grain boundary structure and migration

Ultramicroscopy 29 (1989) 1-8 1 North-Holland, Amsterdam

GRAIN BOUNDARY STRUCTURE AND MIGRATION

David A. SMITH

IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA

Received at Editorial Office October-November 1988; presented at Conference May 1988

Grain boundary dislocations move during grain boundary migration, but only for low-angle boundaries does their motion account satisfactorily for the observed grain boundary motion. The difficulties in the high-angle regime are partly a consequence of the the limitation of high resolution microscopy mainly to simple tilt boundaries and partly attributable to the difficnlties associated with control of contrast conditions during in situ heating.

1. Introduction

One of the earliest pieces of evidence that grain boundary structure was more complicated than might be implied from a binary classification into low-angle and high-angle types came from the investigation of grain boundary migration in the context of the formation of recrystallisation textures. This insight stimulated research into the crystallography of grain boundaries and the nature and behavior of defects in grain boundaries. Nevertheless there is still no accepted atomisti- caUy explicit model of grain boundary migration. To some extent this is because there is also no generally applicable description of grain boundary structure. The purpose of this paper is to relate what is known about grain boundary structure to the body of knowledge concerning grain boundary migration with special reference to the contribu- tions of transmission electron microscopy.

2. Grain boundary structure

There is ample evidence for the view developed by Read and Shockley [1] that grain boundaries can be described as arrays of dislocations provided that the dislocation separation is large relative to the dimensions of the cores of the dislocations [2]. Frank's formula [3] permits the Burgers

vector content of such low-angle grain boundaries to be deduced from knowledge of the axis and angle of rotation and the interface plane. It must be emphasised that the connexion between Burgers vector content and the configuration of dislocations is not always well known even for the decep- tively simple case of the low-angle boundary. It has been found that the Burgers vector content may be partitioned as lattice vectors, usually the shortest possible, or as familiar partial lattice translations of the Frank and Shockley types [4]. The limits to the validity of the low-angle boundary model depend somewhat on definition. One upper limit on rotation angle is that angle above which the dislocation cores are no longer separated by any atoms in a crystal-like coordination shell. Alternatively that angle below which the dislocation separation is such that the dislocations can respond individually to an external stress might be selected as an operational definition. Irrespective of the problem of definition there is a large por- tion of "misorientat ion" space in which Frank's formula has only a formal meaning. This regime is commonly assigned to high-angle grain boundaries, although this definition contains no in- formation other than that the boundary is not comprised of individual lattice or partial dislocations separated by "good" crystal. High resolution transmission electron microscopy, computer mod- elling and X-ray diffraction are all limited to the

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2 D.A. Smith / Grain boundary structure and migration

Fig. 1. (a) A twinning dislocation with associated step in a (111} first order twin in GaAs. The lines indicate the change of level resulting from the twinning dislocation which is mowed. (b) Line diagram clarifying the configuration of the lattice planes in (a).

characterisation of a very small subset of the totality of the grain boundary structures possible. The structures of a few short-period pure tilt [5-7] and pure twist [8] boundaries have been de- termined experimentally and a larger number of related structures has been simulated [9-11]. The discussion which follows is accordingly limited by the present knowledge of grain boundary structure. Extensive investigations by transmission electron microscopy showed that dislocations were common features of high-angle grain boundaries although the dislocations observed were invariably insufficient to accommodate the total grain misorientation [12-14]. These observations were ex- plained by postulating that there existed sub- sidiary minima in the energy-misorientation function similar to but shallower than that associated with the zero misorientation, i.e. single crystal configuration. If two grains were constrained to a misorientation which deviated from this local minimum the observation of dislocation-like features was interpreted as implying that high-angle boundaries close in misorientation to local energy minima were structurally similar to low-angle boundaries [15]. Explicitly this means that the structure of such high-angle boundaries consists of perfect grain boundary dislocations which bound patches of "good" grain boundary. The term "good" has the same connotations as when it is used in the definition of the Burgers circuit for a crystal lattice dislocation. All the local minima in grain boundary energy (for cubic materials) corre- spond to misorientations where a coincidence site lattice (CSL) exists. To each CSL there is a corresponding DSC (dislocation shift complete) lattice the translation vectors of which are the possible

