 twist component agreed with the PM theory. FIM experiments on [110] tilt grain

boundaries in tungsten [57] showed matching of (110) rings apart from periodic spirals with the same spacings as lines observed with electron microscopy. From these experiments it has to be concluded that for a number of grain boundaries the PM model predicts structures which are in agreement with observations. Generally these boundaries belong to the group of boundaries which has been classified as random boundaries in section II.2.2.1.

II.2.4. Near-coincidence model

In near-coincidence boundaries a high density of coincidence sites is present in the boundary plane while the three-dimensional coincidence site density is low. This is identical to a low value of X in the boundary plane i^^y.) and a high value of the three-dimensional E. This results in a short periodicity in the boundary plane and a long periodicity out of the boundary plane (see also section V.2.2). In a general situation the long periodicity in the direction out of the boundary plane has such a large dimension, compared to the specimen thickness, that the NC model describes the interface structure only two-dimensionally. Bishop and Chalmers[58] pointed out that a two-dimensional

commensurate atomic net in the interface may result in short periodicities, which can be compared with the short periodicities of CSL's with small values

of E. Small deviations from parallelism of the two-dimensional nets may, according to Balluffi and Tan [59] be taken up by a dislocation network analogous to those proposed for deviations from CSL boundaries. Since the NC model generally is a two-dimensional model it should be noted that the Burgers vectors of these dislocations may differ from those obtained with a

three-dimensional model. An analogy with the CSL model should therefore be treated carefully.

The number of significant NC boundaries is not known. However it has been noted [60] that some computeres structures of [111] tilt boundaries all showed a relatively low value of 2 . This may indicate that there are so many NC boundaries that there is no need to maintain any specific boundary periodicity by the introduction of a defect network. The NC model may therefore be more important in non-cubic crystals or at interphase interfaces where relatively few three-dimensional CSL's with small values of I are possible.

Experimental information on two-dimensional matching at incoherent twin

boundaries has been presented by Sargent [61] and Vaughan [62]. There have also been several reports on micron-sized steps formed during the solidification of non-CSL bicrystals. For example Weins and Weins [63] showed that the boundary paths chosen in 25 and 31 [100] tilt boundaries in silver correspond to short period units. However, until now there have been no reports of misfit

dislocations in short period NC boundaries.

II.2.5. Structural unit model

The structural unit model (SUM) emphasizes that it is the atomic configuration of the interface that matters and that coincident sites far removed from it are of no physical significance. With this model Bishop and Chalmers [64] interpreted a CSL boundary in terms of small periodic atomic ledges with a size characteristic for the particular CSL. The existence of coincident sites in the boundary plane has been questioned by Chalmers and Gleiter [65] since from ball models it can be shown that a rigid translation of one grain with respect to the other may result in a better atomic fit. Two types of rigid translations have to be distinguished. The first type is the group of DSC translations. Such translations do not modify the boundary periodicity but translate the periodic structure with respect to the boundary plane. All other rigid translations parallel to the boundary plane preserve the periodicity in the boundary plane but give no coincident sites. They also preserve the three-dimensional periodicity, but the coincident sites no longer coincide with atoms. Therefore this periodicity can be identified with an

O-lattice. Recent electron microscopic observations [66, 67] on e.g. stacking fault like defects on incoherent E = 3 CSL facets, using diffraction vectors common to both grains, have confirmed the existence of such translations. Computer simulations which are based on the thermodynamic principle of minimum free energy of the system showed that translation of the two crystals results in a decrease of the boundary energy, although several assumptions(e.g. the form of the interaction potential between the atoms) in addition to relaxation of individual atoms are involved. Computer simulations also showed that due to translations and other relaxations no coincidence sites exist in the boundary and therefore coincidence sites can play no role in the actual boundary

structure.

Recent developments [68, 69] showed that exact coincidence boundaries can also be interpreted in terms of polyhedral groups of atoms. This description is complementary to the SUM, Its main attractions are its simplicity and the insight it offers into certain properties (e,g. symmetry) of the boundary, So far we have considered only boundaries at an exact coincidence orientation relationship. If the orientation relationship departs from such an ideal value, two structures are geometrically conceivable. Firstly, each orientation relationship may have its own structure related to the size of the periodically repeating unit, which would be very large for most angles. The second

conceivable structure would be that the departure from an ideal orientation relationship does not drastically change the structure of the boundary, but that it is gradually modified. Since it is shown experimentally that the special properties present at coincidence orientations are preserved up to several degrees away from the ideal orientations, the second model is most likely. This structure, which can be compared with the superposition of a dislocation array on an exact coincidence grain boundary, is obtained by adding periodically a unit characteristic of a CSL different from the existing

boundary ledge structure [64].