Burger vectors for perfect grain boundary dislocations. Such perfect grain boundary dislocations have been characterized in a wide range of (100) twist boundaries for gold and magnesium oxide [12,16]. Similar observations have been made in more general boundaries and in specimens pre- pared from conventionally processed bulk materials [13,14]. These grain boundary dislocations are visible under two-beam or weak-beam conditions in the transmission electron microscope because of their elastic displacement fields which do not cancel, at least over distances of the same order as the extinction distance. X-ray scattering experiments and high resolution transmission electron microscopy have revealed the atomic structure of the grain boundary core region in a few instances. The emerging understanding from both types of investigations is that the coordination at grain boundaries is as crystal-like as the incompatibility resulting from rotation will permit. In metals this means that excess volume is minimized, and the corollary is that high coordination is maintained. However, in the present context the key result is the verification of the crystallographically derived result that appropriately oriented DSC dislocations have steps at their cores [15]. Thus when DSC dislocations move grain boundary migration also occurs. It is established that grain boundary dislocations can move and multiply and that they retain compact cores at elevated temperatures [17,18]. A familiar example of the association between grain boundary dislocations and steps is the

(112) twinning dislocation in fcc materials. Such a dislocation moves by glide in {111} planes and the twin plane advances by one {111} planar spacing with the passage of each dislocation. Sup-

D.A. Smith / Grain boundary structure and migration 3

pose that the interface plane is ( lmn}; then the dislocation cannot glide but a combined glide and climb motion is possible and will result in migration of the twin boundary by the component of the step height in the direction normal to ( lmn}. Although this crystallographic result has not been verified for the general case, it has been shown to hold for {111} twins in gold [4] and gallium arsenide and the related case of a cubic-hexagonal interphase boundary [19]. Fig. 1 is a high resolution electron micrograph which illustrates the dislocation and step character of a ~(112) twinning dislocation in GaAs. Such dual nature is a prop- erty of DSC dislocations in general, according to theory, and has also been observed for a ~ ( 4 1 0 ) dislocation in an ~ = 17 tilt boundary in gold.

3. Grain boundary migration

Movement of grain boundaries is a thermally activated process which involves the.' transfer of an atom or possibly a group of atoms from one grain to a neighbor across a grain boundary. The kinetics of grain boundary migration can be analyzed in terms of absolute rate theory [20]. On a one-di- mensional argument the velocity, v, of the interface is give by the expression:

v = r A v ( & z / k T ) exp(-Ag/kT), (1)

where r is the distance moved per activated event, A is a geometric factor, A/~ the chemical potential change driving the process, Ag the free energy of activation and kT has its usual meaning; r, A, v and A g are all, in principle, structure dependent. The above expression is in fact valid for the movement of any interface providing the kinetics of attachment are rate limiting. The rate theory approach can readily be extended to a two-dimen- sional situation in which it is supposed that migration occurs by the attachment of individual atoms or groups of atoms to interfacial defects. Then an equation similar to eq. (1) describes the thermally activated motion of an individual defect, and the motion of the interface through the collective motion of a density p of defects each moving at a

velocity o and having a step height h results in an interface velocity:

v = hpv. (2)

These concepts are not satisfactorily explicit about the nature of the sites to which atoms are attached during the migration process. The closely related theory of diffusional phase transformations is framed in terms of attachment of atoms to steps and ledges [21]. However, it is becoming clear that steps and dislocations are not separate entities but two aspects of the character of a rather general kind of defect in partially coherent interfaces.

An extensive body of experiments provides a reference against which a detailed theory of grain boundary migration can be assessed. The key observations are enumerated below. (1) The ~ = 5, 13 and 17 [001] boundaries migrate with activation energies which are local minima [221. (2) The mobility of random boundaries is sensi- tive to impurities whilst that of special boundaries is not [23]. (3) The mobility of (111} boundaries exhibits a maximum at a rnisorientation angle of 40 o [24]. (4) The mobility of the ~ = 7 tilt boundary is greater than that of the ~ = 7 twist boundary [25]. (5) The mobility of low-angle boundaries is less than that of high-angle boundaries [25]. (6) A vacancy flux or supersaturation enhances grain boundary mobility [26,27]. (7) A twin boundary migrates at high velocity in response to a shear stress and at a low velocity during recrystallization [24,28].

The remainder of this paper is devoted to a discussion of the use of transmission electron microscopy in the observation and characterization of grain boundary defects and the elucidation of their role in grain boundary migration.