It has been suggested [64] that different combinations of units may be used to build up any high-angle boundary. This idea is supported by computations [70] of the structure of the 31.9 [100] symmetric tilt boundary, that lies between the 28.1° [100], l^^ = 17 and 36.9° [100], Z^^ = 5 coincidence boundaries. The computed structure may indeed be regarded as the inter-penetration of ledges characteristic for the two coincidence boundaries. This

implies that every boundary is related to at least one special boundary and no truly random boundaries exist. Thus with the SUM it is possible to predict a boundary structure for those boundaries that were expected to be random in the CSL model. In order to describe a general boundary with small structural

units, the interface generally has to be faceted. Each of these facets will be symmetrical to the two crystals and consists, in general, of various structural units that correspond to the neighbouring coincidence orientation relationships. Only in the very special case of a symmetrical boundary between two crystals of exact coincidence orientation relationship, the boundary is plane on an atomic scale and is formed by only one kind of structural units. A number of studies have been published in which energies of grain boundaries have been computed [71-73]. Using an interatomic potential, boundary atoms are allowed to relax into their minimum energy positions.

Such calculations [73] have shown significant energy variations for various symmetrical [100] tilt CSL boundaries in gold. Hasson and Goux [71] calculated the energies for Al[100] and [110] symmetrical tilt boundaries. Probably their structures are not those of minimum energy since they did not allow relaxation of coincident sites parallel to the boundary plane. For the [100] boundaries there is a very small ('^'10%) variation of energy, outside the low angle region. For the [110] rotation axis the energy shows minima at E = 3 and E = 11

coincidence boundaries. For fixed orientations the computation of boundary energy as a function of interfacial plane indicates that short period boundaries have low energies, Hasson et al. [74] showed that the boundary energy increases if the boundary period increases as a result of deviations from the {111}

symmetry plane for the 70.5 [110], E = 3 CSL and from the {013} symmetry plane for the 36.9 [100], E = 5 CSL. By contrast, deviations from the symmetry position in a 32.2 [111] exact coincidence boundary with E = 39 result in a reduction of boundary energy since the coincidence period is reduced [75] . For all these calculations it should be kept in mind that calculated energies are sensitive to the type of the interatomic potential used.

Except from the calculated energies, which present strong evidence for the SUM and CSL model, information can be obtained from energy measurements. Several energy measurements have been performed [76-79]. For example measurements of Hasson and Goux [78] on [110] symmetrical tilt boundaries in Al showed deep cusps near 70,5 and 129,5 which correspond to the E = 3 and E = 11 CSL's, These results are in excellent agreement with the authors earlier energy computations. The existence of cusps at the E = 3 and E = 11 coincidence boundaries has been confirmed in copper [79], by the measurement of surface grooves formed by heating at 1050 C. The energy of the E = 3 coherent twin was 0.06 E (E is the energy of the random high-angle boundary) and that of the incoherent twin boundary with no special boundary plane was 0.66 E . The

R

energy of the E = 11 boundary (rotation of 50.5 about <110>) with no special boundary plane was 0.81 E .

The results of these energy experiments show qualitative agreement with the CSL model and the SUM. However the variation in the depths of the energy

cusps can not be explained at the moment by these models. Due to the inaccuracy of energy measurements so far no such evidence for the PM model has been published.

From the information presented in this chapter it may be concluded that the CSL model and the O-lattice concept give a proper description of the boundary structure at exact low E coincidence orientations and small deviations from these orientations. Applying the SUM to boundaries that deviate slightly from the exact coincidence orientation does not result in a dislocation

description of the interface. Since SGBD's have been experimentally observed the value of the SUM is doubtful for this boundary type. The predictions of the SUM for general interfaces (combinations of ledges that correspond to the closest exact low E coincidence orientations)have not been verified

experimentally. Rigid translations of the boundary forming grains, which have been calculated with the SUM, have been detected. For rotations about low-index axes the PM model has been shown to be valuable.

III. EXPERIMENTAL TECHNIQUES

III.l, Single and bicrystal preparation

In order to perform a systematic TEM study on twist boundary structures and periodicities, thin well oriented bicrystals have been prepared. Such

bicrystals were obtained by careful manipulation of very thin gel-grown single crystals with known orientation. After the discussion of an earlier method to prepare bicrystals, the preparation of such single crystals will be described, In the second part of this section the formation of unrelaxed bicrystals from single crystals will be presented together with some characteristic relaxation features of these specimens.

In 1969 a method to produce thin film bicrystal specimens of gold under controlled conditions was developed by Schober and Balluffi [80] . Here thin single crystal films of the desired orientation were grown epitaxially on appropriate substrates from the vapour phase. Two films, while still on their substrate, were then welded together face to face at any desired

mis-orientation. In the welding operation two coated substrates were mounted (with the gold films face to face) in a stainless steel clamp under a moderate pressure which could be adjusted by means of four tightening screws. By an anneal of 300 s at 670K in air the clamped sandwich was welded. After removing the welded films from the substrates and mounting of the films on EM grids a final anneal in vacuum for 180 s at 620K was given, in order to promote further sintering at the interface. There are several important differences between this technique and the technique that is described below. The first difference concerns the perfection of the single crystals employed. The evaporated films contain many defects (dislocations, stacking faults) while the gel-grown films are almost perfect single crystals. Secondly during welding of the films it is very likely that plastic deformation will occur. In the technique employed by us no plastic deformation will be introduced initially. A third difference is in the initial state of relaxation of the bicrystals. The interfaces of the welded bicrystals are already in a relaxed