4. Observations

The fundamental challenge to the microscopist in the context of the study of interface migration is to deduce an atomistic mechanism from observations which do not have true single-atom


sensitivity and in additional to justify the extension of this mechanism to systems which, at least by present techniques, are inaccessible to observation. With these rather severe caveats in mind some observations of interface migration will be discussed.

It is well known that a symmetrical, low-angle tilt boundary comprised of a single set of dislocations can migrate in a plane perpendicular to itself by glide of its constituent dislocations. Movement of more general boundaries requires diffusion. It is possible to cause low-angle boundaries to move in situ by heating. These boundaries can move in response to a variety of sources of chemical potential which, it must be noted, are difficult to quan- tify satisfactorily for a particular experiment. In a thin foil heated in a transmission electron microscope image forces, temperature gradients and surface grooving in addition to the usual driving forces need to be considered. In fact, provided the objective is to elucidate a mechanism rather than to make a study of kinetics, the magnitude of the driving force is a secondary consideration. Quali- tatively it is clear from in situ studies that motion of a low-angle grain boundary occurs by the con- secutive motion of the constituent dislocations; this is illustrated in fig. 2 which is a series of micrographs showing the motion of a low-angle grain boundary at 600 °C in gold. It can be seen that the boundary does not simply translate but changes its conformation. This process involves the movement of individual lattice dislocations which are separated by good crystal. The pure translational motion of a grain boundary leaves both the Burgers vector content and the Burgers vector distribution invariant. Consequently there is zero net flux to the boundary. However, the individual dislocations move by a mixture of glide and climb so that local diffusional fluxes may flow between dislocations in the boundary; note that this is lattice diffusion. This process has been observed and analysed similarly by Lanxner and Bauer [29]. Some long-range diffusion seems necessary when a boundary changes from tilt to twist character; this is to accommodate the excess volume associated with the dilatational component of the displacement fields of edge dislocations.

Fig. 2. A series of transmission electron micrographs showing the motion of low-angle boundaries by the loosely coupled movement of individual lattice dislocations; the observations were made in situ at 600 o in a gold specimen. The twist boundary configuration seen in (a) rotates into a partial tilt

configuration in (b) and (c).


The extension of the description developed above to high-angle boundaries appears trivial in theory but in practice is confronted by numerous difficulties. The case of the ~ = 3 twin is considered first. The paradox of this example is that deformation twinning is known as a very rapid process and yet the (111) annealing twin, which of course has the same lattice rotation and consequently the same underlying structure, is a boundary of exceptionally low mobility. The resolution of this paradox begins with further ex- amination of fig. 1. It is clear that the hard sphere model of the (111) twin boundary structure is substantially correct. The atomic environment at this interface can be deduced to be virtually un- changed relative to that of the crystal interior at the first nearest-neighbor level. This defect-free interface is sessile for the same fundamental rea- son that close-packed surfaces are slow growing. In order to grow it is necessary to overcome the nucleation barrier associated with the formation of a step. Similarly the motion of a twin requires the formation of defects which are not part of the equilibrium structure; Shockley dislocations, of which one is arrowed in fig. 1, have a high mobility and thus once they are formed a twin can propagate rapidly. In the context of plastic de-

formation the applied stress, especially in the pres- ence of a stress raiser, can readily overcome the nucleation barrier. However, defect nucleation during thermally activated motion in response to the modest driving forces found in recrystallization must rely on heterogeneous effects such as threading dislocations, and consequently once the supply is exhausted annealing twins adopt a characteristic lamellar habit bounded by low-mobility (111} surfaces. This is characteristic of the (multiple) twins which are formed during conventional recrystallization and diffusion-induced recrystallization. An example of the latter behavior for the case of copper diffusing into gold is illustrated in fig. 3. It is, in fact, commonplace for the defects which are necessary to mediate the motion of an interface not to be part of the (local) equilibrium structure of the interface [38]. In addition, an equilibrated dislocation network resists the inde- pendent motion of individual constituent dislocations or the generation of further dislocations so that there is a threshold which must be exceeded before a dislocation-mediated process can occur in a grain boundary [39].

It is clear that the so-called coherent {111} twin defines a deep minimum in the T surface. The commonly observed facetting of twins implies

Fig. 3. A transmission electron micrograph illustrating the morphology of the multiple twins formed during diffusion induced recrystallization in gold. The direction of motion of the recrystallization interface is from right to left.


Fig. 4. A transmission electron micrograph showing some but not all of the geometrically necessary dislocations in a migrating boundary in gold. The rotation axis is near [001] and the rotation angle is measured as 31 ° from selected area diffraction patterns. The image is dominated by those dislocations with large Burgers vectors and separations even though the remaining dislocations are

not invisible according to the g . b criterion.

that there are other deep minima in the -/surface. These, too, resist the nucleation of extrinsic defects to permit growth.

From a crystallographic point of view the description of twin boundary migration seems satis- factory. In principle eq. (2) is applicable to the description of the growth kinetics of deformation and annealing twins but thus far has not been used successfully. Indeed the known values of V imply extremely large, possibly implausibly large, values of v. A different problem arises in the case of annealing twins and actually other grain boundaries; it is that there has never been an observation of the number of defects expected in order to balance eq. (2). It remains to be seen whether or not this is a deficiency of the model or a problem of observation.

Some of the practical problems attending the in situ verification of the dislocation model for grain boundary migration become clear from consideration of fig. 4 which is a view of a migrating boundary in a 31 ° [001] mixed grain boundary in a gold bicrystal. The visible dislocations are those of the so-called b 3 type and accommodate a small misalignment of the [001] axes; a denser array of geometrically necessary dislocations having smaller Burgers vectors is not resolved. The proper control

of contrast conditions for observing all the dislocations present together with operation of the microscope at sufficiently high magnification seem incompatible with the constraints imposed by high temperature and the single tilt normally available on heating stages. Even so, it is clear than when a grain boundary moves that part of the dislocation structure which is visible also moves. At present it remains unclear on the basis of in situ observations by transmission electron microscopy whether the motion of grain boundary dislocations is a mechanism or merely a byproduct of grain boundary migration.

5. Discussion

The observations and overview of the behavior of grain boundaries during migration highlight the difficulties of understanding the properties of a defect which has as many degrees of freedom as a grain boundary. It has long been recognized that the existence of 5 macroscopic degrees of freedom renders it impracticable to characterise the whole spectrum of boundary structures even for a particular material. Instead, the objective of research efforts has been to elucidate general rules through


the study of particular boundaries. This approach has enjoyed only limited success. For low-angle grain boundaries the principles are established; even though it is not always quite obvious what dislocations will occur, it is incontrovertible that a dislocation network in which the overall Burgers vector is normal to the rotation axis will be the essence of the grain boundary structure. Further- more, it is relatively straightforward to deduce the structure of a mixed boundary from those of the corresponding tilt and twist boundaries. This is done by projecting the dislocation structures of the simple boundaries onto the desired plane and then reconstructing unstable nodes [2]. No such procedure can be developed for high-angle grain boundaries. This is because in contrast to the low-angle case the variation of the boundary structure between dislocations as the orientation of the boundary plane changes is not known. High resolution transmission electron microscopy and computer simulation are both limited to relatively simple structures so that the nature of the arbitrary boundary remains inaccessible to structural characterisation. For the present the most explicit experimentally based general statement that can be made is that the grain boundary coordination is as crystal-like as possible. Furthermore, the widespread observations of grain boundary dislocations and the generality of the crystal rotation phenomenon suggest that grain boundary structures structures resist perturbation; a corollary of this result is that all boundaries are in as yet undefined way related to special boundaries. With the additional hypothesis that the unifying factor is grain boundary dislocations, i.e. all grain boundaries may be described physically in terms of arrays of primary or secondary dislocations the movement of which is the mechanism of migration, much of the phenomenology of grain boundary migration can be rationalised. Since the dislocations required for the motion of particular boundaries are characteristic of that boundary as is the activation energy for the climb component of the dislocation motion, so it is expected that grain boundary mobility will be a function of the crystallography of the boundary. Present understanding does not extend to making this connec- tion explicit. It is clear how solute drag and the

enhancement of grain boundary mobility by fluxes of point defects could both be the grain boundary dislocation analogs of well known behavior of crystal dislocations.

6. Conclusions

Movement of grain boundary dislocations provides a plausible mechanism for grain boundary migration and its dependence on grain boundary crystallography, solutes and excess point defects.

Verification of the proposed role for grain boundary dislocations by transmission electron microscopy requires the development of techniques for characterising arbitrary boundaries and the refinement of methods for in situ observation of dynamic processes at high resolution.

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[29] M. Lanxner and C.L. Bauer, in: Grain Boundary Struc- ture and Related Phenomena [Suppl. Trans. Japan. Inst. Metals 27 (1968) 411].

grain boundary structure and migration

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